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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 19:05:54 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 19:05:54 -0700 |
| commit | 9638e58f45d4e1a4277028e9bcf9651e334bec4f (patch) | |
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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/47464-h.zip b/47464-h.zip Binary files differnew file mode 100644 index 0000000..23f100c --- /dev/null +++ b/47464-h.zip diff --git a/47464-h/47464-h.htm b/47464-h/47464-h.htm new file mode 100644 index 0000000..c8406d0 --- /dev/null +++ b/47464-h/47464-h.htm @@ -0,0 +1,6272 @@ +<!DOCTYPE html> +<html lang="en"> +<head> + <meta charset="UTF-8"> + <title> + The Theory of Spectra And Atomic Constitution | Project Gutenberg + </title> + <link rel="icon" href="images/cover.jpg" type="image/x-cover"> + <style> + + #pg-header div, #pg-footer div { + all: initial; + display: block; + margin-top: 1em; + margin-bottom: 1em; + margin-left: 2em; + } + #pg-footer div.agate { + font-size: 90%; + margin-top: 0; + margin-bottom: 0; + text-align: center; + } + #pg-footer li { + all: initial; + display: block; + margin-top: 1em; 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+ text-align: right; + padding-bottom: 1em;} + +.caption p +{ + text-align: center; + text-indent: 0; + margin: 0.25em 0; +} + +/* Footnotes */ + +.footnote {margin-left: 10%; margin-right: 10%; font-size: 0.9em;} + +.footnote .label {position: absolute; right: 84%; text-align: right;} + +.fnanchor { + vertical-align: super; + font-size: .8em; + text-decoration: + none; +} + +/* Transcriber's notes */ +.transnote {background-color: #E6E6FA; + color: black; + font-size:small; + padding:0.5em; + margin-bottom:5em; + font-family:sans-serif, serif; +} + + </style> +</head> +<body> +<section class='pg-boilerplate pgheader' id='pg-header' lang='en'> +<h2 id='pg-header-heading' title=''>The Project Gutenberg eBook of The Theory of Spectra and Atomic Constitution: Three Essays by Niels Bohr</h2> + +<div>This ebook is for the use of anyone anywhere in the United States and +most other parts of the world at no cost and with almost no restrictions +whatsoever. You may copy it, give it away or re-use it under the terms +of the Project Gutenberg License included with this ebook or online +at <a class="reference external" href="https://www.gutenberg.org">www.gutenberg.org</a>. If you are not located in the United States, +you will have to check the laws of the country where you are located +before using this eBook.</div> + + +<div class='container' id='pg-machine-header'> +<p><strong>Title: </strong>The Theory of Spectra and Atomic Constitution: Three Essays</p> +<div id='pg-header-authlist'> +<p><strong>Author: </strong>Niels Bohr</p> +</div> + +<p><strong>Release Date: </strong>September 26, 2023 [eBook #47464]</p> +<p><strong>Language: </strong>English</p> +<p><strong>Credits: </strong>Andrew D. Hwang. HTML version by Laura Natal. (This ebook was produced using scanned images and OCR text generously provided by the Brandeis University Library through the Internet Archive.)</p> +</div> +<div id='pg-start-separator'> +<span>*** START OF THE PROJECT GUTENBERG EBOOK THE THEORY OF SPECTRA AND ATOMIC CONSTITUTION: THREE ESSAYS ***</span> +</div> +</section> +<p><span class="pagenum" id="Page_i">[Pg i]</span></p> + + +<p class="center">THE THEORY OF SPECTRA<br> +AND<br> +ATOMIC CONSTITUTION</p> +<p><span class="pagenum" id="Page_ii">[Pg ii]</span></p> + +<p class="center space-above3 space-below2">CAMBRIDGE UNIVERSITY PRESS<br> +C. F. CLAY, Manager<br> +LONDON: FETTER LANE, E.C. 4</p> + + +<p class="center">LONDON: H. K. LEWIS AND CO., <span class="allsmcap">LTD.,</span><br> +136 Gower Street, W.C. 1<br> +<img style="vertical-align: -3.507ex; width: 50.968ex; height: 8.145ex;" src="images/65.svg" alt=" " data-tex="\left. +\begin{aligned} +&\qquad\text{BOMBAY}\\ +&\qquad\text{CALCUTTA}\\ +&\qquad\text{MADRAS}\\ +\end{aligned} +\right\} +\text{MACMILLAN AND CO., LTD.}"><br> +TORONTO: THE MACMILLAN CO. OF<br> +CANADA, <span class="allsmcap">LTD.</span><br> +TOKYO: MARUZEN-KABUSHIKI-KAISHA</p> + +<p class="center space-above3 space-below2">ALL RIGHTS RESERVED +</p> + +<p><span class="pagenum" id="Page_iii">[Pg iii]</span></p> + +<div class="figcenter width500"> +<img src="images/cover.jpg" width="1600" alt="cover"> +</div> + + +<h1>THE THEORY OF SPECTRA<br> +AND<br> +ATOMIC CONSTITUTION</h1> + +<p class="center space-above3 space-below2">THREE ESSAYS</p> + + +<p class="center space-above3 space-below2 fontsize_80">BY</p> + +<div style="text-align:center; font-size:1.2em;">NIELS BOHR</div> + +<p class="center">Professor of Theoretical Physics in the University of Copenhagen</p> + + +<p class="center space-above3 space-below2">CAMBRIDGE<br> +AT THE UNIVERSITY PRESS<br> +1922 +</p> + +<p><span class="pagenum" id="Page_iv">[Pg iv]</span></p> + +<p class="center">PRINTED IN GREAT BRITAIN<br> +AT THE CAMBRIDGE UNIVERSITY PRESS +</p> + +<p><span class="pagenum" id="Page_v">[Pg v]</span></p> + + + +<p class="center">PREFACE</p> + + +<p class="nind"> +THE three essays which here appear in English all deal with the +application of the quantum theory to problems of atomic structure, and +refer to the different stages in the development of this theory.</p> + +<p>The first essay "On the spectrum of hydrogen" is a translation of a +Danish address given before the Physical Society of Copenhagen on the +20th of December 1913, and printed in <i>Fysisk Tidsskrift</i>, XII. p. +97, 1914. Although this address was delivered at a time when the formal +development of the quantum theory was only at its beginning, the reader +will find the general trend of thought very similar to that expressed +in the later addresses, which form the other two essays. As emphasized +at several points the theory does not attempt an "explanation" in the +usual sense of this word, but only the establishment of a connection +between facts which in the present state of science are unexplained, +that is to say the usual physical conceptions do not offer sufficient +basis for a detailed description.</p> + +<p>The second essay "On the series spectra of the elements" is a +translation of a German address given before the Physical Society +of Berlin on the 27th of April 1920, and printed in <i>Zeitschrift +für Physik</i>, VI. p. 423, 1920. This address falls into two main +parts. The considerations in the first part are closely related to +the contents of the first essay; especially no use is made of the +new formal conceptions established through the later development of +the quantum theory. The second part contains a survey of the results +reached by this development. An attempt is made to elucidate the +problems by means of a general principle which postulates a formal +correspondence between the fundamentally different conceptions of the +classical electrodynamics and those of the quantum theory. The first +germ of this correspondence principle may be found in the first essay +in the deduction of the expression for the constant of the hydrogen +spectrum in terms of Planck's constant and of the quantities which in +Rutherford's atomic model are necessary for the description of the +<span class="pagenum" id="Page_vi">[Pg vi]</span> +hydrogen atom.</p> + +<p>The third essay "The structure of the atom and the physical and +chemical properties of the elements" is based on a Danish address, +given before a joint meeting of the Physical and Chemical Societies +of Copenhagen on the 18th of October 1921, and printed in <i>Fysisk +Tidsskrift</i>, XIX. p. 153, 1921. While the first two essays form +verbal translations of the respective addresses, this essay differs +from the Danish original in certain minor points. Besides the addition +of a few new figures with explanatory text, certain passages dealing +with problems discussed in the second essay are left out, and some +remarks about recent contributions to the subject are inserted. Where +such insertions have been introduced will clearly appear from the +text. This essay is divided into four parts. The first two parts +contain a survey of previous results concerning atomic problems and a +short account of the theoretical ideas of the quantum theory. In the +following parts it is shown how these ideas lead to a view of atomic +constitution which seems to offer an explanation of the observed +physical and chemical properties of the elements, and especially to +bring the characteristic features of the periodic table into close +connection with the interpretation of the optical and high frequency +spectra of the elements.</p> + +<p>For the convenience of the reader all three essays are subdivided into +smaller paragraphs, each with a headline. Conforming to the character +of the essays there is, however, no question of anything like a full +account or even a proportionate treatment of the subject stated in +these headlines, the principal object being to emphasize certain +general views in a freer form than is usual in scientific treatises +or text books. For the same reason no detailed references to the +literature are given, although an attempt is made to mention the main +contributions to the development of the subject. As regards further +information the reader in the case of the second essay is referred to +a larger treatise "On the quantum theory of line spectra," two parts +of which have appeared in the Transactions of the Copenhagen Academy +(<i>D. Kgl. Danske Vidensk. Selsk. Skrifter</i>, 8. Række, IV. 1, I and +II, 1918), where full references to the literature may be found. The +proposed continuation of this treatise, mentioned at several places +<span class="pagenum" id="Page_vii">[Pg vii]</span> +in the second essay, has for various reasons been delayed, but in the +near future the work will be completed by the publication of a third +part. It is my intention to deal more fully with the problems discussed +in the third essay by a larger systematic account of the application of +the quantum theory to atomic problems, which is under preparation.</p> + +<p>As mentioned both in the beginning and at the end of the third essay, +the considerations which it contains are clearly still incomplete in +character. This holds not only as regards the elaboration of details, +but also as regards the development of the theoretical ideas. It may be +useful once more to emphasize, that—although the word "explanation" +has been used more liberally than for instance in the first essay—we +are not concerned with a description of the phenomena, based on a +well-defined physical picture. It may rather be said that hitherto +every progress in the problem of atomic structure has tended to +emphasize the well-known "mysteries" of the quantum theory more and +more. I hope the exposition in these essays is sufficiently clear, +nevertheless, to give the reader an impression of the peculiar charm +which the study of atomic physics possesses just on this account.</p> + +<p>I wish to express my best thanks to Dr A. D. Udden, University of +Pennsylvania, who has undertaken the translation of the original +addresses into English, and to Mr C. D. Ellis, Trinity College, +Cambridge, who has looked through the manuscript and suggested many +valuable improvements in the exposition of the subject.</p> + +<p style="text-align:right">N. BOHR.</p> + +<p class="nind"> +<span class="allsmcap">COPENHAGEN,</span><br> +<i>May</i> 1922.</p> + +<p><span class="pagenum" id="Page_viii">[Pg viii]</span></p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="CONTENTS">CONTENTS</h2> +</div> + +<table class="autotable" > +<tbody><tr> +<td class="tdl"></td> +<td class="tdc">ESSAY I</td> +<td class="tdr"></td> +</tr><tr> +<td class="tdl"></td> +<td class="tdc">ON THE SPECTRUM OF HYDROGEN</td> +<td class="tdr"></td> +</tr><tr> +<td class="tdl"></td> +<td class="tdc"></td> +<td class="tdr">PAGE</td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Empirical Spectral Laws</span></td> +<td class="tdr"><a href="#Page_1">1</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Laws of Temperature Radiation</span></td> +<td class="tdr"><a href="#Page_4">4</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">The Nuclear Theory of the Atom</span></td> +<td class="tdr"><a href="#Page_7">7</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Quantum Theory of Spectra</span></td> +<td class="tdr"><a href="#Page_10">10</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Hydrogen Spectrum</span></td> +<td class="tdr"><a href="#Page_12">12</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">The Pickering Lines</span></td> +<td class="tdr"><a href="#Page_15">15</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Other Spectra</span></td> +<td class="tdr"><a href="#Page_18">18</a></td> +</tr><tr> +<td class="tdl"></td> +<td class="tdc">ESSAY II</td> +<td class="tdr"></td> +</tr><tr> +<td class="tdl"></td> +<td class="tdc">ON THE SERIES SPECTRA OF THE ELEMENTS</td> +<td class="tdr"></td> +</tr><tr> +<td class="tdr">I.</td> +<td class="tdl">INTRODUCTION</td> +<td class="tdr"><a href="#Page_20">20</a></td> +</tr><tr> +<td class="tdr">II.</td> +<td class="tdl">GENERAL PRINCIPLES OF THE QUANTUM THEORY OF SPECTRA</td> +<td class="tdr"><a href="#Page_23">23</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Hydrogen Spectrum</span></td> +<td class="tdr"><a href="#Page_24">24</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">The Correspondence Principle</span></td> +<td class="tdr"><a href="#Page_27">27</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">General Spectral Laws</span></td> +<td class="tdr"><a href="#Page_29">29</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Absorption and Excitation of Radiation</span></td> +<td class="tdr"><a href="#Page_32">32</a></td> +</tr><tr> +<td class="tdr">III.</td> +<td class="tdl">DEVELOPMENT OF THE QUANTUM THEORY OF SPECTRA</td> +<td class="tdr"><a href="#Page_36">36</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Effect of External Forces on the Hydrogen Spectrum</span></td> +<td class="tdr"><a href="#Page_37">37</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">The Stark Effect</span></td> +<td class="tdr"><a href="#Page_39">39</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">The Zeeman Effect</span></td> +<td class="tdr"><a href="#Page_42">42</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Central Perturbations</span></td> +<td class="tdr"><a href="#Page_44">44</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Relativity Effect on Hydrogen Lines</span></td> +<td class="tdr"><a href="#Page_46">46</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Theory of Series Spectra</span></td> +<td class="tdr"><a href="#Page_48">48</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Correspondence Principle and Conservation of Angular Momentum</span></td> +<td class="tdr"><a href="#Page_50">50</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">The Spectra of Helium and Lithium</span></td> +<td class="tdr"><a href="#Page_54">54</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Complex Structure of Series Lines</span></td> +<td class="tdr"><a href="#Page_58">58</a></td> +</tr><tr> +<td class="tdr">IV.</td> +<td class="tdl">CONCLUSION</td> +<td class="tdr"><a href="#Page_59">59</a><span class="pagenum" id="Page_ix">[Pg ix]</span></td> +</tr><tr> +<td class="tdl"></td> +<td class="tdc">ESSAY III</td> +<td class="tdr"></td> +</tr><tr> +<td class="tdl"></td> +<td class="tdc">THE STRUCTURE OF THE ATOM AND THE PHYSICAL +AND CHEMICAL PROPERTIES OF THE ELEMENTS</td> +<td class="tdr"></td> +</tr><tr> +<td class="tdr">I.</td> +<td class="tdl">PRELIMINARY</td> +<td class="tdr"><a href="#Page_61">61</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">The Nuclear Atom</span></td> +<td class="tdr"><a href="#Page_61">61</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">The Postulates of the Quantum +Theory</span></td> +<td class="tdr"><a href="#Page_62">62</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Hydrogen Atom</span></td> +<td class="tdr"><a href="#Page_63">63</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Hydrogen Spectrum and X-ray +Spectra</span></td> +<td class="tdr"><a href="#Page_65">65</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">The Fine Structure of the +Hydrogen Lines</span></td> +<td class="tdr"><a href="#Page_67">67</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Periodic Table</span></td> +<td class="tdr"><a href="#Page_69">69</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Recent Atomic Models</span></td> +<td class="tdr"><a href="#Page_74">74</a></td> +</tr><tr> +<td class="tdr">II.</td> +<td class="tdl">SERIES SPECTRA AND THE CAPTURE OF ELECTRONS BY ATOMS</td> +<td class="tdr"><a href="#Page_75">75</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Arc and Spark Spectra</span></td> +<td class="tdr"><a href="#Page_76">76</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Series Diagram</span></td> +<td class="tdr"><a href="#Page_78">78</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Correspondence Principle</span></td> +<td class="tdr"><a href="#Page_81">81</a></td> +</tr><tr> +<td class="tdr">III.</td> +<td class="tdl">FORMATION OF ATOMS AND THE PERIODIC TABLE</td> +<td class="tdr"><a href="#Page_85">85</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">First Period. Hydrogen—Helium</span></td> +<td class="tdr"><a href="#Page_85">85</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Second Period. Lithium—Neon</span></td> +<td class="tdr"><a href="#Page_89">89</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Third Period. Sodium—Argon</span></td> +<td class="tdr"><a href="#Page_95">95</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Fourth Period. Potassium—Krypton</span></td> +<td class="tdr"><a href="#Page_100">100</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Fifth Period. Rubidium—Xenon</span></td> +<td class="tdr"><a href="#Page_108">108</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Sixth Period. Caesium—Niton</span></td> +<td class="tdr"><a href="#Page_109">109</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Seventh Period</span></td> +<td class="tdr"><a href="#Page_111">111</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Survey of the Periodic Table</span></td> +<td class="tdr"><a href="#Page_113">113</a></td> +</tr><tr> +<td class="tdr">IV.</td> +<td class="tdl">REORGANIZATION OF ATOMS AND X-RAY SPECTRA</td> +<td class="tdr"><a href="#Page_116">116</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Absorption and Emission of X-rays and +Correspondence Principle</span></td> +<td class="tdr"><a href="#Page_117">117</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">X-ray Spectra and Atomic Structure</span></td> +<td class="tdr"><a href="#Page_119">119</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Classification of X-ray Spectra</span></td> +<td class="tdr"><a href="#Page_121">121</a></td> +</tr><tr> +<td class="tdc"></td> +<td class="tdl"><span class="smcap">Conclusion</span></td> +<td class="tdr"><a href="#Page_125">125</a></td> +</tr> +</tbody> +</table> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_1">[Pg 1]</span></p> + +<h2 class="nobreak" id="ESSAY_I">ESSAY I +<br><br> +ON THE SPECTRUM OF HYDROGEN<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">[1]</a></h2> +</div> + +<p class="space-above3"> +<b>Empirical spectral laws.</b> Hydrogen possesses not only the +smallest atomic weight of all the elements, but it also occupies a +peculiar position both with regard to its physical and its chemical +properties. One of the points where this becomes particularly apparent +is the hydrogen line spectrum.</p> + +<p>The spectrum of hydrogen observed in an ordinary Geissler tube consists +of a series of lines, the strongest of which lies at the red end of +the spectrum, while the others extend out into the ultra-violet, the +distance between the various lines, as well as their intensities, +constantly decreasing. In the ultra-violet the series converges to a +limit.</p> + +<p>Balmer, as we know, discovered (1885) that it was possible to represent +the wave lengths of these lines very accurately by the simple law +<span class="align-center"><img style="vertical-align: -2.148ex; width: 28.559ex; height: 5.428ex;" src="images/1.svg" alt=" " data-tex=" +\frac{1}{\lambda_{n}} = R \left(\frac{1}{4} - \frac{1}{n^{2}}\right), +\qquad\text{(1)} +"></span> +where <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/66.svg" alt=" " data-tex="R"> is a constant and <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> is a whole number. The wave lengths +of the five strongest hydrogen lines, corresponding to +<span class="nowrap"><img style="vertical-align: -0.439ex; width: 15.566ex; height: 1.971ex;" src="images/68.svg" alt=" " data-tex="n = 3,\, 4,\, 5,\, 6,\, 7">,</span> measured in air at ordinary pressure and +temperature, and the values of these wave lengths multiplied by +<img style="vertical-align: -2.148ex; width: 11.563ex; height: 5.428ex;" src="images/69.svg" alt=" " data-tex="\left(\dfrac{1}{4} - \dfrac{1}{n^{2}}\right)"> are given in the +following table:</p> + +<table class="autotable"> + <thead><tr> + <th class="tdc"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"><img style="vertical-align: 0; width: 4.525ex; height: 0.036ex;" src="images/70.svg" alt=" " data-tex="\qquad"></th> + <th class="tdc"><img style="vertical-align: 0; width: 0.378ex; height: 0.036ex;" src="images/71.svg" alt=" " data-tex="\,"><img style="vertical-align: -0.05ex; width: 6.204ex; height: 2.005ex;" src="images/72.svg" alt=" " data-tex="\lambda · 10^{8}"><img style="vertical-align: 0; width: 4.525ex; height: 0.036ex;" src="images/70.svg" alt=" " data-tex="\qquad"></th> + <th class="tdc"><img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/73.svg" alt=" " data-tex="\lambda"> · <img style="vertical-align: -2.148ex; width: 17.248ex; height: 5.428ex;" src="images/74.svg" alt=" " data-tex="\left(\dfrac{1}{4} - \dfrac{1}{n^{2}}\right) · 10^{10}"></th> + </tr> + </thead> + <tbody><tr> + <td class="tdl">3</td> + <td class="tdl">6563.04 </td> + <td class="tdl"> 91153.3</td> + </tr><tr> + <td class="tdl">4</td> + <td class="tdl">4861.49</td> + <td class="tdl"> 91152.9</td> + </tr><tr> + <td class="tdl">5</td> + <td class="tdl">4340.66 </td> + <td class="tdl"> 91153.9</td> + </tr><tr> + <td class="tdl">6</td> + <td class="tdl">4101.85 </td> + <td class="tdl"> 91152.2</td> + </tr><tr> + <td class="tdl">7</td> + <td class="tdl">3970.25 </td> + <td class="tdl"> 91153.7</td> +</tr> + </tbody> +</table> + +<p class="nind"> +The table shows that the product is nearly constant, while the +deviations are not greater than might be ascribed to experimental +errors.</p> + +<p>As you already know, Balmer's discovery of the law relating to the +hydrogen spectrum led to the discovery of laws applying to the spectra +of other elements. The most important work in this connection was +<span class="pagenum" id="Page_2">[Pg 2]</span> +done by Rydberg (1890) and Ritz (1908). Rydberg pointed out that the +spectra of many elements contain series of lines whose wave lengths are +given approximately by the formula +<span class="align-center"><img style="vertical-align: -2.194ex; width: 20.885ex; height: 5.269ex;" src="images/2.svg" alt=" " data-tex=" +\frac{1}{\lambda_{n}} = A - \frac{R}{(n + \alpha)^{2}}, +"></span> +where <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/75.svg" alt=" " data-tex="A"> and <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/76.svg" alt=" " data-tex="\alpha"> are constants having different values for +the various series, while <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/66.svg" alt=" " data-tex="R"> is a universal constant equal to the +constant in the spectrum of hydrogen. If the wave lengths are measured +in vacuo Rydberg calculated the value of <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/66.svg" alt=" " data-tex="R"> to be <span class="nowrap"><img style="vertical-align: -0.05ex; width: 6.787ex; height: 1.579ex;" src="images/77.svg" alt=" " data-tex="109675">.</span> In +the spectra of many elements, as opposed to the simple spectrum of +hydrogen, there are several series of lines whose wave lengths are to +a close approximation given by Rydberg's formula if different values +are assigned to the constants <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/75.svg" alt=" " data-tex="A"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/76.svg" alt=" " data-tex="\alpha">.</span> Rydberg showed, +however, in his earliest work, that certain relations existed between +the constants in the various series of the spectrum of one and the +same element. These relations were later very successfully generalized +by Ritz through the establishment of the "combination principle." +According to this principle, the wave lengths of the various lines in +the spectrum of an element may be expressed by the formula +<span class="align-center"><img style="vertical-align: -1.579ex; width: 29.487ex; height: 4.615ex;" src="images/3.svg" alt=" " data-tex=" +\frac{1}{\lambda} = F_{r}(n_{1}) - F_{s}(n_{2}). +\qquad\text{(2)} +"></span> +In this formula <img style="vertical-align: -0.339ex; width: 2.345ex; height: 1.339ex;" src="images/78.svg" alt=" " data-tex="n_{1}"> and <img style="vertical-align: -0.339ex; width: 2.345ex; height: 1.339ex;" src="images/79.svg" alt=" " data-tex="n_{2}"> are whole numbers, and +<img style="vertical-align: -0.566ex; width: 15.91ex; height: 2.262ex;" src="images/80.svg" alt=" " data-tex="F_{1}(n),\, F_{2}(n)\, \dots"> is a series of functions of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">,</span> +which may be written approximately +<span class="align-center"><img style="vertical-align: -2.194ex; width: 19.352ex; height: 5.269ex;" src="images/4.svg" alt=" " data-tex=" +F_{r}(n) = \frac{R}{(n + \alpha_{r})^{2}}, +"></span> +where <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/66.svg" alt=" " data-tex="R"> is Rydberg's universal constant and <img style="vertical-align: -0.357ex; width: 2.357ex; height: 1.357ex;" src="images/81.svg" alt=" " data-tex="\alpha_{r}"> is a +constant which is different for the different functions. A particular +spectral line will, according to this principle, correspond to each +combination of <span class="nowrap"><img style="vertical-align: -0.339ex; width: 9.093ex; height: 1.91ex;" src="images/82.svg" alt=" " data-tex="n_{1}\, \text{and}\, n_{2}">,</span> as well as to the +functions <span class="nowrap"><img style="vertical-align: -0.439ex; width: 10.304ex; height: 1.977ex;" src="images/83.svg" alt=" " data-tex="F_{1},\, F_{2},\, \dots">.</span> The establishment of this +principle led therefore to the prediction of a great number of lines +which were not included in the spectral formulae previously considered, +and in a large number of cases the calculations were found to be in +close agreement with the experimental observations. In the case of +hydrogen Ritz assumed that formula (1) was a special case of the +general formula +<span class="align-center"><img style="vertical-align: -2.827ex; width: 28.879ex; height: 6.785ex;" src="images/5.svg" alt=" " data-tex=" +\frac{1}{\lambda} = R\left(\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}\right), +\qquad\text{(3)} +"></span> +<span class="pagenum" id="Page_3">[Pg 3]</span> +and therefore predicted among other things a series of lines in the +infra-red given by the formula +<span class="align-center"><img style="vertical-align: -2.148ex; width: 19.618ex; height: 5.428ex;" src="images/6.svg" alt=" " data-tex=" +\frac{1}{\lambda} = R\left(\frac{1}{9} - \frac{1}{n^{2}}\right). +"></span> +In 1909 Paschen succeeded in observing the first two lines of this +series corresponding to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 15.414ex; height: 1.756ex;" src="images/84.svg" alt=" " data-tex="n = 4\, \text{and}\, n = 5">.</span></p> + +<p>The part played by hydrogen in the development of our knowledge of the +spectral laws is not solely due to its ordinary simple spectrum, but it +can also be traced in other less direct ways. At a time when Rydberg's +laws were still in want of further confirmation Pickering (1897) found +in the spectrum of a star a series of lines whose wave lengths showed a +very simple relation to the ordinary hydrogen spectrum, since to a very +close approximation they could be expressed by the formula +<span class="align-center"><img style="vertical-align: -2.827ex; width: 26.193ex; height: 6.785ex;" src="images/7.svg" alt=" " data-tex=" +\frac{1}{\lambda} = R\left(\frac{1}{4} - \frac{1}{(n + \frac{1}{2})^{2}}\right). +"></span> +Rydberg considered these lines to represent a new series of lines in +the spectrum of hydrogen, and predicted according to his theory the +existence of still another series of hydrogen lines the wave lengths of +which would be given by +<span class="align-center"><img style="vertical-align: -2.827ex; width: 23.284ex; height: 6.785ex;" src="images/8.svg" alt=" " data-tex=" +\frac{1}{\lambda} = R\left(\frac{1}{(\frac{3}{2})^{2}} - \frac{1}{n^{2}}\right). +"></span> +By examining earlier observations it was actually found that a line had +been observed in the spectrum of certain stars which coincided closely +with the first line in this series (corresponding to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.506ex; height: 1.692ex;" src="images/85.svg" alt=" " data-tex="n = 2">)</span>; +from analogy with other spectra it was also to be expected that this +would be the strongest line. This was regarded as a great triumph for +Rydberg's theory and tended to remove all doubt that the new spectrum +was actually due to hydrogen. Rydberg's view has therefore been +generally accepted by physicists up to the present moment. Recently +however the question has been reopened and Fowler (1912) has succeeded +in observing the Pickering lines in ordinary laboratory experiments. We +shall return to this question again later.</p> + +<p>The discovery of these beautiful and simple laws concerning the line +spectra of the elements has naturally resulted in many attempts at a +theoretical explanation. Such attempts are very alluring because +<span class="pagenum" id="Page_4">[Pg 4]</span> +the simplicity of the spectral laws and the exceptional accuracy with +which they apply appear to promise that the correct explanation will +be very simple and will give valuable information about the properties +of matter. I should like to consider some of these theories somewhat +more closely, several of which are extremely interesting and have been +developed with the greatest keenness and ingenuity, but unfortunately +space does not permit me to do so here. I shall have to limit myself to +the statement that not one of the theories so far proposed appears to +offer a satisfactory or even a plausible way of explaining the laws of +the line spectra. Considering our deficient knowledge of the laws which +determine the processes inside atoms it is scarcely possible to give +an explanation of the kind attempted in these theories. The inadequacy +of our ordinary theoretical conceptions has become especially apparent +from the important results which have been obtained in recent years +from the theoretical and experimental study of the laws of temperature +radiation. You will therefore understand that I shall not attempt +to propose an explanation of the spectral laws; on the contrary I +shall try to indicate a way in which it appears possible to bring +the spectral laws into close connection with other properties of the +elements, which appear to be equally inexplicable on the basis of the +present state of the science. In these considerations I shall employ +the results obtained from the study of temperature radiation as well as +the view of atomic structure which has been reached by the study of the +radioactive elements.</p> + +<p class="space-above3"> +<b>Laws of temperature radiation.</b> I shall commence by mentioning +the conclusions which have been drawn from experimental and theoretical +work on temperature radiation.</p> + +<p>Let us consider an enclosure surrounded by bodies which are in +temperature equilibrium. In this space there will be a certain amount +of energy contained in the rays emitted by the surrounding substances +and crossing each other in every direction. By making the assumption +that the temperature equilibrium will not be disturbed by the mutual +radiation of the various bodies Kirchhoff (1860) showed that the +amount of energy per unit volume as well as the distribution of this +energy among the various wave lengths is independent of the form +and size of the space and of the nature of the surrounding bodies +<span class="pagenum" id="Page_5">[Pg 5]</span> +and depends only on the temperature. Kirchhoff's result has been +confirmed by experiment, and the amount of energy and its distribution +among the various wave lengths and the manner in which it depends +on the temperature are now fairly well known from a great amount of +experimental work; or, as it is usually expressed, we have a fairly +accurate experimental knowledge of the "laws of temperature radiation."</p> + +<p>Kirchhoff's considerations were only capable of predicting the +existence of a law of temperature radiation, and many physicists +have subsequently attempted to find a more thorough explanation of +the experimental results. You will perceive that the electromagnetic +theory of light together with the electron theory suggests a method +of solving this problem. According to the electron theory of matter +a body consists of a system of electrons. By making certain definite +assumptions concerning the forces acting on the electrons it is +possible to calculate their motion and consequently the energy radiated +from the body per second in the form of electromagnetic oscillations +of various wave lengths. In a similar manner the absorption of rays of +a given wave length by a substance can be determined by calculating +the effect of electromagnetic oscillations upon the motion of the +electrons. Having investigated the emission and absorption of a body at +all temperatures, and for rays of all wave lengths, it is possible, as +Kirchhoff has shown, to determine immediately the laws of temperature +radiation. Since the result is to be independent of the nature of the +body we are justified in expecting an agreement with experiment, even +though very special assumptions are made about the forces acting upon +the electrons of the hypothetical substance. This naturally simplifies +the problem considerably, but it is nevertheless sufficiently difficult +and it is remarkable that it has been possible to make any advance at +all in this direction. As is well known this has been done by Lorentz +(1903). He calculated the emissive as well as the absorptive power of +a metal for long wave lengths, using the same assumptions about the +motions of the electrons in the metal that Drude (1900) employed in his +calculation of the ratio of the electrical and thermal conductivities. +Subsequently, by calculating the ratio of the emissive to the +<span class="pagenum" id="Page_6">[Pg 6]</span> +absorptive power, Lorentz really obtained an expression for the law of +temperature radiation which for long wave lengths agrees remarkably +well with experimental facts. In spite of this beautiful and promising +result, it has nevertheless become apparent that the electromagnetic +theory is incapable of explaining the law of temperature radiation. +For, it is possible to show, that, if the investigation is not confined +to oscillations of long wave lengths, as in Lorentz's work, but is also +extended to oscillations corresponding to small wave lengths, results +are obtained which are contrary to experiment. This is especially +evident from Jeans' investigations (1905) in which he employed a very +interesting statistical method first proposed by Lord Rayleigh.</p> + +<p>We are therefore compelled to assume, that the classical +electrodynamics does not agree with reality, or expressed more +carefully, that it cannot be employed in calculating the absorption and +emission of radiation by atoms. Fortunately, the law of temperature +radiation has also successfully indicated the direction in which the +necessary changes in the electrodynamics are to be sought. Even before +the appearance of the papers by Lorentz and Jeans, Planck (1900) had +derived theoretically a formula for the black body radiation which was +in good agreement with the results of experiment. Planck did not limit +himself exclusively to the classical electrodynamics, but introduced +the further assumption that a system of oscillating electrical +particles (elementary resonators) will neither radiate nor absorb +energy continuously, as required by the ordinary electrodynamics, but +on the contrary will radiate and absorb discontinuously. The energy +contained within the system at any moment is always equal to a whole +multiple of the so-called quantum of energy the magnitude of which is +equal to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.502ex; height: 1.595ex;" src="images/86.svg" alt=" " data-tex="h\nu">,</span> where <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;" src="images/87.svg" alt=" " data-tex="h"> is Planck's constant and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/88.svg" alt=" " data-tex="\nu"> is the +frequency of oscillation of the system per second. In formal respects +Planck's theory leaves much to be desired; in certain calculations +the ordinary electrodynamics is used, while in others assumptions +distinctly at variance with it are introduced without any attempt being +made to show that it is possible to give a consistent explanation of +the procedure used. Planck's theory would hardly have acquired general +recognition merely on the ground of its agreement with experiments +on black body radiation, but, as you know, the theory has also +contributed quite remarkably to the elucidation of many different +<span class="pagenum" id="Page_7">[Pg 7]</span> +physical phenomena, such as specific heats, photoelectric effect, +X-rays and the absorption of heat rays by gases. These explanations +involve more than the qualitative assumption of a discontinuous +transformation of energy, for with the aid of Planck's constant <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;" src="images/87.svg" alt=" " data-tex="h"> +it seems to be possible, at least approximately, to account for a great +number of phenomena about which nothing could be said previously. It +is therefore hardly too early to express the opinion that, whatever +the final explanation will be, the discovery of "energy quanta" must +be considered as one of the most important results arrived at in +physics, and must be taken into consideration in investigations of the +properties of atoms and particularly in connection with any explanation +of the spectral laws in which such phenomena as the emission and +absorption of electromagnetic radiation are concerned.</p> + +<p class="space-above3"> +<b>The nuclear theory of the atom.</b> We shall now consider the +second part of the foundation on which we shall build, namely the +conclusions arrived at from experiments with the rays emitted by +radioactive substances. I have previously here in the Physical Society +had the opportunity of speaking of the scattering of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/76.svg" alt=" " data-tex="\alpha"> rays +in passing through thin plates, and to mention how Rutherford (1911) +has proposed a theory for the structure of the atom in order to +explain the remarkable and unexpected results of these experiments. I +shall, therefore, only remind you that the characteristic feature of +Rutherford's theory is the assumption of the existence of a positively +charged nucleus inside the atom. A number of electrons are supposed +to revolve in closed orbits around the nucleus, the number of these +electrons being sufficient to neutralize the positive charge of the +nucleus. The dimensions of the nucleus are supposed to be very small +in comparison with the dimensions of the orbits of the electrons, and +almost the entire mass of the atom is supposed to be concentrated in +the nucleus.</p> + +<p>According to Rutherford's calculation the positive charge of the +nucleus corresponds to a number of electrons equal to about half the +atomic weight. This number coincides approximately with the number +of the particular element in the periodic system and it is therefore +natural to assume that the number of electrons in the atom is +<span class="pagenum" id="Page_8">[Pg 8]</span> +exactly equal to this number. This hypothesis, which was first stated +by van den Broek (1912), opens the possibility of obtaining a simple +explanation of the periodic system. This assumption is strongly +confirmed by experiments on the elements of small atomic weight. In +the first place, it is evident that according to Rutherford's theory +the <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/76.svg" alt=" " data-tex="\alpha"> particle is the same as the nucleus of a helium atom. +Since the <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/76.svg" alt=" " data-tex="\alpha"> particle has a double positive charge it follows +immediately that a neutral helium atom contains two electrons. Further +the concordant results obtained from calculations based on experiments +as different as the diffuse scattering of X-rays and the decrease +in velocity of <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/76.svg" alt=" " data-tex="\alpha"> rays in passing through matter render the +conclusion extremely likely that a hydrogen atom contains only a single +electron. This agrees most beautifully with the fact that J. J. Thomson +in his well-known experiments on rays of positive electricity has never +observed a hydrogen atom with more than a single positive charge, while +all other elements investigated may have several charges.</p> + +<p>Let us now assume that a hydrogen atom simply consists of an electron +revolving around a nucleus of equal and opposite charge, and of a mass +which is very large in comparison with that of the electron. It is +evident that this assumption may explain the peculiar position already +referred to which hydrogen occupies among the elements, but it appears +at the outset completely hopeless to attempt to explain anything at all +of the special properties of hydrogen, still less its line spectrum, on +the basis of considerations relating to such a simple system.</p> + +<p>Let us assume for the sake of brevity that the mass of the nucleus +is infinitely large in proportion to that of the electron, and that +the velocity of the electron is very small in comparison with that +of light. If we now temporarily disregard the energy radiation, +which, according to the ordinary electrodynamics, will accompany the +accelerated motion of the electron, the latter in accordance with +Kepler's first law will describe an ellipse with the nucleus in one of +the foci. Denoting the frequency of revolution by <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega">,</span> and the +major axis of the ellipse by <img style="vertical-align: -0.023ex; width: 2.328ex; height: 1.529ex;" src="images/90.svg" alt=" " data-tex="2a"> we find that +<span class="align-center"><img style="vertical-align: -1.671ex; width: 33.116ex; height: 5.087ex;" src="images/9.svg" alt=" " data-tex=" +\omega^{2} = \frac{2W^{3}}{\pi^{2} e^{4} m},\quad +2a = \frac{e^{2}}{W}, +\qquad\text{(4)} +"></span> +<span class="pagenum" id="Page_9">[Pg 9]</span> +where <img style="vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;" src="images/91.svg" alt=" " data-tex="e"> is the charge of the electron and <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/92.svg" alt=" " data-tex="m"> its mass, while +<img style="vertical-align: -0.05ex; width: 2.371ex; height: 1.595ex;" src="images/93.svg" alt=" " data-tex="W"> is the work which must be added to the system in order to remove +the electron to an infinite distance from the nucleus.</p> + +<p>These expressions are extremely simple and they show that the magnitude +of the frequency of revolution as well as the length of the major +axis depend only on <span class="nowrap"><img style="vertical-align: -0.05ex; width: 2.371ex; height: 1.595ex;" src="images/93.svg" alt=" " data-tex="W">,</span> and are independent of the eccentricity +of the orbit. By varying <img style="vertical-align: -0.05ex; width: 2.371ex; height: 1.595ex;" src="images/93.svg" alt=" " data-tex="W"> we may obtain all possible values for +<img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> and <span class="nowrap"><img style="vertical-align: -0.023ex; width: 2.328ex; height: 1.529ex;" src="images/90.svg" alt=" " data-tex="2a">.</span> This condition shows, however, that it is not +possible to employ the above formulae directly in calculating the orbit +of the electron in a hydrogen atom. For this it will be necessary to +assume that the orbit of the electron cannot take on all values, and in +any event, the line spectrum clearly indicates that the oscillations +of the electron cannot vary continuously between wide limits. The +impossibility of making any progress with a simple system like the one +considered here might have been foretold from a consideration of the +dimensions involved; for with the aid of <img style="vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;" src="images/91.svg" alt=" " data-tex="e"> and <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/92.svg" alt=" " data-tex="m"> alone it is +impossible to obtain a quantity which can be interpreted as a diameter +of an atom or as a frequency.</p> + +<p>If we attempt to account for the radiation of energy in the manner +required by the ordinary electrodynamics it will only make matters +worse. As a result of the radiation of energy <img style="vertical-align: -0.05ex; width: 2.371ex; height: 1.595ex;" src="images/93.svg" alt=" " data-tex="W"> would continually +increase, and the above expressions (4) show that at the same time +the frequency of revolution of the system would increase, and the +dimensions of the orbit decrease. This process would not stop until the +particles had approached so closely to one another that they no longer +attracted each other. The quantity of energy which would be radiated +away before this happened would be very great. If we were to treat +these particles as geometrical points this energy would be infinitely +great, and with the dimensions of the electrons as calculated from +their mass (about <span class="nowrap"><img style="vertical-align: -0.05ex; width: 9.191ex; height: 2.005ex;" src="images/94.svg" alt=" " data-tex="10^{-13}\, \text{cm}.">)</span>, and of the nucleus as +calculated by Rutherford (about <span class="nowrap"><img style="vertical-align: -0.05ex; width: 9.191ex; height: 2.005ex;" src="images/95.svg" alt=" " data-tex="10^{-12}\, \text{cm}.">)</span>, this energy +would be many times greater than the energy changes with which we are +familiar in ordinary atomic processes.</p> + +<p>It can be seen that it is impossible to employ Rutherford's atomic +model so long as we confine ourselves exclusively to the ordinary +electrodynamics. But this is nothing more than might have been +expected. As I have mentioned we may consider it to be an established +fact that it is impossible to obtain a satisfactory explanation +<span class="pagenum" id="Page_10">[Pg 10]</span> +of the experiments on temperature radiation with the aid of +electrodynamics, no matter what atomic model be employed. The fact +that the deficiencies of the atomic model we are considering stand out +so plainly is therefore perhaps no serious drawback; even though the +defects of other atomic models are much better concealed they must +nevertheless be present and will be just as serious.</p> + +<p class="space-above3"> +<b>Quantum theory of spectra.</b> Let us now try to overcome these +difficulties by applying Planck's theory to the problem.</p> + +<p>It is readily seen that there can be no question of a direct +application of Planck's theory. This theory is concerned with the +emission and absorption of energy in a system of electrical particles, +which oscillate with a given frequency per second, dependent only +on the nature of the system and independent of the amount of energy +contained in the system. In a system consisting of an electron and +a nucleus the period of oscillation corresponds to the period of +revolution of the electron. But the formula (4) for <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> shows +that the frequency of revolution depends upon <span class="nowrap"><img style="vertical-align: -0.05ex; width: 2.371ex; height: 1.595ex;" src="images/93.svg" alt=" " data-tex="W">,</span> i.e. on the +energy of the system. Still the fact that we cannot immediately apply +Planck's theory to our problem is not as serious as it might seem to +be, for in assuming Planck's theory we have manifestly acknowledged +the inadequacy of the ordinary electrodynamics and have definitely +parted with the coherent group of ideas on which the latter theory is +based. In fact in taking such a step we cannot expect that all cases of +disagreement between the theoretical conceptions hitherto employed and +experiment will be removed by the use of Planck's assumption regarding +the quantum of the energy momentarily present in an oscillating system. +We stand here almost entirely on virgin ground, and upon introducing +new assumptions we need only take care not to get into contradiction +with experiment. Time will have to show to what extent this can be +avoided; but the safest way is, of course, to make as few assumptions +as possible.</p> + +<p>With this in mind let us first examine the experiments on temperature +radiation. The subject of direct observation is the distribution of +radiant energy over oscillations of the various wave lengths. Even +though we may assume that this energy comes from systems of oscillating +particles, we know little or nothing about these systems. No one has +<span class="pagenum" id="Page_11">[Pg 11]</span> +ever seen a Planck's resonator, nor indeed even measured its frequency +of oscillation; we can observe only the period of oscillation of the +radiation which is emitted. It is therefore very convenient that it is +possible to show that to obtain the laws of temperature radiation it is +not necessary to make any assumptions about the systems which emit the +radiation except that the amount of energy emitted each time shall be +equal to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.502ex; height: 1.595ex;" src="images/86.svg" alt=" " data-tex="h\nu">,</span> where <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;" src="images/87.svg" alt=" " data-tex="h"> is Planck's constant and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/88.svg" alt=" " data-tex="\nu"> is the +frequency of the radiation. Indeed, it is possible to derive Planck's +law of radiation from this assumption alone, as shown by Debye, who +employed a method which is a combination of that of Planck and of +Jeans. Before considering any further the nature of the oscillating +systems let us see whether it is possible to bring this assumption +about the emission of radiation into agreement with the spectral laws.</p> + +<p>If the spectrum of some element contains a spectral line corresponding +to the frequency <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/88.svg" alt=" " data-tex="\nu"> it will be assumed that one of the atoms of +the element (or some other elementary system) can emit an amount of +energy <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.502ex; height: 1.595ex;" src="images/86.svg" alt=" " data-tex="h\nu">.</span> Denoting the energy of the atom before and after the +emission of the radiation by <img style="vertical-align: -0.339ex; width: 9.717ex; height: 1.91ex;" src="images/96.svg" alt=" " data-tex="E_{1}\, \text{and}\, E_{2}"> we have +<span class="align-center"><img style="vertical-align: -1.577ex; width: 41.721ex; height: 4.645ex;" src="images/10.svg" alt=" " data-tex=" +h\nu = E_{1} - E_{2} \text{ or }\quad +\nu = \frac{E_{1}}{h} - \frac{E_{2}}{h}. +\qquad\text{(5)} +"></span></p> + +<p>During the emission of the radiation the system may be regarded as +passing from one state to another; in order to introduce a name for +these states, we shall call them "stationary" states, simply indicating +thereby that they form some kind of waiting places between which occurs +the emission of the energy corresponding to the various spectral +lines. As previously mentioned the spectrum of an element consists of +a series of lines whose wave lengths may be expressed by the formula +(2). By comparing this expression with the relation given above it +is seen that—since <span class="nowrap"><img style="vertical-align: -1.579ex; width: 6.531ex; height: 4.109ex;" src="images/97.svg" alt=" " data-tex="\nu = \dfrac{c}{\lambda}">,</span> where <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/98.svg" alt=" " data-tex="c"> is the +velocity of light—each of the spectral lines may be regarded as being +emitted by the transition of a system between two stationary states in +which the energy apart from an additive arbitrary constant is given +by <img style="vertical-align: -0.566ex; width: 26.462ex; height: 2.262ex;" src="images/99.svg" alt=" " data-tex="-ch F_{r}(n_{1})\, \text{and}\, -ch F_{s}(n_{2})"> respectively. +Using this interpretation the combination principle asserts that a +series of stationary states exists for the given system, and that it +can pass from one to any other of these states with the emission +<span class="pagenum" id="Page_12">[Pg 12]</span> +of a monochromatic radiation. We see, therefore, that with a simple +extension of our first assumption it is possible to give a formal +explanation of the most general law of line spectra.</p> + +<p class="space-above3"> +<b>Hydrogen spectrum.</b> This result encourages us to make an attempt +to obtain a clear conception of the stationary states which have so far +only been regarded as formal. With this end in view, we naturally turn +to the spectrum of hydrogen. The formula applying to this spectrum is +given by the expression +<span class="align-center"><img style="vertical-align: -2.448ex; width: 15.407ex; height: 5.522ex;" src="images/11.svg" alt=" " data-tex=" +\frac{1}{\lambda} = \frac{R}{n_{1}^{2}} - \frac{R}{n_{2}^{2}}. +"></span> +According to our assumption this spectrum is produced by transitions +between a series of stationary states of a system, concerning which +we can for the present only say that the energy of the system in +the <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">th state, apart from an additive constant, is given by +<span class="nowrap"><img style="vertical-align: -1.654ex; width: 6.756ex; height: 4.753ex;" src="images/100.svg" alt=" " data-tex="-\dfrac{Rhc}{n^{2}}">.</span> Let us now try to find a connection between +this and the model of the hydrogen atom. We assume that in the +calculation of the frequency of revolution of the electron in the +stationary states of the atom it will be possible to employ the above +formula for <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega">.</span> It is quite natural to make this assumption; +since, in trying to form a reasonable conception of the stationary +states, there is, for the present at least, no other means available +besides the ordinary mechanics.</p> + +<p>Corresponding to the <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">th stationary state in formula (4) for +<span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega">,</span> let us by way of experiment put <span class="nowrap"><img style="vertical-align: -1.654ex; width: 10.384ex; height: 4.753ex;" src="images/101.svg" alt=" " data-tex="W = \dfrac{Rhc}{n^{2}}">.</span> +This gives us +<span class="align-center"><img style="vertical-align: -1.671ex; width: 25.603ex; height: 5.086ex;" src="images/12.svg" alt=" " data-tex=" +\omega_{n}^{2} = \frac{2}{\pi^{2}}\, \frac{R^{3} h^{3} c^{3}}{e^{4} mn^{6}}. +\qquad\text{(6)} +"></span></p> + +<p>The radiation of light corresponding to a particular spectral line +is according to our assumption emitted by a transition between two +stationary states, corresponding to two different frequencies of +revolution, and we are not justified in expecting any simple relation +between these frequencies of revolution of the electron and the +frequency of the emitted radiation. You understand, of course, that +I am by no means trying to give what might ordinarily be described +as an explanation; nothing has been said here about how or why the +<span class="pagenum" id="Page_13">[Pg 13]</span> +radiation is emitted. On one point, however, we may expect a connection +with the ordinary conceptions; namely, that it will be possible to +calculate the emission of slow electromagnetic oscillations on the +basis of the classical electrodynamics. This assumption is very +strongly supported by the result of Lorentz's calculations which have +already been described. From the formula for <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> it is seen that +the frequency of revolution decreases as <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> increases, and that the +expression <img style="vertical-align: -2.023ex; width: 5.595ex; height: 4.554ex;" src="images/102.svg" alt=" " data-tex="\dfrac{\omega_{n}}{\omega_{n+1}}"> approaches the value +<span class="nowrap"><img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/103.svg" alt=" " data-tex="1">.</span></p> + +<p>According to what has been said above, the frequency of the radiation +corresponding to the transition between the <img style="vertical-align: -0.566ex; width: 7.015ex; height: 2.262ex;" src="images/104.svg" alt=" " data-tex="(n + 1)">th and the +<img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">th stationary state is given by +<span class="align-center"><img style="vertical-align: -2.194ex; width: 26.353ex; height: 5.474ex;" src="images/13.svg" alt=" " data-tex=" +\nu = Rc \left(\frac{1}{n^{2}} - \frac{1}{(n + 1)^{2}}\right). +"></span> +If <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> is very large this expression is approximately equal to +<span class="align-center"><img style="vertical-align: -0.566ex; width: 12.15ex; height: 2.564ex;" src="images/14.svg" alt=" " data-tex=" +\nu = 2Rc/n^{3}. +"></span> +In order to obtain a connection with the ordinary electrodynamics let +us now place this frequency equal to the frequency of revolution, that +is +<span class="align-center"><img style="vertical-align: -0.566ex; width: 13.505ex; height: 2.564ex;" src="images/15.svg" alt=" " data-tex=" +\omega_{n} = 2Rc/n^{3}. +"></span> +Introducing this value of <img style="vertical-align: -0.357ex; width: 2.555ex; height: 1.359ex;" src="images/105.svg" alt=" " data-tex="\omega_{n}"> in (6) we see that <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> +disappears from the equation, and further that the equation will be +satisfied only if +<span class="align-center"><img style="vertical-align: -1.652ex; width: 21.589ex; height: 5.086ex;" src="images/16.svg" alt=" " data-tex=" +R = \frac{2\pi^{2} e^{4} m}{ch^{3}}. +\qquad\text{(7)} +"></span> +The constant <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/66.svg" alt=" " data-tex="R"> is very accurately known, and is, as I have said +before, equal to <span class="nowrap"><img style="vertical-align: -0.05ex; width: 6.787ex; height: 1.579ex;" src="images/77.svg" alt=" " data-tex="109675">.</span> By introducing the most recent values for +<span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;" src="images/91.svg" alt=" " data-tex="e">,</span> <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/92.svg" alt=" " data-tex="m"> and <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;" src="images/87.svg" alt=" " data-tex="h"> the expression on the right-hand side of the +equation becomes equal to <span class="nowrap"><img style="vertical-align: -0.05ex; width: 8.907ex; height: 2.005ex;" src="images/106.svg" alt=" " data-tex="1.09 · 10^{5}">.</span> The agreement is as good +as could be expected, considering the uncertainty in the experimental +determination of the constants <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;" src="images/91.svg" alt=" " data-tex="e">,</span> <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/92.svg" alt=" " data-tex="m"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;" src="images/87.svg" alt=" " data-tex="h">.</span> The agreement +between our calculations and the classical electrodynamics is, +therefore, fully as good as we are justified in expecting.</p> + +<p>We cannot expect to obtain a corresponding explanation of the frequency +values of the other stationary states. Certain simple formal relations +apply, however, to all the stationary states. By introducing the +expression, which has been found for <span class="nowrap"><img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/66.svg" alt=" " data-tex="R">,</span> we get for the <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">th +<span class="pagenum" id="Page_14">[Pg 14]</span> +state <span class="nowrap"><img style="vertical-align: -0.781ex; width: 13.311ex; height: 2.737ex;" src="images/107.svg" alt=" " data-tex="W_{n} = \frac{1}{2}nh\omega_{n}">.</span> This equation is entirely +analogous to Planck's assumption concerning the energy of a resonator. +<img style="vertical-align: -0.05ex; width: 2.371ex; height: 1.595ex;" src="images/93.svg" alt=" " data-tex="W"> in our system is readily shown to be equal to the average value +of the kinetic energy of the electron during a single revolution. +The energy of a resonator was shown by Planck you may remember to be +always equal to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 3.86ex; height: 1.595ex;" src="images/108.svg" alt=" " data-tex="nh\nu">.</span> Further the average value of the kinetic +energy of Planck's resonator is equal to its potential energy, so that +the average value of the kinetic energy of the resonator, according +to Planck, is equal to <span class="nowrap"><img style="vertical-align: -0.781ex; width: 5.863ex; height: 2.737ex;" src="images/109.svg" alt=" " data-tex="\frac{1}{2}nh\omega">.</span> This analogy suggests +another manner of presenting the theory, and it was just in this way +that I was originally led into these considerations. When we consider +how differently the equation is employed here and in Planck's theory +it appears to me misleading to use this analogy as a foundation, and +in the account I have given I have tried to free myself as much as +possible from it.</p> + +<p>Let us continue with the elucidation of the calculations, and in the +expression for <img style="vertical-align: -0.023ex; width: 2.328ex; height: 1.529ex;" src="images/90.svg" alt=" " data-tex="2a"> introduce the value of <img style="vertical-align: -0.05ex; width: 2.371ex; height: 1.595ex;" src="images/93.svg" alt=" " data-tex="W"> which corresponds to +the <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">th stationary state. This gives us +<span class="align-center"><img style="vertical-align: -1.654ex; width: 54.189ex; height: 5.07ex;" src="images/17.svg" alt=" " data-tex=" +2a = n^{2} · \frac{e^{2}}{chR} + = n^{2} · \frac{h^{2}}{2\pi^{2} me^{2}} + = n^{2} · 1.1 · 10^{-8}. +\qquad\text{(8)} +"></span> +</p> + +<p>It is seen that for small values of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">,</span> we obtain values for the +major axis of the orbit of the electron which are of the same order +of magnitude as the values of the diameters of the atoms calculated +from the kinetic theory of gases. For large values of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">,</span> <img style="vertical-align: -0.023ex; width: 2.328ex; height: 1.529ex;" src="images/90.svg" alt=" " data-tex="2a"> +becomes very large in proportion to the calculated dimensions of the +atoms. This, however, does not necessarily disagree with experiment. +Under ordinary circumstances a hydrogen atom will probably exist +only in the state corresponding to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.506ex; height: 1.692ex;" src="images/110.svg" alt=" " data-tex="n = 1">.</span> For this state <img style="vertical-align: -0.05ex; width: 2.371ex; height: 1.595ex;" src="images/93.svg" alt=" " data-tex="W"> +will have its greatest value and, consequently, the atom will have +emitted the largest amount of energy possible; this will therefore +represent the most stable state of the atom from which the system +cannot be transferred except by adding energy to it from without. +The large values for <img style="vertical-align: -0.023ex; width: 2.328ex; height: 1.529ex;" src="images/90.svg" alt=" " data-tex="2a"> corresponding to large <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> need not, +therefore, be contrary to experiment; indeed, we may in these large +values seek an explanation of the fact, that in the laboratory it has +hitherto not been possible to observe the hydrogen lines corresponding +to large values of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> in Balmer's formula, while they have been +observed in the spectra of certain stars. In order that the large +orbits of the electrons may not be disturbed by electrical forces +<span class="pagenum" id="Page_15">[Pg 15]</span> +from the neighbouring atoms the pressure will have to be very low, so +low, indeed, that it is impossible to obtain sufficient light from a +Geissler tube of ordinary dimensions. In the stars, however, we may +assume that we have to do with hydrogen which is exceedingly attenuated +and distributed throughout an enormously large region of space.</p> + +<p class="space-above3"> +<b>The Pickering lines.</b> You have probably noticed that we have not +mentioned at all the spectrum found in certain stars which according +to the opinion then current was assigned to hydrogen, and together +with the ordinary hydrogen spectrum was considered by Rydberg to form +a connected system of lines completely analogous to the spectra of +other elements. You have probably also perceived that difficulties +would arise in interpreting this spectrum by means of the assumptions +which have been employed. If such an attempt were to be made it would +be necessary to give up the simple considerations which lead to the +expression (7) for the constant <span class="nowrap"><img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/66.svg" alt=" " data-tex="R">.</span> We shall see, however, that it +appears possible to explain the occurrence of this spectrum in another +way. Let us suppose that it is not due to hydrogen, but to some other +simple system consisting of a single electron revolving about a nucleus +with an electrical charge <span class="nowrap"><img style="vertical-align: -0.025ex; width: 3.063ex; height: 1.57ex;" src="images/111.svg" alt=" " data-tex="Ne">.</span> The expression for <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> becomes +then +<span class="align-center"><img style="vertical-align: -1.671ex; width: 17.835ex; height: 5.086ex;" src="images/18.svg" alt=" " data-tex=" +\omega^{2} = \frac{2}{\pi^{2}}\, \frac{W^{3}}{N^{2} e^{4} m}. +"></span> +Repeating the same calculations as before only in the inverse order we +find, that this system will emit a line spectrum given by the expression +<span class="align-center"><img style="vertical-align: -3.07ex; width: 62.839ex; height: 7.028ex;" src="images/19.svg" alt=" " data-tex=" +\frac{1}{\lambda} + = \frac{2\pi^{2} N^{2} e^{4} m}{ch^{3}}\left(\frac{1}{n_{1}^{2}} +- \frac{1}{n_{2}^{2}}\right) + = R \Biggl({\frac{1}{\left(\tfrac{n_{1}}{N}\right)^{2}} +- \frac{1}{\left(\tfrac{n_{2}}{N}\right)^{2}}\Biggr)}. +\qquad\text{(9)} +"></span> +</p> + +<p>By comparing this formula with the formula for Pickering's and +Rydberg's series, we see that the observed lines can be explained on +the basis of the theory, if it be assumed that the spectrum is due +to an electron revolving about a nucleus with a charge <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.186ex; height: 1.532ex;" src="images/112.svg" alt=" " data-tex="2e">,</span> or +according to Rutherford's theory around the nucleus of a helium atom. +The fact that the spectrum in question is not observed in an ordinary +helium tube, but only in stars, may be accounted for by the high +<span class="pagenum" id="Page_16">[Pg 16]</span> +degree of ionization which is required for the production of this +spectrum; a neutral helium atom contains of course two electrons while +the system under consideration contains only one.</p> + +<p>These conclusions appear to be supported by experiment. Fowler, as I +have mentioned, has recently succeeded in observing Pickering's and +Rydberg's lines in a laboratory experiment. By passing a very heavy +current through a mixture of hydrogen and helium Fowler observed not +only these lines but also a new series of lines. This new series was of +the same general type, the wave length being given approximately by +<span class="align-center"><img style="vertical-align: -2.827ex; width: 29.605ex; height: 6.785ex;" src="images/20.svg" alt=" " data-tex=" +\frac{1}{\lambda} + = R\left(\frac{1}{(\frac{3}{2})^{2}} +- \frac{1}{(n + \frac{1}{2})^{2}}\right). +"></span> +Fowler interpreted all the observed lines as the hydrogen spectrum +sought for. With the observation of the latter series of lines, +however, the basis of the analogy between the hypothetical hydrogen +spectrum and the other spectra disappeared, and thereby also the +foundation upon which Rydberg had founded his conclusions; on the +contrary it is seen, that the occurrence of the lines was exactly what +was to be expected on our view.</p> + +<p>In the following table the first column contains the wave lengths +measured by Fowler, while the second contains the limiting +values of the experimental errors given by him; in the third +column we find the products of the wave lengths by the quantity +<span class="nowrap"><img style="vertical-align: -2.827ex; width: 18.715ex; height: 6.785ex;" src="images/113.svg" alt=" " data-tex="\left(\dfrac{1}{n_{1}^{2}} - \dfrac{1}{n_{2}^{2}}\right) · 10^{10}">;</span> +the values employed for <img style="vertical-align: -0.339ex; width: 2.345ex; height: 1.339ex;" src="images/78.svg" alt=" " data-tex="n_{1}"> and <img style="vertical-align: -0.339ex; width: 2.345ex; height: 1.339ex;" src="images/79.svg" alt=" " data-tex="n_{2}"> are enclosed in +parentheses in the last column.</p> + +<table class="autotable"> + <thead><tr> + <th class="tdc"><img style="vertical-align: -0.05ex; width: 6.204ex; height: 2.005ex;" src="images/72.svg" alt=" " data-tex="\lambda · 10^{8}"><img style="vertical-align: 0; width: 4.525ex; height: 0.036ex;" src="images/70.svg" alt=" " data-tex="\qquad"></th> + <th class="tdc"><img style="vertical-align: -0.025ex; width: 13.188ex; height: 1.62ex;" src="images/114.svg" alt=" " data-tex="\text{Limit of error}"><img style="vertical-align: 0; width: 4.525ex; height: 0.036ex;" src="images/70.svg" alt=" " data-tex="\qquad"></th> + <th class="tdc"><img style="vertical-align: -2.827ex; width: 21.669ex; height: 6.785ex;" src="images/115.svg" alt=" " data-tex="\lambda · \left(\dfrac{1}{n_{1}^{2}} - \dfrac{1}{n_{2}^{2}}\right) · 10^{10}"><img style="vertical-align: 0; width: 4.525ex; height: 0.036ex;" src="images/70.svg" alt=" " data-tex="\qquad"></th> + <th class="tdc"> </th> + </tr> + </thead> + <tbody><tr> + <td class="tdl">4685.98 </td> + <td class="tdl"> 0.01 </td> + <td class="tdl"> 22779.1 </td> + <td class="tdl">(3 : 4)</td> + </tr><tr> + <td class="tdl">3203.30 </td> + <td class="tdl"> 0.05 </td> + <td class="tdl"> 22779.0 </td> + <td class="tdl">(3 : 5)</td> + </tr><tr> + <td class="tdl">2733.34 </td> + <td class="tdl"> 0.05 </td> + <td class="tdl"> 22777.8 </td> + <td class="tdl">(3 : 6)</td> + </tr><tr> + <td class="tdl">2511.31 </td> + <td class="tdl"> 0.05 </td> + <td class="tdl"> 22778.3 </td> + <td class="tdl">(3 : 7)</td> + </tr><tr> + <td class="tdl">2385.47 </td> + <td class="tdl"> 0.05 </td> + <td class="tdl"> 22777.9 </td> + <td class="tdl">(3 : 8)</td> + </tr><tr> + <td class="tdl">2306.20 </td> + <td class="tdl"> 0.10 </td> + <td class="tdl"> 22777.3 </td> + <td class="tdl">(3 : 9)</td> + </tr><tr> + <td class="tdl">2252.88 </td> + <td class="tdl"> 0.10 </td> + <td class="tdl"> 22779.1 </td> + <td class="tdl">(3 : 10)</td> + </tr><tr> + <td class="tdl">5410.5 </td> + <td class="tdl"> 1.0 </td> + <td class="tdl"> 22774 </td> + <td class="tdl">(4 : 7)</td> + </tr><tr> + <td class="tdl">4541.3 </td> + <td class="tdl"> 0.25 </td> + <td class="tdl"> 22777 </td> + <td class="tdl">(4 : 9)</td> + </tr><tr> + <td class="tdl">4200.3 </td> + <td class="tdl"> 0.5 </td> + <td class="tdl"> 22781 </td> + <td class="tdl">(4 : 11)</td> +</tr> + </tbody> +</table> + +<p><span class="pagenum" id="Page_17">[Pg 17]</span></p> + +<p>The values of the products are seen to be very nearly equal, while +the deviations are of the same order of magnitude as the limits of +experimental error. The value of the product +<span class="align-center"><img style="vertical-align: -2.827ex; width: 14.727ex; height: 6.785ex;" src="images/21.svg" alt=" " data-tex=" +\lambda \left(\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}\right) +"></span> +should for this spectrum, according to the formula (9), be exactly +<img style="vertical-align: -0.781ex; width: 1.795ex; height: 2.737ex;" src="images/116.svg" alt=" " data-tex="\frac{1}{4}"> of the corresponding product for the hydrogen spectrum. +From the tables on pages <a href="#Page_1">1</a> and <a href="#Page_16">16</a> we find for these products <img style="vertical-align: -0.05ex; width: 5.656ex; height: 1.557ex;" src="images/117.svg" alt=" " data-tex="91153"> +and <span class="nowrap"><img style="vertical-align: -0.05ex; width: 5.656ex; height: 1.579ex;" src="images/118.svg" alt=" " data-tex="22779">,</span> and dividing the former by the latter we get <span class="nowrap"><img style="vertical-align: -0.05ex; width: 6.285ex; height: 1.581ex;" src="images/119.svg" alt=" " data-tex="4.0016">.</span> +This value is very nearly equal to <span class="nowrap"><img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">;</span> the deviation is, however, +much greater than can be accounted for in any way by the errors of +the experiments. It has been easy, however, to find a theoretical +explanation of this point. In all the foregoing calculations we have +assumed that the mass of the nucleus is infinitely great compared to +that of the electron. This is of course not the case, even though it +holds to a very close approximation; for a hydrogen atom the ratio of +the mass of the nucleus to that of the electron will be about <img style="vertical-align: -0.05ex; width: 4.525ex; height: 1.557ex;" src="images/121.svg" alt=" " data-tex="1850"> +and for a helium atom four times as great.</p> + +<p>If we consider a system consisting of an electron revolving about a +nucleus with a charge <img style="vertical-align: -0.025ex; width: 3.063ex; height: 1.57ex;" src="images/111.svg" alt=" " data-tex="Ne"> and a mass <span class="nowrap"><img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/122.svg" alt=" " data-tex="M">,</span> we find the following +expression for the frequency of revolution of the system: +<span class="align-center"><img style="vertical-align: -1.671ex; width: 23.061ex; height: 5.163ex;" src="images/22.svg" alt=" " data-tex=" +\omega^{2} = \frac{2}{\pi^{2}}\, \frac{W^{3} (M + m)}{N^{2} e^{4} Mm}. +"></span></p> + +<p>From this formula we find in a manner quite similar to that previously +employed that the system will emit a line spectrum, the wave lengths of +which are given by the formula +<span class="align-center"><img style="vertical-align: -2.827ex; width: 41.845ex; height: 6.785ex;" src="images/23.svg" alt=" " data-tex=" +\frac{1}{\lambda} + = \frac{2\pi^{2} N^{2} e^{4} mM}{ch^{3} (M + m)} + \left(\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}\right). +\qquad\text{(10)} +"></span> +</p> + +<p>If with the aid of this formula we try to find the ratio of the product +for the hydrogen spectrum, to that of the hypothetical helium spectrum +we get the value <img style="vertical-align: -0.05ex; width: 7.416ex; height: 1.581ex;" src="images/123.svg" alt=" " data-tex="4.00163"> which is in complete agreement with the +preceding value calculated from the experimental observations.</p> + +<p>I must further mention that Evans has made some experiments to +determine whether the spectrum in question is due to hydrogen or +helium. He succeeded in observing one of the lines in very pure +<span class="pagenum" id="Page_18">[Pg 18]</span> +helium; there was, at any rate, not enough hydrogen present to enable +the hydrogen lines to be observed. Since in any event Fowler does not +seem to consider such evidence as conclusive it is to be hoped that +these experiments will be continued. There is, however, also another +possibility of deciding this question. As is evident from the formula +(10), the helium spectrum under consideration should contain, besides +the lines observed by Fowler, a series of lines lying close to the +ordinary hydrogen lines. These lines may be obtained by putting +<span class="nowrap"><img style="vertical-align: -0.439ex; width: 20.532ex; height: 1.971ex;" src="images/124.svg" alt=" " data-tex="n_{1} = 4,\, n_{2} = 6,\, 8,\, 10">,</span> etc. Even if these lines were +present, it would be extremely difficult to observe them on account of +their position with regard to the hydrogen lines, but should they be +observed this would probably also settle the question of the origin +of the spectrum, since no reason would seem to be left to assume the +spectrum to be due to hydrogen.</p> + +<p class="space-above3"> +<b>Other spectra.</b> For the spectra of other elements the problem +becomes more complicated, since the atoms contain a larger number of +electrons. It has not yet been possible on the basis of this theory to +explain any other spectra besides those which I have already mentioned. +On the other hand it ought to be mentioned that the general laws +applying to the spectra are very simply interpreted on the basis of +our assumptions. So far as the combination principle is concerned its +explanation is obvious. In the method we have employed our point of +departure was largely determined by this particular principle. But a +simple explanation can be also given of the other general law, namely, +the occurrence of Rydberg's constant in all spectral formulae. Let +us assume that the spectra under consideration, like the spectrum of +hydrogen, are emitted by a neutral system, and that they are produced +by the binding of an electron previously removed from the system. If +such an electron revolves about the nucleus in an orbit which is large +in proportion to that of the other electrons it will be subjected to +forces much the same as the electron in a hydrogen atom, since the +inner electrons individually will approximately neutralize the effect +of a part of the positive charge of the nucleus. We may therefore +assume that for this system there will exist a series of stationary +<span class="pagenum" id="Page_19">[Pg 19]</span> +states in which the motion of the outermost electron is approximately +the same as in the stationary states of a hydrogen atom. I shall not +discuss these matters any further, but shall only mention that they +lead to the conclusion that Rydberg's constant is not exactly the +same for all elements. The expression for this constant will in fact +contain the factor <span class="nowrap"><img style="vertical-align: -1.738ex; width: 8.125ex; height: 4.812ex;" src="images/125.svg" alt=" " data-tex="\dfrac{M}{M + m}">,</span> where <img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/122.svg" alt=" " data-tex="M"> is the mass of +the nucleus. The correction is exceedingly small for elements of +large atomic weight, but for hydrogen it is, from the point of view +of spectrum analysis, very considerable. If the procedure employed +leads to correct results, it is not therefore permissible to calculate +Rydberg's constant directly from the hydrogen spectrum; the value of +the universal constant should according to the theory be <img style="vertical-align: -0.05ex; width: 6.787ex; height: 1.579ex;" src="images/126.svg" alt=" " data-tex="109735"> and +not <span class="nowrap"><img style="vertical-align: -0.05ex; width: 6.787ex; height: 1.579ex;" src="images/77.svg" alt=" " data-tex="109675">.</span></p> + +<p>I shall not tire you any further with more details; I hope to return to +these questions here in the Physical Society, and to show how, on the +basis of the underlying ideas, it is possible to develop a theory for +the structure of atoms and molecules. Before closing I only wish to say +that I hope I have expressed myself sufficiently clearly so that you +have appreciated the extent to which these considerations conflict with +the admirably coherent group of conceptions which have been rightly +termed the classical theory of electrodynamics. On the other hand, by +emphasizing this conflict, I have tried to convey to you the impression +that it may be also possible in the course of time to discover a +certain coherence in the new ideas.</p> + + +<p><span class="pagenum" id="Page_20">[Pg 20]</span> + +</p><div class="footnote"> + +<p class="nind"> +<a id="Footnote_1" href="#FNanchor_1" class="label">[1]</a> +Address delivered before the Physical Society in +Copenhagen, Dec. 20, 1913.</p> + +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="ESSAY_II">ESSAY II<a id="FNanchor_2" href="#Footnote_2" class="fnanchor">[2]</a> +<br><br> +ON THE SERIES SPECTRA OF THE ELEMENTS +</h2> +</div> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="I_INTRODUCTION"> +I.INTRODUCTION</h2> +</div> + +<p>The subject on which I have the honour to speak here, at the kind +invitation of the Council of your society, is very extensive and it +would be impossible in a single address to give a comprehensive survey +of even the most important results obtained in the theory of spectra. +In what follows I shall try merely to emphasize some points of view +which seem to me important when considering the present state of the +theory of spectra and the possibilities of its development in the near +future. I regret in this connection not to have time to describe the +history of the development of spectral theories, although this would be +of interest for our purpose. No difficulty, however, in understanding +this lecture need be experienced on this account, since the points +of view underlying previous attempts to explain the spectra differ +fundamentally from those upon which the following considerations rest. +This difference exists both in the development of our ideas about the +structure of the atom and in the manner in which these ideas are used +in explaining the spectra.</p> + +<p>We shall assume, according to Rutherford's theory, that an atom +consists of a positively charged nucleus with a number of electrons +revolving about it. Although the nucleus is assumed to be very small +in proportion to the size of the whole atom, it will contain nearly +the entire mass of the atom. I shall not state the reasons which led +to the establishment of this <i>nuclear theory of the atom</i>, nor +describe the very strong support which this theory has received from +very different sources. I shall mention only that result which lends +such charm and simplicity to the modern development of the atomic +theory. I refer to the idea that the number of electrons in a neutral +atom is exactly equal to the number, giving the position of the +element in the periodic table, the so-called "atomic number." This +assumption, which was first proposed by van den Broek, immediately +suggests the possibility ultimately of deriving the explanation +<span class="pagenum" id="Page_21">[Pg 21]</span> +of the physical and chemical properties of the elements from their +atomic numbers. If, however, an explanation of this kind is attempted +on the basis of the classical laws of mechanics and electrodynamics, +insurmountable difficulties are encountered. These difficulties become +especially apparent when we consider the spectra of the elements. In +fact, the difficulties are here so obvious that it would be a waste of +time to discuss them in detail. It is evident that systems like the +nuclear atom, if based upon the usual mechanical and electrodynamical +conceptions, would not even possess sufficient stability to give a +spectrum consisting of sharp lines.</p> + +<p>In this lecture I shall use the ideas of the quantum theory. It will +not be necessary, particularly here in Berlin, to consider in detail +how Planck's fundamental work on temperature radiation has given rise +to this theory, according to which the laws governing atomic processes +exhibit a definite element of discontinuity. I shall mention only +Planck's chief result about the properties of an exceedingly simple +kind of atomic system, the Planck "oscillator." This consists of an +electrically charged particle which can execute harmonic oscillations +about its position of equilibrium with a frequency independent of the +amplitude. By studying the statistical equilibrium of a number of +such systems in a field of radiation Planck was led to the conclusion +that the emission and absorption of radiation take place in such a +manner, that, so far as a statistical equilibrium is concerned only +certain distinctive states of the oscillator are to be taken into +consideration. In these states the energy of the system is equal to a +whole multiple of a so-called "energy quantum," which was found to be +proportional to the frequency of the oscillator. The particular energy +values are therefore given by the well-known formula +<span class="align-center"><img style="vertical-align: -0.566ex; width: 18.325ex; height: 2.262ex;" src="images/24.svg" alt=" " data-tex=" +E_{n} = nh\omega, +\qquad\text{(1)} +"></span> +where <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> is a whole number, <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> the frequency of vibration of +the oscillator, and <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;" src="images/87.svg" alt=" " data-tex="h"> is Planck's constant.</p> + +<p>If we attempt to use this result to explain the spectra of the +elements, however, we encounter difficulties, because the motion of +the particles in the atom, in spite of its simple structure, is in +general exceedingly complicated compared with the motion of a Planck +<span class="pagenum" id="Page_22">[Pg 22]</span> +oscillator. The question then arises, how Planck's result ought to be +generalized in order to make its application possible. Different points +of view immediately suggest themselves. Thus we might regard this +equation as a relation expressing certain characteristic properties +of the distinctive motions of an atomic system and try to obtain the +general form of these properties. On the other hand, we may also +regard equation (1) as a statement about a property of the process of +radiation and inquire into the general laws which control this process.</p> + +<p>In Planck's theory it is taken for granted that the frequency of the +radiation emitted and absorbed by the oscillator is equal to its own +frequency, an assumption which may be written +<span class="align-center"><img style="vertical-align: -0.566ex; width: 14.046ex; height: 2.262ex;" src="images/25.svg" alt=" " data-tex=" +\nu \equiv \omega, +\qquad\text{(2)} +"></span> +if in order to make a sharp distinction between the frequency of the +emitted radiation and the frequency of the particles in the atoms, +we here and in the following denote the former by <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/88.svg" alt=" " data-tex="\nu"> and the +latter by <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega">.</span> We see, therefore, that Planck's result may be +interpreted to mean, that the oscillator can emit and absorb radiation +only in "radiation quanta" of magnitude +<span class="align-center"><img style="vertical-align: -0.566ex; width: 17.555ex; height: 2.262ex;" src="images/26.svg" alt=" " data-tex=" +\Delta E = h\nu. +\qquad\text{(3)} +"></span> +<span class="pagenum" id="Page_23">[Pg 23]</span> +It is well known that ideas of this kind led Einstein to a theory +of the photoelectric effect. This is of great importance, since it +represents the first instance in which the quantum theory was applied +to a phenomenon of non-statistical character. I shall not here discuss +the familiar difficulties to which the "hypothesis of light quanta" +leads in connection with the phenomena of interference, for the +explanation of which the classical theory of radiation has shown itself +to be so remarkably suited. Above all I shall not consider the problem +of the nature of radiation, I shall only attempt to show how it has +been possible in a purely formal manner to develop a spectral theory, +the essential elements of which may be considered as a simultaneous +rational development of the two ways of interpreting Planck's result.</p> + + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="II_GENERAL_PRINCIPLES_OF_THE_QUANTUM_THEORY"> +II. GENERAL PRINCIPLES OF THE QUANTUM THEORY +SPECTRA</h2> +</div> + + +<p>In order to explain the appearance of line spectra we are compelled +to assume that the emission of radiation by an atomic system takes +place in such a manner that it is not possible to follow the emission +in detail by means of the usual conceptions. Indeed, these do not +even offer us the means of calculating the frequency of the emitted +radiation. We shall see, however, that it is possible to give a very +simple explanation of the general empirical laws for the frequencies +of the spectral lines, if for each emission of radiation by the atom +we assume the fundamental law to hold, that during the entire period +of the emission the radiation possesses one and the same frequency +<span class="nowrap"><img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/88.svg" alt=" " data-tex="\nu">,</span> connected with the total energy emitted by the <i>frequency +relation</i> +<span class="align-center"><img style="vertical-align: -0.566ex; width: 22.031ex; height: 2.262ex;" src="images/27.svg" alt=" " data-tex=" +h\nu = E′ - E″. +\qquad\text{(4)} +"></span> +Here <img style="vertical-align: 0; width: 2.351ex; height: 1.538ex;" src="images/127.svg" alt=" " data-tex="E′"> and <img style="vertical-align: 0; width: 2.973ex; height: 1.538ex;" src="images/128.svg" alt=" " data-tex="E″"> represent the energy of the system before and +after the emission.</p> + +<p>If this law is assumed, the spectra do not give us information about +the motion of the particles in the atom, as is supposed in the usual +theory of radiation, but only a knowledge of the energy changes in +the various processes which can occur in the atom. From this point of +view the spectra show the existence of certain, definite energy values +corresponding to certain distinctive states of the atoms. These states +will be called the <i>stationary states</i> of the atoms, since we +shall assume that the atom can remain a finite time in each state, +and can leave this state only by a process of transition to another +stationary state. Notwithstanding the fundamental departure from the +ordinary mechanical and electrodynamical conceptions, we shall see, +however, that it is possible to give a rational interpretation of the +evidence provided by the spectra on the basis of these ideas.</p> + +<p>Although we must assume that the ordinary mechanics cannot be used to +describe the transitions between the stationary states, nevertheless, +it has been found possible to develop a consistent theory on the +assumption that the motion in these states can be described by the +use of the ordinary mechanics. Moreover, although the process of +radiation cannot be described on the basis of the ordinary theory of +<span class="pagenum" id="Page_24">[Pg 24]</span> +electrodynamics, according to which the nature of the radiation emitted +by an atom is directly related to the harmonic components occurring +in the motion of the system, there is found, nevertheless, to exist +a far-reaching <i>correspondence</i> between the various types of +possible transitions between the stationary states on the one hand and +the various harmonic components of the motion on the other hand. This +correspondence is of such a nature, that the present theory of spectra +is in a certain sense to be regarded as a rational generalization of +the ordinary theory of radiation.</p> + +<p class="space-above3"> +<b>Hydrogen spectrum.</b> In order that the principal points may +stand out as clearly as possible I shall, before considering the more +complicated types of series spectra, first consider the simplest +spectrum, namely, the series spectrum of hydrogen. This spectrum +consists of a number of lines whose frequencies are given with great +exactness by Balmer's formula +<span class="align-center"><img style="vertical-align: -2.194ex; width: 27.472ex; height: 5.269ex;" src="images/28.svg" alt=" " data-tex=" +\nu = \frac{K}{(n″)^{2}} - \frac{K}{(n′)^{2}}, +\qquad\text{(5)} +"></span> +where <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K"> is a constant, and <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> and <img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″"> are whole numbers. +If we put <img style="vertical-align: -0.186ex; width: 6.75ex; height: 1.692ex;" src="images/132.svg" alt=" " data-tex="n″=2"> and give to <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> the values <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">,</span> <span class="nowrap"><img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">,</span> etc., +we get the well-known Balmer series of hydrogen. If we put <img style="vertical-align: -0.186ex; width: 6.75ex; height: 1.692ex;" src="images/134.svg" alt=" " data-tex="n″=1"> +or <img style="vertical-align: -0.186ex; width: 6.75ex; height: 1.69ex;" src="images/135.svg" alt=" " data-tex="n″=3"> we obtain respectively the ultra-violet and infra-red +series. We shall assume the hydrogen atom simply to consist of a +positively charged nucleus with a single electron revolving about it. +For the sake of simplicity we shall suppose the mass of the nucleus to +be infinite in comparison with the mass of the electron, and further we +shall disregard the small variations in the motion due to the change +in mass of the electron with its velocity. With these simplifications +the electron will describe a closed elliptical orbit with the nucleus +at one of the foci. The frequency of revolution <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> and the +major axis <img style="vertical-align: -0.023ex; width: 2.328ex; height: 1.529ex;" src="images/90.svg" alt=" " data-tex="2a"> of the orbit will be connected with the energy of the +system by the following equations: +<span class="align-center"><img style="vertical-align: -2.345ex; width: 34.436ex; height: 6.923ex;" src="images/29.svg" alt=" " data-tex=" +\omega = \sqrt{\frac{2W^{3}}{\pi^{2} e^{4} m}},\quad +2a = \frac{e^{2}}{W}. +\qquad\text{(6)} +"></span> +Here <img style="vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;" src="images/91.svg" alt=" " data-tex="e"> is the charge of the electron and <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/92.svg" alt=" " data-tex="m"> its mass, while +<img style="vertical-align: -0.05ex; width: 2.371ex; height: 1.595ex;" src="images/93.svg" alt=" " data-tex="W"> is the work required to remove the electron to infinity.</p> + +<p>The simplicity of these formulae suggests the possibility of using +them in an attempt to explain the spectrum of hydrogen. This, +<span class="pagenum" id="Page_25">[Pg 25]</span> +however, is not possible so long as we use the classical theory of +radiation. It would not even be possible to understand how hydrogen +could emit a spectrum consisting of sharp lines; for since <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> +varies with <span class="nowrap"><img style="vertical-align: -0.05ex; width: 2.371ex; height: 1.595ex;" src="images/93.svg" alt=" " data-tex="W">,</span> the frequency of the emitted radiation would vary +continuously during the emission. We can avoid these difficulties if +we use the ideas of the quantum theory. If for each line we form the +product <img style="vertical-align: -0.025ex; width: 2.502ex; height: 1.595ex;" src="images/86.svg" alt=" " data-tex="h\nu"> by multiplying both sides of (5) by <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;" src="images/87.svg" alt=" " data-tex="h">,</span> then, since +the right-hand side of the resulting relation may be written as the +difference of two simple expressions, we are led by comparison with +formula (4) to the assumption that the separate lines of the spectrum +will be emitted by transitions between two stationary states, forming +members of an infinite series of states, in which the energy in the +<img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">th state apart from an arbitrary additive constant is determined +by the expression +<span class="align-center"><img style="vertical-align: -1.654ex; width: 20.327ex; height: 4.753ex;" src="images/30.svg" alt=" " data-tex=" +E_{n} = -\frac{Kh}{n^{2}}. +\qquad\text{(7)} +"></span> +The negative sign has been chosen because the energy of the atom +will be most simply characterized by the work <img style="vertical-align: -0.05ex; width: 2.371ex; height: 1.595ex;" src="images/93.svg" alt=" " data-tex="W"> required to +remove the electron completely from the atom. If we now substitute +<img style="vertical-align: -1.654ex; width: 4.31ex; height: 4.753ex;" src="images/136.svg" alt=" " data-tex="\dfrac{Kh}{n^{2}}"> for <img style="vertical-align: -0.05ex; width: 2.371ex; height: 1.595ex;" src="images/93.svg" alt=" " data-tex="W"> in formula (6), we obtain the following +expression for the frequency and the major axis in the <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">th +stationary state: +<span class="align-center"><img style="vertical-align: -2.345ex; width: 42.321ex; height: 6.923ex;" src="images/31.svg" alt=" " data-tex=" +\omega_{n} = \frac{1}{n^{3}} \sqrt{\frac{2h^{3} K^{3}}{\pi^{2} e^{4} m}},\quad +2a_{n} = \frac{n^{2} e^{2}}{hK}. +\qquad\text{(8)} +"></span> +A comparison between the motions determined by these equations and +the distinctive states of a Planck resonator may be shown to offer a +theoretical determination of the constant <span class="nowrap"><img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K">.</span> Instead of doing this +I shall show how the value of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K"> can be found by a simple comparison +of the spectrum emitted with the motion in the stationary states, a +comparison which at the same time will lead us to the principle of +correspondence.</p> + +<p>We have assumed that each hydrogen line is the result of a transition +between two stationary states of the atom corresponding to different +values of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">.</span> Equations (8) show that the frequency of revolution +and the major axis of the orbit can be entirely different in the two +states, since, as the energy decreases, the major axis of the orbit +becomes smaller and the frequency of revolution increases. In +<span class="pagenum" id="Page_26">[Pg 26]</span> +general, therefore, it will be impossible to obtain a relation between +the frequency of revolution of the electrons and the frequency of the +radiation as in the ordinary theory of radiation. If, however, we +consider the ratio of the frequencies of revolution in two stationary +states corresponding to given values of <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″">,</span> we see that +this ratio approaches unity as <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> and <img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″"> gradually increase, +if at the same time the difference <img style="vertical-align: -0.186ex; width: 7.347ex; height: 1.505ex;" src="images/137.svg" alt=" " data-tex="n′ - n″"> remains unchanged. By +considering transitions corresponding to large values of <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> and +<img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″"> we may therefore hope to establish a certain connection with +the ordinary theory. For the frequency of the radiation emitted by a +transition, we get according to (5) +<span class="align-center"><img style="vertical-align: -2.194ex; width: 53.059ex; height: 5.269ex;" src="images/32.svg" alt=" " data-tex=" +\nu = \frac{K}{(n″)^{2}} - \frac{K}{(n′)^{2}} + = (n′ - n″) K\, \frac{n′ + n″}{(n′)^{2} (n″)^{2}}. +\qquad\text{(9)} +"></span> +If now the numbers <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> and <img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″"> are large in proportion to their +difference, we see that by equations (8) this expression may be written +approximately, +<span class="align-center"><img style="vertical-align: -2.326ex; width: 35.78ex; height: 6.923ex;" src="images/33.svg" alt=" " data-tex=" +\nu \sim (n′ - n″)\, \omega\, \sqrt{\frac{2\pi^{2} e^{4} m}{Kh^{3}}}, +\qquad\text{(10)} +"></span> +where <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> represents the frequency of revolution in the one or +the other of the two stationary states. Since <img style="vertical-align: -0.186ex; width: 7.347ex; height: 1.505ex;" src="images/137.svg" alt=" " data-tex="n′ - n″"> is a whole +number, we see that the first part of this expression, i.e. +<span class="nowrap"><img style="vertical-align: -0.566ex; width: 10.515ex; height: 2.262ex;" src="images/138.svg" alt=" " data-tex="(n′ - n″)\omega">,</span> is the same as the frequency of one of the harmonic +components into which the elliptical motion may be decomposed. This +involves the well-known result that for a system of particles having a +periodic motion of frequency <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega">,</span> the displacement <img style="vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;" src="images/139.svg" alt=" " data-tex="\xi"> of +the particles in a given direction in space may be represented as a +function of the time by a trigonometric series of the form +<span class="align-center"><img style="vertical-align: -1.018ex; width: 35.954ex; height: 3.167ex;" src="images/34.svg" alt=" " data-tex=" +\xi = \sum C_{\tau} \cos 2\pi(\tau\omega t + c_{\tau}), +\qquad\text{(11)} +"></span> +where the summation is to be extended over all positive integral values +of <span class="nowrap"><img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/140.svg" alt=" " data-tex="\tau">.</span></p> + +<p>We see, therefore, that the frequency of the radiation emitted by a +transition between two stationary states, for which the numbers <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> +and <img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″"> are large in proportion to their difference, will coincide +with the frequency of one of the components of the radiation, which +according to the ordinary ideas of radiation would be expected from the +motion of the atom in these states, provided the last factor on the +<span class="pagenum" id="Page_27">[Pg 27]</span> +right-hand side of equation (10) is equal to <span class="nowrap"><img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/103.svg" alt=" " data-tex="1">.</span> This condition, +which is identical to the condition +<span class="align-center"><img style="vertical-align: -1.652ex; width: 23.014ex; height: 5.086ex;" src="images/35.svg" alt=" " data-tex=" +K = \frac{2\pi^{2} e^{4} m}{h^{3}}, +\qquad\text{(12)} +"></span> +is in fact fulfilled, if we give to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K"> its value as found from +measurements on the hydrogen spectrum, and if for <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;" src="images/91.svg" alt=" " data-tex="e">,</span> <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/92.svg" alt=" " data-tex="m"> and +<img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;" src="images/87.svg" alt=" " data-tex="h"> we use the values obtained directly from experiment. This +agreement clearly gives us a <i>connection between the spectrum and the +atomic model of hydrogen</i>, which is as close as could reasonably be +expected considering the fundamental difference between the ideas of +the quantum theory and of the ordinary theory of radiation.</p> + +<p class="space-above2"> +<b>The correspondence principle.</b> Let us now consider somewhat more +closely this relation between the spectra one would expect on the +basis of the quantum theory, and on the ordinary theory of radiation. +The frequencies of the spectral lines calculated according to both +methods agree completely in the region where the stationary states +deviate only little from one another. We must not forget, however, that +the mechanism of emission in both cases is different. The different +frequencies corresponding to the various harmonic components of the +motion are emitted simultaneously according to the ordinary theory +of radiation and with a relative intensity depending directly upon +the ratio of the amplitudes of these oscillations. But according to +the quantum theory the various spectral lines are emitted by entirely +distinct processes, consisting of transitions from one stationary state +to various adjacent states, so that the radiation corresponding to the +<img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/140.svg" alt=" " data-tex="\tau">th "harmonic" will be emitted by a transition for which +<span class="nowrap"><img style="vertical-align: -0.186ex; width: 11.534ex; height: 1.505ex;" src="images/141.svg" alt=" " data-tex="n′ - n″ = \tau">.</span> The relative intensity with which each particular line +is emitted depends consequently upon the relative probability of the +occurrence of the different transitions.</p> + +<p>This correspondence between the frequencies determined by the two +methods must have a deeper significance and we are led to anticipate +that it will also apply to the intensities. This is equivalent to +the statement that, when the quantum numbers are large, the relative +probability of a particular transition is connected in a simple manner +with the amplitude of the corresponding harmonic component in the +motion.</p> + +<p><span class="pagenum" id="Page_28">[Pg 28]</span></p> + +<p>This peculiar relation suggests a <i>general law for the occurrence of +transitions between stationary states</i>. Thus we shall assume that +even when the quantum numbers are small the possibility of transition +between two stationary states is connected with the presence of a +certain harmonic component in the motion of the system. If the numbers +<img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> and <img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″"> are not large in proportion to their difference, +the numerical value of the amplitudes of these components in the two +stationary states may be entirely different. We must be prepared to +find, therefore, that the exact connection between the probability of +a transition and the amplitude of the corresponding harmonic component +in the motion is in general complicated like the connection between the +frequency of the radiation and that of the component. From this point +of view, for example, the green line <img style="vertical-align: -0.65ex; width: 2.973ex; height: 2.195ex;" src="images/142.svg" alt=" " data-tex="H_{\beta}"> of the hydrogen +spectrum which corresponds to a transition from the fourth to the +second stationary state may be considered in a certain sense to be an +"octave" of the red line <span class="nowrap"><img style="vertical-align: -0.357ex; width: 3.092ex; height: 1.902ex;" src="images/143.svg" alt=" " data-tex="H_{\alpha}">,</span> corresponding to a transition +from the third to the second state, even though the frequency of the +first line is by no means twice as great as that of the latter. In +fact, the transition giving rise to <img style="vertical-align: -0.65ex; width: 2.973ex; height: 2.195ex;" src="images/142.svg" alt=" " data-tex="H_{\beta}"> may be regarded as +due to the presence of a harmonic oscillation in the motion of the +atom, which is an octave higher than the oscillation giving rise to the +emission of <span class="nowrap"><img style="vertical-align: -0.357ex; width: 3.092ex; height: 1.902ex;" src="images/143.svg" alt=" " data-tex="H_{\alpha}">.</span></p> + +<p>Before considering other spectra, where numerous opportunities will be +found to use this point of view, I shall briefly mention an interesting +application to the Planck oscillator. If from (1) and (4) we calculate +the frequency, which would correspond to a transition between two +particular states of such an oscillator, we find +<span class="align-center"><img style="vertical-align: -0.566ex; width: 24.662ex; height: 2.262ex;" src="images/36.svg" alt=" " data-tex=" +\nu = (n′ - n″)\,\omega, +\qquad\text{(13)} +"></span> +where <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> and <img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″"> are the numbers characterizing the states. It +was an essential assumption in Planck's theory that the frequency of +the radiation emitted and absorbed by the oscillator is always equal to +<span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega">.</span> We see that this assumption is equivalent to the assertion +that transitions occur only between two successive stationary states in +sharp contrast to the hydrogen atom. According to our view, however, +this was exactly what might have been expected, for we must assume that +the essential difference between the oscillator and the hydrogen atom +is that the motion of the oscillator is simple harmonic. We can see +<span class="pagenum" id="Page_29">[Pg 29]</span> +that it is possible to develop a formal theory of radiation, in which +the spectrum of hydrogen and the simple spectrum of a Planck oscillator +appear completely analogous. This theory can only be formulated by +one and the same condition for a system as simple as the oscillator. +In general this condition breaks up into two parts, one concerning +the fixation of the stationary states, and the other relating to the +frequency of the radiation emitted by a transition between these states.</p> + +<p class="space-above2"> +<b>General spectral laws.</b> Although the series spectra of the +elements of higher atomic number have a more complicated structure +than the hydrogen spectrum, simple laws have been discovered showing a +remarkable analogy to the Balmer formula. Rydberg and Ritz showed that +the frequencies in the series spectra of many elements can be expressed +by a formula of the type +<span class="align-center"><img style="vertical-align: -0.566ex; width: 30.217ex; height: 2.262ex;" src="images/37.svg" alt=" " data-tex=" +\nu = f_{k″}(n″) - f_{k′}(n′), +\qquad\text{(14)} +"></span> +where <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> and <img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″"> are two whole numbers and <img style="vertical-align: -0.464ex; width: 2.57ex; height: 2.059ex;" src="images/144.svg" alt=" " data-tex="f_{k′}"> and +<img style="vertical-align: -0.464ex; width: 3.01ex; height: 2.059ex;" src="images/145.svg" alt=" " data-tex="f_{k″}"> are two functions belonging to a series of functions +characteristic of the element. These functions vary in a simple manner +with <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> and in particular converge to zero for increasing values +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">.</span> The various series of lines are obtained from this formula +by allowing the first term <img style="vertical-align: -0.566ex; width: 7.372ex; height: 2.262ex;" src="images/146.svg" alt=" " data-tex="f_{k″}(n″)"> to remain constant, while +a series of consecutive whole numbers are substituted for <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> in +the second term <span class="nowrap"><img style="vertical-align: -0.566ex; width: 6.31ex; height: 2.262ex;" src="images/147.svg" alt=" " data-tex="f_{k′}(n′)">.</span> According to the Ritz <i>combination +principle</i> the entire spectrum may then be obtained by forming +every possible combination of two values among all the quantities +<span class="nowrap"><img style="vertical-align: -0.566ex; width: 5.248ex; height: 2.262ex;" src="images/148.svg" alt=" " data-tex="f_{k}(n)">.</span></p> + +<p>The fact that the frequency of each line of the spectrum may be +written as the difference of two simple expressions depending upon +whole numbers suggests at once that the terms on the right-hand side +multiplied by <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;" src="images/87.svg" alt=" " data-tex="h"> may be placed equal to the energy in the various +stationary states of the atom. The existence in the spectra of the +other elements of a number of separate functions of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> compels us to +assume the presence not of one but of a number of series of stationary +states, the energy of the <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">th state of the <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">th series apart +from an arbitrary additive constant being given by +<span class="align-center"><img style="vertical-align: -0.566ex; width: 26.69ex; height: 2.262ex;" src="images/38.svg" alt=" " data-tex=" +E_{k}(n) = -h f_{k}(n). +\qquad\text{(15)} +"></span> +This complicated character of the ensemble of stationary states of +atoms of higher atomic number is exactly what was to be expected +<span class="pagenum" id="Page_30">[Pg 30]</span> +from the relation between the spectra calculated on the quantum +theory, and the decomposition of the motions of the atoms into +harmonic oscillations. From this point of view we may regard the +simple character of the stationary states of the hydrogen atom as +intimately connected with the simple periodic character of this atom. +Where the neutral atom contains more than one electron, we find much +more complicated motions with correspondingly complicated harmonic +components. We must therefore expect a more complicated ensemble of +stationary states, if we are still to have a corresponding relation +between the motions in the atom and the spectrum. In the course of the +lecture we shall trace this correspondence in detail, and we shall +be led to a simple explanation of the apparent capriciousness in the +occurrence of lines predicted by the combination principle.</p> + +<p>The following figure gives a survey of the stationary states of the +sodium atom deduced from the series terms.</p> + +<div class="figcenter"> +<img src="images/001.jpg" width="400" alt="fig01"> +<div class="caption"> +<p>Diagram of the series spectrum of sodium.</p> +</div></div> + +<p>The stationary states are represented by black dots whose distance from +the vertical line a—a is proportional to the numerical value of the +energy in the states. The arrows in the figure indicate the transitions +giving those lines of the sodium spectrum which appear under the usual +conditions of excitation. The arrangement of the states in horizontal +rows corresponds to the ordinary arrangement of the "spectral terms" +in the spectroscopic tables. Thus, the states in the first row (<span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;" src="images/150.svg" alt=" " data-tex="S">)</span> +correspond to the variable term in the "sharp series," the lines of +which are emitted by transitions from these states to the first state +in the second row. The states in the second row (<span class="nowrap"><img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P">)</span> correspond +<span class="pagenum" id="Page_31">[Pg 31]</span> +to the variable term in the "principal series" which is emitted by +transitions from these states to the first state in the <img style="vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;" src="images/150.svg" alt=" " data-tex="S"> row. The +<img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/152.svg" alt=" " data-tex="D"> states correspond to the variable term in the "diffuse series," +which like the sharp series is emitted by transitions to the first +state in the <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P"> row, and finally the <img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/153.svg" alt=" " data-tex="B"> states correspond to +the variable term in the "Bergmann" series (fundamental series), in +which transitions take place to the first state in the <img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/152.svg" alt=" " data-tex="D"> row. The +manner in which the various rows are arranged with reference to one +another will be used to illustrate the more detailed theory which will +be discussed later. The apparent capriciousness of the combination +principle, which I mentioned, consists in the fact that under the +usual conditions of excitation not all the lines belonging to possible +combinations of the terms of the sodium spectrum appear, but only those +indicated in the figure by arrows.</p> + +<p>The general question of the fixation of the stationary states of an +atom containing several electrons presents difficulties of a profound +character which are perhaps still far from completely solved. It is +possible, however, to obtain an immediate insight into the stationary +states involved in the emission of the series spectra by considering +the empirical laws which have been discovered about the spectral terms. +According to the well-known law discovered by Rydberg for the spectra +of elements emitted under the usual conditions of excitation the +functions <img style="vertical-align: -0.566ex; width: 5.248ex; height: 2.262ex;" src="images/148.svg" alt=" " data-tex="f_{k}(n)"> appearing in formula (14) can be written in the +form +<span class="align-center"><img style="vertical-align: -1.654ex; width: 26.646ex; height: 4.728ex;" src="images/39.svg" alt=" " data-tex=" +f_{k}(n) = \frac{K}{n^{2}} \phi_{k}(n), +\qquad\text{(16)} +"></span> +<span class="pagenum" id="Page_32">[Pg 32]</span> +where <img style="vertical-align: -0.566ex; width: 5.487ex; height: 2.262ex;" src="images/154.svg" alt=" " data-tex="\phi_{k}(n)"> represents a function which converges to unity +for large values of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">.</span> <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K"> is the same constant which appears in +formula (5) for the spectrum of hydrogen. This result must evidently +be explained by supposing the atom to be electrically neutral in these +states and one electron to be moving round the nucleus in an orbit +the dimensions of which are very large in proportion to the distance +of the other electrons from the nucleus. We see, indeed, that in this +case the electric force acting on the outer electron will to a first +approximation be the same as that acting upon the electron in the +hydrogen atom, and the approximation will be the better the larger the +orbit.</p> + +<p>On account of the limited time I shall not discuss how this explanation +of the universal appearance of Rydberg's constant in the arc spectra +is convincingly supported by the investigation of the "spark spectra." +These are emitted by the elements under the influence of very strong +electrical discharges, and come from ionized not neutral atoms. It is +important, however, that I should indicate briefly how the fundamental +ideas of the theory and the assumption that in the states corresponding +to the spectra one electron moves in an orbit around the others, are +both supported by investigations on selective absorption and the +excitation of spectral lines by bombardment by electrons.</p> + +<p class="space-above2"> +<b>Absorption and excitation of radiation.</b> Just as we have +assumed that each emission of radiation is due to a transition from +a stationary state of higher to one of lower energy, so also we must +assume absorption of radiation by the atom to be due to a transition in +the opposite direction. For an element to absorb light corresponding to +a given line in its series spectrum, it is therefore necessary for the +atom of this element to be in that one of the two states connected with +the line possessing the smaller energy value. If we now consider an +element whose atoms in the gaseous state do not combine into molecules, +it will be necessary to assume that under ordinary conditions nearly +all the atoms exist in that stationary state in which the value +of the energy is a minimum. This state I shall call the <i>normal +state</i>. We must therefore expect that the absorption spectrum of a +monatomic gas will contain only those lines of the series spectrum, +whose emission corresponds to transitions to the normal state. This +expectation is completely confirmed by the spectra of the alkali +metals. The absorption spectrum of sodium vapour, for example, exhibits +lines corresponding only to the principal series, which as mentioned in +the description of the figure corresponds with transitions to the state +of minimum energy. Further confirmation of this view of the process of +absorption is given by experiments on <i>resonance radiation</i>. Wood +first showed that sodium vapour subjected to light corresponding to the +first line of the principal series—the familiar yellow line—acquires +the ability of again emitting a radiation consisting only of the light +of this line. We can explain this by supposing the sodium atom to +<span class="pagenum" id="Page_33">[Pg 33]</span> +have been transferred from the normal state to the first state in the +second row. The fact that the resonance radiation does not exhibit +the same degree of polarization as the incident light is in perfect +agreement with our assumption that the radiation from the excited +vapour is not a resonance phenomenon in the sense of the ordinary +theory of radiation, but on the contrary depends on a process which is +not directly connected with the incident radiation.</p> + +<p>The phenomenon of the resonance radiation of the yellow sodium line is, +however, not quite so simple as I have indicated, since, as you know, +this line is really a doublet. This means that the variable terms of +the principal series are not simple but are represented by two values +slightly different from one another. According to our picture of the +origin of the sodium spectrum this means that the <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P"> states in the +second row in the figure—as opposed to the <img style="vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;" src="images/150.svg" alt=" " data-tex="S"> states in the first +row—are not simple, but that for each place in this row there are two +stationary states. The energy values differ so little from one another +that it is impossible to represent them in the figure as separate dots. +The emission (and absorption) of the two components of the yellow +line are, therefore, connected with two different processes. This was +beautifully shown by some later researches of Wood and Dunoyer. They +found that if sodium vapour is subjected to radiation from only one +of the two components of the yellow line, the resonance radiation, +at least at low pressures, consists only of this component. These +experiments were later continued by Strutt, and were extended to the +case where the exciting line corresponded to the second line in the +principal series. Strutt found that the resonance radiation consisted +apparently only to a small extent of light of the same frequency as the +incident light, while the greater part consisted of the familiar yellow +line. This result must appear very astonishing on the ordinary ideas of +resonance, since, as Strutt pointed out, no rational connection exists +between the frequencies of the first and second lines of the principal +series. It is however easily explained from our point of view. From the +figure it can be seen that when an atom has been transferred into the +second state in the second row, in addition to the direct return to the +normal state, there are still two other transitions which may give rise +to radiation, namely the transitions to the second state in the first +row and to the first state in the third row. The experiments seem to +<span class="pagenum" id="Page_34">[Pg 34]</span> +indicate that the second of these three transitions is most probable, +and I shall show later that there is some theoretical justification +for this conclusion. By this transition, which results in the emission +of an infra-red line which could not be observed with the experimental +arrangement, the atom is taken to the second state of the first row, +and from this state only one transition is possible, which again gives +an infra-red line. This transition takes the atom to the first state +in the second row, and the subsequent transition to the normal state +then gives rise to the yellow line. Strutt discovered another equally +surprising result, that this yellow resonance radiation seemed to +consist of both components of the first line of the principal series, +even when the incident light consisted of only one component of the +second line of the principal series. This is in beautiful agreement +with our picture of the phenomenon. We must remember that the states in +the first row are simple, so when the atom has arrived in one of these +it has lost every possibility of later giving any indication from which +of the two states in the second row it originally came.</p> + +<p>Sodium vapour, in addition to the absorption corresponding to the +lines of the principal series, exhibits a <i>selective absorption in a +continuous spectral region</i> beginning at the limit of this series +and extending into the ultra-violet. This confirms in a striking manner +our assumption that the absorption of the lines of the principal series +of sodium results in final states of the atom in which one of the +electrons revolves in larger and larger orbits. For we must assume that +this continuous absorption corresponds to transitions from the normal +state to states in which the electron is in a position to remove itself +infinitely far from the nucleus. This phenomenon exhibits a complete +analogy with the <i>photoelectric effect</i> from an illuminated metal +plate in which, by using light of a suitable frequency, electrons of +any velocity can be obtained. The frequency, however, must always lie +above a certain limit connected according to Einstein's theory in a +simple manner with the energy necessary to bring an electron out of the +metal.</p> + +<p>This view of the origin of the emission and absorption spectra has +been confirmed in a very interesting manner by experiments on the +<i>excitation of spectral lines and production of ionization by +electron bombardment</i>. The chief advance in this field is due to +the well-known experiments of Franck and Hertz. These investigators +<span class="pagenum" id="Page_35">[Pg 35]</span> +obtained their first important results from their experiments on +mercury vapour, whose properties particularly facilitate such +experiments. On account of the great importance of the results, these +experiments have been extended to most gases and metals that can be +obtained in a gaseous state. With the aid of the figure I shall briefly +illustrate the results for the case of sodium vapour. It was found +that the electrons upon colliding with the atoms were thrown back with +undiminished velocity when their energy was less than that required +to transfer the atom from the normal state to the next succeeding +stationary state of higher energy value. In the case of sodium vapour +this means from the first state in the first row to the first state +in the second row. As soon, however, as the energy of the electron +reaches this critical value, a new type of collision takes place, in +which the electron loses all its kinetic energy, while at the same +time the vapour is excited and emits a radiation corresponding to +the yellow line. This is what would be expected, if by the collision +the atom was transferred from the normal state to the first one in +the second row. For some time it was uncertain to what extent this +explanation was correct, since in the experiments on mercury vapour it +was found that, together with the occurrence of non-elastic impacts, +ions were always formed in the vapour. From our figure, however, we +would expect ions to be produced only when the kinetic energy of the +electrons is sufficiently great to bring the atom out of the normal +state to the common limit of the states. Later experiments, especially +by Davis and Goucher, have settled this point. It has been shown that +ions can only be directly produced by collisions when the kinetic +energy of the electrons corresponds to the limit of the series, and +that the ionization found at first was an indirect effect arising from +the photoelectric effect produced at the metal walls of the apparatus +by the radiation arising from the return of the mercury atoms to the +normal state. These experiments provide a direct and independent +proof of the reality of the distinctive stationary states, whose +existence we were led to infer from the series spectra. At the same +time we get a striking impression of the insufficiency of the ordinary +electrodynamical and mechanical conceptions for the description of +atomic processes, not only as regards the emission of radiation but +<span class="pagenum" id="Page_36">[Pg 36]</span> +also in such phenomena as the collision of free electrons with atoms.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="III_DEVELOPMENT_OF_THE_QUANTUM_THEORY"> +III. DEVELOPMENT OF THE QUANTUM THEORY +OF SPECTRA</h2> +</div> + + +<p>We see that it is possible by making use of a few simple ideas to +obtain a certain insight into the origin of the series spectra. But +when we attempt to penetrate more deeply, difficulties arise. In +fact, for systems which are not simply periodic it is not possible to +obtain sufficient information about the motions of these systems in +the stationary states from the numerical values of the energy alone; +more determining factors are required for the fixation of the motion. +We meet the same difficulties when we try to explain in detail the +characteristic effect of external forces upon the spectrum of hydrogen. +A foundation for further advances in this field has been made in +recent years through a development of the quantum theory, which allows +a fixation of the stationary states not only in the case of simple +periodic systems, but also for certain classes of non-periodic systems. +These are the <i>conditionally periodic systems</i> whose equations of +motion can be solved by a "separation of the variables." If generalized +coordinates are used the description of the motion of these systems can +be reduced to the consideration of a number of generalized "components +of motion." Each of these corresponds to the change of only one of +the coordinates and may therefore in a certain sense be regarded +as "independent." The method for the fixation of the stationary +states consists in fixing the motion of each of these components by +a condition, which can be considered as a direct generalization of +condition (1) for a Planck oscillator, so that the stationary states +are in general characterized by as many whole numbers as the number +of the degrees of freedom which the system possesses. A considerable +number of physicists have taken part in this development of the quantum +theory, including Planck himself. I also wish to mention the important +contribution made by Ehrenfest to this subject on the limitations of +the applicability of the laws of mechanics to atomic processes. The +decisive advance in the application of the quantum theory to spectra, +however, is due to Sommerfeld and his followers. However, I shall not +further discuss the systematic form in which these authors have +<span class="pagenum" id="Page_37">[Pg 37]</span> +presented their results. In a paper which appeared some time ago in +the Transactions of the Copenhagen Academy, I have shown that the +spectra, calculated with the aid of this method for the fixation of +the stationary states, exhibit a correspondence with the spectra which +should correspond to the motion of the system similar to that which we +have already considered in the case of hydrogen. With the aid of this +general correspondence I shall try in the remainder of this lecture to +show how it is possible to present the theory of series spectra and +the effects produced by external fields of force upon these spectra in +a form which may be considered as the natural generalization of the +foregoing considerations. This form appears to me to be especially +suited for future work in the theory of spectra, since it allows of an +immediate insight into problems for which the methods mentioned above +fail on account of the complexity of the motions in the atom.</p> + +<p class="space-above2"> +<b>Effect of external forces on the hydrogen spectrum.</b> We shall +now proceed to investigate the effect of small perturbing forces upon +the spectrum of the simple system consisting of a single electron +revolving about a nucleus. For the sake of simplicity we shall for the +moment disregard the variation of the mass of the electron with its +velocity. The consideration of the small changes in the motion due +to this variation has been of great importance in the development of +Sommerfeld's theory which originated in the explanation of the <i>fine +structure of the hydrogen lines</i>. This fine structure is due to the +fact, that taking into account the variation of mass with velocity +the orbit of the electron deviates a little from a simple ellipse +and is no longer exactly periodic. This deviation from a Keplerian +motion is, however, very small compared with the perturbations due to +the presence of external forces, such as occur in experiments on the +Zeeman and Stark effects. In atoms of higher atomic number it is also +negligible compared with the disturbing effect of the inner electrons +on the motion of the outer electron. The neglect of the change in mass +will therefore have no important influence upon the explanation of the +Zeeman and Stark effects, or upon the explanation of the difference +between the hydrogen spectrum and the spectra of other elements. +<span class="pagenum" id="Page_38">[Pg 38]</span> +</p> + +<p>We shall therefore as before consider the motion of the unperturbed +hydrogen atom as simply periodic and inquire in the first place about +the stationary states corresponding to this motion. The energy in these +states will then be determined by expression (7) which was derived from +the spectrum of hydrogen. The energy of the system being given, the +major axis of the elliptical orbit of the electron and its frequency +of revolution are also determined. Substituting in formulae (7) and +(8) the expression for <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K"> given in (12), we obtain for the energy, +major axis and frequency of revolution in the <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">th state of the +unperturbed atom the expressions +<span class="align-center"><img style="vertical-align: -1.654ex; width: 79.268ex; height: 5.087ex;" src="images/40.svg" alt=" " data-tex=" +E_{n} = -W_{n} = -\frac{1}{n^{2}}\, \frac{2\pi^{2} e^{4} m}{h^{2}},\quad +2a_{n} = n^{2}\, \frac{h^{2}}{2\pi^{2} e^{2} m},\quad +\omega_{n} = \frac{1}{n^{3}}\, \frac{4\pi^{2} e^{4} m}{h^{3}}. +\qquad\text{(17)} +"></span> +</p> + +<p>We must further assume that in the stationary states of the +unperturbed system the form of the orbit is so far undetermined that +the eccentricity can vary continuously. This is not only immediately +indicated by the principle of correspondence,—since the frequency +of revolution is determined only by the energy and not by the +eccentricity,—but also by the fact that the presence of any small +external forces will in general, in the course of time, produce a +finite change in the position as well as in the eccentricity of the +periodic orbit, while in the major axis it can produce only small +changes proportional to the intensity of the perturbing forces.</p> + +<p>In order to fix the stationary states of systems in the presence +of a given conservative external field of force, we shall have to +investigate, on the basis of the principle of correspondence, how +these forces affect the decomposition of the motion into harmonic +oscillations. Owing to the external forces the form and position of the +orbit will vary continuously. In the general case these changes will be +so complicated that it will not be possible to decompose the perturbed +motion into discrete harmonic oscillations. In such a case we must +expect that the perturbed system will not possess any sharply separated +stationary states. Although each emission of radiation must be assumed +to be monochromatic and to proceed according to the general frequency +condition we shall therefore expect the final effect to be a broadening +of the sharp spectral lines of the unperturbed system. In certain +cases, however, the perturbations will be of such a regular character +<span class="pagenum" id="Page_39">[Pg 39]</span> +that the perturbed system can be decomposed into harmonic oscillations, +although the ensemble of these oscillations will naturally be of a +more complicated kind than in the unperturbed system. This happens, +for example, when the variations of the orbit with respect to time are +periodic. In this case harmonic oscillations will appear in the motion +of the system the frequencies of which are equal to whole multiples +of the period of the orbital perturbations, and in the spectrum to be +expected on the basis of the ordinary theory of radiation we would +expect components corresponding to these frequencies. According to the +principle of correspondence we are therefore immediately led to the +conclusion, that to each stationary state in the unperturbed system +there corresponds a number of stationary states in the perturbed system +in such a manner, that for a transition between two of these states a +radiation is emitted, whose frequency stands in the same relationship +to the periodic course of the variations in the orbit, as the spectrum +of a simple periodic system does to its motion in the stationary states.</p> + +<p class="space-above2"> +<b>The Stark effect.</b> An instructive example of the appearance +of periodic perturbations is obtained when hydrogen is subjected to +the effect of a homogeneous electric field. The eccentricity and the +position of the orbit vary continuously under the influence of the +field. During these changes, however, it is found that the centre of +the orbit remains in a plane perpendicular to the direction of the +electric force and that its motion in this plane is simply periodic. +When the centre has returned to its starting point, the orbit will +resume its original eccentricity and position, and from this moment +the entire cycle of orbits will be repeated. In this case the +determination of the energy of the stationary states of the disturbed +system is extremely simple, since it is found that the period of the +disturbance does not depend upon the original configuration of the +orbits nor therefore upon the position of the plane in which the centre +of the orbit moves, but only upon the major axis and the frequency of +revolution. From a simple calculation it is found that the period a is +given by the following formula +<span class="align-center"><img style="vertical-align: -1.679ex; width: 22.857ex; height: 4.746ex;" src="images/41.svg" alt=" " data-tex=" +\sigma = \frac{3eF}{8\pi^{2} ma\omega}, +\qquad\text{(18)} +"></span> +<span class="pagenum" id="Page_40">[Pg 40]</span> +where <img style="vertical-align: 0; width: 1.695ex; height: 1.538ex;" src="images/155.svg" alt=" " data-tex="F"> is the intensity of the external electric field. From +analogy with the fixation of the distinctive energy values of a Planck +oscillator we must therefore expect that the energy difference between +two different states, corresponding to the same stationary state of the +unperturbed system, will simply be equal to a whole multiple of the +product of <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;" src="images/87.svg" alt=" " data-tex="h"> by the period <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/156.svg" alt=" " data-tex="\sigma"> of the perturbations. We are +therefore immediately led to the following expression for the energy of +the stationary states of the perturbed system, +<span class="align-center"><img style="vertical-align: -0.566ex; width: 23.656ex; height: 2.262ex;" src="images/42.svg" alt=" " data-tex=" +E = E_{n} + kh\sigma, +\qquad\text{(19)} +"></span> +where <img style="vertical-align: -0.357ex; width: 2.817ex; height: 1.895ex;" src="images/157.svg" alt=" " data-tex="E_{n}"> depends only upon the number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> characterizing the +stationary state of the unperturbed system, while <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> is a new whole +number which in this case may be either positive or negative. As we +shall see below, consideration of the relation between the energy and +the motion of the system shows that <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> must be numerically less than +<span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">,</span> if, as before, we place the quantity <img style="vertical-align: -0.357ex; width: 2.817ex; height: 1.895ex;" src="images/157.svg" alt=" " data-tex="E_{n}"> equal to the +energy <img style="vertical-align: -0.357ex; width: 5.044ex; height: 1.902ex;" src="images/158.svg" alt=" " data-tex="-W_{n}"> of the <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">th stationary state of the undisturbed +atom. Substituting the values of <img style="vertical-align: -0.439ex; width: 6.844ex; height: 1.984ex;" src="images/159.svg" alt=" " data-tex="W_{n},\omega_{n}"> and <img style="vertical-align: -0.357ex; width: 2.344ex; height: 1.355ex;" src="images/160.svg" alt=" " data-tex="a_{n}"> +given by (17) in formula (19) we get +<span class="align-center"><img style="vertical-align: -1.679ex; width: 41.335ex; height: 5.112ex;" src="images/43.svg" alt=" " data-tex=" +E = -\frac{1}{n^{2}}\, \frac{2\pi^{2} e^{4} m}{h^{2}} + + nk\, \frac{3h^{2} F}{8\pi^{2} em}. +\qquad\text{(20)} +"></span> +To find the effect of an electric field upon the lines of the hydrogen +spectrum, we use the frequency condition (4) and obtain for the +frequency <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/88.svg" alt=" " data-tex="\nu"> of the radiation emitted by a transition between two +stationary states defined by the numbers <img style="vertical-align: -0.439ex; width: 4.787ex; height: 2.009ex;" src="images/161.svg" alt=" " data-tex="n′, k′"> and <img style="vertical-align: -0.439ex; width: 6.031ex; height: 2.009ex;" src="images/162.svg" alt=" " data-tex="n″, k″"> +<span class="align-center"><img style="vertical-align: -2.194ex; width: 64.53ex; height: 5.628ex;" src="images/44.svg" alt=" " data-tex=" +\nu = \frac{2\pi^{2} e^{4} m}{h^{3}} \left(\frac{1}{(n″)^{2}} + - \frac{1}{(n″)^{2}}\right) + + \frac{3h · F}{8\pi^{2} em} (n′k′ - n″k″). +\qquad\text{(21)} +"></span> +</p> + +<p>It is well known that this formula provides a complete explanation of +the Stark effect of the hydrogen lines. It corresponds exactly with the +one obtained by a different method by Epstein and Schwarzschild. They +used the fact that the hydrogen atom in a homogeneous electric field is +a conditionally periodic system permitting a separation of variables by +the use of parabolic coordinates. The stationary states were fixed by +applying quantum conditions to each of these variables.</p> + +<p>We shall now consider more closely the correspondence between the +changes in the spectrum of hydrogen due to the presence of an +<span class="pagenum" id="Page_41">[Pg 41]</span> +electric field and the decomposition of the perturbed motion of the +atom into its harmonic components. Instead of the simple decomposition +into harmonic components corresponding to a simple Kepler motion, the +displacement<img style="vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;" src="images/139.svg" alt=" " data-tex="\xi"> of the electron in a given direction in space can +be expressed in the present case by the formula +<span class="align-center"><img style="vertical-align: -1.018ex; width: 46.686ex; height: 3.167ex;" src="images/45.svg" alt=" " data-tex=" +\xi = \sum C_{\tau,\kappa} \cos 2\pi \bigl\{t(\tau\omega + \kappa\sigma) + + c_{\tau,\kappa}\bigr\}, +\qquad\text{(22)} +"></span> +where <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> is the average frequency of revolution in the +perturbed orbit and <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/156.svg" alt=" " data-tex="\sigma"> is the period of the orbital +perturbations, while <img style="vertical-align: -0.65ex; width: 3.999ex; height: 2.245ex;" src="images/163.svg" alt=" " data-tex="C_{\tau,\kappa}"> and <img style="vertical-align: -0.65ex; width: 3.361ex; height: 1.65ex;" src="images/164.svg" alt=" " data-tex="c_{\tau,\kappa}"> are +constants. The summation is to be extended over all integral values for +<img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/140.svg" alt=" " data-tex="\tau"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/165.svg" alt=" " data-tex="\kappa">.</span></p> + +<p>If we now consider a transition between two stationary states +characterized by certain numbers <img style="vertical-align: -0.439ex; width: 4.787ex; height: 2.009ex;" src="images/161.svg" alt=" " data-tex="n′, k′"> and <span class="nowrap"><img style="vertical-align: -0.439ex; width: 6.031ex; height: 2.009ex;" src="images/162.svg" alt=" " data-tex="n″, k″">,</span> we +find that in the region where these numbers are large compared with +their differences <img style="vertical-align: -0.186ex; width: 7.347ex; height: 1.505ex;" src="images/137.svg" alt=" " data-tex="n′ - n″"> and <span class="nowrap"><img style="vertical-align: -0.186ex; width: 6.99ex; height: 1.756ex;" src="images/166.svg" alt=" " data-tex="k′ - k″">,</span> the frequency of the +spectral line which is emitted will be given approximately by the +formula +<span class="align-center"><img style="vertical-align: -0.566ex; width: 37.847ex; height: 2.262ex;" src="images/46.svg" alt=" " data-tex=" +\nu \sim (n′ - n″)\,\omega + (k′ - k″)\,\sigma. +\qquad\text{(23)} +"></span> +We see, therefore, that we have obtained a relation between the +spectrum and the motion of precisely the same character as in +the simple case of the unperturbed hydrogen atom. We have here a +similar correspondence between the harmonic component in the motion, +corresponding to definite values for <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/140.svg" alt=" " data-tex="\tau"> and <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/165.svg" alt=" " data-tex="\kappa"> in formula +(22), and the transition between two stationary states for which +<img style="vertical-align: -0.186ex; width: 11.534ex; height: 1.505ex;" src="images/141.svg" alt=" " data-tex="n′ - n″ = \tau"> and <span class="nowrap"><img style="vertical-align: -0.186ex; width: 11.31ex; height: 1.756ex;" src="images/167.svg" alt=" " data-tex="k′ - k″ = \kappa">.</span></p> + +<p>A number of interesting results can be obtained from this +correspondence by considering the motion in more detail. Each harmonic +component in expression (22) for which <img style="vertical-align: -0.186ex; width: 5.239ex; height: 1.505ex;" src="images/168.svg" alt=" " data-tex="\tau + \kappa"> is an even +number corresponds to a linear oscillation parallel to the direction of +the electric field, while each component for which <img style="vertical-align: -0.186ex; width: 5.239ex; height: 1.505ex;" src="images/168.svg" alt=" " data-tex="\tau + \kappa"> +is odd corresponds to an elliptical oscillation perpendicular to +this direction. The correspondence principle suggests at once that +these facts are connected with the <i>characteristic polarization</i> +observed in the Stark effect. We would anticipate that a transition +for which <img style="vertical-align: -0.566ex; width: 20.623ex; height: 2.262ex;" src="images/169.svg" alt=" " data-tex="(n′ - n″) + (k′ - k″)"> is even would give rise to +a component with an electric vector parallel to the field, while +a transition for which <img style="vertical-align: -0.566ex; width: 20.623ex; height: 2.262ex;" src="images/169.svg" alt=" " data-tex="(n′ - n″) + (k′ - k″)"> is odd would +correspond to a component with an electric vector perpendicular +<span class="pagenum" id="Page_42">[Pg 42]</span> +to the field. These results have been fully confirmed by experiment +and correspond to the empirical rule of polarization, which Epstein +proposed in his first paper on the Stark effect.</p> + +<p>The applications of the correspondence principle that have so far been +described have been purely qualitative in character. It is possible +however to obtain a quantitative estimate of the relative intensity of +the various components of the Stark effect of hydrogen, by correlating +the numerical values of the coefficients <img style="vertical-align: -0.65ex; width: 3.999ex; height: 2.245ex;" src="images/163.svg" alt=" " data-tex="C_{\tau,\kappa}"> in formula +(22) with the probability of the corresponding transitions between the +stationary states. This problem has been treated in detail by Kramers +in a recently published dissertation. In this he gives a thorough +discussion of the application of the correspondence principle to the +question of the intensity of spectral lines.</p> + +<p class="space-above2"> +<b>The Zeeman effect.</b> The problem of the effect of a homogeneous +magnetic field upon the hydrogen lines may be treated in an entirely +analogous manner. The effect on the motion of the hydrogen atom +consists simply of the superposition of a uniform rotation upon the +motion of the electron in the unperturbed atom. The axis of rotation is +parallel with the direction of the magnetic force, while the frequency +of revolution is given by the formula +<span class="align-center"><img style="vertical-align: -1.577ex; width: 20.245ex; height: 4.652ex;" src="images/47.svg" alt=" " data-tex=" +\sigma = \frac{eH}{4\pi mc}, +\qquad\text{(24)} +"></span> +where <img style="vertical-align: 0; width: 2.009ex; height: 1.545ex;" src="images/170.svg" alt=" " data-tex="H"> is the intensity of the field and <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/98.svg" alt=" " data-tex="c"> the velocity of light.</p> + +<p>Again we have a case where the perturbations are simply periodic and +where the period of the perturbations is independent of the form and +position of the orbit, and in the present case, even of the major axis. +Similar considerations apply therefore as in the case of the Stark +effect, and we must expect that the energy in the stationary states +will again be given by formula (19), if we substitute for <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/156.svg" alt=" " data-tex="\sigma"> +the value given in expression (24). This result is also in complete +agreement with that obtained by Sommerfeld and Debye. The method they +used involved the solution of the equations of motion by the method of +the separation of the variables. The appropriate coordinates are polar +ones about an axis parallel to the field.</p> + +<p>If we try, however, to calculate directly the effect of the field +by means of the frequency condition (4), we immediately meet +<span class="pagenum" id="Page_43">[Pg 43]</span> +with an apparent disagreement which for some time was regarded as +a grave difficulty for the theory. As both Sommerfeld and Debye +have pointed out, lines are not observed corresponding to every +transition between the stationary states included in the formula. We +overcome this difficulty, however, as soon as we apply the principle +of correspondence. If we consider the harmonic components of the +motion we obtain a simple explanation both of the non-occurrence +of certain transitions and of the observed polarization. In the +magnetic field each elliptic harmonic component having the frequency +<img style="vertical-align: -0.029ex; width: 2.577ex; height: 1.032ex;" src="images/171.svg" alt=" " data-tex="\tau\omega"> splits up into three harmonic components owing to +the uniform rotation of the orbit. Of these one is rectilinear with +frequency <img style="vertical-align: -0.029ex; width: 2.577ex; height: 1.032ex;" src="images/171.svg" alt=" " data-tex="\tau\omega"> oscillating parallel to the magnetic field, +and two are circular with frequencies <img style="vertical-align: -0.186ex; width: 6.634ex; height: 1.505ex;" src="images/172.svg" alt=" " data-tex="\tau\omega + \sigma"> and +<img style="vertical-align: -0.186ex; width: 6.634ex; height: 1.505ex;" src="images/173.svg" alt=" " data-tex="\tau\omega - \sigma"> oscillating in opposite directions in a +plane perpendicular to the direction of the field. Consequently the +motion represented by formula (22) contains no components for which +<img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/165.svg" alt=" " data-tex="\kappa"> is numerically greater than <span class="nowrap"><img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/103.svg" alt=" " data-tex="1">,</span> in contrast to the Stark +effect, where components corresponding to all values of <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/165.svg" alt=" " data-tex="\kappa"> are +present. Now formula (23) again applies for large values of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> and +<span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">,</span> and shows the asymptotic agreement between the frequency of the +radiation and the frequency of a harmonic component in the motion. We +arrive, therefore, at the conclusion that transitions for which <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> +changes by more than unity cannot occur. The argument is similar to +that by which transitions between two distinctive states of a Planck +oscillator for which the values of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> in (1) differ by more than +unity are excluded. We must further conclude that the various possible +transitions consist of two types. For the one type corresponding +to the rectilinear component, <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> remains unchanged, and in the +emitted radiation which possesses the same frequency <img style="vertical-align: -0.375ex; width: 2.105ex; height: 1.375ex;" src="images/174.svg" alt=" " data-tex="\nu_{0}"> +as the original hydrogen line, the electric vector will oscillate +parallel with the field. For the second type, corresponding to the +circular components, <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> will increase or decrease by unity, +and the radiation viewed in the direction of the field will +be circularly polarized and have frequencies +<img style="vertical-align: -0.375ex; width: 6.163ex; height: 1.694ex;" src="images/175.svg" alt=" " data-tex="\nu_{0} + \sigma"> and <img style="vertical-align: -0.375ex; width: 6.163ex; height: 1.694ex;" src="images/176.svg" alt=" " data-tex="\nu_{0} - \sigma"> respectively. +These results agree with those of the familiar Lorentz theory. The +similarity in the two theories is remarkable, when we recall the +fundamental difference between the ideas of the quantum theory and the +ordinary theories of radiation.</p> + +<p><span class="pagenum" id="Page_44">[Pg 44]</span></p> +<p class="space-above2"> +<b>Central perturbations.</b> An illustration based on similar +considerations which will throw light upon the spectra of other +elements consists in finding the effect of a small perturbing field +of force radially symmetrical with respect to the nucleus. In this +case neither the form of the orbit nor the position of its plane will +change with time, and the perturbing effect of the field will simply +consist of a uniform rotation of the major axis of the orbit. The +perturbations are periodic, so that we may assume that to each energy +value of a stationary state of the unperturbed system there belongs a +series of discrete energy values of the perturbed system, characterized +by different values of a whole number <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">.</span> The frequency <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/156.svg" alt=" " data-tex="\sigma"> +of the perturbations is equal to the frequency of rotation of the major +axis. For a given law of force for the perturbing field we find that +<img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/156.svg" alt=" " data-tex="\sigma"> depends both on the major axis and on the eccentricity. The +change in the energy of the stationary states, therefore, will not be +given by an expression as simple as the second term in formula (19), +but will be a function of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">,</span> which is different for different +fields. It is possible, however, to characterize by one and the same +condition the motion in the stationary states of a hydrogen atom +which is perturbed by any central field. In order to show this we +must consider more closely the fixation of the motion of a perturbed +hydrogen atom.</p> + +<p>In the stationary states of the unperturbed hydrogen atom only +the major axis of the orbit is to be regarded as fixed, while the +eccentricity may assume any value. Since the change in the energy of +the atom due to the external field of force depends upon the form and +position of its orbit, the fixation of the energy of the atom in the +presence of such a field naturally involves a closer determination of +the orbit of the perturbed system.</p> + +<p>Consider, for the sake of illustration, the change in the hydrogen +spectrum due to the presence of homogeneous electric and magnetic +fields which was described by equation (19). It is found that this +energy condition can be given a simple geometrical interpretation. In +the case of an electric field the distance from the nucleus to the +plane in which the centre of the orbit moves determines the change +in the energy of the system due to the presence of the field. In the +stationary states this distance is simply equal to <img style="vertical-align: -1.577ex; width: 2.353ex; height: 4.676ex;" src="images/177.svg" alt=" " data-tex="\dfrac{k}{n}"> +<span class="pagenum" id="Page_45">[Pg 45]</span> +times half the major axis of the orbit. In the case of a magnetic +field it is found that the quantity which determines the change of +energy of the system is the area of the projection of the orbit upon a +plane perpendicular to the magnetic force. In the various stationary +states this area is equal to <img style="vertical-align: -1.577ex; width: 2.353ex; height: 4.676ex;" src="images/177.svg" alt=" " data-tex="\dfrac{k}{n}"> times the area of a +circle whose radius is equal to half the major axis of the orbit. In +the case of a perturbing central force the correspondence between +the spectrum and the motion which is required by the quantum theory +leads now to the simple condition that in the stationary states of the +perturbed system the minor axis of the rotating orbit is simply equal +to <img style="vertical-align: -1.577ex; width: 2.353ex; height: 4.676ex;" src="images/177.svg" alt=" " data-tex="\dfrac{k}{n}"> times the major axis. This condition was first +derived by Sommerfeld from his general theory for the determination of +the stationary states of a central motion. It is easily shown that this +fixation of the value of the minor axis is equivalent to the statement +that the parameter <img style="vertical-align: -0.439ex; width: 2.269ex; height: 1.946ex;" src="images/178.svg" alt=" " data-tex="2p"> of the elliptical orbit is given by an +expression of exactly the same form as that which gives the major axis +<img style="vertical-align: -0.023ex; width: 2.328ex; height: 1.529ex;" src="images/90.svg" alt=" " data-tex="2a"> in the unperturbed atom. The only difference from the expression +for <img style="vertical-align: -0.357ex; width: 3.476ex; height: 1.864ex;" src="images/179.svg" alt=" " data-tex="2a_{n}"> in (17) is that <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> is replaced by <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">,</span> so that the +value of the parameter in the stationary states of the perturbed atom +is given by +<span class="align-center"><img style="vertical-align: -1.654ex; width: 26.838ex; height: 5.07ex;" src="images/48.svg" alt=" " data-tex=" +2p_{k} = k^{2}\, \frac{h^{2}}{2\pi^{2} e^{2} m}. +\qquad\text{(25)} +"></span> +The frequency of the radiation emitted by a transition between two +stationary states determined in this way for which <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> and <img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″"> +are large in proportion to their difference is given by an expression +which is the same as that in equation (23), if in this case <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> +is the frequency of revolution of the electron in the slowly rotating +orbit and <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/156.svg" alt=" " data-tex="\sigma"> represents the frequency of rotation of the major +axis.</p> + +<p>Before proceeding further, it might be of interest to note that this +fixation of the stationary states of the hydrogen atom perturbed by +external electric and magnetic forces does not coincide in certain +respects with the theories of Sommerfeld, Epstein and Debye. According +to the theory of conditionally periodic systems the stationary states +for a system of three degrees of freedom will in general be determined +by three conditions, and therefore in these theories each state +<span class="pagenum" id="Page_46">[Pg 46]</span> +is characterized by three whole numbers. This would mean that the +stationary states of the perturbed hydrogen atom corresponding to a +certain stationary state of the unperturbed hydrogen atom, fixed by +one condition, should be subject to two further conditions and should +therefore be characterized by two new whole numbers in addition to +the number <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">.</span> But the perturbations of the Keplerian motion are +simply periodic and the energy of the perturbed atom will therefore +be fixed completely by one additional condition. The introduction of +a second condition will add nothing further to the explanation of the +phenomenon, since with the appearance of new perturbing forces, even if +these are too small noticeably to affect the observed Zeeman and Stark +effects, the forms of motion characterized by such a condition may be +entirely changed. This is completely analogous to the fact that the +hydrogen spectrum as it is usually observed is not noticeably affected +by small forces, even when they are large enough to produce a great +change in the form and position of the orbit of the electron.</p> + +<p class="space-above2"> +<b>Relativity effect on hydrogen lines.</b> Before leaving the hydrogen +spectrum I shall consider briefly the effect of the variation of the +mass of the electron with its velocity. In the preceding sections I +have described how external fields of force split up the hydrogen lines +into several components, but it should be noticed that these results +are only accurate when the perturbations are large in comparison with +the small deviations from a pure Keplerian motion due to the variation +of the mass of the electron with its velocity. When the variation of +the mass is taken into account the motion of the unperturbed atom will +not be exactly periodic. Instead we obtain a motion of precisely the +same kind as that occurring in the hydrogen atom perturbed by a small +central field. According to the correspondence principle an intimate +connection is to be expected between the frequency of revolution of +the major axis of the orbit and the difference of the frequencies +of the fine structure components, and the stationary states will be +those orbits whose parameters are given by expression (25). If we +now consider the effect of external forces upon the fine structure +components of the hydrogen lines it is necessary to keep in mind +that this fixation of the stationary states only applies to the +<span class="pagenum" id="Page_47">[Pg 47]</span> +unperturbed hydrogen atom, and that, as mentioned, the orbits in these +states are in general already strongly influenced by the presence +of external forces, which are small compared with those with which +we are concerned in experiments on the Stark and Zeeman effects. In +general the presence of such forces will lead to a great complexity of +perturbations, and the atom will no longer possess a group of sharply +defined stationary states. The fine structure components of a given +hydrogen line will therefore become diffuse and merged together. There +are, however, several important cases where this does not happen on +account of the simple character of the perturbations. The simplest +example is a hydrogen atom perturbed by a central force acting from +the nucleus. In this case it is evident that the motion of the system +will retain its centrally symmetrical character, and that the perturbed +motion will differ from the unperturbed motion only in that the +frequency of rotation of the major axis will be different for different +values of this axis and of the parameter. This point is of importance +in the theory of the spectra of elements of higher atomic number, +since, as we shall see, the effect of the forces originating from the +inner electrons may to a first approximation be compared with that of +a perturbing central field. We cannot therefore expect these spectra +to exhibit a separate effect due to the variation of the mass of the +electron of the same kind as that found in the case of the hydrogen +lines. This variation will not give rise to a splitting up into +separate components but only to small displacements in the position of +the various lines.</p> + +<p>We obtain still another simple example in which the hydrogen atom +possesses sharp stationary states, although the change of mass +of the electron is considered, if we take an atom subject to a +homogeneous magnetic field. The effect of such a field will consist +in the superposition of a rotation of the entire system about an +axis through the nucleus and parallel with the magnetic force. It +follows immediately from this result according to the principle of +correspondence that each fine structure component must be expected to +split up into a normal Zeeman effect (Lorentz triplet). The problem +may also be solved by means of the theory of conditionally periodic +systems, since the equations of motion in the presence of a magnetic +<span class="pagenum" id="Page_48">[Pg 48]</span> +field, even when the change in the mass is considered, will allow of a +separation of the variables using polar coordinates in space. This has +been pointed out by Sommerfeld and Debye.</p> + +<p>A more complicated case arises when the atom is exposed to a +homogeneous electric field which is not so strong that the effect due +to the change in the mass may be neglected. In this case there is no +system of coordinates by which the equations of motion can be solved +by separation of the variables, and the problem, therefore, cannot +be treated by the theory of the stationary states of conditionally +periodic systems. A closer investigation of the perturbations, however, +shows them to be of such a character that the motion of the electrons +may be decomposed into a number of separate harmonic components. These +fall into two groups for which the direction of oscillation is either +parallel with or perpendicular to the field. According to the principle +of correspondence, therefore, we must expect that also in this case in +the presence of the field each hydrogen line will consist of a number +of sharp, polarized components. In fact by means of the principles +I have described, it is possible to give a unique fixation of the +stationary states. The problem of the effect of a homogeneous electric +field upon the fine structure components of the hydrogen lines has been +treated in detail from this point of view by Kramers in a paper which +will soon be published. In this paper it will be shown how it appears +possible to predict in detail the manner in which the fine structure of +the hydrogen lines gradually changes into the ordinary Stark effect as +the electric intensity increases.</p> + +<p class="space-above2"> +<b>Theory of series spectra.</b> Let us now turn our attention once +more to the problem of the series spectra of elements of higher atomic +number. The general appearance of the Rydberg constant in these spectra +is to be explained by assuming that the atom is neutral and that one +electron revolves in an orbit the dimensions of which are large in +comparison with the distance of the inner electrons from the nucleus. +In a certain sense, therefore, the motion of the outer electron may be +compared with the motion of the electron of the hydrogen atom perturbed +by external forces, and the appearance of the various series in the +spectra of the other elements is from this point of view to be +<span class="pagenum" id="Page_49">[Pg 49]</span> +regarded as analogous to the splitting up of the hydrogen lines into +components on account of such forces.</p> + +<p>In his theory of the structure of series spectra of the type exhibited +by the alkali metals, Sommerfeld has made the assumption that the orbit +of the outer electron to a first approximation possesses the same +character as that produced by a simple perturbing central field whose +intensity diminishes rapidly with increasing distance from the nucleus. +He fixed the motion of the external electron by means of his general +theory for the fixation of the stationary states of a central motion. +The application of this method depends on the possibility of separating +the variables in the equations of motion. In this manner Sommerfeld +was able to calculate a number of energy values which can be arranged +in rows just like the empirical spectral terms shown in the diagram of +the sodium spectrum (<a href="#Page_30">p. 30</a>). The states grouped together by Sommerfeld +in the separate rows are exactly those which were characterized by one +and the same value of <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> in our investigation of the hydrogen atom +perturbed by a central force. The states in the first row of the figure +(row <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;" src="images/150.svg" alt=" " data-tex="S">)</span> correspond to the value <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.327ex; height: 1.756ex;" src="images/180.svg" alt=" " data-tex="k = 1">,</span> those of the second row +(<span class="nowrap"><img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P">)</span> correspond to <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.327ex; height: 1.756ex;" src="images/181.svg" alt=" " data-tex="k = 2">,</span> etc. The states corresponding to one +and the same value of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> are connected by dotted lines which are +continued so that their vertical asymptotes correspond to the energy +value of the stationary states of the hydrogen atom. The fact that +for a constant <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> and increasing values of <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> the energy values +approach the corresponding values for the unperturbed hydrogen atom is +immediately evident from the theory since the outer electron, for large +values of the parameter of its orbit, remains at a great distance from +the inner system during the whole revolution. The orbit will become +almost elliptical and the period of rotation of the major axis will be +very large. It can be seen, therefore, that the effect of the inner +system on the energy necessary to remove this electron from the atom +must become less for increasing values of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">.</span></p> + +<p>These beautiful results suggest the possibility of finding laws of +force for the perturbing central field which would account for the +spectra observed. Although Sommerfeld in this way has in fact succeeded +in deriving formulae for the spectral terms which vary with <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> +for a constant <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> in agreement with Rydberg's formulae, it has +<span class="pagenum" id="Page_50">[Pg 50]</span> +not been possible to explain the simultaneous variation with both +<img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> and <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> in any actual case. This is not surprising, since it +is to be anticipated that the effect of the inner electrons on the +spectrum could not be accounted for in such a simple manner. Further +consideration shows that it is necessary to consider not only the +forces which originate from the inner electrons but also to consider +the effect of the presence of the outer electron upon the motion of the +inner electrons.</p> + +<p>Before considering the series spectra of elements of low atomic number +I shall point out how the occurrence or non-occurrence of certain +transitions can be shown by the correspondence principle to furnish +convincing evidence in favour of Sommerfeld's assumption about the +orbit of the outer electron. For this purpose we must describe the +motion of the outer electron in terms of its harmonic components. +This is easily performed if we assume that the presence of the inner +electrons simply produces a uniform rotation of the orbit of the outer +electron in its plane. On account of this rotation, the frequency +of which we will denote by <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/156.svg" alt=" " data-tex="\sigma">,</span> two circular rotations with +the periods <img style="vertical-align: -0.186ex; width: 6.634ex; height: 1.505ex;" src="images/172.svg" alt=" " data-tex="\tau\omega + \sigma"> and <img style="vertical-align: -0.186ex; width: 6.634ex; height: 1.505ex;" src="images/173.svg" alt=" " data-tex="\tau\omega - \sigma"> will +appear in the motion of the perturbed electron, instead of each of +the harmonic elliptical components with a period <img style="vertical-align: -0.029ex; width: 2.577ex; height: 1.032ex;" src="images/171.svg" alt=" " data-tex="\tau\omega"> in +the unperturbed motion. The decomposition of the perturbed motion +into harmonic components consequently will again be represented by a +formula of the type (22), in which only such terms appear for which +<img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/165.svg" alt=" " data-tex="\kappa"> is equal to <img style="vertical-align: -0.186ex; width: 2.891ex; height: 1.692ex;" src="images/182.svg" alt=" " data-tex="+1"> or <span class="nowrap"><img style="vertical-align: -0.186ex; width: 2.891ex; height: 1.692ex;" src="images/183.svg" alt=" " data-tex="-1">.</span> Since the frequency of the +emitted radiation in the regions where <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> and <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> are large is +again given by the asymptotic formula (23), we at once deduce from the +correspondence principle that the only transitions which can take place +are those for which the values of <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> differ by unity. A glance at +the figure for the sodium spectrum shows that this agrees exactly with +the experimental results. This fact is all the more remarkable, since +in Sommerfeld's theory the arrangement of the energy values of the +stationary states in rows has no special relation to the possibility of +transition between these states.</p> + +<p class="space-above2"> +<b>Correspondence principle and conservation of angular momentum.</b> +Besides these results the correspondence principle suggests that +the radiation emitted by the perturbed atom must exhibit circular +<span class="pagenum" id="Page_51">[Pg 51]</span> +polarization. On account of the indeterminateness of the plane of the +orbit, however, this polarization cannot be directly observed. The +assumption of such a polarization is a matter of particular interest +for the theory of radiation emission. On account of the general +correspondence between the spectrum of an atom and the decomposition +of its motion into harmonic components, we are led to compare the +radiation emitted during the transition between two stationary states +with the radiation which would be emitted by a harmonically oscillating +electron on the basis of the classical electrodynamics. In particular +the radiation emitted according to the classical theory by an electron +revolving in a circular orbit possesses an angular momentum and the +energy <img style="vertical-align: 0; width: 3.613ex; height: 1.62ex;" src="images/184.svg" alt=" " data-tex="\Delta E"> and the angular momentum <img style="vertical-align: 0; width: 3.584ex; height: 1.62ex;" src="images/185.svg" alt=" " data-tex="\Delta P"> of the +radiation emitted during a certain time are connected by the relation +<span class="align-center"><img style="vertical-align: -0.566ex; width: 25.23ex; height: 2.262ex;" src="images/49.svg" alt=" " data-tex=" +\Delta E = 2\pi\omega · \Delta P. +\qquad\text{(26)} +"></span> +</p> + +<p>Here <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> represents the frequency of revolution of the electron, +and according to the classical theory this is equal to the frequency +<img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/88.svg" alt=" " data-tex="\nu"> of the radiation. If we now assume that the total energy +emitted is equal to <img style="vertical-align: -0.025ex; width: 2.502ex; height: 1.595ex;" src="images/86.svg" alt=" " data-tex="h\nu"> we obtain for the total angular momentum +of the radiation +<span class="align-center"><img style="vertical-align: -1.577ex; width: 19.571ex; height: 4.676ex;" src="images/50.svg" alt=" " data-tex=" +\Delta P = \frac{h}{2\pi}. +\qquad\text{(27)} +"></span> +</p> + +<p>It is extremely interesting to note that this expression is equal +to the change in the angular momentum which the atom suffers in a +transition where <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> varies by unity. For in Sommerfeld's theory +the general condition for the fixation of the stationary states of +a central system, which in the special case of an approximately +Keplerian motion is equivalent to the relation (25), asserts that the +angular momentum of the system must be equal to a whole multiple of +<span class="nowrap"><img style="vertical-align: -1.577ex; width: 3.416ex; height: 4.676ex;" src="images/186.svg" alt=" " data-tex="\dfrac{h}{2\pi}">,</span> a condition which may be written in our notation +<span class="align-center"><img style="vertical-align: -1.577ex; width: 19.243ex; height: 4.676ex;" src="images/51.svg" alt=" " data-tex=" +P = k\, \frac{h}{2\pi}. +\qquad\text{(28)} +"></span> +We see, therefore, that this condition has obtained direct support from +a simple consideration of the conservation of angular momentum during +the emission of the radiation. I wish to emphasize that this equation +is to be regarded as a rational generalization of Planck's original +<span class="pagenum" id="Page_52">[Pg 52]</span> +statement about the distinctive states of a harmonic oscillator. It may +be of interest to recall that the possible significance of the angular +momentum in applications of the quantum theory to atomic processes +was first pointed out by Nicholson on the basis of the fact that for +a circular motion the angular momentum is simply proportional to the +ratio of the kinetic energy to the frequency of revolution.</p> + +<p>In a previous paper which I presented to the Copenhagen Academy I +pointed out that these results confirm the conclusions obtained by +the application of the correspondence principle to atomic systems +possessing radial or axial symmetry. Rubinowicz has independently +indicated the conclusions which may be obtained directly from a +consideration of conservation of angular momentum during the radiation +process. In this way he has obtained several of our results concerning +the various types of possible transitions and the polarization of +the emitted radiation. Even for systems possessing radial or axial +symmetry, however, the conclusions which we can draw by means of the +correspondence principle are of a more detailed character than can be +obtained solely from a consideration of the conservation of angular +momentum. For example, in the case of the hydrogen atom perturbed by +a central force we can only conclude that <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> cannot change by more +than unity, while the correspondence principle requires that <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> +shall vary by unity for every possible transition and that its value +cannot remain unchanged. Further, this principle enables us not only +to exclude certain transitions as being impossible—and can from this +point of view be considered as a "selection principle"—but it also +enables us to draw conclusions about the relative probabilities of the +various possible types of transitions from the values of the amplitudes +of the harmonic components. In the present case, for example, the fact +that the amplitudes of those circular components which rotate in the +same sense as the electron are in general greater than the amplitudes +of those which rotate in the opposite sense leads us to expect that +lines corresponding to transitions for which <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> decreases by unity +will in general possess greater intensity than lines during the +emission of which <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> increases by unity. Simple considerations +like this, however, apply only to spectral lines corresponding to +transitions from one and the same stationary state. In other cases +<span class="pagenum" id="Page_53">[Pg 53]</span> +when we wish to estimate the relative intensities of two spectral +lines it is clearly necessary to take into consideration the relative +number of atoms which are present in each of the two stationary states +from which the transitions start. While the intensity naturally cannot +depend upon the number of atoms in the final state, it is to be +noticed, however, that in estimating the probability of a transition +between two stationary states it is necessary to consider the character +of the motion in the final as well as in the initial state, since the +values of the amplitudes of the components of oscillation of both +states are to be regarded as decisive for the probability.</p> + +<p>To show how this method can be applied I shall return for a moment to +the problem which I mentioned in connection with Strutt's experiment on +the resonance radiation of sodium vapour. This involved the discussion +of the relative probability of the various possible transitions which +can start from that state corresponding to the second term in the +second row of the figure on <a href="#Page_30">p. 30</a>. These were transitions to the first +and second states in the first row and to the first state in the third +row, and the results of experiment indicate, as we saw, that the +probability is greatest for the second transitions. These transitions +correspond to those harmonic components having frequencies +<span class="nowrap"><img style="vertical-align: -0.186ex; width: 6.596ex; height: 1.692ex;" src="images/187.svg" alt=" " data-tex="2\omega + \sigma">,</span> <img style="vertical-align: -0.186ex; width: 5.465ex; height: 1.505ex;" src="images/188.svg" alt=" " data-tex="\omega + \sigma"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/156.svg" alt=" " data-tex="\sigma">,</span> +and it is seen that only for the second transition do the amplitudes of +the corresponding harmonic component differ from zero in the initial +as well as in the final state. [In the next essay the reader will find +that the values of quantum number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> assigned in <a href="#Page_70">Fig. 1</a> to the +various stationary states must be altered. While this correction in no +way influences the other conclusions in this essay it involves that the +reasoning in this passage cannot be maintained.]</p> + +<p>I have shown how the correspondence between the spectrum of an +element and the motion of the atom enables us to understand the +limitations in the direct application of the combination principle in +the prediction of spectral lines. The same ideas give an immediate +explanation of the interesting discovery made in recent years by Stark +and his collaborators, that certain <i>new series of combination +line</i> appear with considerable intensity when the radiating atoms +are subject to a strong external electric field. This phenomenon is +entirely analogous to the appearance of the so-called combination +<span class="pagenum" id="Page_54">[Pg 54]</span> +tones in acoustics. It is due to the fact that the perturbation of +the motion will not only consist in an effect upon the components +originally present, but in addition will give rise to new components. +The frequencies of these new components may be +<span class="nowrap"><img style="vertical-align: -0.186ex; width: 7.938ex; height: 1.505ex;" src="images/189.svg" alt=" " data-tex="\tau\omega + \kappa\sigma">,</span> where <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/165.svg" alt=" " data-tex="\kappa"> is different +from <span class="nowrap"><img style="vertical-align: 0; width: 2.891ex; height: 1.507ex;" src="images/190.svg" alt=" " data-tex="±1">.</span> According to the correspondence principle we must +therefore expect that the electric field will not only influence the +lines appearing under ordinary circumstances, but that it will also +render possible new types of transitions which give rise to the "new" +combination lines observed. From an estimate of the amplitudes of the +particular components in the initial and final states it has even been +found possible to account for the varying facility with which the new +lines are brought up by the external field.</p> + +<p>The general problem of the effect of an electric field on the spectra +of elements of higher atomic number differs essentially from the simple +Stark effect of the hydrogen lines, since we are here concerned not +with the perturbation of a purely periodic system, but with the effect +of the field on a periodic motion already subject to a perturbation. +The problem to a certain extent resembles the effect of a weak +electric force on the fine structure components of the hydrogen atom. +In much the same way the effect of an electric field upon the series +spectra of the elements may be treated directly by investigating the +perturbations of the external electron. A continuation of my paper in +the Transactions of the Copenhagen Academy will soon appear in which +I shall show how this method enables us to understand the interesting +observations Stark and others have made in this field.</p> + +<p class="space-above2"> +<b>The spectra of helium and lithium.</b> We see that it has been +possible to obtain a certain general insight into the origin of +the series spectra of a type like that of sodium. The difficulties +encountered in an attempt to give a detailed explanation of the +spectrum of a particular element, however, become very serious, even +when we consider the spectrum of helium whose neutral atom contains +only two electrons. The spectrum of this element has a simple structure +in that it consists of single lines or at any rate of double lines +whose components are very close together. We find, however, that the +lines fall into two groups each of which can be described by a +<span class="pagenum" id="Page_55">[Pg 55]</span> +formula of the type (14). These are usually called the (ortho) helium +and parhelium spectra. While the latter consists of simple lines, the +former possesses narrow doublets. The discovery that helium, as opposed +to the alkali metals, possesses two complete spectra of the Rydberg +type which do not exhibit any mutual combinations was so surprising +that at times there has been a tendency to believe that helium +consisted of two elements. This way out of the difficulty is no longer +open, since there is no room for another element in this region of the +periodic system, or more correctly expressed, for an element possessing +a new spectrum. The existence of the two spectra can, however, be +traced back to the fact that in the stationary states corresponding +to the series spectra we have to do with a system possessing only one +inner electron and in consequence the motion of the inner system, +in the absence of the outer electron, will be simply periodic and +therefore easily perturbed by external forces.</p> + +<p>In order to illustrate this point we shall have to consider more +carefully the stationary states connected with the origin of a series +spectrum. We must assume that in these states one electron revolves +in an orbit outside the nucleus and the other electrons. We might now +suppose that in general a number of different groups of such states +might exist, each group corresponding to a different stationary state +of the inner system considered by itself. Further consideration shows, +however, that under the usual conditions of excitation those groups +have by far the greatest probability for which the motion of the inner +electrons corresponds to the "normal" state of the inner system, i.e. +to that stationary state having the least energy. Further the energy +required to transfer the inner system from its normal state to another +stationary state is in general very large compared with the energy +which is necessary to transfer an electron from the normal state of +the neutral atom to a stationary orbit of greater dimensions. Lastly +the inner system is in general capable of a permanent existence only +in its normal state. Now, the configuration of an atomic system in its +stationary states and also in the normal state will, in general, be +completely determined. We may therefore expect that the inner system +under the influence of the forces arising from the presence of the +outer electron can in the course of time suffer only small changes. For +this reason we must assume that the influence of the inner system +<span class="pagenum" id="Page_56">[Pg 56]</span> +upon the motion of the external electron will, in general, be of the +same character as the perturbations produced by a constant external +field upon the motion of the electron in the hydrogen atom. We must +therefore expect a spectrum consisting of an ensemble of spectral +terms, which in general form a connected group, even though in the +absence of external perturbing forces not every combination actually +occurs. The case of the helium spectrum, however, is quite different +since here the inner system contains only one electron the motion +of which in the absence of the external electron is simple periodic +provided the small changes due to the variation in the mass of the +electron with its velocity are neglected. For this reason the form of +the orbit in the stationary states of the inner system considered by +itself will not be determined. In other words, the stability of the +orbit is so slight, even if the variation in the mass is taken into +account, that small external forces are in a position to change the +eccentricity in the course of time to a finite extent. In this case, +therefore, it is possible to have several groups of stationary states, +for which the energy of the inner system is approximately the same +while the form of the orbit of the inner electron and its position +relative to the motion of the other electrons are so essentially +different, that no transitions between the states of different groups +can occur even in the presence of external forces. It can be seen that +these conclusions summarize the experimental observations on the helium +spectra.</p> + +<p>These considerations suggest an investigation of the nature of the +perturbations in the orbit of the inner electron of the helium atom, +due to the presence of the external electron. A discussion of the +helium spectrum from this point of view has recently been given by +Landé. The results of this work are of great interest particularly in +the demonstration of the large back effect on the outer electron due +to the perturbations of the inner orbit which themselves arise from +the presence of the outer electron. Nevertheless, it can scarcely be +regarded as a satisfactory explanation of the helium spectrum. Apart +from the serious objections which may be raised against his calculation +of the perturbations, difficulties arise if we try to apply the +correspondence principle to Landé's results in order to account for the +occurrence of two distinct spectra showing no mutual combinations. To +<span class="pagenum" id="Page_57">[Pg 57]</span> +explain this fact it seems necessary to base the discussion on a more +thorough investigation of the mutual perturbations of the outer and the +inner orbits. As a result of these perturbations both electrons move +in such an extremely complicated way that the stationary states cannot +be fixed by the methods developed for conditionally periodic systems. +Dr Kramers and I have in the last few years been engaged in such an +investigation, and in an address on atomic problems at the meeting of +the Dutch Congress of Natural and Medical Sciences held in Leiden, +April 1919, I gave a short communication of our results. For various +reasons we have up to the present time been prevented from publishing, +but in the very near future we hope to give an account of these results +and of the light which they seem to throw upon the helium spectrum.</p> + +<p>The problem presented by the spectra of elements of higher atomic +number is simpler, since the inner system is better defined in its +normal state. On the other hand the difficulty of the mechanical +problem of course increases with the number of the particles in the +atom. We obtain an example of this in the case of lithium with three +electrons. The differences between the spectral terms of the lithium +spectrum and the corresponding spectral terms of hydrogen are very +small for the variable term of the principal series (<span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.327ex; height: 1.756ex;" src="images/181.svg" alt=" " data-tex="k = 2">)</span> and +for the diffuse series (<span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.327ex; height: 1.756ex;" src="images/191.svg" alt=" " data-tex="k = 3">)</span>, on the other hand it is very +considerable for the variable term of the sharp series (<span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.327ex; height: 1.756ex;" src="images/180.svg" alt=" " data-tex="k = 1">)</span>. +This is very different from what would be expected if it were possible +to describe the effect of the inner electron by a central force +varying in a simple manner with the distance. This must be because the +parameter of the orbit of the outer electron in the stationary states +corresponding to the terms of the sharp series is not much greater than +the linear dimensions of the orbits of the inner electrons. According +to the principle of correspondence the frequency of rotation of the +major axis of the orbit of the outer electron is to be regarded as a +measure of the deviation of the spectral terms from the corresponding +hydrogen terms. In order to calculate this frequency it appears +necessary to consider in detail the mutual effect of all three +electrons, at all events for that part of the orbit where the outer +electron is very close to the other two electrons. Even if we assumed +that we were fully acquainted with the normal state of the inner +<span class="pagenum" id="Page_58">[Pg 58]</span> +system in the absence of the outer electron—which would be expected +to be similar to the normal state of the neutral helium atom—the +exact calculation of this mechanical problem would evidently form an +exceedingly difficult task.</p> + +<p class="space-above2"> +<b>Complex structure of series lines.</b> For the spectra of elements +of still higher atomic number the mechanical problem which has to +be solved in order to describe the motion in the stationary states +becomes still more difficult. This is indicated by the extraordinarily +complicated structure of many of the observed spectra. The fact that +the series spectra of the alkali metals, which possess the simplest +structure, consist of double lines whose separation increases with +the atomic number, indicates that here we have to do with systems in +which the motion of the outer electron possesses in general a somewhat +more complicated character than that of a simple central motion. This +gives rise to a more complicated ensemble of stationary states. It +would, however, appear that in the sodium atom the major axis and +the parameter of the stationary states corresponding to each pair of +spectral terms are given approximately by formulae (17) and (25). This +is indicated not only by the similar part played by the two states +in the experiments on the resonance radiation of sodium vapour, but +is also shown in a very instructive manner by the peculiar effect +of magnetic fields on the doublets. For small fields each component +splits up into a large number of sharp lines instead of into the normal +Lorentz triplet. With increasing field strength Paschen and Back found +that this <i>anomalous Zeeman effect</i> changed into the normal +Lorentz triplet of a single line by a gradual fusion of the components.</p> + +<p>This effect of a magnetic field upon the doublets of the alkali +spectrum is of interest in showing the intimate relation of the +components and confirms the reality of the simple explanation of the +general structure of the spectra of the alkali metals. If we may +again here rely upon the correspondence principle we have unambiguous +evidence that the effect of a magnetic field on the motion of the +electrons simply consists in the superposition of a uniform rotation +with a frequency given by equation (24) as in the case of the hydrogen +atom. For if this were the case the correspondence principle would +indicate under all conditions a normal Zeeman effect for each +<span class="pagenum" id="Page_59">[Pg 59]</span> +component of the doublets. I want to emphasize that the difference +between the simple effect of a magnetic field, which the theory +predicts for the fine structure of components of the hydrogen lines, +and the observed effect on the alkali doublets is in no way to be +considered as a contradiction. The fine structure components are not +analogous to the individual doublet components, but each single fine +structure component corresponds to the ensemble of components (doublet, +triplet) which makes up one of the series lines in Rydberg's scheme. +The occurrence in strong fields of the effect observed by Paschen and +Back must therefore be regarded as a strong support for the theoretical +prediction of the effect of a magnetic field on the fine structure +components of the hydrogen lines.</p> + +<p>It does not appear necessary to assume the "anomalous" effect of small +fields on the doublet components to be due to a failure of the ordinary +electrodynamical laws for the description of the motion of the outer +electron, but rather to be connected with an effect of the magnetic +field on that intimate interaction between the motion of the inner +and outer electrons which is responsible for the occurrence of the +doublets. Such a view is probably not very different from the "coupling +theory" by which Voigt was able to account formally for the details of +the anomalous Zeeman effect. We might even expect it to be possible +to construct a theory of these effects which would exhibit a formal +analogy with the Voigt theory similar to that between the quantum +theory of the normal Zeeman effect and the theory originally developed +by Lorentz. Time unfortunately does not permit me to enter further into +this interesting problem, so I must refer you to the continuation of my +paper in the Transactions of the Copenhagen Academy, which will contain +a general discussion of the origin of series spectra and of the effects +of electric and magnetic fields.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="IV_CONCLUSION">IV. CONCLUSION</h2> +</div> + + +<p>In this lecture I have purposely not considered the question of the +structure of atoms and molecules although this is of course most +intimately connected with the kind of spectral theory I have developed. +We are encouraged to use results obtained from the spectra, since even +the simple theory of the hydrogen spectrum gives a value for the major +axis of the orbit of the electron in the normal state (<span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.506ex; height: 1.692ex;" src="images/110.svg" alt=" " data-tex="n = 1">)</span> of +<span class="pagenum" id="Page_60">[Pg 60]</span> +the same order of magnitude as that derived from the kinetic theory of +gases. In my first paper on the subject I attempted to sketch a theory +of the structure of atoms and of molecules of chemical compounds. This +theory was based on a simple generalization of the results for the +stationary states of the hydrogen atom. In several respects the theory +was supported by experiment, especially in the general way in which +the properties of the elements change with increasing atomic number, +shown most clearly by Moseley's results. I should like, however, to +use this occasion to state, that in view of the recent development of +the quantum theory, many of the special assumptions will certainly +have to be changed in detail. This has become clear from various sides +by the lack of agreement of the theory with experiment. It appears no +longer possible to justify the assumption that in the normal states +the electrons move in orbits of special geometrical simplicity, like +"electronic rings." Considerations relating to the stability of +atoms and molecules against external influences and concerning the +possibility of the formation of an atom by successive addition of the +individual electrons compel us to claim, first that the configurations +of electrons are not only in mechanical equilibrium but also possess +a certain stability in the sense required by ordinary mechanics, and +secondly that the configurations employed must be of such a nature +that transitions to these from other stationary states of the atom +are possible. These requirements are not in general fulfilled by such +simple configurations as electronic rings and they force us to look +about for possibilities of more complicated motions. It will not be +possible here to consider further these still open questions and I must +content myself by referring to the discussion in my forthcoming paper. +In closing, however, I should like to emphasize once more that in this +lecture I have only intended to bring out certain general points of +view lying at the basis of the spectral theory. In particular it was +my intention to show that, in spite of the fundamental differences +between these points of view and the ordinary conceptions of the +phenomena of radiation, it still appears possible on the basis of the +general correspondence between the spectrum and the motion in the +atom to employ these conceptions in a certain sense as guides in the +investigation of the spectra.</p> + + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_2" href="#FNanchor_2" class="label">[2]</a> +Address delivered before the Physical Society in Berlin, +April 27, 1920.</p> + +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_61">[Pg 61]</span></p> +<h2 class="nobreak" id="Essay_III">Essay III<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">[3]</a> +<br><br> +THE STRUCTURE OF THE ATOM AND THE PHYSICAL +AND CHEMICAL PROPERTIES OF THE ELEMENTS +</h2> +</div> + +<h2 class="nobreak" id="I_PRELIMINARY">I. PRELIMINARY</h2> + + +<p>In an address which I delivered to you about a year ago I described the +main features of a theory of atomic structure which I shall attempt +to develop this evening. In the meantime this theory has assumed more +definite form, and in two recent letters to <i>Nature</i> I have given +a somewhat further sketch of the development<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">[4]</a>. The results which +I am about to present to you are of no final character; but I hope +to be able to show you how this view renders a correlation of the +various properties of the elements in such a way, that we avoid the +difficulties which previously appeared to stand in the way of a simple +and consistent explanation. Before proceeding, however, I must ask your +forbearance if initially I deal with matters already known to you, but +in order to introduce you to the subject it will first be necessary +to give a brief description of the most important results which have +been obtained in recent years in connection with the work on atomic +structure.</p> + +<p class="space-above2"> +<b>The nuclear atom.</b> The conception of atomic structure which will +form the basis of all the following remarks is the so-called nuclear +atom according to which an atom is assumed to consist of a nucleus +surrounded by a number of electrons whose distances from one another +and from the nucleus are very large compared to the dimensions of the +particles themselves. The nucleus possesses almost the entire mass +of the atom and has a positive charge of such a magnitude that the +number of electrons in a neutral atom is equal to the number of the +element in the periodic system, the so-called <i>atomic number</i>. +This idea of the atom, which is due principally to Rutherford's +fundamental researches on radioactive substances, exhibits extremely +simple features, but just this simplicity appears at first sight to +present difficulties in explaining the properties of the elements. When +we treat this question on the basis of the ordinary mechanical and +<span class="pagenum" id="Page_62">[Pg 62]</span> +electrodynamical theories it is impossible to find a starting point +for an explanation of the marked properties exhibited by the various +elements, indeed not even of their permanency. On the one hand the +particles of the atom apparently could not be at rest in a state of +stable equilibrium, and on the other hand we should have to expect that +every motion which might be present would give rise to the emission of +electromagnetic radiation which would not cease until all the energy +of the system had been emitted and all the electrons had fallen into +the nucleus. A method of escaping from these difficulties has now been +found in the application of ideas belonging to the quantum theory, +the basis of which was laid by Planck in his celebrated work on the +law of temperature radiation. This represented a radical departure +from previous conceptions since it was the first instance in which the +assumption of a discontinuity was employed in the formulation of the +general laws of nature.</p> + +<p class="space-above2"> +<b>The postulates of the quantum theory.</b> The quantum theory in +the form in which it has been applied to the problems of atomic +structure rests upon two postulates which have a direct bearing on +the difficulties mentioned above. According to the first postulate +there are certain states in which the atom can exist without emitting +radiation, although the particles are supposed to have an accelerated +motion relative to one another. These <i>stationary states</i> are, +in addition, supposed to possess a peculiar kind of stability, so +that it is impossible either to add energy to or remove energy from +the atom except by a process involving a transition of the atom into +another of these states. According to the second postulate each +emission of radiation from the atom resulting from such a transition +always consists of a train of purely harmonic waves. The frequency of +these waves does not depend directly upon the motion of the atom, but +is determined by a <i>frequency relation</i>, according to which the +frequency multiplied by the universal constant introduced by Planck is +equal to the total energy emitted during the process. For a transition +between two stationary states for which the values of the energy of the +atom before and after the emission of radiation are <img style="vertical-align: 0; width: 2.351ex; height: 1.538ex;" src="images/127.svg" alt=" " data-tex="E′"> and <img style="vertical-align: 0; width: 2.973ex; height: 1.538ex;" src="images/128.svg" alt=" " data-tex="E″"> +respectively, we have therefore +<span class="align-center"><img style="vertical-align: -0.566ex; width: 22.031ex; height: 2.262ex;" src="images/52.svg" alt=" " data-tex=" +h\nu = E′ - E″, +\qquad\text{(1)} +"></span> +<span class="pagenum" id="Page_63">[Pg 63]</span> +where <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;" src="images/87.svg" alt=" " data-tex="h"> is Planck's constant and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/88.svg" alt=" " data-tex="\nu"> is the frequency of the +emitted radiation. Time does not permit me to give a systematic survey +of the quantum theory, the recent development of which has gone hand +in hand with its applications to atomic structure. I shall therefore +immediately proceed to the consideration of those applications of the +theory which are of direct importance in connection with our subject.</p> + +<p class="space-above2"> +<b>Hydrogen atom.</b> We shall commence by considering the simplest +atom conceivable, namely, an atom consisting of a nucleus and one +electron. If the charge on the nucleus corresponds to that of a single +electron and the system consequently is neutral we have a hydrogen +atom. Those developments of the quantum theory which have made possible +its application to atomic structure started with the interpretation +of the well-known simple spectrum emitted by hydrogen. This spectrum +consists of a series of lines, the frequencies of which are given by +the extremely simple Balmer formula +<span class="align-center"><img style="vertical-align: -2.194ex; width: 33.191ex; height: 5.474ex;" src="images/53.svg" alt=" " data-tex=" +\nu = K\left(\frac{1}{(n″)^{2}} - \frac{1}{(n′)^{2}}\right), +\qquad\text{(2)} +"></span> +where <img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″"> and <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> are integers. According to the quantum theory +we shall now assume that the atom possesses a series of stationary +states characterized by a series of integers, and it can be seen how +the frequencies given by formula (2) may be derived from the frequency +relation if it is assumed that a hydrogen line is connected with a +radiation emitted during a transition between two of these states +corresponding to the numbers <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″">,</span> and if the energy in +the <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">th state apart from an arbitrary additive constant is supposed +to be given by the formula +<span class="align-center"><img style="vertical-align: -1.654ex; width: 20.327ex; height: 4.753ex;" src="images/54.svg" alt=" " data-tex=" +E_{n} = -\frac{Kh}{n^{2}}. +\qquad\text{(3)} +"></span> +The negative sign is used because the energy of the atom is measured +most simply by the work required to remove the electron to infinite +distance from the nucleus, and we shall assume that the numerical value +of the expression on the right-hand side of formula (3) is just equal +to this work.</p> + +<p>As regards the closer description of the stationary states we find that +the electron will very nearly describe an ellipse with the nucleus +in the focus. The major axis of this ellipse is connected with the +<span class="pagenum" id="Page_64">[Pg 64]</span> +energy of the atom in a simple way, and corresponding to the energy +values of the stationary states given by formula (3) there are a series +of values for the major axis <img style="vertical-align: -0.023ex; width: 2.328ex; height: 1.529ex;" src="images/90.svg" alt=" " data-tex="2a"> of the orbit of the electron given +by the formula +<span class="align-center"><img style="vertical-align: -1.577ex; width: 20.298ex; height: 4.993ex;" src="images/55.svg" alt=" " data-tex=" +2a_{n} = \frac{n^{2} e^{2}}{hK}, +\qquad\text{(4)} +"></span> +where <img style="vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;" src="images/91.svg" alt=" " data-tex="e"> is the numerical value of the charge of the electron and +the nucleus.</p> + +<p>On the whole we may say that the spectrum of hydrogen shows us the +<i>formation of the hydrogen atom</i>, since the stationary states may +be regarded as different stages of a process by which the electron +under the emission of radiation is bound in orbits of smaller and +smaller dimensions corresponding to states with decreasing values +of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">.</span> It will be seen that this view has certain characteristic +features in common with the binding process of an electron to +the nucleus if this were to take place according to the ordinary +electrodynamics, but that our view differs from it in just such a way +that it is possible to account for the observed properties of hydrogen. +In particular it is seen that the final result of the binding process +leads to a quite definite stationary state of the atom, namely that +state for which <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.506ex; height: 1.692ex;" src="images/110.svg" alt=" " data-tex="n = 1">.</span> This state which corresponds to the minimum +energy of the atom will be called the <i>normal state</i> of the atom. +It may be stated here that the values of the energy of the atom and the +major axis of the orbit of the electron which are found if we put +<img style="vertical-align: -0.186ex; width: 5.506ex; height: 1.692ex;" src="images/110.svg" alt=" " data-tex="n = 1"> in formulae (3) and (4) are of the same order of magnitude as the +values of the firmness of binding of electrons and of the dimensions of +the atoms which have been obtained from experiments on the electrical +and mechanical properties of gases. A more accurate check of formulae +(3) and (4) can however not be obtained from such a comparison, because +in such experiments hydrogen is not present in the form of simple atoms +but as molecules.</p> + +<p>The formal basis of the quantum theory consists not only of the +frequency relation, but also of conditions which permit the +determination of the stationary states of atomic systems. The latter +conditions, like that assumed for the frequency, may be regarded as +natural generalizations of that assumption regarding the interaction +between simple electrodynamic systems and a surrounding field of +<span class="pagenum" id="Page_65">[Pg 65]</span> +electromagnetic radiation which forms the basis of Planck's theory of +temperature radiation. I shall not here go further into the nature of +these conditions but only mention that by their means the stationary +states are characterized by a number of integers, the so-called +<i>quantum numbers</i>. For a purely periodic motion like that assumed +in the case of the hydrogen atom only a single quantum number is +necessary for the determination of the stationary states. This number +determines the energy of the atom and also the major axis of the orbit +of the electron, but not its eccentricity. The energy in the various +stationary states, if the small influence of the motion of the nucleus +is neglected, is given by the following formula: +<span class="align-center"><img style="vertical-align: -1.654ex; width: 27.569ex; height: 5.087ex;" src="images/56.svg" alt=" " data-tex=" +E_{n} = -\frac{2\pi^{2} N^{2} e^{4} m}{n^{2} h^{2}}, +\qquad\text{(5)} +"></span> +where <img style="vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;" src="images/91.svg" alt=" " data-tex="e"> and <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/92.svg" alt=" " data-tex="m"> are respectively the charge and the mass of the +electron, and where for the sake of subsequent applications the charge +on the nucleus has been designated by <span class="nowrap"><img style="vertical-align: -0.025ex; width: 3.063ex; height: 1.57ex;" src="images/111.svg" alt=" " data-tex="Ne">.</span></p> + +<p>For the atom of hydrogen <span class="nowrap"><img style="vertical-align: -0.186ex; width: 6.157ex; height: 1.731ex;" src="images/192.svg" alt=" " data-tex="N = 1">,</span> and a comparison with equation (3) +leads to the following theoretical expression for <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K"> in formula (2), +namely +<span class="align-center"><img style="vertical-align: -1.652ex; width: 21.883ex; height: 5.086ex;" src="images/57.svg" alt=" " data-tex=" +K = \frac{2\pi^{2} e^{4} m}{h^{3}}. +\qquad\text{(6)} +"></span> +This agrees with the empirical value of the constant for the spectrum +of hydrogen within the limit of accuracy with which the various +quantities can be determined.</p> +<p><span class="pagenum" id="Page_66">[Pg 66]</span></p> + +<p class="space-above2"> +<b>Hydrogen spectrum and X-ray spectra.</b> If in the above formula we +put <img style="vertical-align: -0.186ex; width: 6.157ex; height: 1.731ex;" src="images/193.svg" alt=" " data-tex="N = 2"> which corresponds to an atom consisting of an electron +revolving around a nucleus with a double charge, we get values for the +energies in the stationary states, which are four times larger than +the energies in the corresponding states of the hydrogen atom, and we +obtain the following formula for the spectrum which would be emitted by +such an atom: +<span class="align-center"><img style="vertical-align: -2.194ex; width: 34.322ex; height: 5.474ex;" src="images/58.svg" alt=" " data-tex=" +\nu = 4K \left(\frac{1}{(n″)^{2}} - \frac{1}{(n′)^{2}}\right). +\qquad\text{(7)} +"></span> +This formula represents certain lines which have been known for +some time and which had been attributed to hydrogen on account of +the great similarity between formulae (2) and (7) since it had +never been anticipated that two different substances could exhibit +properties so closely resembling each other. According to the theory +we may, however, expect that the emission of the spectrum given by +(7) corresponds to the <i>first stage of the formation of the helium +atom</i>, i.e. to the binding of a first electron by the doubly charged +nucleus of this atom. This interpretation has been found to agree with +more recent information. For instance it has been possible to obtain +this spectrum from pure helium. I have dwelt on this point in order +to show how this intimate connection between the properties of two +elements, which at first sight might appear quite surprising, is to +be regarded as an immediate expression of the characteristic simple +structure of the nuclear atom. A short time after the elucidation of +this question, new evidence of extraordinary interest was obtained of +such a similarity between the properties of the elements. I refer to +Moseley's fundamental researches on the X-ray spectra of the elements. +Moseley found that these spectra varied in an extremely simple manner +from one element to the next in the periodic system. It is well +known that the lines of the X-ray spectra may be divided into groups +corresponding to the different characteristic absorption regions for +X-rays discovered by Barkla. As regards the <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K"> group which contains +the most penetrating X-rays, Moseley found that the strongest line for +all elements investigated could be represented by a formula which with +a small simplification can be written +<span class="align-center"><img style="vertical-align: -2.148ex; width: 30.471ex; height: 5.428ex;" src="images/59.svg" alt=" " data-tex=" +\nu = N^{2} K \left(\frac{1}{1^{2}} - \frac{1}{2^{2}}\right). +\qquad\text{(8)} +"></span> +<img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K"> is the same constant as in the hydrogen spectrum, and <img style="vertical-align: 0; width: 2.009ex; height: 1.545ex;" src="images/194.svg" alt=" " data-tex="N"> the +atomic number. The great significance of this discovery lies in the +fact that it would seem firmly to establish the view that this atomic +number is equal to the number of electrons in the atom. This assumption +had already been used as a basis for work on atomic structure and was +first stated by van den Broek. While the significance of this aspect of +Moseley's discovery was at once clear to all, it has on the other hand +been more difficult to understand the very great similarity between +the spectrum of hydrogen and the X-ray spectra. This similarity is +shown, not only by the lines of the <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K"> group, but also by groups of +<span class="pagenum" id="Page_67">[Pg 67]</span> +less penetrating X-rays. Thus Moseley found for all the elements he +investigated that the frequencies of the strongest line in the <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/195.svg" alt=" " data-tex="L"> +group may be represented by a formula which with a simplification +similar to that employed in formula (8) can be written +<span class="align-center"><img style="vertical-align: -2.148ex; width: 30.471ex; height: 5.428ex;" src="images/60.svg" alt=" " data-tex=" +\nu = N^{2} K \left(\frac{1}{2^{2}} - \frac{1}{3^{2}}\right). +\qquad\text{(9)} +"></span> +Here again we obtain an expression for the frequency which corresponds +to a line in the spectrum which would be emitted by the <i>binding of +an electron to a nucleus, whose charge is</i> <span class="nowrap"><img style="vertical-align: -0.025ex; width: 3.063ex; height: 1.57ex;" src="images/111.svg" alt=" " data-tex="Ne">.</span></p> + +<p class="space-above2"> +<b>The fine structure of the hydrogen lines.</b> This similarity +between the structure of the X-ray spectra and the hydrogen spectrum +was still further extended in a very interesting manner by Sommerfeld's +important theory of the fine structure of the hydrogen lines. The +calculation given above of the energy in the stationary states of the +hydrogen system, where each state is characterized by a single quantum +number, rests upon the assumption that the orbit of the electron in +the atom is simply periodic. This is, however, only approximately +true. It is found that if the change in the mass of the electron +due to its velocity is taken into consideration the orbit of the +electron no longer remains a simple ellipse, but its motion may be +described as a <i>central motion</i> obtained by superposing a slow +and uniform rotation upon a simple periodic motion in a very nearly +elliptical orbit. For a central motion of this kind the stationary +states are characterized by <i>two quantum numbers</i>. In the case +under consideration one of these may be so chosen that to a very +close approximation it will determine the energy of the atom in the +same manner as the quantum number previously used determined the +energy in the case of a simple elliptical orbit. This quantum number +which will always be denoted by <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> will therefore be called the +"principal quantum number." Besides this condition, which to a very +close approximation determines the major axis in the rotating and +almost elliptical orbit, a second condition will be imposed upon the +stationary states of a central orbit, namely that the angular momentum +of the electron about the centre shall be equal to a whole multiple +of Planck's constant divided by <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.421ex; height: 1.532ex;" src="images/196.svg" alt=" " data-tex="2\pi">.</span> The whole number, which +occurs as a factor in this expression, may be regarded as the second +quantum number and will be denoted by <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">.</span> The latter condition +<span class="pagenum" id="Page_68">[Pg 68]</span> +fixes the eccentricity of the rotating orbit which in the case of a +simple periodic orbit was undetermined. It should be mentioned that +the possible importance of the angular momentum in the quantum theory +was pointed out by Nicholson before the application of this theory to +the spectrum of hydrogen, and that a determination of the stationary +states for the hydrogen atom similar to that employed by Sommerfeld was +proposed almost simultaneously by Wilson, although the latter did not +succeed in giving a physical application to his results.</p> + +<p>The simplest description of the form of the rotating nearly elliptical +electronic orbit in the hydrogen atom is obtained by considering the +chord which passes through the focus and is perpendicular to the major +axis, the so-called "parameter." The length <img style="vertical-align: -0.439ex; width: 2.269ex; height: 1.946ex;" src="images/178.svg" alt=" " data-tex="2p"> of this parameter +is given to a very close approximation by an expression of exactly the +same form as the expression for the major axis, except that <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> takes +the place of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">.</span> Using the same notation as before we have therefore +<span class="align-center"><img style="vertical-align: -1.654ex; width: 49.604ex; height: 5.07ex;" src="images/61.svg" alt=" " data-tex=" +2a = n^{2}\, \frac{h^{2}}{2\pi^{2} N e^{2} m},\quad +2p = k^{2}\, \frac{h^{2}}{2\pi^{2} N e^{2} m}. +\qquad\text{(10)} +"></span> +For each of the stationary states which had previously been denoted by +a given value of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">,</span> we obtain therefore a set of stationary states +corresponding to values of <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> from <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/103.svg" alt=" " data-tex="1"> to <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">.</span> Instead of the +simple formula (5) Sommerfeld found a more complicated expression for +the energy in the stationary states which depends on <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> as well as +<span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">.</span> Taking the variation of the mass of the electron with velocity +into account and neglecting terms of higher order of magnitude he +obtained +<span class="align-center"><img style="vertical-align: -2.148ex; width: 61.689ex; height: 5.582ex;" src="images/62.svg" alt=" " data-tex=" +E_{n,k} = -\frac{2\pi^{2} N^{2} e^{4} m}{n^{2} h^{2}} + \left[1 + \frac{4\pi^{2} N^{2} e^{4}}{h^{2} c^{2}} +\left(-\frac{3}{4n^{2}} + \frac{1}{nk}\right)\right], +\qquad\text{(11)} +"></span> +where <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/98.svg" alt=" " data-tex="c"> is the velocity of light.</p> + +<p>Corresponding to each of the energy values for the stationary states +of the hydrogen atom given by the simple formula (5) we obtain <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> +values differing only very little from one another, since the second +term within the bracket is very small. With the aid of the general +frequency relation (1) we therefore obtain a number of components with +nearly coincident frequencies instead of each hydrogen line given by +the simple formula (2). Sommerfeld has now shown that this calculation +actually agrees with measurements of the fine structure. This +<span class="pagenum" id="Page_69">[Pg 69]</span> +agreement applies not only to the fine structure of the hydrogen lines +which is very difficult to measure on account of the extreme proximity +of the components, but it is also possible to account in detail for +the fine structure of the helium lines given by formula (7) which has +been very carefully investigated by Paschen. Sommerfeld in connection +with this theory also pointed out that formula (11) could be applied to +the X-ray spectra. Thus he showed that in the <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K"> and <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/195.svg" alt=" " data-tex="L"> groups +pairs of lines appeared the differences of whose frequencies could be +determined by the expression (11) for the energy in the stationary +states which correspond to the binding of a single electron by a +nucleus of charge <span class="nowrap"><img style="vertical-align: -0.025ex; width: 3.063ex; height: 1.57ex;" src="images/111.svg" alt=" " data-tex="Ne">.</span></p> + +<p class="space-above2"> +<b>Periodic table.</b> In spite of the great formal similarity between +the X-ray spectra and the hydrogen spectrum indicated by these +results a far-reaching difference must be assumed to exist between +the processes which give rise to the appearance of these two types of +spectra. While the emission of the hydrogen spectrum, like the emission +of the ordinary optical spectra of other elements, may be assumed to be +connected with the binding of an electron by an atom, observations on +the appearance and absorption of X-ray spectra clearly indicate that +these spectra are connected with a process which may be described as a +<i>reorganization of the electronic arrangement</i> after a disturbance +within the atom due to the effect of external agencies. We should +therefore expect that the appearance of the X-ray spectra would depend +not only upon the direct interaction between a single electron and the +nucleus, but also on the manner in which the electrons are arranged in +the completely formed atom.</p> + +<p>The peculiar manner in which the properties of the elements vary with +the atomic number, as expressed in the periodic system, provides a +guide of great value in considering this latter problem. A simple +survey of this system is given in <a href="#Page_70">Fig. 1</a>. The number preceding each +element indicates the atomic number, and the elements within the +various vertical columns form the different "periods" of the system. +The lines, which connect pairs of elements in successive columns, +indicate homologous properties of such elements. Compared with usual +representations of the periodic system, this method, proposed more +<span class="pagenum" id="Page_70">[Pg 70]</span> +than twenty years ago by Julius Thomsen, of indicating the periodic +variations in the properties of the elements is more suited for +comparison with theories of atomic constitution. The meaning of the +frames round certain sequences of elements within the later periods of +the table will be explained later. They refer to certain characteristic +features of the theory of atomic constitution.</p> + +<div class="figcenter"> +<img src="images/002.jpg" width="400" alt="fig02"> +<div class="caption"> +<p>Fig. 1.</p> +</div></div> + +<p>In an explanation of the periodic system it is natural to assume a +division of the electrons in the atom into distinct groups in such a +manner that the grouping of the elements in the system is attributed +to the gradual formation of the groups of electrons in the atoms as +the atomic number increases. Such a grouping of the electrons in +<span class="pagenum" id="Page_71">[Pg 71]</span> +the atom has formed a prominent part of all more detailed views of +atomic structure ever since J. J. Thomson's famous attempt to explain +the periodic system on the basis of an investigation of the stability +of various electronic configurations. Although Thomson's assumption +regarding the distribution of the positive electricity in the atom is +not consistent with more recent experimental evidence, nevertheless +his work has exerted great influence upon the later development of the +atomic theory on account of the many original ideas which it contained.</p> + +<p>With the aid of the information concerning the binding of electrons +by the nucleus obtained from the theory of the hydrogen spectrum I +attempted in the same paper in which this theory was set forth to +sketch in broad outlines a picture of the structure of the nucleus +atom. In this it was assumed that each electron in its normal state +moved in a manner analogous to the motion in the last stages of the +binding of a single electron by a nucleus. As in Thomson's theory, +it was assumed that the electrons moved in circular orbits and that +the electrons in each separate group during this motion occupied +positions with reference to one another corresponding to the vertices +of plane regular polygons. Such an arrangement is frequently described +as a distribution of the electrons in "rings." By means of these +assumptions it was possible to account for the orders of magnitude of +the dimensions of the atoms as well as the firmness with which the +electrons were bound by the atom, a measure of which may be obtained by +means of experiments on the excitation of the various types of spectra. +It was not possible, however, in this way to arrive at a detailed +explanation of the characteristic properties of the elements even after +it had become apparent from the results of Moseley and the work of +Sommerfeld and others that this simple picture ought to be extended +to include orbits in the fully formed atom characterized by higher +quantum numbers corresponding to previous stages in the formation of +the hydrogen atom. This point has been especially emphasized by Vegard.</p> + +<p>The difficulty of arriving at a satisfactory picture of the atom +is intimately connected with the difficulty of accounting for the +pronounced "stability" which the properties of the elements demand. As +I emphasized when considering the formation of the hydrogen atom, +<span class="pagenum" id="Page_72">[Pg 72]</span> +the postulates of the quantum theory aim directly at this point, but +the results obtained in this way for an atom containing a single +electron do not permit of a direct elucidation of problems like that +of the distribution in groups of the electrons in an atom containing +several electrons. If we imagine that the electrons in the groups +of the atom are orientated relatively to one another at any moment, +like the vertices of regular polygons, and rotating in either circles +or ellipses, the postulates do not give sufficient information to +determine the difference in the stability of electronic configurations +with different numbers of electrons in the groups.</p> + +<p>The peculiar character of stability of the atomic structure, demanded +by the properties of the elements, is brought out in an interesting way +by Kossel in two important papers. In the first paper he shows that a +more detailed explanation of the origin of the high frequency spectra +can be obtained on the basis of the group structure of the atom. He +assumes that a line in the X-ray spectrum is due to a process which +may be described as follows: an electron is removed from the atom by +some external action after which an electron in one of the other groups +takes its place; this exchange of place may occur in as many ways as +there are groups of more loosely bound electrons. This view of the +origin of the characteristic X-rays afforded a simple explanation of +the peculiar absorption phenomena observed. It has also led to the +prediction of certain simple relations between the frequencies of +the X-ray lines from one and the same element and has proved to be a +suitable basis for the classification of the complete spectrum. However +it has not been possible to develop a theory which reconciles in a +satisfactory way Sommerfeld's work on the fine structure of the X-ray +lines with Kossel's general scheme. As we shall see later the adoption +of a new point of view when considering the stability of the atom +renders it possible to bring the different results in a natural way in +connection with one another.</p> + +<p>In his second paper Kossel investigates the possibilities for an +explanation of the periodic system on the basis of the atomic +theory. Without entering further into the problem of the causes of +the division of the electrons into groups, or the reasons for the +different stability of the various electronic configurations, he +points out in connection with ideas which had already played a part +in Thomson's theory, how the periodic system affords evidence of a +<span class="pagenum" id="Page_73">[Pg 73]</span> +periodic appearance of especially stable configurations of electrons. +These configurations appear in the neutral atoms of elements occupying +the final position in each period in <a href="#Page_70">Fig. 1</a>, and the stability in +question is assumed in order to explain not only the inactive chemical +properties of these elements but also the characteristic active +properties of the immediately preceding or succeeding elements. If we +consider for instance an inactive gas like argon, the atomic number +of which is <span class="nowrap"><img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18">,</span> we must assume that the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> electrons in the +atom are arranged in an exceedingly regular configuration possessing a +very marked stability. The pronounced electronegative character of the +preceding element, chlorine, may then be explained by supposing the +neutral atom which contains only <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.579ex;" src="images/198.svg" alt=" " data-tex="17"> electrons to possess a tendency +to capture an additional electron. This gives rise to a negative +chlorine ion with a configuration of <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> electrons similar to that +occurring in the neutral argon atom. On the other hand the marked +electropositive character of potassium may be explained by supposing +one of the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/199.svg" alt=" " data-tex="19"> electrons in the neutral atom to be as it were +superfluous, and that this electron therefore is easily lost; the rest +of the atom forming a positive ion of potassium having a constitution +similar to that of the argon atom. In a corresponding manner it is +possible to account for the electronegative and electropositive +character of elements like sulphur and calcium, whose atomic numbers +are <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/200.svg" alt=" " data-tex="16"> and <span class="nowrap"><img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/201.svg" alt=" " data-tex="20">.</span> In contrast to chlorine and potassium these +elements are divalent, and the stable configuration of <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> electrons +is formed by the addition of two electrons to the sulphur atom and +by the loss of two electrons from the calcium atom. Developing these +ideas Kossel has succeeded not only in giving interesting explanations +of a large number of chemical facts, but has also been led to certain +general conclusions about the grouping of the electrons in elements +belonging to the first periods of the periodic system, which in +certain respects are in conformity with the results to be discussed +in the following paragraphs. Kossel's work was later continued in an +interesting manner by Ladenburg with special reference to the grouping +of the electrons in atoms of elements belonging to the later periods of +the periodic table. It will be seen that Ladenburg's conclusions also +exhibit points of similarity with the results which we shall discuss later. +<span class="pagenum" id="Page_74">[Pg 74]</span> +</p> + +<p class="space-above2"> +<b>Recent atomic models.</b> Up to the present time it has not been +possible to obtain a satisfactory account based upon a consistent +application of the quantum theory to the nuclear atom of the ultimate +cause of the pronounced stability of certain arrangements of electrons. +Nevertheless it has been apparent for some time that the solution +should be sought for by investigating the possibilities of a <i>spatial +distribution of the electronic orbits</i> in the atom instead of +limiting the investigation to configurations in which all electrons +belonging to a particular group move in the same plane as was assumed +for simplicity in my first papers on the structure of the atom. The +necessity of assuming a spatial distribution of the configurations +of electrons has been drawn attention to by various writers. Born +and Landé, in connection with their investigations of the structure +and properties of crystals, have pointed out that the assumption of +spatial configurations appears necessary for an explanation of these +properties. Landé has pursued this question still further, and as will +be mentioned later has proposed a number of different "spatial atomic +models" in which the electrons in each separate group of the atom at +each moment form configurations possessing regular polyhedral symmetry. +These models constitute in certain respects a distinct advance, +although they have not led to decisive results on questions of the +stability of atomic structure.</p> + +<p>The importance of spatial electronic configurations has, in addition, +been pointed out by Lewis and Langmuir in connection with their +atomic models. Thus Lewis, who in several respects independently +came to the same conclusions as Kossel, suggested that the number +<img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/202.svg" alt=" " data-tex="8"> characterizing the first groups of the periodic system might +indicate a constitution of the outer atomic groups where the electrons +within each group formed a configuration like the corners of a cube. +He emphasized how a configuration of this kind leads to instructive +models of the molecular structure of chemical combinations. It is +to be remarked, however, that such a "static" model of electronic +configuration will not be possible if we assume the forces within the +atom to be due exclusively to the electric charges of the particles. +Langmuir, who has attempted to develop Lewis' conceptions still further +and to account not only for the occurrence of the first octaves, but +also for the longer periods of the periodic system, supposes therefore +the structure of the atoms to be governed by forces whose nature is +<span class="pagenum" id="Page_75">[Pg 75]</span> +unknown to us. He conceives the atom to possess a "cellular structure," +so that each electron is in advance assigned a place in a cell and +these cells are arranged in shells in such a manner, that the various +shells from the nucleus of the atom outward contain exactly the same +number of places as the periods in the periodic system proceeding +in the direction of increasing atomic number. Langmuir's work has +attracted much attention among chemists, since it has to some extent +thrown light on the conceptions with which empirical chemical science +is concerned. On his theory the explanation of the properties of the +various elements is based on a number of postulates about the structure +of the atoms formulated for that purpose. Such a descriptive theory is +sharply differentiated from one where an attempt is made to explain +the specific properties of the elements with the aid of general laws +applying to the interaction between the particles in each atom. The +principal task of this lecture will consist in an attempt to show +that an advance along these lines appears by no means hopeless, but +on the contrary that with the aid of a consistent application of the +postulates of the quantum theory it actually appears possible to obtain +an insight into the structure and stability of the atom.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="II_SERIES_SPECTRA_AND_THE_CAPTURE_OF_ELECTRONS"> +II. SERIES SPECTRA AND THE CAPTURE OF ELECTRONS +BY ATOMS</h2> +</div> + + +<p>We attack the problem of atomic constitution by asking the question: +"How may an atom be formed by the successive capture and binding of the +electrons one by one in the field of force surrounding the nucleus?"</p> + +<p>Before attempting to answer this question it will first be necessary to +consider in more detail what the quantum theory teaches us about the +general character of the binding process. We have already seen how the +hydrogen spectrum gives us definite information about the course of +this process of binding the electron by the nucleus. In considering the +formation of the atoms of other elements we have also in their spectra +sources for the elucidation of the formation processes, but the direct +information obtained in this way is not so complete as in the case of +the hydrogen atom. For an element of atomic number <img style="vertical-align: 0; width: 2.009ex; height: 1.545ex;" src="images/194.svg" alt=" " data-tex="N"> the process of +formation may be regarded as occurring in <img style="vertical-align: 0; width: 2.009ex; height: 1.545ex;" src="images/194.svg" alt=" " data-tex="N"> stages, corresponding +<span class="pagenum" id="Page_76">[Pg 76]</span> +with the successive binding of <img style="vertical-align: 0; width: 2.009ex; height: 1.545ex;" src="images/194.svg" alt=" " data-tex="N"> electrons in the field of the +nucleus. A spectrum must be assumed to correspond to each of these +binding processes; but only for the first two elements, hydrogen and +helium, do we possess a detailed knowledge of these spectra. For +other elements of higher atomic number, where several spectra will be +connected with the formation of the atom, we are at present acquainted +with only two types, called the "arc" and "spark" spectra respectively, +according to the experimental conditions of excitation. Although these +spectra show a much more complicated structure than the hydrogen +spectrum, given by formula (2) and the helium spectrum given by formula +(7), nevertheless in many cases it has been possible to find simple +laws for the frequencies exhibiting a close analogy with the laws +expressed by these formulae.</p> + +<p class="space-above2"> +<b>Arc and spark spectra.</b> If for the sake of simplicity we +disregard the complex structure shown by the lines of most spectra +(occurrence of doublets, triplets etc.), the frequency of the lines of +many arc spectra can be represented to a close approximation by the +Rydberg formula +<span class="align-center"><img style="vertical-align: -2.194ex; width: 40.393ex; height: 5.269ex;" src="images/63.svg" alt=" " data-tex=" +\nu = \frac{K}{(n″ + \alpha_{k″})^{2}} - \frac{K}{(n′ + \alpha_{k′})^{2}}, +\qquad\text{(12)} +"></span> +where <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> and <img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″"> are integral numbers, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K"> the same constant +as in the hydrogen spectrum, while <img style="vertical-align: -0.357ex; width: 2.909ex; height: 1.357ex;" src="images/203.svg" alt=" " data-tex="\alpha_{k′}"> and <img style="vertical-align: -0.357ex; width: 3.349ex; height: 1.357ex;" src="images/204.svg" alt=" " data-tex="\alpha_{k″}"> +are two constants belonging to a set characteristic of the element. A +spectrum with a structure of this kind is, like the hydrogen spectrum, +called a series spectrum, since the lines can be arranged into series +in which the frequencies converge to definite limiting values. These +series are for example represented by formula (12) if, using two +definite constants for <img style="vertical-align: -0.357ex; width: 3.349ex; height: 1.357ex;" src="images/204.svg" alt=" " data-tex="\alpha_{k″}"> and <span class="nowrap"><img style="vertical-align: -0.357ex; width: 2.909ex; height: 1.357ex;" src="images/203.svg" alt=" " data-tex="\alpha_{k′}">,</span> <img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″"> +remains unaltered, while <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> assumes a series of successive, +gradually increasing integral values.</p> + +<p>Formula (12) applies only approximately, but it is always found +that the frequencies of the spectral lines can be written, as in +formulae (2) and (12), as a difference of two functions of integral +numbers. Thus the latter formula applies accurately, if the quantities +<img style="vertical-align: -0.357ex; width: 2.469ex; height: 1.357ex;" src="images/205.svg" alt=" " data-tex="\alpha_{k}"> are not considered as constants, but as representatives +of a set of series of numbers <img style="vertical-align: -0.566ex; width: 5.587ex; height: 2.262ex;" src="images/206.svg" alt=" " data-tex="\alpha_{k}(n)"> characteristic of the +element, whose values for increasing <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> within each series quickly +approach a constant limiting value. The fact that the frequencies of +<span class="pagenum" id="Page_77">[Pg 77]</span> +the spectra always appear as the difference of two terms, the so-called +"spectral terms," from the combinations of which the complete spectrum +is formed, has been pointed out by Ritz, who with the establishment +of the combination principle has greatly advanced the study of the +spectra. The quantum theory offers an immediate interpretation of this +principle, since, according to the frequency relation we are led to +consider the lines as due to transitions between stationary states of +the atom, just as in the hydrogen spectrum, only in the spectra of the +other elements we have to do not with a single series of stationary +states, but with a set of such series. From formula (12) we thus obtain +for an arc spectrum—if we temporarily disregard the structure of the +individual lines—information about an ensemble of stationary states, +for which the energy of the atom in the <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">th state of the <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">th +series is given by +<span class="align-center"><img style="vertical-align: -2.194ex; width: 29.469ex; height: 5.294ex;" src="images/64.svg" alt=" " data-tex=" +E_{k}(n) = -\frac{Kh}{(n + \alpha_{k})^{2}} +\qquad\text{(13)} +"></span> +very similar to the simple formula (3) for the energy in the stationary +states of the hydrogen atom.</p> + +<p>As regards the spark spectra, the structure of which has been cleared +up mainly by Fowler's investigations, it has been possible in the case +of many elements to express the frequencies approximately by means of a +formula of exactly the same type as (12), only with the difference that +<span class="nowrap"><img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K">,</span> just as in the helium spectrum given by formula (7), is replaced +by a constant, which is four times as large. For the spark spectra, +therefore, the energy values in the corresponding stationary states of +the atom will be given by an expression of the same type as (13), only +with the difference that <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/129.svg" alt=" " data-tex="K"> is replaced by <span class="nowrap"><img style="vertical-align: 0; width: 3.143ex; height: 1.545ex;" src="images/207.svg" alt=" " data-tex="4K">.</span></p> + +<p>This remarkable similarity between the structure of these types of +spectra and the simple spectra given by (2) and (7) is explained simply +by assuming the arc spectra to be connected with the <i>last stage in +the formation of the neutral atom</i> consisting in the capture and +binding of the <img style="vertical-align: 0; width: 2.009ex; height: 1.545ex;" src="images/194.svg" alt=" " data-tex="N">th electron. On the other hand the spark spectra +are connected with the <i>last stage but one in the formation of the +atom</i>, namely the binding of the <img style="vertical-align: -0.566ex; width: 7.666ex; height: 2.262ex;" src="images/208.svg" alt=" " data-tex="(N - 1)">th electron. In these +cases the field of force in which the electron moves will be much +<span class="pagenum" id="Page_78">[Pg 78]</span> +the same as that surrounding the nucleus of a hydrogen or helium atom +respectively, at least in the earlier stages of the binding process, +where during the greater part of its revolution it moves at a distance +from the nucleus which is large in proportion to the dimensions of the +orbits of the electrons previously bound. From analogy with formula (3) +giving the stationary states of the hydrogen atom, we shall therefore +assume that the numerical value of the expression on the right-hand +side of (13) will be equal to the work required to remove the last +captured electron from the atom, the binding of which gives rise to the +arc spectrum of the element.</p> + +<p class="space-above2"> +<b>Series diagram.</b> While the origin of the arc and spark spectra +was to this extent immediately interpreted on the basis of the original +simple theory of the hydrogen spectrum, it was Sommerfeld's theory +of the fine structure of the hydrogen lines which first gave us a +clear insight into the characteristic difference between the hydrogen +spectrum and the spark spectrum of helium on the one hand, and the arc +and spark spectra of other elements on the other. When we consider +the binding not of the first but of the subsequent electrons in the +atom, the orbit of the electron under consideration—at any rate in +the latter stages of the binding process where the electron last bound +comes into intimate interaction with those previously bound—will no +longer be to a near approximation a closed ellipse, but on the contrary +will to a first approximation be a central orbit of the same type as in +the hydrogen atom, when we take into account the change with velocity +in the mass of the electron. This motion, as we have seen, may be +resolved into a plane periodic motion upon which a uniform rotation +is superposed in the plane of the orbit; only the superposed rotation +will in this case be comparatively much more rapid and the deviation +of the periodic orbit from an ellipse much greater than in the case of +the hydrogen atom. For an orbit of this type the stationary states, +just as in the theory of the fine structure, will be determined by two +quantum numbers which we shall denote by <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">,</span> connected +in a very simple manner with the kinematic properties of the orbit. +For brevity I shall only mention that while the quantum number <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> +is connected with the value of the constant angular momentum of the +electron about the centre in the simple manner previously indicated, +<span class="pagenum" id="Page_79">[Pg 79]</span> +the determination of the principal quantum number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> requires an +investigation of the whole course of the orbit and for an arbitrary +central orbit will not be related in a simple way to the dimensions +of the rotating periodic orbit if this deviates essentially from a +Keplerian ellipse.</p> + +<div class="figcenter"> +<img src="images/003.jpg" width="400" alt="fig03"> +<div class="caption"> +<p>Fig. 2.</p> +</div></div> + +<p>These results are represented in <a href="#Page_79">Fig. 2</a> which is a reproduction of +an illustration I have used on a previous occasion (see Essay II, <a href="#Page_30">p. +30</a>), and which gives a survey of the origin of the sodium spectrum. +The black dots represent the stationary states corresponding to the +various series of spectral terms, shown on the right by the letters +<span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;" src="images/150.svg" alt=" " data-tex="S">,</span> <span class="nowrap"><img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P">,</span> <img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/152.svg" alt=" " data-tex="D"> and <span class="nowrap"><img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/153.svg" alt=" " data-tex="B">.</span> These letters correspond to the usual +notations employed in spectroscopic literature and indicate the nature +of the series (sharp series, principal series, diffuse series, etc.) +obtained by combinations of the corresponding spectral terms. The +distances of the separate points from the vertical line at the right +of the figure are proportional to the numerical value of the energy of +the atom given by equation (13). The oblique, black arrows indicate +finally the transitions between the stationary states giving rise to +the appearance of the lines in the commonly observed sodium spectrum. +The values of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> and <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> attached to the various states indicate +the quantum numbers, which, according to Sommerfeld's theory, from a +preliminary consideration might be regarded as characterizing the orbit +of the outer electron. For the sake of convenience the states which +were regarded as corresponding to the same value of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> are connected +by means of dotted lines, and these are so drawn that their vertical +asymptotes correspond to the terms in the hydrogen spectrum which +<span class="pagenum" id="Page_80">[Pg 80]</span> +belong to the same value of the principal quantum number. The course of +the curves illustrates how the deviation from the hydrogen terms may +be expected to decrease with increasing values of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">,</span> corresponding +to states, where the minimum distance between the electron in its +revolution and the nucleus constantly increases.</p> + +<p>It should be noted that even though the theory represents the principal +features of the structure of the series spectra it has not yet been +possible to give a detailed account of the spectrum of any element +by a closer investigation of the electronic orbits which may occur +in a simple field of force possessing central symmetry. As I have +mentioned already the lines of most spectra show a complex structure. +In the sodium spectrum for instance the lines of the principal series +are doublets indicating that to each <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P">-term not one stationary +state, but two such states correspond with slightly different values +of the energy. This difference is so little that it would not be +recognizable in a diagram on the same scale as <a href="#Page_79">Fig. 2</a>. The appearance +of these doublets is undoubtedly due to the small deviations from +central symmetry of the field of force originating from the inner +system in consequence of which the general type of motion of the +external electron will possess a more complicated character than that +of a simple central motion. As a result the stationary states must +be characterized by more than two quantum numbers, in the same way +that the occurrence of deviations of the orbit of the electron in the +hydrogen atom from a simple periodic orbit requires that the stationary +states of this atom shall be characterized by more than one quantum +number. Now the rules of the quantum theory lead to the introduction of +a third quantum number through the condition that the resultant angular +momentum of the atom, multiplied by <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.421ex; height: 1.532ex;" src="images/196.svg" alt=" " data-tex="2\pi">,</span> is equal to an entire +multiple of Planck's constant. This determines the orientation of the +orbit of the outer electron relative to the axis of the inner system.</p> + +<p>In this way Sommerfeld, Landé and others have shown that it is +possible not only to account in a formal way for the complex structure +of the lines of the series spectra, but also to obtain a promising +interpretation of the complicated effect of external magnetic fields +on this structure. We shall not enter here on these problems but +<span class="pagenum" id="Page_81">[Pg 81]</span> +shall confine ourselves to the problem of the fixation of the two +quantum numbers <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">,</span> which to a first approximation +describe the orbit of the outer electron in the stationary states, and +whose determination is a matter of prime importance in the following +discussion of the formation of the atom. In the determination of +these numbers we at once encounter difficulties of a profound nature, +which—as we shall see—are intimately connected with the question +of the remarkable stability of atomic structure. I shall here only +remark that the values of the quantum number <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">,</span> given in the +figure, undoubtedly cannot be retained, neither for the <img style="vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;" src="images/150.svg" alt=" " data-tex="S"> nor the +<img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P"> series. On the other hand, so far as the values employed for the +quantum number <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> are concerned, it may be stated with certainty, +that the interpretation of the properties of the orbits, which they +indicate, is correct. A starting point for the investigation of this +question has been obtained from considerations of an entirely different +kind from those previously mentioned, which have made it possible to +establish a close connection between the motion in the atom and the +appearance of spectral lines.</p> + +<p class="space-above2"> +<b>Correspondence principle.</b> So far as the principles of the +quantum theory are concerned, the point which has been emphasized +hitherto is the radical departure of these principles from our usual +conceptions of mechanical and electrodynamical phenomena. As I have +attempted to show in recent years, it appears possible, however, to +adopt a point of view which suggests that the quantum theory may, +nevertheless, be regarded as a rational generalization of our ordinary +conceptions. As may be seen from the postulates of the quantum theory, +and particularly the frequency relation, a direct connection between +the spectra and the motion of the kind required by the classical +dynamics is excluded, but at the same time the form of these postulates +leads us to another relation of a remarkable nature. Let us consider +an electrodynamic system and inquire into the nature of the radiation +which would result from the motion of the system on the basis of +the ordinary conceptions. We imagine the motion to be decomposed +into purely harmonic oscillations, and the radiation is assumed to +consist of the simultaneous emission of series of electromagnetic +<span class="pagenum" id="Page_82">[Pg 82]</span> +waves possessing the same frequency as these harmonic components +and intensities which depend upon the amplitudes of the components. +An investigation of the formal basis of the quantum theory shows us +now, that it is possible to trace the question of the origin of the +radiation processes which accompany the various transitions back to an +investigation of the various harmonic components, which appear in the +motion of the atom. The possibility, that a particular transition shall +occur, may be regarded as being due to the presence of a definitely +assignable "corresponding" component in the motion. This principle of +correspondence at the same time throws light upon a question mentioned +several times previously, namely the relation between the number +of quantum numbers, which must be used to describe the stationary +states of an atom, and the types to which the orbits of the electrons +belong. The classification of these types can be based very simply +on a decomposition of the motion into its harmonic components. Time +does not permit me to consider this question any further, and I shall +confine myself to a statement of some simple conclusions, which the +correspondence principle permits us to draw concerning the occurrence +of transitions between various pairs of stationary states. These +conclusions are of decisive importance in the subsequent argument.</p> + +<p>The simplest example of such a conclusion is obtained by considering +an atomic system, which contains a particle describing a <i>purely +periodic orbit</i>, and where the stationary states are characterized +by a single quantum number <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">.</span> In this case the motion can +according to Fourier's theorem be decomposed into a simple series of +harmonic oscillations whose frequency may be written <span class="nowrap"><img style="vertical-align: -0.029ex; width: 2.577ex; height: 1.032ex;" src="images/171.svg" alt=" " data-tex="\tau\omega">,</span> +where <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/140.svg" alt=" " data-tex="\tau"> is a whole number, and <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> is the frequency of +revolution in the orbit. It can now be shown that a transition between +two stationary states, for which the values of the quantum number are +respectively equal to <img style="vertical-align: -0.025ex; width: 1.98ex; height: 1.292ex;" src="images/130.svg" alt=" " data-tex="n′"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 2.602ex; height: 1.292ex;" src="images/131.svg" alt=" " data-tex="n″">,</span> will correspond to a harmonic +component, for which <span class="nowrap"><img style="vertical-align: -0.186ex; width: 11.534ex; height: 1.505ex;" src="images/209.svg" alt=" " data-tex="\tau = n′ - n″">.</span> This throws at once light +upon the remarkable difference which exists between the possibilities +of transitions between the stationary states of a hydrogen atom on the +one hand and of a simple system consisting of an electric particle +capable of executing simple harmonic oscillations about a position of +equilibrium on the other. For the latter system, which is frequently +<span class="pagenum" id="Page_83">[Pg 83]</span> +called a Planck oscillator, the energy in the stationary states is +determined by the familiar formula <span class="nowrap"><img style="vertical-align: -0.186ex; width: 8.813ex; height: 1.756ex;" src="images/210.svg" alt=" " data-tex="E = nh\omega">,</span> and with the aid +of the frequency relation we obtain therefore for the radiation which +will be emitted during a transition between two stationary states +<span class="nowrap"><img style="vertical-align: -0.566ex; width: 15.109ex; height: 2.262ex;" src="images/211.svg" alt=" " data-tex="\nu = (n′ - n″)\, \omega">.</span> Now, an important assumption, which is not +only essential in Planck's theory of temperature radiation, but which +also appears necessary to account for the molecular absorption in the +infra-red region of radiation, states that a harmonic oscillator will +only emit and absorb radiation, for which the frequency <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/88.svg" alt=" " data-tex="\nu"> is +equal to the frequency of oscillation <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> of the oscillator. We +are therefore compelled to assume that in the case of the oscillator +transitions can occur only between stationary states which are +characterized by quantum numbers differing by only one unit, while +in the hydrogen spectrum represented by formula (2) all possible +transitions could take place between the stationary states given by +formula (5). From the point of view of the principle of correspondence +it is seen, however, that this apparent difficulty is explained by the +occurrence in the motion of the hydrogen atom, as opposed to the motion +of the oscillator, of harmonic components corresponding to values of +<span class="nowrap"><img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/140.svg" alt=" " data-tex="\tau">,</span> which are different from <span class="nowrap"><img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/103.svg" alt=" " data-tex="1">;</span> or using a terminology well +known from acoustics, there appear overtones in the motion of the +hydrogen atom.</p> + +<p>Another simple example of the application of the correspondence +principle is afforded by a <i>central motion</i>, to the investigation +of which the explanation of the series spectra in the first +approximation may be reduced. Referring once more to the figure of the +sodium spectrum, we see that the black arrows, which correspond to the +spectral lines appearing under the usual conditions of excitation, +only connect pairs of points in consecutive rows. Now it is found that +this remarkable limitation of the occurrence of combinations between +spectral terms may quite naturally be explained by an investigation of +the harmonic components into which a central motion can be resolved. +It can readily be shown that such a motion can be decomposed into two +series of harmonic components, whose frequencies can be expressed by +<img style="vertical-align: -0.186ex; width: 6.634ex; height: 1.505ex;" src="images/172.svg" alt=" " data-tex="\tau\omega + \sigma"> and <img style="vertical-align: -0.186ex; width: 6.634ex; height: 1.505ex;" src="images/173.svg" alt=" " data-tex="\tau\omega - \sigma"> respectively, where +<img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/140.svg" alt=" " data-tex="\tau"> is a whole number, <img style="vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;" src="images/89.svg" alt=" " data-tex="\omega"> the frequency of revolution +in the rotating periodic orbit and <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/156.svg" alt=" " data-tex="\sigma"> the frequency of the +superposed rotation. These components correspond with transitions +<span class="pagenum" id="Page_84">[Pg 84]</span> +where the principal number <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> decreases by <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/140.svg" alt=" " data-tex="\tau"> units, while +the quantum number <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> decreases or increases, respectively, by one +unit, corresponding exactly with the transitions indicated by the +black arrows in the figure. This may be considered as a very important +result, because we may say, that the quantum theory, which for the +first time has offered a simple interpretation of the fundamental +principle of combination of spectral lines has at the same time removed +the mystery which has hitherto adhered to the application of this +principle on account of the apparent capriciousness of the appearance +of predicted combination lines. Especially attention may be drawn +to the simple interpretation which the quantum theory offers of the +appearance observed by Stark and his collaborators of certain new +series of lines, which do not appear under ordinary circumstances, +but which are excited when the emitting atoms are subject to intense +external electric fields. In fact, on the correspondence principle +this is immediately explained from an examination of the perturbations +in the motion of the outer electron which give rise to the appearance +in this motion—besides the harmonic components already present in a +simple central orbit—of a number of constituent harmonic vibrations +of new type and of amplitudes proportional to the intensity of the +external forces.</p> + +<p>It may be of interest to note that an investigation of the limitation +of the possibility of transitions between stationary states, based +upon a simple consideration of conservation of angular momentum during +the process of radiation, does not, contrary to what has previously +been supposed (compare Essay II, <a href="#Page_62">p. 62</a>), suffice to throw light on +the remarkably simple structure of series spectra illustrated by the +figure. As mentioned above we must assume that the "complexity" of +the spectral terms, corresponding to given values of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">,</span> +which we witness in the fine structure of the spectral lines, may be +ascribed to states, corresponding to different values of this angular +momentum, in which the plane of the electronic orbit is orientated in +a different manner, relative to the configuration of the previously +bound electrons in the atom. Considerations of conservation of angular +momentum can, in connection with the series spectra, therefore only +contribute to an understanding of the limitation of the possibilities +<span class="pagenum" id="Page_85">[Pg 85]</span> +of combination observed in the peculiar laws applying to the number of +components in the complex structure of the lines. So far as the last +question is concerned, such considerations offer a direct support for +the consequences of the correspondence principle.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="III_FORMATION_OF_ATOMS_AND_THE_PERIODIC_TABLE"> +III. FORMATION OF ATOMS AND THE PERIODIC TABLE</h2> +</div> + + +<p>A correspondence has been shown to exist between the motion of the +electron last captured and the occurrence of transitions between the +stationary states corresponding to the various stages of the binding +process. This fact gives a point of departure for a choice between +the numerous possibilities which present themselves when considering +the formation of the atoms by the successive capture and binding of +the electrons. Among the processes which are conceivable and which +according to the quantum theory might occur in the atom we shall +reject those whose occurrence cannot be regarded as consistent with a +correspondence of the required nature.</p> + +<p class="space-above2"> +<b>First Period. Hydrogen—Helium.</b> It will not be necessary to +concern ourselves long with the question of the constitution of the +hydrogen atom. From what has been said previously we may assume that +the final result of the process of <i>binding of the first electron</i> +in any atom will be a stationary state, where the energy of the atom +is given by (5), if we put <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.506ex; height: 1.692ex;" src="images/110.svg" alt=" " data-tex="n = 1">,</span> or more precisely by formula +(11), if we put <img style="vertical-align: -0.186ex; width: 5.506ex; height: 1.692ex;" src="images/110.svg" alt=" " data-tex="n = 1"> and <span class="nowrap"><img style="vertical-align: -0.186ex; width: 5.327ex; height: 1.756ex;" src="images/180.svg" alt=" " data-tex="k = 1">.</span> The orbit of the electron +will be a circle whose radius will be given by formulae (10), if <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> +and <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> are each put equal to <span class="nowrap"><img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/103.svg" alt=" " data-tex="1">.</span> Such an orbit will be called a +<img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/103.svg" alt=" " data-tex="1">-quantum orbit, and in general an orbit for which the principal +quantum number has a given value <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> will be called an <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">-quantum +orbit. Where it is necessary to differentiate between orbits +corresponding to various values of the quantum number <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">,</span> a central +orbit, characterized by given values of the quantum numbers <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> and +<span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">,</span> will be referred to as an <img style="vertical-align: -0.357ex; width: 2.379ex; height: 1.357ex;" src="images/212.svg" alt=" " data-tex="n_{k}"> orbit.</p> + +<p>In the question of the constitution of the helium atom we meet the much +more complicated problem of the <i>binding of the second electron</i>. +Information about this binding process may, however, be obtained +from the arc spectrum of helium. This spectrum, as opposed to most +other simple spectra, consists of two complete systems of lines with +frequencies given by formulae of the type (12). On this account +<span class="pagenum" id="Page_86">[Pg 86]</span> +helium was at first assumed to be a mixture of two different gases, +"orthohelium" and "parhelium," but now we know that the two spectra +simply mean that the binding of the second electron can occur in two +different ways. A theoretical explanation of the main features of the +helium spectrum has recently been attempted in an interesting paper by +Landé. He supposes the emission of the orthohelium spectrum to be due +to transitions between stationary states where both electrons move in +the same plane and revolve in the same sense. The parhelium spectrum, +on the other hand, is ascribed by him to stationary states where the +planes of the orbits form an angle with each other. Dr Kramers and +I have made a closer investigation of the interaction between the +two orbits in the different stationary states. The results of our +investigation which was begun several years before the appearance of +Landé's work have not yet been published. Without going into details +I may say, that even though our results in several respects differ +materially from those of Landé (compare Essay II, <a href="#Page_56">p. 56</a>), we agree with +his general conclusions concerning the origin of the orthohelium and +parhelium spectra.</p> + +<p>The final result of the binding of the second electron is intimately +related to the origin of the two helium spectra. Important information +on this point has been obtained recently by Franck and his co-workers. +As is well known he has thrown light upon many features of the +structure of the atom and of the origin of spectra by observing the +effect of bombarding atoms by electrons of various velocities. A short +time ago these experiments showed that the impact of electrons could +bring helium into a "metastable" state from which the atom cannot +return to its normal state by means of a simple transition accompanied +by the emission of radiation, but only by means of a process analogous +to a chemical reaction involving interaction with atoms of other +elements. This result is closely connected with the fact that the +binding of the second electron can occur in two different ways, as is +shown by the occurrence of two distinct spectra. Thus it is evident +from Franck's experiments that the normal state of the atom is the last +stage in the binding process involving the emission of the parhelium +spectrum by which the electron last captured as well as the one first +<span class="pagenum" id="Page_87">[Pg 87]</span> +captured will be bound in a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/213.svg" alt=" " data-tex="1_{1}"> orbit. The metastable state, on +the contrary, is the final stage of the process giving the orthohelium +spectrum. In this case the second electron, as opposed to the first, +will move in a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbit. This corresponds to a firmness of +binding which is about six times less than for the electron in the +normal state of the atom.</p> + +<p>If we now consider somewhat more closely this apparently surprising +result, it is found that a clear grasp of it may be obtained from the +point of view of correspondence. It can be shown that the coherent +class of motions to which the orthohelium orbits belong does not +contain a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/213.svg" alt=" " data-tex="1_{1}"> orbit. If on the whole we would claim the existence +of a state where the two electrons move in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/213.svg" alt=" " data-tex="1_{1}"> orbits in the same +plane, and if in addition it is claimed that the motion should possess +the periodic properties necessary for the definition of stationary +states, then there seems that no possibility is afforded other than the +assumption that the two electrons move around the nucleus in one and +the same orbit, in such a manner that at each moment they are situated +at the ends of a diameter. This extremely simple ring-configuration +might be expected to correspond to the firmest possible binding of the +electrons in the atom, and it was on this account proposed as a model +for the helium atom in my first paper on atomic structure. If, however, +we inquire about the possibility of a transition from one of the +orthohelium states to a configuration of this type we meet conditions +which are very different from those which apply to transitions between +two of the orthohelium orbits. In fact, the occurrence of each of these +transitions is due to the existence of well-defined corresponding +constituent harmonic vibration in the central orbits which the outer +electron describes in the class of motions to which the stationary +states belong. The transition we have to discuss, on the other hand, +is one by which the last captured electron is transferred from a state +in which it is moving "outside" the other to a state in which it moves +round the nucleus on equal terms with the other electron. Now it is +impossible to find a series of simple intermediate forms for the motion +of those two electrons in which the orbit of the last captured electron +exhibits a sufficient similarity to a central motion that for this +transition there could be a correspondence of the necessary kind. It +is therefore evident, that where the two electrons move in the same +plane, the electron captured last cannot be bound firmer than in a +<span class="pagenum" id="Page_88">[Pg 88]</span> +<img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbit. If, on the other hand, we consider the binding process +which accompanies the emission of the parhelium spectrum and where the +electrons in the stationary states move in orbits whose planes form +angles with one another we meet essentially different conditions. A +corresponding intimate change in the interaction between the electron +last captured and the one previously bound is not required here for +the two electrons in the atom to become equivalent. We may therefore +imagine the last stage of the binding process to take place in a manner +similar to those stages corresponding to transitions between orbits +characterized by greater values of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">.</span></p> + +<p>In the <i>normal state of the helium atom</i> the two electrons +must be assumed to move in equivalent <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/213.svg" alt=" " data-tex="1_{1}"> orbits. As a first +approximation these may be described as two circular orbits, whose +planes make an angle of <img style="vertical-align: -0.05ex; width: 4.525ex; height: 1.667ex;" src="images/215.svg" alt=" " data-tex="120°"> with one another, in agreement with +the conditions which the angular momentum of an atom according to the +quantum theory must satisfy. On account of the interaction between the +two electrons these planes at the same time turn slowly around the +fixed impulse axis of the atom. Starting from a distinctly different +point of view Kemble has recently suggested a similar model for the +helium atom. He has at the same time directed attention to a possible +type of motion of very marked symmetry in which the electrons during +their entire revolution assume symmetrical positions with reference +to a fixed axis. Kemble has not, however, investigated this motion +further. Previous to the appearance of this paper Kramers had commenced +a closer investigation of precisely this type of motion in order +to find out to what extent it was possible from such a calculation +to account for the firmness with which the electrons are bound in +the helium atom, that is to account for the ionization potential. +Early measurements of this potential had given values corresponding +approximately to that which would result from the ring-configuration +already mentioned. This requires <img style="vertical-align: -0.566ex; width: 4.525ex; height: 2.262ex;" src="images/216.svg" alt=" " data-tex="17/8"> as much work to remove a +single electron as is necessary to remove an electron from the hydrogen +atom in its normal state. As the theoretical value for the latter +amount of work—which for the sake of simplicity will be represented +by <img style="vertical-align: -0.05ex; width: 2.371ex; height: 1.595ex;" src="images/93.svg" alt=" " data-tex="W">—corresponds to an ionization potential of <img style="vertical-align: -0.05ex; width: 5.154ex; height: 1.557ex;" src="images/217.svg" alt=" " data-tex="13.53"> +volts, the ionization potential of helium would be expected to be +<img style="vertical-align: -0.05ex; width: 4.023ex; height: 1.557ex;" src="images/218.svg" alt=" " data-tex="28.8"> volts. Recent and more accurate determinations, however, +<span class="pagenum" id="Page_89">[Pg 89]</span> +have given a value for the ionization potential of helium which is +considerably lower and lies in the neighbourhood of <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/219.svg" alt=" " data-tex="25"> volts. This +showed therefore the untenability of the ring-configuration quite +independently of any other considerations. A careful investigation of +the spatial atomic configuration requires elaborate calculation, and +Kramers has not yet obtained final results. With the approximation +to which they have been so far completed the calculations point to +the possibility of an agreement with the experimental results. The +final result may be awaited with great interest, since it offers in +the simplest case imaginable a test of the principles by which we are +attempting to determine stationary states of atoms containing more than +one electron.</p> + +<p>Hydrogen and helium, as seen in the survey of the periodic system given +in <a href="#Page_70">Fig. 1</a>, together form the first period in the system of elements, +since helium is the first of the inactive gases. The great difference +in the chemical properties of hydrogen and helium is closely related +to the great difference in the nature of the binding of the electron. +This is directly indicated by the spectra and ionization potentials. +While helium possesses the highest known ionization potential of +all the elements, the binding of the electron in the hydrogen atom +is sufficiently loose to account for the tendency of hydrogen to +form positive ions in aqueous solutions and chemical combinations. +Further consideration of this particular question requires, however, +a comparison between the nature and firmness of the electronic +configurations of other atoms, and it can therefore not be discussed at +the moment.</p> + +<p class="space-above2"> +<b>Second Period. Lithium—Neon.</b> When considering the atomic +structure of elements which contain more than two electrons in the +neutral atom, we shall assume first of all that what has previously +been said about the formation of the helium atom will in the main +features also apply to the capture and binding of the first two +electrons. These electrons may, therefore, in the normal state of +the atom be regarded as moving in equivalent orbits characterized by +the quantum symbol <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/213.svg" alt=" " data-tex="1_{1}">.</span> We obtain direct information about the +<i>binding of the third electron</i> from the spectrum of lithium. This +spectrum shows the existence of a number of series of stationary +<span class="pagenum" id="Page_90">[Pg 90]</span> +states, where the firmness with which the last captured electron is +bound is very nearly the same as in the stationary states of the +hydrogen atom. These states correspond to orbits where <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> is greater +than or equal to <span class="nowrap"><img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">,</span> and where the last captured electron moves +entirely outside the region where the first two electrons move. But in +addition this spectrum gives us information about a series of states +corresponding to <img style="vertical-align: -0.186ex; width: 5.327ex; height: 1.756ex;" src="images/180.svg" alt=" " data-tex="k = 1"> in which the energy differs essentially +from the corresponding stationary states of the hydrogen atom. In +these states the last captured electron, even if it remains at a +considerable distance from the nucleus during the greater part of its +revolution, will at certain moments during the revolution approach to +a distance from the nucleus which is of the same order of magnitude +as the dimensions of the orbits of the previously bound electrons. +On this account the electrons will be bound with a firmness which is +considerably greater than that with which the electrons are bound in +the stationary states of the hydrogen atom corresponding to the same +value of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n">.</span></p> + +<p>Now as regards the lithium spectrum as well as the other alkali spectra +we are so fortunate (see <a href="#Page_32">p. 32</a>) as to possess definite evidence about +the normal state of the atom from experiments on selective absorption. +In fact these experiments tell us that the first member of the sequence +of <img style="vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;" src="images/150.svg" alt=" " data-tex="S">-terms corresponds to this state. This term corresponds to a +strength of binding which is only a little more than a third of that of +the hydrogen atom. We must therefore conclude that the outer electron +in the normal state of the lithium atom moves in a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbit, +just as the outer electron in the metastable state of the helium atom. +The reason why the binding of the outer electron cannot proceed to an +orbit characterized by a smaller value for the total quantum number may +also be considered as analogous in the two cases. In fact, a transition +by which the third electron in the lithium atom was ultimately bound +in a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/213.svg" alt=" " data-tex="1_{1}"> orbit would lead to a state in the atom in which +the electron would play an equivalent part with the two electrons +previously bound. Such a process would be of a type entirely different +from the transitions between the stationary states connected with the +emission of the lithium spectrum, and would, contrary to these, not +exhibit a correspondence with a harmonic component in the motion of the +atom. +<span class="pagenum" id="Page_91">[Pg 91]</span> +</p> + +<p>We obtain, therefore, a picture of the formation and structure of +the lithium atom which offers a natural explanation of the great +difference of the chemical properties of lithium from those of helium +and hydrogen. This difference is at once explained by the fact that the +firmness by which the last captured electron is bound in its <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> +orbit in the lithium atom is only about a third of that with which the +electron in the hydrogen atom is held, and almost five times smaller +than the firmness of the binding of the electrons in the helium atom.</p> + +<p>What has been said here applies not alone to the formation of the +lithium atom, but may also be assumed to apply to the binding of the +third electron in every atom, so that in contrast to the first two +electrons which move in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/213.svg" alt=" " data-tex="1_{1}"> orbits this may be assumed to move +in a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbit. As regards the <i>binding of the fourth, fifth +and sixth electrons</i> in the atom, we do not possess a similar +guide as no simple series spectra are known of beryllium, boron and +carbon. Although conclusions of the same degree of certainty cannot be +reached it seems possible, however, to arrive at results consistent +with general physical and chemical evidence by proceeding by means of +considerations of the same kind as those applied to the binding of +the first three electrons. In fact, we shall assume that the fourth, +fifth and sixth electrons will be bound in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbits. The +reason why the binding of a first electron in an orbit of this type +will not prevent the capture of the others in two quanta orbits may +be ascribed to the fact that <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbits are not circular but +very eccentric. For example, the <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">rd electron cannot keep the +remaining electrons away from the inner system in the same way in +which the first two electrons bound in the lithium atom prevent the +third from being bound in a <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/103.svg" alt=" " data-tex="1">-quantum orbit. Thus we shall expect +that the <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">th, <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/221.svg" alt=" " data-tex="5">th and <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/222.svg" alt=" " data-tex="6">th electrons in a similar way to +the <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">rd will at certain moments of their revolution enter into the +region where the first two bound electrons move. We must not imagine, +however, that these visits into the inner system take place at the +same time, but that the four electrons visit the nucleus separately at +equal intervals of time. In earlier work on atomic structure it was +supposed that the electrons in the various groups in the atom moved in +separate regions within the atom and that at each moment the electrons +within each separate group were arranged in configurations possessing +<span class="pagenum" id="Page_92">[Pg 92]</span> +symmetry like that of a regular polygon or polyhedron. Among other +things this involved that the electrons in each group were supposed +to be at the point of the orbit nearest the nucleus at the same time. +A structure of this kind may be described as one where the motions of +the electrons within the groups are coupled together in a manner which +is largely independent of the interaction between the various groups. +On the contrary, the characteristic feature of a structure like that I +have suggested is the <i>intimate coupling between the motions of the +electrons in the various groups</i> characterized by different quantum +numbers, as well as the <i>greater independence in the mode of binding +within one and the same group of electrons</i> the orbits of which are +characterized by the same quantum number. In emphasizing this last +feature I have two points in mind. Firstly the smaller effect of the +presence of previously bound electrons on the firmness of binding of +succeeding electrons in the same group. Secondly the way in which the +motions of the electrons within the group reflect the independence both +of the processes by which the group can be formed and by which it can +be reorganized by change of position of the different electrons in the +atom after a disturbance by external forces. The last point will be +considered more closely when we deal with the origin and nature of the +X-ray spectra; for the present we shall continue the consideration of +the structure of the atom to which we are led by the investigation of +the processes connected with the successive capture of the electrons.</p> + +<p>The preceding considerations enable us to understand the fact that the +two elements beryllium and boron immediately succeeding lithium can +appear electropositively with <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2"> and <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3"> valencies respectively +in combination with other substances. For like the third electron in +the lithium atom, the last captured electrons in these elements will +be much more lightly bound than the first two electrons. At the same +time we understand why the electropositive character of these elements +is less marked than in the case of lithium, since the electrons in the +<img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">-quanta orbits will be much more firmly bound on account of the +stronger field in which they are moving. New conditions arise, however, +in the case of the next element, carbon, as this element in its typical +chemical combinations cannot be supposed to occur as an ion, but +rather as a neutral atom. This must be assumed to be due not only to +<span class="pagenum" id="Page_93">[Pg 93]</span> +the great firmness in the binding of the electrons but also to be an +essential consequence of the symmetrical configuration of the electrons.</p> + +<p>With the binding of the <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">th, <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/221.svg" alt=" " data-tex="5">th and <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/222.svg" alt=" " data-tex="6">th electrons in +<img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbits, the spatial symmetry of the regular configuration +of the orbits must be regarded as steadily increasing, until with +the binding of the <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/222.svg" alt=" " data-tex="6">th electron the orbits of the four last +bound electrons may be expected to form an exceptionally symmetrical +configuration in which the normals to the planes of the orbits occupy +positions relative to one another nearly the same as the lines from the +centre to the vertices of a regular tetrahedron. Such a configuration +of groups of <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">-quanta orbits in the carbon atom seems capable +of furnishing a suitable foundation for explaining the structure of +organic compounds. I shall not discuss this question any further, +for it would require a thorough study of the interaction between +the motions of the electrons in the atoms forming the molecule. I +might mention, however, that the types of molecular models to which +we are led are very different from the molecular models which were +suggested in my first papers. In these the chemical "valence bonds" +were represented by "electron rings" of the same type as those which +were assumed to compose the groups of electrons within the individual +atoms. It is nevertheless possible to give a general explanation of +the chemical properties of the elements without touching on those +matters at all. This is largely due to the fact that the structures of +combinations of atoms of the same element and of many organic compounds +do not have the same significance for our purpose as those molecular +structures in which the individual atoms occur as electrically charged +ions. The latter kind of compounds, to which the greater number of +simple inorganic compounds belong, is frequently called "heteropolar" +and possesses a far more typical character than the first compounds +which are called "homoeopolar," and whose properties to quite a +different degree exhibit the individual peculiarities of the elements. +My main purpose will therefore be to consider the fitness which the +configurations of the electrons in the various atoms offer for the +formation of ions.</p> + +<p>Before leaving the carbon atom I should mention, that a model of this +atom in which the orbits of the four most lightly bound electrons +<span class="pagenum" id="Page_94">[Pg 94]</span> +possess a pronounced tetrahedric symmetry had already been suggested +by Landé. In order to agree with the measurements of the size of the +atoms he also assumed that these electrons moved in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbits. +There is, however, this difference between Landé's view and that given +here, that while Landé deduced the characteristic properties of the +carbon atom solely from an investigation of the simplest form of motion +which four electrons can execute employing spatial symmetry, our view +originates from a consideration of the stability of the whole atom. For +our assumptions about the orbits of the electrons are based directly +on an investigation of the interaction between these electrons and the +first two bound electrons. The result is that our model of the carbon +atom has dynamic properties which are essentially different from the +properties of Landé's model.</p> + +<p>In order to account for the properties of <i>the elements in the second +half of the second period</i> it will first of all be necessary to show +why the configuration of ten electrons occurring in the neutral atom +of neon possesses such a remarkable degree of stability. Previously it +has been assumed that the properties of this configuration were due +to the interaction between eight electrons which moved in equivalent +orbits outside the nucleus and an inner group of two electrons like +that in the helium atom. It will be seen, however, that the solution +must be sought in an entirely different direction. It cannot be +expected that <i>the <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.579ex;" src="images/223.svg" alt=" " data-tex="7">th electron</i> will be bound in a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> +orbit equivalent to the orbits of the four preceding electrons. The +occurrence of five such orbits would so definitely destroy the symmetry +in the interaction of these electrons that it is inconceivable that +a process resulting in the accession of a fifth electron to this +group would be in agreement with the correspondence principle. On the +contrary it will be necessary to assume that the four electrons in +their exceptionally symmetrical orbital configuration will keep out +later captured electrons with the result that these electrons will be +bound in orbits of other types.</p> + +<p>The orbits which come into consideration for the <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.579ex;" src="images/223.svg" alt=" " data-tex="7">th electron +in the nitrogen atom and the <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.579ex;" src="images/223.svg" alt=" " data-tex="7">th, <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/202.svg" alt=" " data-tex="8">th, <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/224.svg" alt=" " data-tex="9">th and <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/225.svg" alt=" " data-tex="10">th +electrons in the atoms of the immediately following elements will be +circular orbits of the type <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/226.svg" alt=" " data-tex="2_{2}">.</span> The diameters of these orbits +are considerably larger than those of the <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/227.svg" alt=" " data-tex="l_{1}"> orbits of the +first two electrons; on the other hand the outermost part of the +<span class="pagenum" id="Page_95">[Pg 95]</span> +eccentric <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbits will extend some distance beyond these +circular <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/226.svg" alt=" " data-tex="2_{2}"> orbits. I shall not here discuss the capture and +binding of these electrons. This requires a further investigation +of the interaction between the motions of the electrons in the two +types of <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">-quanta orbits. I shall simply mention, that in the +atom of neon in which we will assume that there are four electrons +in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/226.svg" alt=" " data-tex="2_{2}"> orbits the planes of these orbits must be regarded not +only as occupying a position relative to one another characterized +by a high degree of spatial symmetry, but also as possessing a +configuration harmonizing with the four elliptical <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbits. An +interaction of this kind in which the orbital planes do not coincide +can be attained only if the configurations in both subgroups exhibit +a systematic deviation from tetrahedral symmetry. This will have the +result that the electron groups with <img style="vertical-align: -0.023ex; width: 2.328ex; height: 1.529ex;" src="images/90.svg" alt=" " data-tex="2a">-quanta orbits in the neon +atom will have only a single axis of symmetry which must be supposed +to coincide with the axis of symmetry of the innermost group of two +electrons.</p> + +<p>Before leaving the description of the elements within the second period +it may be pointed out that the above considerations offer a basis for +interpreting that tendency of the neutral atoms of oxygen and fluorine +for capturing further electrons which is responsible for the marked +electronegative character of these elements. In fact, this tendency +may be ascribed to the fact that the orbits of the last captured +electrons will find their place within the region, in which the +previously captured electrons move in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbits. This suggests +an explanation of the great difference between the properties of the +elements in the latter half of the second period of the periodic system +and those of the elements in the first half, in whose atoms there is +only a single type of <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">-quanta orbits.</p> + +<p class="space-above2"> +<b>Third Period. Sodium—Argon.</b> We shall now consider the structure +of atoms of elements in the third period of the periodic system. +This brings us immediately to the question of <i>the binding of the +<img style="vertical-align: 0; width: 2.262ex; height: 1.507ex;" src="images/228.svg" alt=" " data-tex="11">th electron</i> in the atom. Here we meet conditions which in +some respects are analogous to those connected with the binding of +the <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.579ex;" src="images/223.svg" alt=" " data-tex="7">th electron. The same type of argument that applied to the +carbon atom shows that the symmetry of the configuration in the neon +<span class="pagenum" id="Page_96">[Pg 96]</span> +atom would be essentially, if not entirely, destroyed by the addition +of another electron in an orbit of the same type as that in which the +last captured electrons were bound. Just as in the case of the <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">rd +and <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.579ex;" src="images/223.svg" alt=" " data-tex="7">th electrons we may therefore expect to meet a new type of +orbit for the <img style="vertical-align: 0; width: 2.262ex; height: 1.507ex;" src="images/228.svg" alt=" " data-tex="11">th electron in the atom, and the orbits which +present themselves this time are the <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/229.svg" alt=" " data-tex="3_{1}"> orbits. An electron in +such an orbit will for the greater part of the time remain outside +the orbits of the first ten electrons. But at certain moments during +the revolution it will penetrate not only into the region of the +<img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">-quanta orbits, but like the <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbits it will penetrate +to distances from the nucleus which are smaller than the radii of the +<img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/103.svg" alt=" " data-tex="1">-quantum orbits of the two electrons first bound. This fact, which +has a most important bearing on the stability of the atom, leads to a +peculiar result as regards the binding of the <img style="vertical-align: 0; width: 2.262ex; height: 1.507ex;" src="images/228.svg" alt=" " data-tex="11">th electron. In +the sodium atom this electron will move in a field which so far as the +outer part of the orbit is concerned deviates only very little from +that surrounding the nucleus in the hydrogen atom, but the dimensions +of this part of the orbit will, nevertheless, be essentially different +from the dimensions of the corresponding part of a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/229.svg" alt=" " data-tex="3_{1}"> orbit in +the hydrogen atom. This arises from the fact, that even though the +electron only enters the inner configuration of the first ten electrons +for short intervals during its revolution, this part of the orbit will +nevertheless exert an essential influence upon the determination of the +principal quantum number. This is directly related to the fact that +the motion of the electron in the first part of the orbit deviates +only a little from the motion which each of the previously bound +electrons in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbits executes during a complete revolution. +The uncertainty which has prevailed in the determination of the quantum +numbers for the stationary states corresponding to a spectrum like that +of sodium is connected with this. This question has been discussed by +several physicists. From a comparison of the spectral terms of the +various alkali metals, Roschdestwensky has drawn the conclusion that +the normal state does not, as we might be inclined to expect a priori, +correspond to a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/213.svg" alt=" " data-tex="1_{1}"> orbit as shown in Fig. 2 on <a href="#Page_79">p. 79</a>, but that +this state corresponds to a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbit. Schrödinger has arrived +at a similar result in an attempt to account for the great difference +between the <img style="vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;" src="images/150.svg" alt=" " data-tex="S"> terms and the terms in the <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P"> and <img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/152.svg" alt=" " data-tex="D"> series of +the alkali spectra. He assumes that the "outer" electron in the states +<span class="pagenum" id="Page_97">[Pg 97]</span> +corresponding to the <img style="vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;" src="images/150.svg" alt=" " data-tex="S"> terms—in contrast to those corresponding +to the <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P"> and <img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/152.svg" alt=" " data-tex="D"> terms—penetrates partly into the region of the +orbits of the inner electrons during the course of its revolution. +These investigations contain without doubt important hints, but in +reality the conditions must be very different for the different alkali +spectra. Instead of a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbit as in lithium we must thus assume +for the spectrum of sodium not only that the first spectral term in +the <img style="vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;" src="images/150.svg" alt=" " data-tex="S"> series corresponds to a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/229.svg" alt=" " data-tex="3_{1}"> orbit, but also, as a more +detailed consideration shows, that the first term in the <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P"> series +corresponds not to a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/226.svg" alt=" " data-tex="2_{2}"> orbit as indicated in <a href="#Page_79">Fig. 2</a>, but to a +<img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/230.svg" alt=" " data-tex="3_{2}"> orbit. If the numbers in this figure were correct, it would +require among other things that the <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P"> terms should be smaller than +the hydrogen terms corresponding to the same principal quantum number.</p> + +<div class="figcenter"> +<img class="w100" src="images/004.jpg" width="400" alt="fig04"> +<div class="caption"> +<p>Fig. 3.</p> +</div></div> + +<p>This would mean that the average effect of the inner electrons could +be described as a repulsion greater than would occur if their total +electrical charge were united in the nucleus. This, however, cannot +be expected from our view of atomic structure. The fact that the last +captured electron, at any rate for low values of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">,</span> revolves +partly inside the orbits of the previously bound electrons will on the +contrary involve that the presence of these electrons will give rise to +a virtual repulsion which is considerably smaller than that which would +be due to their combined charges. Instead of the curves drawn between +points in <a href="#Page_79">Fig. 2</a> which represent stationary states corresponding to +the same value of the principal quantum number running from right to +left, we obtain curves which run from left to right, as is indicated +in <a href="#Page_97">Fig. 3</a>. The stationary states are labelled with quantum numbers +<span class="pagenum" id="Page_98">[Pg 98]</span> +corresponding to the structure I have described. According to the view +underlying <a href="#Page_79">Fig. 2</a> the sodium spectrum might be described simply as a +distorted hydrogen spectrum, whereas according to <a href="#Page_97">Fig. 3</a> there is not +only distortion but also complete disappearance of certain terms of +low quantum numbers. It may be stated, that this view not only appears +to offer an explanation of the magnitude of the terms, but that the +complexity of the terms in the <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P"> and <img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/152.svg" alt=" " data-tex="D"> series finds a natural +explanation in the deviation of the configuration of the ten electrons +first bound from a purely central symmetry. This lack of symmetry +has its origin in the configuration of the two innermost electrons +and "transmits" itself to the outer parts of the atomic structure, +since the <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbits penetrate partly into the region of these +electrons.</p> + +<p>This view of the sodium spectrum provides at the same time an immediate +explanation of the pronounced electropositive properties of sodium, +since the last bound electron in the sodium atom is still more loosely +bound than the last captured electron in the lithium atom. In this +connection it might be mentioned that the increase in atomic volume +with increasing atomic number in the family of the alkali metals finds +a simple explanation in the successively looser binding of the valency +electrons. In his work on the X-ray spectra Sommerfeld at an earlier +period regarded this increase in the atomic volumes as supporting +the assumption that the principal quantum number of the orbit of the +valency electrons increases by unity as we pass from one metal to the +next in the family. His later investigations on the series spectra +have led him, however, definitely to abandon this assumption. At +first sight it might also appear to entail a far greater increase in +the atomic volume than that actually observed. A simple explanation +of this fact is however afforded by realizing that the orbit of the +electron will run partly inside the region of the inner orbit and that +therefore the "effective" quantum number which corresponds to the +outer almost elliptical loop will be much smaller than the principal +quantum number, by which the whole central orbit is described. It may +be mentioned that Vegard in his investigations on the X-ray spectra has +also proposed the assumption of successively increasing quantum numbers +for the electronic orbits in the various groups of the atom, reckoned +from the nucleus outward. He has introduced assumptions about the +<span class="pagenum" id="Page_99">[Pg 99]</span> +relations between the numbers of electrons in the various groups of +the atom and the lengths of the periods in the periodic system which +exhibit certain formal similarities with the results presented here. +But Vegard's considerations do not offer points of departure for a +further consideration of the evolution and stability of the groups, and +consequently no basis for a detailed interpretation of the properties +of the elements.</p> + +<p>When we consider the elements following sodium in the third period of +the periodic system we meet in <i>the binding of the <img style="vertical-align: 0; width: 2.262ex; height: 1.507ex;" src="images/231.svg" alt=" " data-tex="12">th, <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/232.svg" alt=" " data-tex="13">th +and <img style="vertical-align: 0; width: 2.262ex; height: 1.532ex;" src="images/233.svg" alt=" " data-tex="14">th electrons</i> conditions which are analogous to those we +met in the binding of the <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">th, <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/221.svg" alt=" " data-tex="5">th and <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/222.svg" alt=" " data-tex="6">th electrons. In +the elements of the third periods, however, we possess a far more +detailed knowledge of the series spectra. Too little is known about +the beryllium spectrum to draw conclusions about the binding of the +fourth electron, but we may infer directly from the well-known arc +spectrum of magnesium that the <img style="vertical-align: 0; width: 2.262ex; height: 1.507ex;" src="images/231.svg" alt=" " data-tex="12">th electron in the atom of this +element is bound in a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/229.svg" alt=" " data-tex="3_{1}"> orbit. As regards the binding of the +<img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/232.svg" alt=" " data-tex="13">th electron we meet in aluminium an absorption spectrum different +in structure to that of the alkali metals. In fact here not the lines +of the principal series but the lines of the sharp and diffuse series +are absorption lines. Consequently it is the first member of the <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P"> +terms and not of the <img style="vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;" src="images/150.svg" alt=" " data-tex="S"> terms which corresponds to the normal state +of the aluminium atom, and we must assume that the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/232.svg" alt=" " data-tex="13">th electron +is bound in a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/230.svg" alt=" " data-tex="3_{2}"> orbit. This, however, would hardly seem to be +a general property of the binding of the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/232.svg" alt=" " data-tex="13">th electron in atoms, +but rather to arise from the special conditions for the binding of the +last electron in an atom, where already there are two other electrons +bound as loosely as the valency electron of aluminium. At the present +state of the theory it seems best to assume that in the silicon atom +the four last captured electrons will move in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/229.svg" alt=" " data-tex="3_{1}"> orbits forming +a configuration possessing symmetrical properties similar to the outer +configuration of the four electrons in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> orbits in carbon. Like +what we assumed for the latter configuration we shall expect that the +configuration of the <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/229.svg" alt=" " data-tex="3_{1}"> orbits occurring for the first time in +silicon possesses such a completion, that the addition of a further +electron in a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/229.svg" alt=" " data-tex="3_{1}"> orbit to the atom of the following elements +is impossible, and that <i>the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/234.svg" alt=" " data-tex="15">th electron</i> in the elements +of higher atomic number will be bound in a new type of orbit. In this +<span class="pagenum" id="Page_100">[Pg 100]</span> +case, however, the orbits with which we meet will not be circular, as +in the capture of the <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.579ex;" src="images/223.svg" alt=" " data-tex="7">th electron, but will be rotating eccentric +orbits of the type <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/230.svg" alt=" " data-tex="3_{2}">.</span> This is very closely related to the fact, +mentioned above, that the non-circular orbits will correspond to a +firmer binding than the circular orbits having the same value for the +principal quantum number, since the electrons will at certain moments +penetrate much farther into the interior of the atom. Even though a +<img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/230.svg" alt=" " data-tex="3_{2}"> orbit will not penetrate into the innermost configuration +of <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/213.svg" alt=" " data-tex="1_{1}"> orbits, it will penetrate to distances from the nucleus +which are considerably less than the radii of the circular <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/226.svg" alt=" " data-tex="2_{2}"> +orbits. In the case of the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/200.svg" alt=" " data-tex="16">th, <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.579ex;" src="images/198.svg" alt=" " data-tex="17">th and <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18">th electrons +the conditions are similar to those for the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/234.svg" alt=" " data-tex="15">th. So for argon we +may expect a configuration in which the ten innermost electrons move +in orbits of the same type as in the neon atom while the last eight +electrons will form a configuration of four <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/229.svg" alt=" " data-tex="3_{1}"> orbits and four +<img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/230.svg" alt=" " data-tex="3_{2}"> orbits, whose symmetrical properties must be regarded as +closely corresponding to the configuration of <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">-quanta orbits in +the neon atom. At the same time, as this picture suggests a qualitative +explanation of the similarity of the chemical properties of the +elements in the latter part of the second and third periods, it also +opens up the possibility of a natural explanation of the conspicuous +difference from a quantitative aspect.</p> + +<p class="space-above2"> +<b>Fourth Period. Potassium—Krypton.</b> In the fourth period we meet +at first elements which resemble chemically those at the beginning of +the two previous periods. This is also what we should expect. We must +thus assume that <i>the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/199.svg" alt=" " data-tex="19">th electron</i> is bound in a new type of +orbit, and a closer consideration shows that this will be a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> +orbit. The points which were emphasized in connection with the binding +of the last electron in the sodium atom will be even more marked here +on account of the larger quantum number by which the orbits of the +inner electrons are characterized. In fact, in the potassium atom the +<img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> orbit of the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/199.svg" alt=" " data-tex="19">th electron will, as far as inner loops +are concerned, coincide closely with the shape of a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/229.svg" alt=" " data-tex="3_{1}"> orbit. +On this account, therefore, the dimensions of the outer part of the +orbit will not only deviate greatly from the dimensions of a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> +<span class="pagenum" id="Page_101">[Pg 101]</span> +orbit in the hydrogen atom, but will coincide closely with a hydrogen +orbit of the type <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}">,</span> the dimensions of which are about four +times smaller than the <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> hydrogen orbit. This result allows an +immediate explanation of the main features of the chemical properties +and the spectrum of potassium. Corresponding results apply to calcium, +in the neutral atom of which there will be two valency electrons in +equivalent <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> orbits.</p> + +<p>After calcium the properties of the elements in the fourth period +of the periodic system deviate, however, more and more from the +corresponding elements in the previous periods, until in the family +of the iron metals we meet elements whose properties are essentially +different. Proceeding to still higher atomic numbers we again meet +different conditions. Thus we find in the latter part of the fourth +period a series of elements whose chemical properties approach more +and more to the properties of the elements at the end of the preceding +periods, until finally with atomic number <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/236.svg" alt=" " data-tex="36"> we again meet one of +the inactive gases, namely krypton. This is exactly what we should +expect. The formation and stability of the atoms of the elements in the +first three periods require that each of the first <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> electrons +in the atom shall be bound in each succeeding element in an orbit of +the same principal quantum number as that possessed by the particular +electron, when it first appeared. It is readily seen that this is no +longer the case for the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/199.svg" alt=" " data-tex="19">th electron. With increasing nuclear +charge and the consequent decrease in the difference between the fields +of force inside and outside the region of the orbits of the first +<img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> bound electrons, the dimensions of those parts of a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> +orbit which fall outside will approach more and more to the dimensions +of a <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quantum orbit calculated on the assumption that the +interaction between the electrons in the atom may be neglected. <i>With +increasing atomic number a point will therefore be reached where a +<img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.879ex;" src="images/237.svg" alt=" " data-tex="3_{3}"> orbit will correspond to a firmer binding of the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/199.svg" alt=" " data-tex="19">th +electron than a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> orbit</i>, and this occurs as early as at +the beginning of the fourth period. This cannot only be anticipated +from a simple calculation but is confirmed in a striking way from an +examination of the series spectra. While the spectrum of potassium +indicates that the <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> orbit corresponds to a binding which is +more than twice as firm as in a <img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.879ex;" src="images/237.svg" alt=" " data-tex="3_{3}"> orbit corresponding to the +first spectral term in the <img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/152.svg" alt=" " data-tex="D"> series, the conditions are entirely +different as soon as calcium is reached. We shall not consider the +<span class="pagenum" id="Page_102">[Pg 102]</span> +arc spectrum which is emitted during the capture of the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/201.svg" alt=" " data-tex="20">th +electron but the spark spectrum which corresponds to the capture and +binding of the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/199.svg" alt=" " data-tex="19">th electron. While the spark spectrum of magnesium +exhibits great similarity with the sodium spectrum as regards the +values of the spectral terms in the various series—apart from the fact +that the constant appearing in formula (12) is four times as large as +the Rydberg constant—we meet in the spark spectrum of calcium the +remarkable condition that the first term of the <img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/152.svg" alt=" " data-tex="D"> series is larger +than the first term of the <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P"> series and is only a little smaller +than the first term of the <img style="vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;" src="images/150.svg" alt=" " data-tex="S"> series, which may be regarded as +corresponding to the binding of the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/199.svg" alt=" " data-tex="19">th electron in the normal +state of the calcium atom.</p> + +<div class="figcenter"> +<img src="images/005.jpg" width="400" alt="fig05"> +<div class="caption"> +<p>Fig. 4.</p> +</div></div> + +<p>These facts are shown in <a href="#Page_102">figure 4</a> which gives a survey of the +stationary states corresponding to the arc spectra of sodium and +potassium. As in figures <a href="#Page_79">2</a> and <a href="#Page_97">3</a> of the sodium spectrum, we have +disregarded the complexity of the spectral terms, and the numbers +characterizing the stationary states are simply the quantum numbers +<span class="pagenum" id="Page_103">[Pg 103]</span> +<img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">.</span> For the sake of comparison the scale in which the +energy of the different states is indicated is chosen four times as +small for the spark spectra as for the arc spectra. Consequently the +vertical lines indicated with various values of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> correspond for +the arc spectra to the spectral terms of hydrogen, for the spark +spectra to the terms of the helium spectrum given by formula (7). +Comparing the change in the relative firmness in the binding of the +<img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/199.svg" alt=" " data-tex="19">th electron in a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> and <img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.879ex;" src="images/237.svg" alt=" " data-tex="3_{3}"> orbit for potassium +and calcium we see that we must be prepared already for the next +element, scandium, to find that the <img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.879ex;" src="images/237.svg" alt=" " data-tex="3_{3}"> orbit will correspond +to a stronger binding of this electron than a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> orbit. On the +other hand it follows from previous remarks that the binding will be +much lighter than for the first <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> electrons which agrees that in +chemical combinations scandium appears electropositively with three +valencies.</p> + +<p>If we proceed to the following elements, a still larger number of +<img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.879ex;" src="images/237.svg" alt=" " data-tex="3_{3}"> orbits will occur in the normal state of these atoms, since +the number of such electron orbits will depend upon the firmness of +their binding compared to the firmness with which an electron is +bound in a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> orbit, in which type of orbit at least the last +captured electron in the atom may be assumed to move. We therefore +meet conditions which are essentially different from those which we +have considered in connection with the previous periods, so that +here we have to do with <i>the successive development of one of the +inner groups of electrons in the atom</i>, in this case with groups +of electrons in <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits. Only when the development +of this group has been completed may we expect to find once more +a corresponding change in the properties of the elements with +increasing atomic number such as we find in the preceding periods. The +properties of the elements in the latter part of the fourth period +show immediately that the group, when completed, will possess <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> +electrons. Thus in krypton, for example, we may expect besides the +groups of <span class="nowrap"><img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/103.svg" alt=" " data-tex="1">,</span> <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2"> and <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits a markedly symmetrical +configuration of <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/202.svg" alt=" " data-tex="8"> electrons in <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quanta orbits consisting of +four <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> orbits and four <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/238.svg" alt=" " data-tex="4_{2}"> orbits.</p> + +<p>The question now arises: In which way will the gradual formation of the +group of electrons having <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits take place? From analogy +with the constitution of the groups of electrons with <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">-quanta +orbits we might at first sight be inclined to suppose that the +<span class="pagenum" id="Page_104">[Pg 104]</span> +complete group of <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits would consist of three subgroups +of four electrons each in orbits of the types <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/229.svg" alt=" " data-tex="3_{1}">,</span> <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.844ex;" src="images/230.svg" alt=" " data-tex="3_{2}"> and +<img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.879ex;" src="images/237.svg" alt=" " data-tex="3_{3}"> respectively, so that the total number of electrons would +be <img style="vertical-align: 0; width: 2.262ex; height: 1.507ex;" src="images/231.svg" alt=" " data-tex="12"> instead of <span class="nowrap"><img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18">.</span> Further consideration shows, however, +that such an expectation would not be justified. The stability of the +configuration of eight electrons with <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">-quanta orbits occurring +in neon must be ascribed not only to the symmetrical configuration of +the electronic orbits in the two subgroups of <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/214.svg" alt=" " data-tex="2_{1}"> and <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/226.svg" alt=" " data-tex="2_{2}"> +orbits respectively, but fully as much to the possibility of bringing +the orbits inside these subgroups into harmonic relation with one +another. The situation is different, however, for the groups of +electrons with <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits. Three subgroups of four orbits +each cannot in this case be expected to come into interaction with one +another in a correspondingly simple manner. On the contrary we must +assume that the presence of electrons in <img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.879ex;" src="images/237.svg" alt=" " data-tex="3_{3}"> orbits will diminish +the harmony of the orbits within the first two <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta subgroups, +at any rate when a point is reached where the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/199.svg" alt=" " data-tex="19">th electron is no +longer, as was the case with scandium, bound considerably more lightly +than the previously bound electrons in <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits, but has +been drawn so far into the atom that it revolves within essentially the +same region of the atom where these electrons move. We shall now assume +that this decrease in the harmony will so to say "open" the previously +"closed" configuration of electrons in orbits of these types. As +regards the final result, the number <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> indicates that after the +group is finally formed there will be three subgroups containing six +electrons each. Even if it has not at present been possible to follow +in detail the various steps in the formation of the group this result +is nevertheless confirmed in an interesting manner by the fact that +it is possible to arrange three configurations having six electrons +each in a simple manner relative to one another. The configuration of +the subgroups does not exhibit a tetrahedral symmetry like the groups +of <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">-quanta orbits in carbon, but a symmetry which, so far as the +relative orientation of the normals to the planes of the orbits is +concerned, may be described as trigonal.</p> + +<p>In spite of the great difference in the properties of the elements +of this period, compared with those of the preceding period, the +completion of the group of <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> electrons in <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits in +the fourth period may to a certain extent be said to have the same +<span class="pagenum" id="Page_105">[Pg 105]</span> +characteristic results as the completion of the group of <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">-quanta +orbits in the second period. As we have seen, this determined not +only the properties of neon as an inactive gas, but in addition +the electronegative properties of the preceding elements and the +electropositive properties of the elements which follow. The fact that +there is no inactive gas possessing an outer group of <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> electrons +is very easily accounted for by the much larger dimensions which a +<img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.879ex;" src="images/237.svg" alt=" " data-tex="3_{3}"> orbit has in comparison with a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/226.svg" alt=" " data-tex="2_{2}"> orbit revolving +in the same field of force. On this account a complete <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta +group cannot occur as the outermost group in a neutral atom, but only +in positively charged ions. The characteristic decrease in valency +which we meet in copper, shown by the appearance of the singly charged +cuprous ions, indicates the same tendency towards the completion of +a symmetrical configuration of electrons that we found in the marked +electronegative character of an element like fluorine. Direct evidence +that a complete group of <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits is present in the cuprous +ion is given by the spectrum of copper which, in contrast to the +extremely complicated spectra of the preceding elements resulting from +the unsymmetrical character of the inner system, possesses a simple +structure very much like that of the sodium spectrum. This may no doubt +be ascribed to a simple symmetrical structure present in the cuprous +ion similar to that in the sodium ion, although the great difference +in the constitution of the outer group of electrons in these ions is +shown both by the considerable difference in the values of the spectral +terms and in the separation of the doublets in the <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/151.svg" alt=" " data-tex="P"> terms of the +two spectra. The occurrence of the cupric compounds shows, however, +that the firmness of binding in the group of <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits in the +copper atom is not as great as the firmness with which the electrons +are bound in the group of <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">-quanta orbits in the sodium atom. Zinc, +which is always divalent, is the first element in which the groups +of the electrons are so firmly bound that they cannot be removed by +ordinary chemical processes.</p> + +<p>The picture I have given of the formation and structure of the +atoms of the elements in the fourth period gives an explanation of +the chemical and spectral properties. In addition it is supported +by evidence of a different nature to that which we have hitherto +<span class="pagenum" id="Page_106">[Pg 106]</span> +used. It is a familiar fact, that the elements in the fourth period +differ markedly from the elements in the preceding periods partly in +their <i>magnetic properties</i> and partly in the <i>characteristic +colours</i> of their compounds. Paramagnetism and colours do occur +in elements belonging to the foregoing periods, but not in simple +compounds where the atoms considered enter as ions. Many elements of +the fourth period, on the contrary, exhibit paramagnetic properties +and characteristic colours even in dissociated aqueous solutions. The +importance of this has been emphasized by Ladenburg in his attempt +to explain the properties of the elements in the long periods of the +periodic system (see p. 73). Langmuir in order to account for the +difference between the fourth period and the preceding periods simply +assumed that the atom, in addition to the layers of cells containing +<img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/202.svg" alt=" " data-tex="8"> electrons each, possesses an outer layer of cells with room for +<img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> electrons which is completely filled for the first time in the +case of krypton. Ladenburg, on the other hand, assumes that for some +reason or other an intermediate layer is developed between the inner +electronic configuration in the atom appearing already in argon, and +the external group of valency electrons. This layer commences with +scandium and is completed exactly at the end of the family of iron +metals. In support of this assumption Ladenburg not only mentions the +chemical properties of the elements in the fourth period, but also +refers to the paramagnetism and colours which occur exactly in the +elements, where this intermediate layer should be in development. It is +seen that Ladenburg's ideas exhibit certain formal similarities with +the interpretation I have given above of the appearance of the fourth +period, and it is interesting to note that our view, based on a direct +investigation of the conditions for the formation of the atoms, enables +us to understand the relation emphasized by Ladenburg.</p> + +<p>Our ordinary electrodynamic conceptions are probably insufficient to +form a basis for an explanation of atomic magnetism. This is hardly to +be wondered at when we remember that they have not proved adequate to +account for the phenomena of radiation which are connected with the +intimate interaction between the electric and magnetic forces arising +from the motion of the electrons. In whatever way these difficulties +may be solved it seems simplest to assume that the occurrence of +<span class="pagenum" id="Page_107">[Pg 107]</span> +magnetism, such as we meet in the elements of the fourth period, +results from a lack of symmetry in the internal structure of the +atom, thus preventing the magnetic forces arising from the motion of +the electrons from forming a system of closed lines of force running +wholly within the atom. While it has been assumed that the ions of the +elements in the previous periods, whether positively or negatively +charged, contain configurations of marked symmetrical character, we +must, however, be prepared to encounter a definite lack of symmetry +in the electronic configurations in ions of those elements within +the fourth period which contain a group of electrons in <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta +orbits in the transition stage between symmetrical configurations of +<img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/202.svg" alt=" " data-tex="8"> and <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> electrons respectively. As pointed out by Kossel, +the experimental results exhibit an extreme simplicity, the magnetic +moment of the ions depending only on the number of electrons in the +ion. Ferric ions, for example, exhibit the same atomic magnetism as +manganous ions, while manganic ions exhibit the same atomic magnetism +as chromous ions. It is in beautiful agreement with what we have +assumed about the structure of the atoms of copper and zinc, that +the magnetism disappears with those ions containing <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/239.svg" alt=" " data-tex="28"> electrons +which, as I stated, must be assumed to contain a complete group of +<img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits. On the whole a consideration of the magnetic +properties of the elements within the fourth period gives us a +vivid impression of how a wound in the otherwise symmetrical inner +structure is first developed and then healed as we pass from element +to element. It is to be hoped that a further investigation of the +magnetic properties will give us a clue to the way in which the group +of electrons in <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits is developed step by step.</p> + +<p>Also the colours of the ions directly support our view of atomic +structure. According to the postulates of the quantum theory absorption +as well as emission of radiation is regarded as taking place during +transitions between stationary states. The occurrence of colours, +that is to say the absorption of light in the visible region of the +spectrum, is evidence of transitions involving energy changes of the +same order of magnitude as those giving the usual optical spectra of +the elements. In contrast to the ions of the elements of the preceding +periods where all the electrons are assumed to be very firmly bound, +the occurrence of such processes in the fourth period is exactly what +<span class="pagenum" id="Page_108">[Pg 108]</span> +we should expect. For the development and completion of the electronic +groups with <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits will proceed, so to say, in competition +with the binding of electrons in orbits of higher quanta, since the +binding of electrons in <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits occurs when the electrons +in these orbits are bound more firmly than electrons in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> +orbits. The development of the group will therefore proceed to the +point where we may say there is equilibrium between the two kinds of +orbits. This condition may be assumed to be intimately connected not +only with the colour of the ions, but also with the tendency of the +elements to form ions with different valencies. This is in contrast +to the elements of the first periods where the charge of the ions in +aqueous solutions is always the same for one and the same element.</p> + +<p class="space-above2"> +<b>Fifth Period. Rubidium—Xenon.</b> The structure of the atoms in +the remaining periods may be followed up in complete analogy with +what has already been said. Thus we shall assume that the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.579ex;" src="images/240.svg" alt=" " data-tex="37">th +and <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/241.svg" alt=" " data-tex="38">th electrons in the elements of the fifth period are bound +in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/242.svg" alt=" " data-tex="5_{1}"> orbits. This is supported by the measurements of the arc +spectrum of rubidium and the spark spectrum of strontium. The latter +spectrum indicates at the same time that <img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.906ex;" src="images/243.svg" alt=" " data-tex="4_{3}"> orbits will soon +appear, and therefore in this period, which like the <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">th contains +<img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> elements, we must assume that we are witnessing a <i>further +stage in the development of the electronic group of <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quanta +orbits</i>. The first stage in the formation of this group may be said +to have been attained in krypton with the appearance of a symmetrical +configuration of eight electrons consisting of two subgroups each of +four electrons in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> and <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/238.svg" alt=" " data-tex="4_{2}"> orbits. A second preliminary +completion must be regarded as having been reached with the appearance +of a symmetrical configuration of <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> electrons in the case of +silver, consisting of three subgroups with six electrons each in orbits +of the types <span class="nowrap"><img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}">,</span> <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/238.svg" alt=" " data-tex="4_{2}"> and <span class="nowrap"><img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.906ex;" src="images/243.svg" alt=" " data-tex="4_{3}">.</span> Everything that has +been said about the successive formation of the group of electrons +with <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits applies unchanged to this stage in the +transformation of the group with <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quanta orbits. For in no case +have we made use of the absolute values of the quantum numbers nor of +assumptions concerning the form of the orbits but only of the number of +possible types of orbits which might come into consideration. At the +<span class="pagenum" id="Page_109">[Pg 109]</span> +same time it may be of interest to mention that the properties of these +elements compared with those of the foregoing period nevertheless show +a difference corresponding exactly to what would be expected from the +difference in the types of orbits. For instance, the divergencies from +the characteristic valency conditions of the elements in the second and +third periods appear later in the fifth period than for elements in +the fourth period. While an element like titanium in the fourth period +already shows a marked tendency to occur with various valencies, on the +other hand an element like zirconium is still quadri-valent like carbon +in the second period and silicon in the third. A simple investigation +of the kinematic properties of the orbits of the electrons shows in +fact that an electron in an eccentric <img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.906ex;" src="images/243.svg" alt=" " data-tex="4_{3}"> orbit of an element +in the fifth period will be considerably more loosely bound than an +electron in a circular <img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.879ex;" src="images/237.svg" alt=" " data-tex="3_{3}"> orbit of the corresponding element in +the fourth period, while electrons which are bound in eccentric orbits +of the types <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/242.svg" alt=" " data-tex="5_{1}"> and <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> respectively will correspond to a +binding of about the same firmness.</p> + +<p>At the end of the fifth period we may assume that xenon, the atomic +number of which is <span class="nowrap"><img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.581ex;" src="images/244.svg" alt=" " data-tex="54">,</span> has a structure which in addition to the +two <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/103.svg" alt=" " data-tex="1">-quantum, eight <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">-quanta, eighteen <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta and +eighteen <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quanta orbits already mentioned contains a symmetrical +configuration of eight electrons in <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/221.svg" alt=" " data-tex="5">-quanta orbits consisting +of two subgroups with four electrons each in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/242.svg" alt=" " data-tex="5_{1}"> and <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/245.svg" alt=" " data-tex="5_{2}"> +orbits respectively.</p> + +<p class="space-above2"> +<b>Sixth Period. Caesium—Niton.</b> If we now consider the atoms of +elements of still higher atomic number, we must first of all assume +that the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/246.svg" alt=" " data-tex="55">th and <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/247.svg" alt=" " data-tex="56">th electrons in the atoms of caesium and +barium are bound in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/248.svg" alt=" " data-tex="6_{1}"> orbits. This is confirmed by the spectra +of these elements. It is clear, however, that we must be prepared +shortly to meet entirely new conditions. With increasing nuclear charge +we shall have to expect not only that an electron in a <img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.881ex;" src="images/249.svg" alt=" " data-tex="5_{3}"> +orbit will be bound more firmly than in a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/248.svg" alt=" " data-tex="6_{1}"> orbit, but we +must also expect that a moment will arrive when during the formation +of the atom a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/250.svg" alt=" " data-tex="4_{4}"> orbit will represent a firmer binding of the +electron than an orbit of <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/221.svg" alt=" " data-tex="5"> or <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/222.svg" alt=" " data-tex="6">-quanta, in much the same way +as in the elements of the fourth period a new stage in the development +of the <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta group was started when a point was reached where +<span class="pagenum" id="Page_110">[Pg 110]</span> +for the first time the <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/199.svg" alt=" " data-tex="19">th electron was bound in a <img style="vertical-align: -0.375ex; width: 2.119ex; height: 1.879ex;" src="images/237.svg" alt=" " data-tex="3_{3}"> +orbit instead of in a <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/235.svg" alt=" " data-tex="4_{1}"> orbit. We shall thus expect in the +sixth period to meet with a new stage in the development of the group +with <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quanta orbits. Once this point has been reached we must be +prepared to find with increasing atomic number a number of elements +following one another, which as in the family of the iron metals have +very nearly the same properties. The similarity will, however, be +still more pronounced, since in this case we are concerned with the +successive transformation of a configuration of electrons which lies +deeper in the interior of the atom. You will have already guessed that +what I have in view is a simple explanation of the occurrence of the +<i>family of rare earths</i> at the beginning of the sixth period. +As in the case of the transformation and completion of the group of +<img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits in the fourth period and the partial completion of +groups of <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quanta orbits in the fifth period, we may immediately +deduce from the length of the sixth period the number of electrons, +namely <span class="nowrap"><img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/251.svg" alt=" " data-tex="32">,</span> which are finally contained in the <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quanta group +of orbits. Analogous to what applied to the group of <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta +orbits it is probable that, when the group is completed, it will +contain eight electrons in each of the four subgroups. Even though it +has not yet been possible to follow the development of the group step +by step, we can even here give some theoretical evidence in favour of +the occurrence of a symmetrical configuration of exactly this number +of electrons. I shall simply mention that it is not possible without +coincidence of the planes of the orbits to arrive at an interaction +between four subgroups of six electrons each in a configuration of +simple trigonal symmetry, which is equally simple as that shown by +three subgroups. The difficulties which we meet make it probable that +a harmonic interaction can be attained precisely by four groups each +containing eight electrons the orbital configurations of which exhibit +axial symmetry.</p> + +<p>Just as in the case of the family of the iron metals in the fourth +period, the proposed explanation of the occurrence of the family of +rare earths in the sixth period is supported in an interesting manner +by an investigation of the magnetic properties of these elements. In +spite of the great chemical similarity the members of this family +exhibit very different magnetic properties, so that while some of them +exhibit but very little magnetism others exhibit a greater magnetic +<span class="pagenum" id="Page_111">[Pg 111]</span> +moment per atom than any other element which has been investigated. +It is also possible to give a simple interpretation of the peculiar +colours exhibited by the compounds of these elements in much the same +way as in the case of the family of iron metals in the fourth period. +The idea that the appearance of the group of the rare earths is +connected with the development of inner groups in the atom is not in +itself new and has for instance been considered by Vegard in connection +with his work on X-ray spectra. The new feature of the present +considerations lies, however, in the emphasis laid on the peculiar +way in which the relative strength of the binding for two orbits of +the same principal quantum number but of different shapes varies with +the nuclear charge and with the number of electrons previously bound. +Due to this fact the presence of a group like that of the rare earths +in the sixth period may be considered as a direct consequence of the +theory and might actually have been predicted on a quantum theory, +adapted to the explanation of the properties of the elements within the +preceding periods in the way I have shown.</p> + +<p>Besides <i>the final development of the group of <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quanta +orbits</i> we observe in the sixth period in the family of the +platinum metals <i>the second stage in the development of the group +of <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/221.svg" alt=" " data-tex="5">-quanta orbits</i>. Also in the radioactive, chemically +inactive gas niton, which completes this period, we observe the first +preliminary step in the development of a group of electrons with +<img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/222.svg" alt=" " data-tex="6">-quanta orbits. In the atom of this element, in addition to the +groups of electrons of two <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/103.svg" alt=" " data-tex="1">-quantum, eight <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">-quanta, eighteen +<img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta, thirty-two <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quanta and eighteen <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/221.svg" alt=" " data-tex="5">-quanta orbits +respectively, there is also an outer symmetrical configuration of eight +electrons in <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/222.svg" alt=" " data-tex="6">-quanta orbits, which we shall assume to consist +of two subgroups with four electrons each in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/248.svg" alt=" " data-tex="6_{1}"> and <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/252.svg" alt=" " data-tex="6_{2}"> +orbits respectively.</p> + +<p class="space-above2"> +<b>Seventh Period.</b> In the seventh and last period of the periodic +system we may expect the appearance of <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.579ex;" src="images/223.svg" alt=" " data-tex="7">-quanta orbits in the +normal state of the atom. Thus in the neutral atom of radium in +addition to the electronic structure of niton there will be two +electrons in <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.869ex;" src="images/253.svg" alt=" " data-tex="7_{1}"> orbits which will penetrate during their +revolution not only into the region of the orbits of electrons +possessing lower values for the principal quantum number, but even +<span class="pagenum" id="Page_112">[Pg 112]</span> +to distances from the nucleus which are less than the radii of the +orbits of the innermost <img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/103.svg" alt=" " data-tex="1">-quantum orbits. The properties of the +elements in the seventh period are very similar to the properties of +the elements in the fifth period. Thus, in contrast to the conditions +in the sixth period, there are no elements whose properties resemble +one another like those of the rare earths. In exact analogy with what +has already been said about the relations between the properties of +the elements in the fourth and fifth periods this may be very simply +explained by the fact that an eccentric <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/254.svg" alt=" " data-tex="5_{4}"> orbit will correspond +to a considerably looser binding of an electron in the atom of an +element of the seventh period than the binding of an electron in a +circular <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.871ex;" src="images/250.svg" alt=" " data-tex="4_{4}"> orbit in the corresponding element of the sixth +period, while there will be a much smaller difference in the firmness +of the binding of these electrons in orbits of the types <img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.869ex;" src="images/253.svg" alt=" " data-tex="7_{1}"> and +<img style="vertical-align: -0.339ex; width: 2.119ex; height: 1.846ex;" src="images/248.svg" alt=" " data-tex="6_{1}"> respectively.</p> + +<p>It is well known that the seventh period is not complete, for no atom +has been found having an atomic number greater than <span class="nowrap"><img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/255.svg" alt=" " data-tex="92">.</span> This is +probably connected with the fact that the last elements in the system +are radioactive and that nuclei of atoms with a total charge greater +than <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/255.svg" alt=" " data-tex="92"> will not be sufficiently stable to exist under conditions +where the elements can be observed. It is tempting to sketch a picture +of the atoms formed by the capture and binding of electrons around +nuclei having higher charges, and thus to obtain some idea of the +properties which the corresponding hypothetical elements might be +expected to exhibit. I shall not develop this matter further, however, +since the general results we should get will be evident to you from +the views I have developed to explain the properties of the elements +actually observed. A survey of these results is given in the following +table, which gives a symbolical representation of the atomic structure +of the inactive gases which complete the first six periods in the +periodic system. In order to emphasize the progressive change the table +includes the probable arrangement of electrons in the next atom which +would possess properties like the inactive gases.</p> + +<p>The view of atomic constitution underlying this table, which involves +configurations of electrons moving with large velocities between each +other, so that the electrons in the "outer" groups penetrate into the +region of the orbits of the electrons of the "inner" groups, is of +course completely different from such statical models of the atom as +<span class="pagenum" id="Page_113">[Pg 113]</span> +are proposed by Langmuir. But quite apart from this it will be seen +that the arrangement of the electronic groups in the atom, to which +we have been lead by tracing the way in which each single electron +has been bound, is essentially different from the arrangement of the +groups in Langmuir's theory. In order to explain the properties of the +elements of the sixth period Langmuir assumes for instance that, in +addition to the inner layers of cells containing <span class="nowrap"><img style="vertical-align: 0; width: 1.131ex; height: 1.507ex;" src="images/220.svg" alt=" " data-tex="2">,</span> <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/202.svg" alt=" " data-tex="8">,</span> <span class="nowrap"><img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/202.svg" alt=" " data-tex="8">,</span> +<img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> and <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/197.svg" alt=" " data-tex="18"> electrons respectively, which are employed to account +for the properties of the elements in the earlier periods, the atom +also possesses a layer of cells with room for <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.557ex;" src="images/251.svg" alt=" " data-tex="32"> electrons which is +just completed in the case of niton.</p> + +<div class="figcenter"> +<img src="images/006.jpg" width="400" alt="fig06"> +</div> + +<p>In this connection it may be of interest to mention a recent paper +by Bury, to which my attention was first drawn after the deliverance +of this address, and which contains an interesting survey of the +chemical properties of the elements based on similar conceptions of +atomic structure as those applied by Lewis and Langmuir. From purely +chemical considerations Bury arrives at conclusions which as regards +the arrangement and completion of the groups in the main coincide with +those of the present theory, the outlines of which were given in my +letters to Nature mentioned in the introduction.</p> +<p><span class="pagenum" id="Page_114">[Pg 114]</span></p> + +<p class="space-above2"> +<b>Survey of the periodic table.</b> The results given in this +address are also illustrated by means of the representation of the +periodic system given in <a href="#Page_70">Fig. 1</a>. In this figure the frames are meant +to indicate such elements in which one of the "inner" groups is in a +stage of development. Thus there will be found in the fourth and +fifth periods a single frame indicating the final completion of the +electronic group with <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.554ex;" src="images/133.svg" alt=" " data-tex="3">-quanta orbits, and the last stage but one +in the development of the group with <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quanta orbits respectively. +In the sixth period it has been necessary to introduce two frames, of +which the inner one indicates the last stage of the evolution of the +group with <img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quanta orbits, giving rise to the rare earths. This +occurs at a place in the periodic system where the third stage in the +development of an electronic group with <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/221.svg" alt=" " data-tex="5">-quanta orbits, indicated +by the outer frame, has already begun. In this connection it will be +seen that the inner frame encloses a smaller number of elements than +is usually attributed to the family of the rare earths. At the end of +this group an uncertainty exists, due to the fact that no element of +atomic number <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.579ex;" src="images/256.svg" alt=" " data-tex="72"> is known with certainty. However, as indicated +in <a href="#Page_70">Fig. 1</a>, we must conclude from the theory that the group with +<img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quanta orbits is finally completed in lutetium (<span class="nowrap"><img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.579ex;" src="images/257.svg" alt=" " data-tex="71">)</span>. This +element therefore ought to be the last in the sequence of consecutive +elements with similar properties in the first half of the sixth +period, and at the place <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.579ex;" src="images/256.svg" alt=" " data-tex="72"> an element must be expected which in +its chemical and physical properties is homologous with zirconium +and thorium. This, which is already indited on Julius Thomsen's old +table, has also been pointed out by Bury. [Quite recently Dauvillier +has in an investigation of the X-ray spectrum excited in preparations +containing rare earths, observed certain faint lines which he ascribes +to an element of atomic number <span class="nowrap"><img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.579ex;" src="images/256.svg" alt=" " data-tex="72">.</span> This element is identified by +him as the element celtium, belonging to the family of rare earths, the +existence of which had previously been suspected by Urbain. Quite apart +from the difficulties which this result, if correct, might entail for +atomic theories, it would, since the rare earths according to chemical +view possess three valencies, imply a rise in positive valency of two +units when passing from the element <img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.579ex;" src="images/256.svg" alt=" " data-tex="72"> to the next element <span class="nowrap"><img style="vertical-align: -0.05ex; width: 2.262ex; height: 1.579ex;" src="images/258.svg" alt=" " data-tex="73">,</span> +tantalum. This would mean an exception from the otherwise general rule, +that the valency never increases by more than one unit when passing +from one element to the next in the periodic table.] In the case of +the incomplete seventh period the full drawn frame indicates the third +stage in the development of the electronic group with <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/222.svg" alt=" " data-tex="6">-quanta +<span class="pagenum" id="Page_115">[Pg 115]</span> +orbits, which must begin in actinium. The dotted frame indicates the +last stage but one in the development of the group with <img style="vertical-align: -0.05ex; width: 1.131ex; height: 1.557ex;" src="images/221.svg" alt=" " data-tex="5">-quanta +orbits, which hitherto has not been observed, but which ought to begin +shortly after uranium, if it has not already begun in this element.</p> + +<p>With reference to the homology of the elements the exceptional position +of the elements enclosed by frames in <a href="#Page_70">Fig. 1</a> is further emphasized +by taking care that, in spite of the large similarity many elements +exhibit, no connecting lines are drawn between two elements which +occupy different positions in the system with respect to framing. +In fact, the large chemical similarity between, for instance, +aluminium and scandium, both of which are trivalent and pronounced +electropositive elements, is directly or indirectly emphasized in the +current representations of the periodic table. While this procedure +is justified by the analogous structure of the trivalent ions of +these elements, our more detailed ideas of atomic structure suggest, +however, marked differences in the physical properties of aluminium +and scandium, originating in the essentially different character of +the way in which the last three electrons in the neutral atom are +bound. This fact gives probably a direct explanation of the marked +difference existing between the spectra of aluminium and scandium. +Even if the spectrum of scandium is not yet sufficiently cleared up, +this difference seems to be of a much more fundamental character than +for instance the difference between the arc spectra of sodium and +copper, which apart from the large difference in the absolute values +of the spectral terms possess a completely analogous structure, as +previously mentioned in this essay. On the whole we must expect that +the spectra of elements in the later periods lying inside a frame will +show new features compared with the spectra of the elements in the +first three periods. This expectation seems supported by recent work on +the spectrum of manganese by Catalan, which appeared just before the +printing of this essay.</p> + +<p>Before I leave the interpretation of the chemical properties by means +of this atomic model I should like to remind you once again of the +fundamental principles which we have used. The whole theory has evolved +from an investigation of the way in which electrons can be captured by +an atom. The formation of an atom was held to consist in the successive +binding of electrons, this binding resulting in radiation according +to the quantum theory. According to the fundamental postulates of +<span class="pagenum" id="Page_116">[Pg 116]</span> +the theory this binding takes place in stages by transitions between +stationary states accompanied by emission of radiation. For the problem +of the stability of the atom the essential problem is at what stage +such a process comes to an end. As regards this point the postulates +give no direct information, but here the correspondence principle is +brought in. Even though it has been possible to penetrate considerably +further at many points than the time has permitted me to indicate +to you, still it has not yet been possible to follow in detail all +stages in the formation of the atoms. We cannot say, for instance, +that the above table of the atomic constitution of the inert gases may +in every detail be considered as the unambiguous result of applying +the correspondence principle. On the other hand it appears that our +considerations already place the empirical data in a light which +scarcely permits of an essentially different interpretation of the +properties of the elements based upon the postulates of the quantum +theory. This applies not only to the series spectra and the close +relationship of these to the chemical properties of the elements, but +also to the X-ray spectra, the consideration of which leads us into +an investigation of interatomic processes of an entirely different +character. As we have already mentioned, it is necessary to assume that +the emission of the latter spectra is connected with processes which +may be described as a reorganization of the completely formed atom +after a disturbance produced in the interior of the atom by the action +of external forces.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="IV_REORGANIZATION_OF_ATOMS_AND_X-RAY_SPECTRA"> +IV. REORGANIZATION OF ATOMS AND X-RAY SPECTRA</h2> +</div> + + +<p>As in the case of the series spectra it has also been possible to +represent the frequency of each line in the X-ray spectrum of an +element as the difference of two of a set of spectral terms. We shall +therefore assume that each X-ray line is due to a transition between +two stationary states of the atom. The values of the atomic energy +corresponding to these states are frequently referred to as the "energy +levels" of the X-ray spectra. The great difference between the origin +of the X-ray and the series spectra is clearly seen, however, in +the difference of the laws applying to the absorption of radiation +in the X-ray and the optical regions of the spectra. The absorption +by non-excited atoms in the latter case is connected with those +<span class="pagenum" id="Page_117">[Pg 117]</span> +lines in the series spectrum which correspond to combinations of the +various spectral terms with the largest of these terms. As has been +shown, especially by the investigations of Wagner and de Broglie, the +absorption in the X-ray region, on the other hand, is connected not +with the X-ray lines but with certain spectral regions commencing +at the so-called "absorption edges." The frequencies of these edges +agree very closely with the spectral terms used to account for the +X-ray lines. We shall now see how the conception of atomic structure +developed in the preceding pages offers a simple interpretation of +these facts. Let us consider the following question: What changes in +the state of the atom can be produced by the absorption of radiation, +and which processes of emission can be initiated by such changes?</p> + +<p class="space-above2"> +<b>Absorption and emission of X-rays and correspondence principle.</b> +The possibility of producing a change at all in the motion of an +electron in the interior of an atom by means of radiation must in the +first place be regarded as intimately connected with the character of +the interaction between the electrons within the separate groups. In +contrast to the forms of motion where at every moment the position of +the electrons exhibits polygonal or polyhedral symmetry, the conception +of this interaction evolved from a consideration of the possible +formation of atoms by successive binding of electrons has such a +character that the harmonic components in the motion of an electron are +in general represented in the resulting electric moment of the atom. As +a result of this it will be possible to release a single electron from +the interaction with the other electrons in the same group by a process +which possesses the necessary analogy with an absorption process +on the ordinary electrodynamic view claimed by the correspondence +principle. The points of view on which we based the interpretation of +the development and completion of the groups during the formation of +an atom imply, on the other hand, that just as no additional electron +can be taken up into a previously completed group in the atom by a +change involving emission of radiation, similarly it will not be +possible for a new electron to be added to such a group, when the state +of the atom is changed by absorption of radiation. This means that +an electron which belongs to one of the inner groups of the atom, +<span class="pagenum" id="Page_118">[Pg 118]</span> +as a consequence of an absorption process—besides the case where it +leaves the atom completely—can only go over either to an incompleted +group, or to an orbit where the electron during the greater part of its +revolution moves at a distance from the nucleus large compared to the +distance of the other electrons. On account of the peculiar conditions +of stability which control the occurrence of incomplete groups in the +interior of the atom, the energy which is necessary to bring about a +transition to such a group will in general differ very little from +that required to remove the particular electron completely from the +atom. We must therefore assume that the energy levels corresponding to +the absorption edges indicate to a first approximation the amount of +work that is required to remove an electron in one of the inner groups +completely from the atom. The correspondence principle also provides a +basis for understanding the experimental evidence about the appearance +of the emission lines of the X-ray spectra due to transitions between +the stationary states corresponding to these energy levels. Thus the +nature of the interaction between the electrons in the groups of the +atom implies that each electron in the atom is so to say prepared, +independently of the other electrons in the same group, to seize any +opportunity which is offered to become more firmly bound by being taken +up into a group of electrons with orbits corresponding to smaller +values of the principal quantum number. It is evident, however, that +on the basis of our views of atomic structure, such an opportunity is +always at hand as soon as an electron has been removed from one of +these groups.</p> + +<p>At the same time that our view of the atom leads to a natural +conception of the phenomena of emission and absorption of X-rays, +agreeing closely with that by which Kossel has attempted to give a +formal explanation of the experimental observations, it also suggests +a simple explanation of those quantitative relations holding for +the frequencies of the lines which have been discovered by Moseley +and Sommerfeld. These researches brought to light a remarkable and +far-reaching similarity between the Röntgen spectrum of a given +element and the spectrum which would be expected to appear upon +the binding of a single electron by the nucleus. This similarity +we immediately understand if we recall that in the normal state of +the atom there are electrons moving in orbits which, with certain +<span class="pagenum" id="Page_119">[Pg 119]</span> +limitations, correspond to all stages of such a binding process and +that, when an electron is removed from its original place in the atom, +processes may be started within the atom which will correspond to +all transitions between these stages permitted by the correspondence +principle. This brings us at once out of those difficulties which +apparently arise, when one attempts to account for the origin of the +X-ray spectra by means of an atomic structure, suited to explain the +periodic system. This difficulty has been felt to such an extent that +it has led Sommerfeld for example in his recent work to assume that +the configurations of the electrons in the various atoms of one and +the same element may be different even under usual conditions. Since, +in contrast to our ideas, he supposed all electrons in the principal +groups of the atom to move in equivalent orbits, he is compelled +to assume that these groups are different in the different atoms, +corresponding to different possible types of orbital shapes. Such an +assumption, however, seems inconsistent with an interpretation of the +definite character of the physical and chemical properties of the +elements, and stands in marked contradiction with the points of view +about the stability of the atoms which form the basis of the view of +atomic structure here proposed.</p> + +<p class="space-above2"> +<b>X-ray spectra and atomic structure.</b> In this connection it is of +interest to emphasize that the group distribution of the electrons in +the atom, on which we have based both the explanation of the periodic +system and the classification of the lines in the X-ray spectra, shows +itself in an entirely different manner in these two phenomena. While +the characteristic change of the chemical properties with atomic +number is due to the gradual development and completion of the groups +of the loosest bound electrons, the characteristic absence of almost +every trace of a periodic change in the X-ray spectra is due to two +causes. Firstly the electronic configuration of the completed groups +is repeated unchanged for increasing atomic number, and secondly the +gradual way in which the incompleted groups are developed implies that +a type of orbit, from the moment when it for the first time appears in +the normal state of the neutral atom, always will occur in this state +<span class="pagenum" id="Page_120">[Pg 120]</span> +and will correspond to a steadily increasing firmness of binding. The +development of the groups in the atom with increasing atomic number, +which governs the chemical properties of the elements shows itself in +the X-ray spectra mainly in the appearance of new lines. Swinne has +already referred to a connection of this kind between the periodic +system and the X-ray spectra in connection with Kossel's theory. We +can only expect a closer connection between the X-ray phenomena and +the chemical properties of the elements, when the conditions on the +surface of the atom are concerned. In agreement with what has been +brought to light by investigations on absorption of X-rays in elements +of lower atomic number, such as have been performed in recent years +in the physical laboratory at Lund, we understand immediately that +the position and eventual structure of the absorption edges will to a +certain degree depend upon the physical and chemical conditions under +which the element investigated exists, while such a dependence does not +appear in the characteristic emission lines.</p> + +<p>If we attempt to obtain a more detailed explanation of the experimental +observations, we meet the question of the influence of the presence of +the other electrons in the atom upon the firmness of the binding of an +electron in a given type of orbit. This influence will, as we at once +see, be least for the inner parts of the atom, where for each electron +the attraction of the nucleus is large in proportion to the repulsion +of the other electrons. It should also be recalled, that while the +relative influence of the presence of the other electrons upon the +firmness of the binding will decrease with increasing charge of the +nucleus, the effect of the variation in the mass of the electron with +the velocity upon the firmness of the binding will increase strongly. +This may be seen from Sommerfeld's formula (11). While we obtain a +fairly good agreement for the levels corresponding to the removal of +one of the innermost electrons in the atom by using the simple formula +(11), it is, however, already necessary to take the influence of the +other electrons into consideration in making an approximate calculation +of the levels corresponding to a removal of an electron from one of the +outer groups in the atom. Just this circumstance offers us, however, +a possibility of obtaining information about the configurations of +the electrons in the interior of the atoms from the X-ray spectra. +Numerous investigations have been directed at this question both by +<span class="pagenum" id="Page_121">[Pg 121]</span> +Sommerfeld and his pupils and by Debye, Vegard and others. It may also +be remarked that de Broglie and Dauvillier in a recent paper have +thought it possible to find support in the experimental material for +certain assumptions about the numbers of electrons in the groups of +the atom to which Dauvillier had been led by considerations about the +periodic system similar to those proposed by Langmuir and Ladenburg. +In calculations made in connection with these investigations it is +assumed that the electrons in the various groups move in separate +concentric regions of the atom, so that the effect of the presence +of the electrons in inner groups upon the motion of the electrons in +outer groups as a first approximation may be expected to consist in a +simple screening of the nucleus. On our view, however, the conditions +are essentially different, since for the calculation of the firmness of +the binding of the electrons it is necessary to take into consideration +that the electrons in the more lightly bound groups in general during +a certain fraction of their revolution will penetrate into the region +of the orbits of electrons in the more firmly bound groups. On account +of this fact, many examples of which we saw in the series spectra, we +cannot expect to give an account of the firmness of the binding of +the separate electrons, simply by means of a "screening correction" +consisting in the subtraction of a constant quantity from the value for +<img style="vertical-align: 0; width: 2.009ex; height: 1.545ex;" src="images/194.svg" alt=" " data-tex="N"> in such formulae as (5) and (11). Furthermore in the calculation +of the work corresponding to the energy levels we must take account not +only of the interaction between, the electrons in the normal state of +the atom, but also of the changes in the configuration and interaction +of the remaining electrons, which establish themselves automatically +without emission of radiation during the removal of the electron. +Even though such calculations have not yet been made very accurately, +a preliminary investigation has already shown that it is possible +approximately to account for the experimental results.</p> + +<p class="space-above2"> +<b>Classification of X-ray spectra.</b> Independently of a definite +view of atomic structure it has been possible by means of a formal +application of Kossel's and Sommerfeld's theories to disentangle the +large amount of experimental material on X-ray spectra. This material +is drawn mainly from the accurate measurements of Siegbahn and +<span class="pagenum" id="Page_122">[Pg 122]</span> +his collaborators. From this disentanglement of the experimental +observations, in which besides Sommerfeld and his students especially +Smekal and Coster have taken part, we have obtained a nearly complete +classification of the energy levels corresponding to the X-ray spectra. +These levels are formally referred to types of orbits characterized +by two quantum numbers <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">,</span> and certain definite rules +for the possibilities of combination between the various levels have +also been found. In this way a number of results of great interest for +the further elucidation of the origin of the X-ray spectra have been +attained. First it has not only been possible to find levels, which +within certain limits correspond to all possible pairs of numbers +for <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/67.svg" alt=" " data-tex="n"> and <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">,</span> but it has been found that in general to each +such pair more than one level must be assigned. This result, which +at first may appear very surprising, upon further consideration can +be given a simple interpretation. We must remember that the levels +depend not only upon the constitution of the atom in the normal state, +but also upon the configurations which appear after the removal of +one of the inner electrons and which in contrast to the normal state +do not possess a uniquely completed character. If we thus consider a +process in which one of the electrons in a group (subgroup) is removed +we must be prepared to find that after the process the orbits of the +remaining electrons in this group may be orientated in more than +one way in relation to one another, and still fulfil the conditions +required of the stationary states by the quantum theory. Such a view +of the "complexity" of the levels, as further consideration shows, +just accounts for the manner in which the energy difference of the two +levels varies with the atomic number. Without attempting to develop a +more detailed picture of atomic structure, Smekal has already discussed +the possibility of accounting for the multiplicity of levels. Besides +referring to the possibility that the separate electrons in the +principal groups do not move in equivalent orbits, Smekal suggests +the introduction of three quantum numbers for the description of the +various groups, but does not further indicate to what extent these +quantum numbers shall be regarded as characterizing a complexity in the +structure of the groups in the normal state itself or on the contrary +characterizing the incompleted groups which appear when an electron is +removed. +<span class="pagenum" id="Page_123">[Pg 123]</span> +</p> + +<p>It will be seen that the complexity of the X-ray levels exhibits a +close analogy with the explanation of the complexity of the terms of +the series spectra. There exists, however, this difference between the +complex structure of the X-ray spectra and the complex structure of the +lines in the series spectra, that in the X-ray spectra there occur not +only combinations between spectral terms, for which <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> varies by +unity, but also between terms corresponding to the same value of <span class="nowrap"><img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k">.</span></p> + +<div class="figcenter"> +<img src="images/007.jpg" width="400" alt="fig07"> +<div class="caption"> +<p>Fig. 5.</p> +</div></div> + +<p>This may be assumed to be due to the fact, that in the X-ray spectra in +contrast to the series spectra we have to do with transitions between +stationary states where, both before and after the transition, the +electron concerned takes part in an intimate interaction with other +electrons in orbits with the same principal quantum number. Even +though this interaction may be assumed to be of such a nature that the +harmonic components which would appear in the motion of an electron in +the absence of the others will in general also appear in the resulting +moment of the atom, we must expect that the interaction between the +electrons will give rise to the appearance in this moment of new types +of harmonic components.</p> + +<p><span class="pagenum" id="Page_124">[Pg 124]</span></p> + +<p>It may be of interest to insert here a few words about a new paper of +Coster which appeared after this address was given, and in which he has +succeeded in obtaining an extended and detailed connection between the +X-ray spectra and the ideas of atomic structure given in this essay. +The classification mentioned above was based on measurements of the +spectra of the heaviest elements, and the results in their complete +form, which were principally due to independent work of Coster and +Wentzel, may be represented by the diagram in <a href="#Page_123">Fig. 5</a>, which refers to +elements in the neighbourhood of niton.</p> + +<div class="figcenter"> +<img src="images/008.jpg" width="400" alt="fig08"> +<div class="caption"> +<p>Fig. 6.</p> +</div></div> + +<p>The vertical arrows represent the observed lines arising from +combinations between the different energy levels which are represented +by horizontal lines. In each group the levels are arranged in the same +succession as their energy values, but their distances do not give +a quantitative picture of the actual energy-differences, since this +would require a much larger figure. The numbers <img style="vertical-align: -0.357ex; width: 2.379ex; height: 1.357ex;" src="images/212.svg" alt=" " data-tex="n_{k}"> attached to +the different levels indicate the type of the corresponding orbit. +The letters <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/259.svg" alt=" " data-tex="a"> and <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/260.svg" alt=" " data-tex="b"> refer to the rules of combination which I +mentioned. According to these rules the possibility of combination is +limited (1) by the exclusion of combinations, for which <img style="vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;" src="images/149.svg" alt=" " data-tex="k"> changes +by more than one unit, (2) by the condition that only combinations +between an <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/259.svg" alt=" " data-tex="a">- and a <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/260.svg" alt=" " data-tex="b">-level can take place. The latter rule was +given in this form by Coster; Wentzel formulated it in a somewhat +<span class="pagenum" id="Page_125">[Pg 125]</span> +different way by the formal introduction of a third quantum number. +In his new paper Coster has established a similar classification for +the lighter elements. For the elements in the neighbourhood of xenon +and krypton he has obtained results illustrated by the diagrams given +in <a href="#Page_124">Fig. 6</a>. Just as in <a href="#Page_123">Fig. 5</a> the levels correspond exactly to those +types of orbits which, as seen from the table on <a href="#Page_113">page 113</a>, according +to the theory will be present in the atoms of these elements. In xenon +several of the levels present in niton have disappeared, and in krypton +still more levels have fallen away. Coster has also investigated in +which elements these particular levels appear for the last time, when +passing from higher to lower atomic number. His results concerning +this point confirm in detail the predictions of the theory. Further +he proves that the change in the firmness of binding of the electrons +in the outer groups in the elements of the family of the rare earths +shows a dependence on the atomic number which strongly supports the +assumption that in these elements a completion of an inner group of +<img style="vertical-align: 0; width: 1.131ex; height: 1.532ex;" src="images/120.svg" alt=" " data-tex="4">-quanta orbits takes place. For details the reader is referred to +Coster's paper in the <i>Philosophical Magazine</i>. Another important +contribution to our systematic knowledge of the X-ray spectra is +contained in a recent paper by Wentzel. He shows that various lines, +which find no place in the classification hitherto considered, can be +ascribed in a natural manner to processes of reorganization, initiated +by the removal of more than one electron from the atom; these lines are +therefore in a certain sense analogous to the enhanced lines in the +optical spectra.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="CONCLUSION">CONCLUSION</h2> +</div> + + +<p>Before bringing this address to a close I wish once more to emphasize +the complete analogy in the application of the quantum theory to the +stability of the atom, used in explaining two so different phenomena +as the periodic system and X-ray spectra. This point is of the +greatest importance in judging the reality of the theory, since the +justification for employing considerations, relating to the formation +of atoms by successive capture of electrons, as a guiding principle for +the investigation of atomic structure might appear doubtful if such +<span class="pagenum" id="Page_126">[Pg 126]</span> +considerations could not be brought into natural agreement with views +on the reorganization of the atom after a disturbance in the normal +electronic arrangement. Even though a certain inner consistency in this +view of atomic structure will be recognized, it is, however, hardly +necessary for me to emphasize the incomplete character of the theory, +not only as regards the elaboration of details, but also so far as the +foundation of the general points of view is concerned. There seems, +however, to be no other way of advance in atomic problems than that +which hitherto has been followed, namely to let the work in these two +directions go hand in hand.</p> + + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_3" href="#FNanchor_3" class="label">[3]</a> +Address delivered before a joint meeting of the Physical +and Chemical Societies in Copenhagen, October 18, 1921.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_4" href="#FNanchor_4" class="label">[4]</a> +<i>Nature</i>, March 24, and October 13, 1921.</p> + +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="TRANSCRIBERS_NOTES">TRANSCRIBER'S NOTES</h2> +</div> + + +<p>This ebook was produced using scanned images and OCR text generously +provided by the Brandeis University Library through the Internet +Archive.</p> + +<p>Minor typographical corrections and presentational changes have been +made without comment.</p> + + +<div style='display:block; margin-top:4em'></div> +<section class='pg-boilerplate pgheader' id='pg-footer' lang='en' > +<div id='pg-end-separator'> +<span>*** END OF THE PROJECT GUTENBERG EBOOK THE THEORY OF SPECTRA AND ATOMIC CONSTITUTION: THREE ESSAYS ***</span> +</div> + +<div> +Updated editions will replace the previous one—the old editions will +be renamed. +</div> +<div> +Creating the works from print editions not protected by U.S. copyright +law means that no one owns a United States copyright in these works, +so the Foundation (and you!) can copy and distribute it in the United +States without permission and without paying copyright +royalties. 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You may copy it, give it away or re-use it under the terms of +% the Project Gutenberg License included with this eBook or online at % +% www.gutenberg.org. If you are not located in the United States, you'll have +% to check the laws of the country where you are located before using this ebook. +% % +% % +% % +% Title: The Theory of Spectra and Atomic Constitution % +% Three Essays % +% % +% Author: Niels (Niels Henrik David) Bohr % +% % +% Release Date: November 26, 2014 [EBook #47464] % +% Most recently updated: June 11, 2021 % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK THEORY OF SPECTRA *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{47464} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Latin-1 text encoding. 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Required. %% +%% %% +%% %% +%% Producer's Comments: %% +%% %% +%% OCR text for this ebook was obtained on Nov. 21, 2014, from %% +%% http://www.archive.org/details/theoryofspectraa00bohr. %% +%% %% +%% Minor changes to the original are noted in this file in two %% +%% ways: %% +%% 1. \Chg{}{} and \Add{} for modernization and consistency: %% +%% * "excentric" modernized to "eccentric". %% +%% * "can not" uniformized to "cannot". %% +%% * "ultra-violet" and "infra-red" hyphenated uniformly. %% +%% %% +%% 2. [** TN: Note]s for other remarks. %% +%% %% +%% Compilation Flags: %% +%% %% +%% The following behavior may be controlled by a boolean flag. %% +%% %% +%% ForPrinting (false by default): %% +%% If false, compile a screen optimized file (one-sided layout, %% +%% blue hyperlinks). 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+\begin{minipage}{\textwidth} +\small +\begin{PGtext} +The Project Gutenberg EBook of The Theory of Spectra and Atomic +Constitution, by Niels (Niels Henrik David) Bohr + +This eBook is for the use of anyone anywhere in the United States and most +other parts of the world at no cost and with almost no restrictions +whatsoever. You may copy it, give it away or re-use it under the terms of +the Project Gutenberg License included with this eBook or online at +www.gutenberg.org. If you are not located in the United States, you'll have +to check the laws of the country where you are located before using this ebook. + + + +Title: The Theory of Spectra and Atomic Constitution + Three Essays + +Author: Niels (Niels Henrik David) Bohr + +Release Date: November 26, 2014 [EBook #47464] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK THEORY OF SPECTRA *** +\end{PGtext} +\end{minipage} +\end{center} +\newpage +%% Credits and transcriber's note %% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang +\end{PGtext} +\end{minipage} +\vfill +\end{center} + +\begin{minipage}{0.85\textwidth} +\small +\BookMark{0}{Transcriber's Note.} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\PageSep{i} +\iffalse +% [** TN: Omitting half-title page and verso] +The Theory of Spectra +and +Atomic Constitution +\PageSep{ii} +%[** TN: Publisher's information] + +CAMBRIDGE UNIVERSITY PRESS +C. F. CLAY, Manager +LONDON : FETTER LANE, E.C. 4 + +%[** TN: Publisher's device] + +LONDON : H. K. LEWIS AND CO., Ltd., +136 Gower Street, W.C. 1 +NEW YORK : THE MACMILLAN CO. +BOMBAY } +CALCUTTA } MACMILLAN AND CO., Ltd. +MADRAS } +TORONTO : THE MACMILLAN CO. OF +CANADA, Ltd. +TOKYO : MARUZEN-KABUSHIKI-KAISHA + +ALL RIGHTS RESERVED +\fi +%[** TN: End of omitted half-title] +\PageSep{iii} +\newpage +\begin{center} +\Huge\bfseries +The Theory of Spectra \\ +and \\ +Atomic Constitution +\bigskip + +\large\normalfont +THREE ESSAYS \\ +BY \\ +\Large +NIELS BOHR +\medskip + +\normalsize +Professor of Theoretical Physics in~the~University~of~Copenhagen +\vfill + +\Large +CAMBRIDGE \\ +AT THE UNIVERSITY PRESS \\ +1922 +\end{center} +\newpage +\PageSep{iv} +\ifthenelse{\boolean{ForPrinting}}{% Publisher's verso +\begin{center} +\null\vfill +\footnotesize +PRINTED IN GREAT BRITAIN \\ +AT THE CAMBRIDGE UNIVERSITY PRESS +\end{center} +}{}% Omit for screen-formatted version +\PageSep{v} + +\FrontMatter + +\Preface + +\First{The} three essays which here appear in English all deal with +the application of the quantum theory to problems of atomic +structure, and refer to the different stages in the development of +this theory. + +The first essay ``On the spectrum of hydrogen'' is a translation of +a Danish address given before the Physical Society of Copenhagen +on the 20th~of December 1913, and printed in \Title{Fysisk Tidsskrift}, +\No{xii.}\ p.~97, 1914. Although this address was delivered at a time +when the formal development of the quantum theory was only at +its beginning, the reader will find the general trend of thought +very similar to that expressed in the later addresses, which +form the other two essays. As emphasized at several points the +theory does not attempt an ``explanation'' in the usual sense of +this word, but only the establishment of a connection between facts +which in the present state of science are unexplained, that is to +say the usual physical conceptions do not offer sufficient basis for +a detailed description. + +The second essay ``On the series spectra of the elements'' is a +translation of a German address given before the Physical Society +of Berlin on the 27th~of April 1920, and printed in \Title{Zeitschrift für +Physik}, \No{vi.}\ p.~423, 1920. This address falls into two main parts. +The considerations in the first part are closely related to the contents +of the first essay; especially no use is made of the new +formal conceptions established through the later development of +the quantum theory. The second part contains a survey of the +results reached by this development. An attempt is made to +elucidate the problems by means of a general principle which postulates +a formal correspondence between the fundamentally different +conceptions of the classical electrodynamics and those of the +quantum theory. The first germ of this correspondence principle +may be found in the first essay in the deduction of the expression +for the constant of the hydrogen spectrum in terms of +Planck's constant and of the quantities which in Rutherford's +\PageSep{vi} +atomic model are necessary for the description of the hydrogen +atom. + +The third essay ``The structure of the atom and the physical +and chemical properties of the elements'' is based on a Danish +address, given before a joint meeting of the Physical and Chemical +Societies of Copenhagen on the 18th~of October 1921, and printed +in \Title{Fysisk Tidsskrift}, \No{xix.}\ p.~153, 1921. While the first two essays +form verbal translations of the respective addresses, this essay +differs from the Danish original in certain minor points. Besides +the addition of a few new figures with explanatory text, certain +passages dealing with problems discussed in the second essay are +left out, and some remarks about recent contributions to the +subject are inserted. Where such insertions have been introduced +will clearly appear from the text. This essay is divided into +four parts. The first two parts contain a survey of previous results +concerning atomic problems and a short account of the theoretical +ideas of the quantum theory. In the following parts it is shown +how these ideas lead to a view of atomic constitution which seems +to offer an explanation of the observed physical and chemical +properties of the elements, and especially to bring the characteristic +features of the periodic table into close connection with the +interpretation of the optical and high frequency spectra of the +elements. + +For the convenience of the reader all three essays are subdivided +into smaller paragraphs, each with a headline. Conforming to the +character of the essays there is, however, no question of anything +like a full account or even a proportionate treatment of the subject +stated in these headlines, the principal object being to emphasize +certain general views in a freer form than is usual in scientific +treatises or text books. For the same reason no detailed references +to the literature are given, although an attempt is made to mention +the main contributions to the development of the subject. As +regards further information the reader in the case of the second +essay is referred to a larger treatise ``On the quantum theory of +line spectra,'' two parts of which have appeared in the Transactions of +the Copenhagen Academy (\Title{D.\ Kgl.\ Danske Vidensk.\ Selsk.\ Skrifter}, +8.\ Række, \No{iv.}~1, I~and~II, 1918),\footnote + {See \href{http://www.gutenberg.org/ebooks/47167}{www.gutenberg.org/ebooks/47167}.---\textit{Trans.}} +where full references to the literature +may be found. The proposed continuation of this treatise, mentioned +\PageSep{vii} +at several places in the second essay, has for various reasons been +delayed, but in the near future the work will be completed by the +publication of a third part. It is my intention to deal more fully +with the problems discussed in the third essay by a larger systematic +account of the application of the quantum theory to atomic +problems, which is under preparation. + +As mentioned both in the beginning and at the end of the +third essay, the considerations which it contains are clearly still +incomplete in character. This holds not only as regards the +elaboration of details, but also as regards the development of the +theoretical ideas. It may be useful once more to emphasize, +that---although the word ``explanation'' has been used more +liberally than for instance in the first essay---we are not concerned +with a description of the phenomena, based on a well-defined +physical picture. It may rather be said that hitherto every +progress in the problem of atomic structure has tended to emphasize +the well-known ``mysteries'' of the quantum theory more and more. +I hope the exposition in these essays is sufficiently clear, nevertheless, +to give the reader an impression of the peculiar charm +which the study of atomic physics possesses just on this account. + +I wish to express my best thanks to Dr~A.~D. Udden, University +of Pennsylvania, who has undertaken the translation of the +original addresses into English, and to Mr~C.~D. Ellis, Trinity +College, Cambridge, who has looked through the manuscript and +suggested many valuable improvements in the exposition of the +subject. +\Signature{N. BOHR.}{Copenhagen,}{May}{1922.} +\PageSep{viii} + +\TableofContents + +\iffalse +%[** TN: Original ToC text (not manually verified)] +CONTENTS + +ESSAY I +ON THE SPECTRUM OF HYDROGEN + +PAGE + +Empirical Spectral Laws 1 +Laws of Temperature Radiation 4 +The Nuclear Theory of the Atom 7 +Quantum Theory of Spectra 10 +Hydrogen Spectrum 12 +The Pickering Lines 15 +Other Spectra 18 + + +ESSAY II +ON THE SERIES SPECTRA OF THE ELEMENTS + +I. Introduction .20 + +II. General Principles of the Quantum Theory of Spectra . 23 +Hydrogen Spectrum 24 +The Correspondence Principle 27 +General Spectral Laws 29 +Absorption and Excitation of Radiation 32 + +III. Development of the Quantum Theory of Spectra . . 36 +Effect of External Forces on the Hydrogen Spectrum . . 37 +The Stark Effect 39 +The Zeeman Effect 42 +Central Perturbations . 44 +Relativity Effect on Hydrogen Lines 46 +Theory of Series Spectra 48 +Correspondence Principle and Conservation of Angular Momentum 50 +The Spectra of Helium and Lithium 54 +Complex Structure of Series Lines 58 + +IV. Conclusion 59 + +\PageSep{ix} +CONTENTS + +ESSAY III + +THE STRUCTURE OF THE ATOM AND THE PHYSICAL +AND CHEMICAL PROPERTIES OF THE ELEMENTS + +PAGE + +I. Preliminary 61 +The Nuclear Atom 61 +The Postulates of the Quantum Theory 62 +Hydrogen Atom 63 +Hydrogen Spectrum and X-ray Spectra 65 +The Fine Structure of the Hydrogen Lines .... 67 +Periodic Table 69 +Recent Atomic Models 74 + +II. Series Spectra and the Capture of Electrons by Atoms . 75 +Arc and Spark Spectra 76 +Series Diagram 78 +Correspondence Principle 81 + +III. Formation of Atoms and the Periodic Table ... 85 +First Period. Hydrogen---Helium 85 +Second Period. Lithium---Neon 89 +Third Period. Sodium---Argon 95 +Fourth Period. Potassium---Krypton 100 +Fifth Period. Rubidium--- Xenon 108 +Sixth Period. Caesium---Niton 109 +Seventh Period 111 +Survey of the Periodic Table 113 + +IV. Reorganization of Atoms and X-ray Spectra . . .116 +Absorption and Emission of X-rays and Correspondence Principle 117 +X-ray Spectra and Atomic Structure 119 +Classification of X-ray Spectra 121 +Conclusion 125 +\fi +%[** TN: End of original ToC text] +\PageSep{1} +\MainMatter + +\Essay{I}{On the Spectrum of Hydrogen} + {Address delivered before the Physical Society in Copenhagen, Dec.~20, 1913.} + +\Section{Empirical spectral laws.} Hydrogen possesses not only the +smallest atomic weight of all the elements, but it also occupies a +peculiar position both with regard to its physical and its chemical +properties. One of the points where this becomes particularly apparent +is the hydrogen line spectrum. + +The spectrum of hydrogen observed in an ordinary Geissler tube +consists of a series of lines, the strongest of which lies at the red +end of the spectrum, while the others extend out into the ultra\Add{-}violet, +the distance between the various lines, as well as their intensities, +constantly decreasing. In the ultra\Add{-}violet the series converges +to a limit. + +Balmer, as we know, discovered (1885) that it was possible to +represent the wave lengths of these lines very accurately by the +simple law +\[ +\frac{1}{\lambda_{n}} = R \left(\frac{1}{4} - \frac{1}{n^{2}}\right), +\Tag{(1)} +\] +where $R$~is a constant and $n$~is a whole number. The wave lengths +of the five strongest hydrogen lines, corresponding to $n = 3$, $4$,~$5$, $6$,~$7$, +measured in air at ordinary pressure and temperature, and the +values of these wave lengths multiplied by $\left(\dfrac{1}{4} - \dfrac{1}{n^{2}}\right)$ are given in +the following table:\Pagelabel{1} +\[ +%[** TN: Original uses a period for multiplication and a center dot as a decimal point] +\begin{array}{*{2}{c<{\qquad\qquad}}c} +n & \lambda · 10^{8} & \lambda · \left(\dfrac{1}{4} - \dfrac{1}{n^{2}}\right) · 10^{10} \\ +3 & 6563.04 & 91153.3 \\ +4 & 4861.49 & 91152.9 \\ +5 & 4340.66 & 91153.9 \\ +6 & 4101.85 & 91152.2 \\ +7 & 3970.25 & 91153.7 \\ +\end{array} +\] +The table shows that the product is nearly constant, while the deviations +are not greater than might be ascribed to experimental errors. + +As you already know, Balmer's discovery of the law relating to +the hydrogen spectrum led to the discovery of laws applying to +the spectra of other elements. The most important work in this +\PageSep{2} +connection was done by Rydberg (1890) and Ritz (1908). Rydberg +pointed out that the spectra of many elements contain series of +lines whose wave lengths are given approximately by the formula +\[ +\frac{1}{\lambda_{n}} = A - \frac{R}{(n + \alpha)^{2}}, +\] +where $A$~and~$\alpha$ are constants having different values for the various +series, while $R$~is a universal constant equal to the constant in the +spectrum of hydrogen. If the wave lengths are measured in vacuo +Rydberg calculated the value of~$R$ to be~$109675$. In the spectra of +many elements, as opposed to the simple spectrum of hydrogen, there +are several series of lines whose wave lengths are to a close approximation +given by Rydberg's formula if different values are assigned to +the constants $A$~and~$\alpha$. Rydberg showed, however, in his earliest +work, that certain relations existed between the constants in the +various series of the spectrum of one and the same element. These +relations were later very successfully generalized by Ritz through +the establishment of the ``combination principle.'' According to +this principle, the wave lengths of the various lines in the spectrum +of an element may be expressed by the formula +\[ +\frac{1}{\lambda} = F_{r}(n_{1}) - F_{s}(n_{2}). +\Tag{(2)} +\] +In this formula $n_{1}$~and~$n_{2}$ are whole numbers, and $F_{1}(n)$, $F_{2}(n)$,~\dots\ is +a series of functions of~$n$, which may be written approximately +\[ +F_{r}(n) = \frac{R}{(n + \alpha_{r})^{2}}, +\] +where $R$~is Rydberg's universal constant and $\alpha_{r}$ is a constant which +is different for the different functions. A particular spectral line will, +according to this principle, correspond to each combination of $n_{1}$~and~$n_{2}$, +as well as to the functions $F_{1}$, $F_{2}$,~\dots. The establishment of +this principle led therefore to the prediction of a great number of +lines which were not included in the spectral formulae previously +considered, and in a large number of cases the calculations were +found to be in close agreement with the experimental observations. +In the case of hydrogen Ritz assumed that formula~\Eq{(1)} was a special +case of the general formula +\[ +\frac{1}{\lambda} = R\left(\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}\right), +\Tag{(3)} +\] +\PageSep{3} +and therefore predicted among other things a series of lines in the +infra\Add{-}red given by the formula +\[ +\frac{1}{\lambda} = R\left(\frac{1}{9} - \frac{1}{n^{2}}\right). +\] +In 1909 Paschen succeeded in observing the first two lines of this +series corresponding to $n = 4$ and $n = 5$. + +The part played by hydrogen in the development of our +knowledge of the spectral laws is not solely due to its ordinary +simple spectrum, but it can also be traced in other less direct +ways. At a time when Rydberg's laws were still in want of +further confirmation Pickering (1897) found in the spectrum of a +star a series of lines whose wave lengths showed a very simple relation +to the ordinary hydrogen spectrum, since to a very close +approximation they could be expressed by the formula +\[ +\frac{1}{\lambda} = R\left(\frac{1}{4} - \frac{1}{(n + \frac{1}{2})^{2}}\right). +\] +Rydberg considered these lines to represent a new series of lines +in the spectrum of hydrogen, and predicted according to his theory +the existence of still another series of hydrogen lines the wave +lengths of which would be given by +\[ +\frac{1}{\lambda} = R\left(\frac{1}{(\frac{3}{2})^{2}} - \frac{1}{n^{2}}\right). +\] +By examining earlier observations it was actually found that a line +had been observed in the spectrum of certain stars which coincided +closely with the first line in this series (corresponding to $n = 2$); +from analogy with other spectra it was also to be expected that this +would be the strongest line. This was regarded as a great triumph +for Rydberg's theory and tended to remove all doubt that the new +spectrum was actually due to hydrogen. Rydberg's view has therefore +been generally accepted by physicists up to the present moment. +Recently however the question has been reopened and Fowler +(1912) has succeeded in observing the Pickering lines in ordinary +laboratory experiments. We shall return to this question again +later. + +The discovery of these beautiful and simple laws concerning the +line spectra of the elements has naturally resulted in many attempts +at a theoretical explanation. Such attempts are very alluring +\PageSep{4} +because the simplicity of the spectral laws and the exceptional accuracy +with which they apply appear to promise that the correct explanation +will be very simple and will give valuable information +about the properties of matter. I should like to consider some of +these theories somewhat more closely, several of which are extremely +interesting and have been developed with the greatest keenness +and ingenuity, but unfortunately space does not permit me to do +so here. I shall have to limit myself to the statement that not +one of the theories so far proposed appears to offer a satisfactory or +even a plausible way of explaining the laws of the line spectra. +Considering our deficient knowledge of the laws which determine +the processes inside atoms it is scarcely possible to give an explanation +of the kind attempted in these theories. The inadequacy of +our ordinary theoretical conceptions has become especially apparent +from the important results which have been obtained in recent years +from the theoretical and experimental study of the laws of temperature +radiation. You will therefore understand that I shall not +attempt to propose an explanation of the spectral laws; on the +contrary I shall try to indicate a way in which it appears possible +to bring the spectral laws into close connection with other properties +of the elements, which appear to be equally inexplicable on +the basis of the present state of the science. In these considerations +I shall employ the results obtained from the study of temperature +radiation as well as the view of atomic structure which has been +reached by the study of the radioactive elements. + +\Section{Laws of temperature radiation.} I shall commence by mentioning +the conclusions which have been drawn from experimental +and theoretical work on temperature radiation. + +Let us consider an enclosure surrounded by bodies which are in +temperature equilibrium. In this space there will be a certain +amount of energy contained in the rays emitted by the surrounding +substances and crossing each other in every direction. By making +the assumption that the temperature equilibrium will not be disturbed +by the mutual radiation of the various bodies Kirchhoff +(1860) showed that the amount of energy per unit volume as well +as the distribution of this energy among the various wave lengths +is independent of the form and size of the space and of the nature +\PageSep{5} +of the surrounding bodies and depends only on the temperature. +Kirchhoff's result has been confirmed by experiment, and the +amount of energy and its distribution among the various wave +lengths and the manner in which it depends on the temperature +are now fairly well known from a great amount of experimental +work; or, as it is usually expressed, we have a fairly +accurate experimental knowledge of the ``laws of temperature +radiation.'' + +Kirchhoff's considerations were only capable of predicting the +existence of a law of temperature radiation, and many physicists +have subsequently attempted to find a more thorough explanation +of the experimental results. You will perceive that the electromagnetic +theory of light together with the electron theory suggests +a method of solving this problem. According to the electron theory +of matter a body consists of a system of electrons. By making +certain definite assumptions concerning the forces acting on the +electrons it is possible to calculate their motion and consequently +the energy radiated from the body per second in the form of +electromagnetic oscillations of various wave lengths. In a similar +manner the absorption of rays of a given wave length by a substance +can be determined by calculating the effect of electromagnetic +oscillations upon the motion of the electrons. Having investigated +the emission and absorption of a body at all temperatures, and for +rays of all wave lengths, it is possible, as Kirchhoff has shown, to +determine immediately the laws of temperature radiation. Since +the result is to be independent of the nature of the body we are +justified in expecting an agreement with experiment, even though +very special assumptions are made about the forces acting upon +the electrons of the hypothetical substance. This naturally +simplifies the problem considerably, but it is nevertheless sufficiently +difficult and it is remarkable that it has been possible +to make any advance at all in this direction. As is well known +this has been done by Lorentz (1903). He calculated the +emissive as well as the absorptive power of a metal for long +wave lengths, using the same assumptions about the motions +of the electrons in the metal that Drude (1900) employed in +his calculation of the ratio of the electrical and thermal conductivities. +Subsequently, by calculating the ratio of the emissive +\PageSep{6} +to the absorptive power, Lorentz really obtained an expression +for the law of temperature radiation which for long wave lengths +agrees remarkably well with experimental facts. In spite of this +beautiful and promising result, it has nevertheless become apparent +that the electromagnetic theory is incapable of explaining the law +of temperature radiation. For, it is possible to show, that, if the +investigation is not confined to oscillations of long wave lengths, +as in Lorentz's work, but is also extended to oscillations corresponding +to small wave lengths, results are obtained which are +contrary to experiment. This is especially evident from Jeans' +investigations (1905) in which he employed a very interesting +statistical method first proposed by Lord Rayleigh. + +We are therefore compelled to assume, that the classical electrodynamics +does not agree with reality, or expressed more carefully, +that it \Chg{can not}{cannot} be employed in calculating the absorption and +emission of radiation by atoms. Fortunately, the law of temperature +radiation has also successfully indicated the direction in which the +necessary changes in the electrodynamics are to be sought. Even +before the appearance of the papers by Lorentz and Jeans, Planck +(1900) had derived theoretically a formula for the black body radiation +which was in good agreement with the results of experiment. +Planck did not limit himself exclusively to the classical electrodynamics, +but introduced the further assumption that a system of +oscillating electrical particles (elementary resonators) will neither +radiate nor absorb energy continuously, as required by the ordinary +electrodynamics, but on the contrary will radiate and absorb discontinuously. +The energy contained within the system at any +moment is always equal to a whole multiple of the so-called +quantum of energy the magnitude of which is equal to~$h\nu$, where +$h$~is Planck's constant and $\nu$~is the frequency of oscillation of the +system per second. In formal respects Planck's theory leaves much +to be desired; in certain calculations the ordinary electrodynamics +is used, while in others assumptions distinctly at variance with it +are introduced without any attempt being made to show that it +is possible to give a consistent explanation of the procedure used. +Planck's theory would hardly have acquired general recognition +merely on the ground of its agreement with experiments on black +body radiation, but, as you know, the theory has also contributed +\PageSep{7} +quite remarkably to the elucidation of many different physical +phenomena, such as specific heats, photoelectric effect, X-rays and +the absorption of heat rays by gases. These explanations involve +more than the qualitative assumption of a discontinuous transformation +of energy, for with the aid of Planck's constant~$h$ it +seems to be possible, at least approximately, to account for a great +number of phenomena about which nothing could be said previously. +It is therefore hardly too early to express the opinion that, whatever +the final explanation will be, the discovery of ``energy quanta'' +must be considered as one of the most important results arrived at +in physics, and must be taken into consideration in investigations +of the properties of atoms and particularly in connection with any +explanation of the spectral laws in which such phenomena as +the emission and absorption of electromagnetic radiation are +concerned. + +\Section{The nuclear theory of the atom.} We shall now consider the +second part of the foundation on which we shall build, namely the +conclusions arrived at from experiments with the rays emitted by +radioactive substances. I have previously here in the Physical +Society had the opportunity of speaking of the scattering of $\alpha$~rays +in passing through thin plates, and to mention how Rutherford +(1911) has proposed a theory for the structure of the atom in +order to explain the remarkable and unexpected results of these +experiments. I shall, therefore, only remind you that the characteristic +feature of Rutherford's theory is the assumption of the +existence of a positively charged nucleus inside the atom. A number +of electrons are supposed to revolve in closed orbits around the +nucleus, the number of these electrons being sufficient to neutralize +the positive charge of the nucleus. The dimensions of the nucleus +are supposed to be very small in comparison with the dimensions +of the orbits of the electrons, and almost the entire mass of the +atom is supposed to be concentrated in the nucleus. + +According to Rutherford's calculation the positive charge of the +nucleus corresponds to a number of electrons equal to about half +the atomic weight. This number coincides approximately with the +number of the particular element in the periodic system and it is +therefore natural to assume that the number of electrons in the +\PageSep{8} +atom is exactly equal to this number. This hypothesis, which was +first stated by van~den Broek (1912), opens the possibility of +obtaining a simple explanation of the periodic system. This assumption +is strongly confirmed by experiments on the elements +of small atomic weight. In the first place, it is evident that according +to Rutherford's theory the $\alpha$~particle is the same as the +nucleus of a helium atom. Since the $\alpha$~particle has a double positive +charge it follows immediately that a neutral helium atom contains +two electrons. Further the concordant results obtained from calculations +based on experiments as different as the diffuse scattering +of X-rays and the decrease in velocity of $\alpha$~rays in passing +through matter render the conclusion extremely likely that a +hydrogen atom contains only a single electron. This agrees most +beautifully with the fact that J.~J. Thomson in his well-known +experiments on rays of positive electricity has never observed a +hydrogen atom with more than a single positive charge, while all +other elements investigated may have several charges. + +Let us now assume that a hydrogen atom simply consists of an +electron revolving around a nucleus of equal and opposite charge, +and of a mass which is very large in comparison with that of the +electron. It is evident that this assumption may explain the peculiar +position already referred to which hydrogen occupies among the +elements, but it appears at the outset completely hopeless to attempt +to explain anything at all of the special properties of hydrogen, +still less its line spectrum, on the basis of considerations relating +to such a simple system. + +Let us assume for the sake of brevity that the mass of the nucleus +is infinitely large in proportion to that of the electron, and that the +velocity of the electron is very small in comparison with that of +light. If we now temporarily disregard the energy radiation, which, +according to the ordinary electrodynamics, will accompany the accelerated +motion of the electron, the latter in accordance with +Kepler's first law will describe an ellipse with the nucleus in one +of the foci. Denoting the frequency of revolution by~$\omega$, and the +major axis of the ellipse by~$2a$ we find that +\[ +\omega^{2} = \frac{2W^{3}}{\pi^{2} e^{4} m},\quad +2a = \frac{e^{2}}{W}, +\Tag{(4)} +\] +\PageSep{9} +where $e$~is the charge of the electron and $m$~its mass, while $W$~is +the work which must be added to the system in order to remove +the electron to an infinite distance from the nucleus. + +These expressions are extremely simple and they show that the +magnitude of the frequency of revolution as well as the length of +the major axis depend only on~$W$, and are independent of the +\Chg{excentricity}{eccentricity} of the orbit. By varying~$W$ we may obtain all possible +values for $\omega$~and~$2a$. This condition shows, however, that it is not +possible to employ the above formulae directly in calculating the +orbit of the electron in a hydrogen atom. For this it will be necessary +to assume that the orbit of the electron \Chg{can not}{cannot} take on all values, +and in any event, the line spectrum clearly indicates that the +oscillations of the electron cannot vary continuously between wide +limits. The impossibility of making any progress with a simple +system like the one considered here might have been foretold from +a consideration of the dimensions involved; for with the aid of $e$~and +$m$~alone it is impossible to obtain a quantity which can be +interpreted as a diameter of an atom or as a frequency. + +If we attempt to account for the radiation of energy in the manner +required by the ordinary electrodynamics it will only make matters +worse. As a result of the radiation of energy~$W$ would continually +increase, and the above expressions~\Eq{(4)} show that at the same time +the frequency of revolution of the system would increase, and the +dimensions of the orbit decrease. This process would not stop until +the particles had approached so closely to one another that they no +longer attracted each other. The quantity of energy which would +be radiated away before this happened would be very great. If we +were to treat these particles as geometrical points this energy would +be infinitely great, and with the dimensions of the electrons as +calculated from their mass (about $10^{-13}$~cm.), and of the nucleus as +calculated by Rutherford (about $10^{-12}$~cm.), this energy would be +many times greater than the energy changes with which we are +familiar in ordinary atomic processes. + +It can be seen that it is impossible to employ Rutherford's atomic +model so long as we confine ourselves exclusively to the ordinary +electrodynamics. But this is nothing more than might have been +expected. As I have mentioned we may consider it to be an +established fact that it is impossible to obtain a satisfactory +\PageSep{10} +explanation of the experiments on temperature radiation with the +aid of electrodynamics, no matter what atomic model be employed. +The fact that the deficiencies of the atomic model we are +considering stand out so plainly is therefore perhaps no serious +drawback; even though the defects of other atomic models are +much better concealed they must nevertheless be present and will +be just as serious. + +\Section{Quantum theory of spectra.} Let us now try to overcome these +difficulties by applying Planck's theory to the problem. + +It is readily seen that there can be no question of a direct application +of Planck's theory. This theory is concerned with the emission +and absorption of energy in a system of electrical particles, which +oscillate with a given frequency per second, dependent only on the +nature of the system and independent of the amount of energy +contained in the system. In a system consisting of an electron and +a nucleus the period of oscillation corresponds to the period of +revolution of the electron. But the formula~\Eq{(4)} for~$\omega$ shows that the +frequency of revolution depends upon~$W$, \ie\ on the energy of the +system. Still the fact that we \Chg{can not}{cannot} immediately apply Planck's +theory to our problem is not as serious as it might seem to be, for +in assuming Planck's theory we have manifestly acknowledged the +inadequacy of the ordinary electrodynamics and have definitely +parted with the coherent group of ideas on which the latter theory +is based. In fact in taking such a step we \Chg{can not}{cannot} expect that all +cases of disagreement between the theoretical conceptions hitherto +employed and experiment will be removed by the use of Planck's +assumption regarding the quantum of the energy momentarily +present in an oscillating system. We stand here almost entirely on +virgin ground, and upon introducing new assumptions we need only +take care not to get into contradiction with experiment. Time will +have to show to what extent this can be avoided; but the safest +way is, of course, to make as few assumptions as possible. + +With this in mind let us first examine the experiments on +temperature radiation. The subject of direct observation is the +distribution of radiant energy over oscillations of the various wave +lengths. Even though we may assume that this energy comes from +systems of oscillating particles, we know little or nothing about +\PageSep{11} +these systems. No one has ever seen a Planck's resonator, nor +indeed even measured its frequency of oscillation; we can observe +only the period of oscillation of the radiation which is emitted. It +is therefore very convenient that it is possible to show that to +obtain the laws of temperature radiation it is not necessary to +make any assumptions about the systems which emit the radiation +except that the amount of energy emitted each time shall be equal +to~$h\nu$, where $h$~is Planck's constant and $\nu$~is the frequency of the +radiation. Indeed, it is possible to derive Planck's law of radiation +from this assumption alone, as shown by Debye, who employed a +method which is a combination of that of Planck and of Jeans. +Before considering any further the nature of the oscillating systems +let us see whether it is possible to bring this assumption about the +emission of radiation into agreement with the spectral laws. + +If the spectrum of some element contains a spectral line corresponding +to the frequency~$\nu$ it will be assumed that one of the +atoms of the element (or some other elementary system) can emit +an amount of energy~$h\nu$. Denoting the energy of the atom before +and after the emission of the radiation by $E_{1}$ and~$E_{2}$ we have +\[ +h\nu = E_{1} - E_{2} \text{ or } +\nu = \frac{E_{1}}{h} - \frac{E_{2}}{h}. +\Tag{(5)} +\] + +During the emission of the radiation the system may be regarded +as passing from one state to another; in order to introduce a name +for these states, we shall call them ``stationary'' states, simply +indicating thereby that they form some kind of waiting places +between which occurs the emission of the energy corresponding to +the various spectral lines. As previously mentioned the spectrum +of an element consists of a series of lines whose wave lengths may +be expressed by the formula~\Eq{(2)}. By comparing this expression +with the relation given above it is seen that---since $\nu = \dfrac{c}{\lambda}$, where $c$~is +the velocity of light---each of the spectral lines may be regarded +as being emitted by the transition of a system between two stationary +states in which the energy apart from an additive arbitrary +constant is given by $-ch F_{r}(n_{1})$ and $-ch F_{s}(n_{2})$ respectively. Using +this interpretation the combination principle asserts that a series +of stationary states exists for the given system, and that it can +\PageSep{12} +pass from one to any other of these states with the emission of +a monochromatic radiation. We see, therefore, that with a simple +extension of our first assumption it is possible to give a formal +explanation of the most general law of line spectra. + +\Section{Hydrogen spectrum.} This result encourages us to make an +attempt to obtain a clear conception of the stationary states which +have so far only been regarded as formal. With this end in view, +we naturally turn to the spectrum of hydrogen. The formula +applying to this spectrum is given by the expression +\[ +\frac{1}{\lambda} = \frac{R}{n_{1}^{2}} - \frac{R}{n_{2}^{2}}. +\] +According to our assumption this spectrum is produced by transitions +between a series of stationary states of a system, concerning +which we can for the present only say that the energy of the system +in the $n$th~state, apart from an additive constant, is given by +$-\dfrac{Rhc}{n^{2}}$. Let us now try to find a connection between this and the +model of the hydrogen atom. We assume that in the calculation +of the frequency of revolution of the electron in the stationary states +of the atom it will be possible to employ the above formula for~$\omega$. +It is quite natural to make this assumption; since, in trying to +form a reasonable conception of the stationary states, there is, for +the present at least, no other means available besides the ordinary +mechanics. + +Corresponding to the $n$th~stationary state in formula~\Eq{(4)} for~$\omega$, +let us by way of experiment put $W = \dfrac{Rhc}{n^{2}}$. This gives us +\[ +\omega_{n}^{2} = \frac{2}{\pi^{2}}\, \frac{R^{3} h^{3} c^{3}}{e^{4} mn^{6}}. +\Tag{(6)} +\] + +The radiation of light corresponding to a particular spectral line +is according to our assumption emitted by a transition between +two stationary states, corresponding to two different frequencies of +revolution, and we are not justified in expecting any simple relation +between these frequencies of revolution of the electron and +the frequency of the emitted radiation. You understand, of course, +that I am by no means trying to give what might ordinarily be +described as an explanation; nothing has been said here about +\PageSep{13} +how or why the radiation is emitted. On one point, however, we +may expect a connection with the ordinary conceptions; namely, +that it will be possible to calculate the emission of slow electromagnetic +oscillations on the basis of the classical electrodynamics. +This assumption is very strongly supported by the result of +Lorentz's calculations which have already been described. From +the formula for~$\omega$ it is seen that the frequency of revolution decreases +as $n$~increases, and that the expression~$\dfrac{\omega_{n}}{\omega_{n+1}}$ approaches the +value~$1$. + +According to what has been said above, the frequency of the +radiation corresponding to the transition between the $(n + 1)$th +and the $n$th~stationary state is given by +\[ +\nu = Rc \left(\frac{1}{n^{2}} - \frac{1}{(n + 1)^{2}}\right). +\] +If $n$~is very large this expression is approximately equal to +\[ +\nu = 2Rc/n^{3}. +\] +In order to obtain a connection with the ordinary electrodynamics +let us now place this frequency equal to the frequency of revolution, +that is +\[ +\omega_{n} = 2Rc/n^{3}. +\] +Introducing this value of~$\omega_{n}$ in~\Eq{(6)} we see that $n$~disappears from +the equation, and further that the equation will be satisfied only if +\[ +R = \frac{2\pi^{2} e^{4} m}{ch^{3}}. +\Tag{(7)} +\] +The constant~$R$ is very accurately known, and is, as I have said +before, equal to~$109675$. By introducing the most recent values +for $e$,~$m$ and~$h$ the expression on the right-hand side of the equation +becomes equal to $1.09 · 10^{5}$. The agreement is as good as +could be expected, considering the uncertainty in the experimental +determination of the constants $e$,~$m$ and~$h$. The agreement between +our calculations and the classical electrodynamics is, therefore, +fully as good as we are justified in expecting. + +We \Chg{can not}{cannot} expect to obtain a corresponding explanation of the +frequency values of the other stationary states. Certain simple +formal relations apply, however, to all the stationary states. By +introducing the expression, which has been found for~$R$, we +get for the $n$th~state $W_{n} = \frac{1}{2}nh\omega_{n}$. This equation is entirely +\PageSep{14} +analogous to Planck's assumption concerning the energy of a +resonator. $W$~in our system is readily shown to be equal to the +average value of the kinetic energy of the electron during a +single revolution. The energy of a resonator was shown by Planck +you may remember to be always equal to~$nh\nu$. Further the average +value of the kinetic energy of Planck's resonator is equal to its +potential energy, so that the average value of the kinetic energy +of the resonator, according to Planck, is equal to~$\frac{1}{2}nh\omega$. This +analogy suggests another manner of presenting the theory, and it +was just in this way that I was originally led into these considerations. +When we consider how differently the equation is +employed here and in Planck's theory it appears to me misleading +to use this analogy as a foundation, and in the account I have +given I have tried to free myself as much as possible from it. + +Let us continue with the elucidation of the calculations, and in +the expression for~$2a$ introduce the value of~$W$ which corresponds +to the $n$th~stationary state. This gives us +\[ +2a = n^{2} · \frac{e^{2}}{chR} + = n^{2} · \frac{h^{2}}{2\pi^{2} me^{2}} + = n^{2} · 1.1 · 10^{-8}. +\Tag{(8)} +\] + +It is seen that for small values of~$n$, we obtain values for the +major axis of the orbit of the electron which are of the same +order of magnitude as the values of the diameters of the atoms +calculated from the kinetic theory of gases. For large values of~$n$, +$2a$~becomes very large in proportion to the calculated dimensions +of the atoms. This, however, does not necessarily disagree with +experiment. Under ordinary circumstances a hydrogen atom will +probably exist only in the state corresponding to $n = 1$. For this +state $W$~will have its greatest value and, consequently, the atom +will have emitted the largest amount of energy possible; this will +therefore represent the most stable state of the atom from which +the system \Chg{can not}{cannot} be transferred except by adding energy to it +from without. The large values for~$2a$ corresponding to large~$n$ need +not, therefore, be contrary to experiment; indeed, we may in these +large values seek an explanation of the fact, that in the laboratory +it has hitherto not been possible to observe the hydrogen lines +corresponding to large values of~$n$ in Balmer's formula, while they +have been observed in the spectra of certain stars. In order that +the large orbits of the electrons may not be disturbed by electrical +\PageSep{15} +forces from the neighbouring atoms the pressure will have to be +very low, so low, indeed, that it is impossible to obtain sufficient +light from a Geissler tube of ordinary dimensions. In the stars, +however, we may assume that we have to do with hydrogen which +is exceedingly attenuated and distributed throughout an enormously +large region of space. + +\Section{The Pickering lines.} You have probably noticed that we have +not mentioned at all the spectrum found in certain stars which +according to the opinion then current was assigned to hydrogen, +and together with the ordinary hydrogen spectrum was considered +by Rydberg to form a connected system of lines completely +analogous to the spectra of other elements. You have probably +also perceived that difficulties would arise in interpreting this +spectrum by means of the assumptions which have been employed. +If such an attempt were to be made it would be necessary to give +up the simple considerations which lead to the expression~\Eq{(7)} for +the constant~$R$. We shall see, however, that it appears possible to +explain the occurrence of this spectrum in another way. Let us +suppose that it is not due to hydrogen, but to some other simple +system consisting of a single electron revolving about a nucleus +with an electrical charge~$Ne$. The expression for~$\omega$ becomes then +\[ +\omega^{2} = \frac{2}{\pi^{2}}\, \frac{W^{3}}{N^{2} e^{4} m}. +\] +Repeating the same calculations as before only in the inverse +order we find, that this system will emit a line spectrum given by +the expression +\[ +\frac{1}{\lambda} + = \frac{2\pi^{2} N^{2} e^{4} m}{ch^{3}}\left(\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}\right) + = R\raisebox{-4pt}{$\Biggl($}\frac{1}{\left(\dfrac{n_{1}}{N}\right)^{2}} - \frac{1}{\left(\dfrac{n_{2}}{N}\right)^{2}}\raisebox{-4pt}{$\Biggr)$}. +\Tag{(9)} +\] + +By comparing this formula with the formula for Pickering's and +Rydberg's series, we see that the observed lines can be explained +on the basis of the theory, if it be assumed that the spectrum is +due to an electron revolving about a nucleus with a charge~$2e$, or +according to Rutherford's theory around the nucleus of a helium +atom. The fact that the spectrum in question is not observed in +an ordinary helium tube, but only in stars, may be accounted for +\PageSep{16} +by the high degree of ionization which is required for the production +of this spectrum; a neutral helium atom contains of course +two electrons while the system under consideration contains +only one. + +These conclusions appear to be supported by experiment. +Fowler, as I have mentioned, has recently succeeded in observing +Pickering's and Rydberg's lines in a laboratory experiment. By +passing a very heavy current through a mixture of hydrogen and +helium Fowler observed not only these lines but also a new series +of lines. This new series was of the same general type, the wave +length being given approximately by +\[ +\frac{1}{\lambda} + = R\left(\frac{1}{(\frac{3}{2})^{2}} - \frac{1}{(n + \frac{1}{2})^{2}}\right). +\] +Fowler interpreted all the observed lines as the hydrogen spectrum +sought for. With the observation of the latter series of lines, +however, the basis of the analogy between the hypothetical +hydrogen spectrum and the other spectra disappeared, and thereby +also the foundation upon which Rydberg had founded his conclusions; +on the contrary it is seen, that the occurrence of the lines +was exactly what was to be expected on our view. + +In the following table the first column contains the wave +lengths measured by Fowler, while the second contains the limiting +values of the experimental errors given by him; in the third +column we find the products of the wave lengths by the quantity +$\left(\dfrac{1}{n_{1}^{2}} - \dfrac{1}{n_{2}^{2}}\right) \Add{·} 10^{10}$; the values employed for $n_{1}$~and~$n_{2}$ are enclosed in +parentheses in the last column. +\begin{table}[hbt] +\Pagelabel{16} +\[ +\begin{array}{l*{2}{>{\qquad}l}l} +\ColHead{\lambda · 10^{8}} & +\ColHead{\text{Limit of error}} & +\ColHead{\lambda · \left(\dfrac{1}{n_{1}^{2}} - \dfrac{1}{n_{2}^{2}}\right) · 10^{10}} & \\ +4685.98 & 0.01 & 22779.1 & (3 : 4) \\ +3203.30 & 0.05 & 22779.0 & (3 : 5) \\ +2733.34 & 0.05 & 22777.8 & (3 : 6) \\ +2511.31 & 0.05 & 22778.3 & (3 : 7) \\ +2385.47 & 0.05 & 22777.9 & (3 : 8) \\ +2306.20 & 0.10 & 22777.3 & (3 : 9) \\ +2252.88 & 0.10 & 22779.1 & (3 : 10) \\ +5410.5 & 1.0 & 22774 & (4 : 7) \\ +4541.3 & 0.25 & 22777 & (4 : 9) \\ +4200.3 & 0.5 & 22781 & (4 : 11) \\ +\end{array} +\] +\end{table} +\PageSep{17} + +The values of the products are seen to be very nearly equal, +while the deviations are of the same order of magnitude as the +limits of experimental error. The value of the product +\[ +\lambda \left(\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}\right) +\] +should for this spectrum, according to the formula~\Eq{(9)}, be exactly +$\frac{1}{4}$~of the corresponding product for the hydrogen spectrum. From +the tables on pages \PageNum{1} and~\PageNum{16} we find for these products $91153$ +and $22779$, and dividing the former by the latter we get $4.0016$. +This value is very nearly equal to~$4$; the deviation is, however, +much greater than can be accounted for in any way by the errors +of the experiments. It has been easy, however, to find a theoretical +explanation of this point. In all the foregoing calculations +we have assumed that the mass of the nucleus is infinitely great +compared to that of the electron. This is of course not the +case, even though it holds to a very close approximation; for a +hydrogen atom the ratio of the mass of the nucleus to that of the +electron will be about $1850$ and for a helium atom four times as +great. + +If we consider a system consisting of an electron revolving about +a nucleus with a charge~$Ne$ and a mass~$M$, we find the following +expression for the frequency of revolution of the system: +\[ +\omega^{2} = \frac{2}{\pi^{2}}\, \frac{W^{3} (M + m)}{N^{2} e^{4} Mm}. +\] + +From this formula we find in a manner quite similar to that +previously employed that the system will emit a line spectrum, +the wave lengths of which are given by the formula +\[ +\frac{1}{\lambda} + = \frac{2\pi^{2} N^{2} e^{4} mM}{ch^{3} (M + m)} + \left(\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}\right). +\Tag{(10)} +\] + +If with the aid of this formula we try to find the ratio of the +product for the hydrogen spectrum, to that of the hypothetical +helium spectrum we get the value $4.00163$ which is in complete +agreement with the preceding value calculated from the experimental +observations. + +I must further mention that Evans has made some experiments +to determine whether the spectrum in question is due to hydrogen +or helium. He succeeded in observing one of the lines in very +\PageSep{18} +pure helium; there was, at any rate, not enough hydrogen present +to enable the hydrogen lines to be observed. Since in any event +Fowler does not seem to consider such evidence as conclusive it is +to be hoped that these experiments will be continued. There is, +however, also another possibility of deciding this question. As is +evident from the formula~\Eq{(10)}, the helium spectrum under consideration +should contain, besides the lines observed by Fowler, a +series of lines lying close to the ordinary hydrogen lines. These +lines may be obtained by putting $n_{1} = 4$, $n_{2} = 6$, $8$, $10$,~etc. Even +if these lines were present, it would be extremely difficult to +observe them on account of their position with regard to the +hydrogen lines, but should they be observed this would probably +also settle the question of the origin of the spectrum, since no +reason would seem to be left to assume the spectrum to be due to +hydrogen. + +\Section{Other spectra.} For the spectra of other elements the problem +becomes more complicated, since the atoms contain a larger +number of electrons. It has not yet been possible on the basis of +this theory to explain any other spectra besides those which I +have already mentioned. On the other hand it ought to be +mentioned that the general laws applying to the spectra are very +simply interpreted on the basis of our assumptions. So far as the +combination principle is concerned its explanation is obvious. In +the method we have employed our point of departure was largely +determined by this particular principle. But a simple explanation +can be also given of the other general law, namely, the occurrence +of Rydberg's constant in all spectral formulae. Let us assume +that the spectra under consideration, like the spectrum of hydrogen, +are emitted by a neutral system, and that they are produced by +the binding of an electron previously removed from the system. +If such an electron revolves about the nucleus in an orbit which +is large in proportion to that of the other electrons it will be +subjected to forces much the same as the electron in a hydrogen +atom, since the inner electrons individually will approximately +neutralize the effect of a part of the positive charge of the nucleus. +We may therefore assume that for this system there will exist a +series of stationary states in which the motion of the outermost +\PageSep{19} +electron is approximately the same as in the stationary states of a +hydrogen atom. I shall not discuss these matters any further, +but shall only mention that they lead to the conclusion that +Rydberg's constant is not exactly the same for all elements. +The expression for this constant will in fact contain the factor +$\dfrac{M}{M + m}$, where $M$~is the mass of the nucleus. The correction is +exceedingly small for elements of large atomic weight, but for +hydrogen it is, from the point of view of spectrum analysis, very +considerable. If the procedure employed leads to correct results, it +is not therefore permissible to calculate Rydberg's constant directly +from the hydrogen spectrum; the value of the universal constant +should according to the theory be~$109735$ and not~$109675$. + +I shall not tire you any further with more details; I hope to +return to these questions here in the Physical Society, and to +show how, on the basis of the underlying ideas, it is possible +to develop a theory for the structure of atoms and molecules. +Before closing I only wish to say that I hope I have expressed +myself sufficiently clearly so that you have appreciated the extent +to which these considerations conflict with the admirably coherent +group of conceptions which have been rightly termed the classical +theory of electrodynamics. On the other hand, by emphasizing +this conflict, I have tried to convey to you the impression that it +may be also possible in the course of time to discover a certain +coherence in the new ideas. +\PageSep{20} + + +\Essay{II}{On the Series Spectra of the Elements} + {Address delivered before the Physical Society in Berlin, April~27, 1920.} + +\Chapter{I.}{Introduction} + +The subject on which I have the honour to speak here, at the +kind invitation of the Council of your society, is very extensive and +it would be impossible in a single address to give a comprehensive +survey of even the most important results obtained in the theory +of spectra. In what follows I shall try merely to emphasize some +points of view which seem to me important when considering the +present state of the theory of spectra and the possibilities of its +development in the near future. I regret in this connection not to +have time to describe the history of the development of spectral +theories, although this would be of interest for our purpose. No +difficulty, however, in understanding this lecture need be experienced +on this account, since the points of view underlying previous +attempts to explain the spectra differ fundamentally from those upon +which the following considerations rest. This difference exists both +in the development of our ideas about the structure of the atom +and in the manner in which these ideas are used in explaining the +spectra. + +We shall assume, according to Rutherford's theory, that an atom +consists of a positively charged nucleus with a number of electrons +revolving about it. Although the nucleus is assumed to be very +small in proportion to the size of the whole atom, it will contain +nearly the entire mass of the atom. I shall not state the reasons +which led to the establishment of this nuclear theory of the atom, +nor describe the very strong support which this theory has received +from very different sources. I shall mention only that result +which lends such charm and simplicity to the modern development +of the atomic theory. I refer to the idea that the number of electrons +in a neutral atom is exactly equal to the number, giving the +position of the element in the periodic table, the so-called ``atomic +number.'' This assumption, which was first proposed by van~den +Broek, immediately suggests the possibility ultimately of deriving +\PageSep{21} +the explanation of the physical and chemical properties of the +elements from their atomic numbers. If, however, an explanation +of this kind is attempted on the basis of the classical laws of +mechanics and electrodynamics, insurmountable difficulties are encountered. +These difficulties become especially apparent when we +consider the spectra of the elements. In fact, the difficulties are +here so obvious that it would be a waste of time to discuss them in +detail. It is evident that systems like the nuclear atom, if based +upon the usual mechanical and electrodynamical conceptions, +would not even possess sufficient stability to give a spectrum consisting +of sharp lines. + +In this lecture I shall use the ideas of the quantum theory. It +will not be necessary, particularly here in Berlin, to consider in +detail how Planck's fundamental work on temperature radiation +has given rise to this theory, according to which the laws governing +atomic processes exhibit a definite element of discontinuity. I shall +mention only Planck's chief result about the properties of an exceedingly +simple kind of atomic system, the Planck ``oscillator.'' +This consists of an electrically charged particle which can execute +harmonic oscillations about its position of equilibrium with a frequency +independent of the amplitude. By studying the statistical +equilibrium of a number of such systems in a field of radiation +Planck was led to the conclusion that the emission and absorption +of radiation take place in such a manner, that, so far as a statistical +equilibrium is concerned only certain distinctive states of the +oscillator are to be taken into consideration. In these states the +energy of the system is equal to a whole multiple of a so-called +``energy quantum,'' which was found to be proportional to the frequency +of the oscillator. The particular energy values are therefore +given by the well-known formula +\[ +E_{n} = nh\omega, +\Tag{(1)} +\] +where $n$~is a whole number, $\omega$~the frequency of vibration of the +oscillator, and $h$~is Planck's constant. + +If we attempt to use this result to explain the spectra of the +elements, however, we encounter difficulties, because the motion of +the particles in the atom, in spite of its simple structure, is in general +exceedingly complicated compared with the motion of a Planck +\PageSep{22} +oscillator. The question then arises, how Planck's result ought to +be generalized in order to make its application possible. Different +points of view immediately suggest themselves. Thus we might +regard this equation as a relation expressing certain characteristic +properties of the distinctive motions of an atomic system and try +to obtain the general form of these properties. On the other hand, +we may also regard equation~\Eq{(1)} as a statement about a property +of the process of radiation and inquire into the general laws which +control this process. + +In Planck's theory it is taken for granted that the frequency of +the radiation emitted and absorbed by the oscillator is equal to its +own frequency, an assumption which may be written +\[ +\nu \equiv \omega, +\Tag{(2)} +\] +if in order to make a sharp distinction between the frequency of +the emitted radiation and the frequency of the particles in the atoms, +we here and in the following denote the former by~$\nu$ and the latter +by~$\omega$. We see, therefore, that Planck's result may be interpreted to +mean, that the oscillator can emit and absorb radiation only in +``radiation quanta'' of magnitude +\[ +\Delta E = h\nu. +\Tag{(3)} +\] +It is well known that ideas of this kind led Einstein to a theory +of the photoelectric effect. This is of great importance, since it +represents the first instance in which the quantum theory was +applied to a phenomenon of non-statistical character. I shall not +here discuss the familiar difficulties to which the ``hypothesis of +light quanta'' leads in connection with the phenomena of interference, +for the explanation of which the classical theory of radiation +has shown itself to be so remarkably suited. Above all I shall not +consider the problem of the nature of radiation, I shall only attempt +to show how it has been possible in a purely formal manner to +develop a spectral theory, the essential elements of which may be +considered as a simultaneous rational development of the two ways +of interpreting Planck's result. +\PageSep{23} + + +\Chapter{II.}{General Principles of the Quantum Theory +of\protect~Spectra} + +In order to explain the appearance of line spectra we are compelled +to assume that the emission of radiation by an atomic system +takes place in such a manner that it is not possible to follow the +emission in detail by means of the usual conceptions. Indeed, these +do not even offer us the means of calculating the frequency of the +emitted radiation. We shall see, however, that it is possible to give +a very simple explanation of the general empirical laws for the +frequencies of the spectral lines, if for each emission of radiation +by the atom we assume the fundamental law to hold, that during +the entire period of the emission the radiation possesses one and +the same frequency~$\nu$, connected with the total energy emitted by +the \emph{frequency relation} +\[ +h\nu = E' - E''. +\Tag{(4)} +\] +Here $E'$~and $E''$ represent the energy of the system before and +after the emission. + +If this law is assumed, the spectra do not give us information +about the motion of the particles in the atom, as is supposed in the +usual theory of radiation, but only a knowledge of the energy +changes in the various processes which can occur in the atom. +From this point of view the spectra show the existence of certain, +definite energy values corresponding to certain distinctive states +of the atoms. These states will be called the \emph{stationary states} of +the atoms, since we shall assume that the atom can remain a finite +time in each state, and can leave this state only by a process of +transition to another stationary state. Notwithstanding the fundamental +departure from the ordinary mechanical and electrodynamical +conceptions, we shall see, however, that it is possible to give a +rational interpretation of the evidence provided by the spectra on +the basis of these ideas. + +Although we must assume that the ordinary mechanics \Chg{can not}{cannot} +be used to describe the transitions between the stationary states, +nevertheless, it has been found possible to develop a consistent +theory on the assumption that the motion in these states can be +described by the use of the ordinary mechanics. Moreover, although +the process of radiation \Chg{can not}{cannot} be described on the basis of the +\PageSep{24} +ordinary theory of electrodynamics, according to which the nature +of the radiation emitted by an atom is directly related to the harmonic +components occurring in the motion of the system, there is +found, nevertheless, to exist a far-reaching \emph{correspondence} between +the various types of possible transitions between the stationary +states on the one hand and the various harmonic components of the +motion on the other hand. This correspondence is of such a nature, +that the present theory of spectra is in a certain sense to be regarded +as a rational generalization of the ordinary theory of radiation. + +\Section{Hydrogen spectrum.} In order that the principal points may +stand out as clearly as possible I shall, before considering the more +complicated types of series spectra, first consider the simplest spectrum, +namely, the series spectrum of hydrogen. This spectrum +consists of a number of lines whose frequencies are given with great +exactness by Balmer's formula +\[ +\nu = \frac{K}{(n'')^{2}} - \frac{K}{(n')^{2}}, +\Tag{(5)} +\] +where $K$~is a constant, and $n'$~and $n''$ are whole numbers. If we put +$n'' = 2$ and give to~$n'$ the values $3$,~$4$,~etc., we get the well-known +Balmer series of hydrogen. If we put $n'' = 1$ or $n'' = 3$ we obtain +respectively the ultra-violet and infra-red series. We shall assume +the hydrogen atom simply to consist of a positively charged nucleus +with a single electron revolving about it. For the sake of simplicity +we shall suppose the mass of the nucleus to be infinite in comparison +with the mass of the electron, and further we shall disregard the +small variations in the motion due to the change in mass of the +electron with its velocity. With these simplifications the electron +will describe a closed elliptical orbit with the nucleus at one of the +foci. The frequency of revolution~$\omega$ and the major axis~$2a$ of the +orbit will be connected with the energy of the system by the following +equations: +\[ +\omega = \sqrt{\frac{2W^{3}}{\pi^{2} e^{4} m}},\quad +2a = \frac{e^{2}}{W}. +\Tag{(6)} +\] +Here $e$~is the charge of the electron and $m$~its mass, while $W$~is the +work required to remove the electron to infinity. + +The simplicity of these formulae suggests the possibility of using +them in an attempt to explain the spectrum of hydrogen. This, +\PageSep{25} +however, is not possible so long as we use the classical theory of +radiation. It would not even be possible to understand how hydrogen +could emit a spectrum consisting of sharp lines; for since $\omega$~varies +with~$W$, the frequency of the emitted radiation would vary continuously +during the emission. We can avoid these difficulties if +we use the ideas of the quantum theory. If for each line we form +the product~$h\nu$ by multiplying both sides of~\Eq{(5)} by~$h$, then, since +the right-hand side of the resulting relation may be written as +the difference of two simple expressions, we are led by comparison +with formula~\Eq{(4)} to the assumption that the separate lines of the +spectrum will be emitted by transitions between two stationary +states, forming members of an infinite series of states, in which the +energy in the $n$th~state apart from an arbitrary additive constant is +determined by the expression +\[ +E_{n} = -\frac{Kh}{n^{2}}. +\Tag{(7)} +\] +The negative sign has been chosen because the energy of the atom +will be most simply characterized by the work~$W$ required to remove +the electron completely from the atom. If we now substitute $\dfrac{Kh}{n^{2}}$ +for~$W$ in formula~\Eq{(6)}, we obtain the following expression for the frequency +and the major axis in the $n$th~stationary state: +\[ +\omega_{n} = \frac{1}{n^{3}} \sqrt{\frac{2h^{3} K^{3}}{\pi^{2} e^{4} m}},\quad +2a_{n} = \frac{n^{2} e^{2}}{hK}. +\Tag{(8)} +\] +A comparison between the motions determined by these equations +and the distinctive states of a Planck resonator may be shown to +offer a theoretical determination of the constant~$K$. Instead of +doing this I shall show how the value of~$K$ can be found by a simple +comparison of the spectrum emitted with the motion in the stationary +states, a comparison which at the same time will lead us to the +principle of correspondence. + +We have assumed that each hydrogen line is the result of a +transition between two stationary states of the atom corresponding +to different values of~$n$. Equations~\Eq{(8)} show that the frequency of +revolution and the major axis of the orbit can be entirely different +in the two states, since, as the energy decreases, the major axis of +the orbit becomes smaller and the frequency of revolution increases. +\PageSep{26} +In general, therefore, it will be impossible to obtain a relation between +the frequency of revolution of the electrons and the frequency +of the radiation as in the ordinary theory of radiation. If, however, +we consider the ratio of the frequencies of revolution in two stationary +states corresponding to given values of $n'$~and~$n''$, we see that this +ratio approaches unity as $n'$~and $n''$ gradually increase, if at the +same time the difference $n' - n''$ remains unchanged. By considering +transitions corresponding to large values of $n'$~and~$n''$ we may +therefore hope to establish a certain connection with the ordinary +theory. For the frequency of the radiation emitted by a transition, +we get according to~\Eq{(5)} +\[ +\nu = \frac{K}{(n'')^{2}} - \frac{K}{(n')^{2}} + = (n' - n'') K\, \frac{n' + n''}{(n')^{2} (n'')^{2}}. +\Tag{(9)} +\] +If now the numbers $n'$~and $n''$ are large in proportion to their difference, +we see that by equations~\Eq{(8)} this expression may be written +approximately, +\[ +\nu \sim (n' - n'') \omega \sqrt{\frac{2\pi^{2} e^{4} m}{Kh^{3}}}, +\Tag{(10)} +\] +where $\omega$~represents the frequency of revolution in the one or the +other of the two stationary states. Since $n' - n''$ is a whole number, +we see that the first part of this expression, \ie\ $(n' - n'')\omega$, is the +same as the frequency of one of the harmonic components into +which the elliptical motion may be decomposed. This involves the +well-known result that for a system of particles having a periodic +motion of frequency~$\omega$, the displacement~$\xi$ of the particles in a given +direction in space may be represented as a function of the time by +a trigonometric series of the form +\[ +\xi = \sum C_{\tau} \cos 2\pi(\tau\omega t + c_{\tau}), +\Tag{(11)} +\] +where the summation is to be extended over all positive integral +values of~$\tau$. + +We see, therefore, that the frequency of the radiation emitted +by a transition between two stationary states, for which the numbers +$n'$~and $n''$ are large in proportion to their difference, will coincide +with the frequency of one of the components of the radiation, which +according to the ordinary ideas of radiation would be expected from +the motion of the atom in these states, provided the last factor on +\PageSep{27} +the right-hand side of equation~\Eq{(10)} is equal to~$1$. This condition, +which is identical to the condition +\[ +K = \frac{2\pi^{2} e^{4} m}{h^{3}}, +\Tag{(12)} +\] +is in fact fulfilled, if we give to~$K$ its value as found from measurements +on the hydrogen spectrum, and if for $e$,~$m$ and~$h$ we use the +values obtained directly from experiment. This agreement clearly +gives us a \emph{connection between the spectrum and the atomic model of +hydrogen}, which is as close as could reasonably be expected considering +the fundamental difference between the ideas of the quantum +theory and of the ordinary theory of radiation. + +\Section{The correspondence principle.} Let us now consider somewhat +more closely this relation between the spectra one would expect on +the basis of the quantum theory, and on the ordinary theory of +radiation. The frequencies of the spectral lines calculated according +to both methods agree completely in the region where the stationary +states deviate only little from one another. We must not forget, +however, that the mechanism of emission in both cases is different. +The different frequencies corresponding to the various harmonic +components of the motion are emitted simultaneously according to +the ordinary theory of radiation and with a relative intensity depending +directly upon the ratio of the amplitudes of these oscillations. +But according to the quantum theory the various spectral +lines are emitted by entirely distinct processes, consisting of transitions +from one stationary state to various adjacent states, so that +the radiation corresponding to the $\tau$th~``harmonic'' will be emitted +by a transition for which $n' - n'' = \tau$. The relative intensity +with which each particular line is emitted depends consequently +upon the relative probability of the occurrence of the different +transitions. + +This correspondence between the frequencies determined by the +two methods must have a deeper significance and we are led to +anticipate that it will also apply to the intensities. This is equivalent +to the statement that, when the quantum numbers are large, +the relative probability of a particular transition is connected in a +simple manner with the amplitude of the corresponding harmonic +component in the motion. +\PageSep{28} + +This peculiar relation suggests a \emph{general law for the occurrence +of transitions between stationary states}. Thus we shall assume that +even when the quantum numbers are small the possibility of +transition between two stationary states is connected with the +presence of a certain harmonic component in the motion of the +system. If the numbers $n'$~and $n''$ are not large in proportion to +their difference, the numerical value of the amplitudes of these +components in the two stationary states may be entirely different. +We must be prepared to find, therefore, that the exact connection +between the probability of a transition and the amplitude of the +corresponding harmonic component in the motion is in general +complicated like the connection between the frequency of the radiation +and that of the component. From this point of view, for +example, the green line~$H_{\beta}$ of the hydrogen spectrum which corresponds +to a transition from the fourth to the second stationary +state may be considered in a certain sense to be an ``octave'' of the +red line~$H_{\alpha}$, corresponding to a transition from the third to the +second state, even though the frequency of the first line is by no +means twice as great as that of the latter. In fact, the transition +giving rise to~$H_{\beta}$ may be regarded as due to the presence of a harmonic +oscillation in the motion of the atom, which is an octave +higher than the oscillation giving rise to the emission of~$H_{\alpha}$. + +Before considering other spectra, where numerous opportunities +will be found to use this point of view, I shall briefly mention an +interesting application to the Planck oscillator. If from \Eq{(1)}~and \Eq{(4)} +we calculate the frequency, which would correspond to a transition +between two particular states of such an oscillator, we find +\[ +\nu = (n' - n'')\omega, +\Tag{(13)} +\] +where $n'$~and $n''$ are the numbers characterizing the states. It was +an essential assumption in Planck's theory that the frequency of +the radiation emitted and absorbed by the oscillator is always equal +to~$\omega$. We see that this assumption is equivalent to the assertion +that transitions occur only between two successive stationary states +in sharp contrast to the hydrogen atom. According to our view, +however, this was exactly what might have been expected, for we +must assume that the essential difference between the oscillator +and the hydrogen atom is that the motion of the oscillator is simple +\PageSep{29} +harmonic. We can see that it is possible to develop a formal theory +of radiation, in which the spectrum of hydrogen and the simple +spectrum of a Planck oscillator appear completely analogous. This +theory can only be formulated by one and the same condition for +a system as simple as the oscillator. In general this condition +breaks up into two parts, one concerning the fixation of the stationary +states, and the other relating to the frequency of the radiation +emitted by a transition between these states. + +\Section{General spectral laws.} Although the series spectra of the +elements of higher atomic number have a more complicated structure +than the hydrogen spectrum, simple laws have been discovered +showing a remarkable analogy to the Balmer formula. Rydberg +and Ritz showed that the frequencies in the series spectra of many +elements can be expressed by a formula of the type +\[ +\nu = f_{k''}(n'') - f_{k'}(n'), +\Tag{(14)} +\] +where $n'$~and $n''$ are two whole numbers and $f_{k'}$~and $f_{k''}$ are two +functions belonging to a series of functions characteristic of the +element. These functions vary in a simple manner with~$n$ and in +particular converge to zero for increasing values of~$n$. The various +series of lines are obtained from this formula by allowing the first +term~$f_{k''}(n'')$ to remain constant, while a series of consecutive whole +numbers are substituted for~$n'$ in the second term~$f_{k'}(n')$. According +to the Ritz \emph{combination principle} the entire spectrum may then +be obtained by forming every possible combination of two values +among all the quantities~$f_{k}(n)$. + +The fact that the frequency of each line of the spectrum may be +written as the difference of two simple expressions depending upon +whole numbers suggests at once that the terms on the right-hand +side multiplied by~$h$ may be placed equal to the energy in the +various stationary states of the atom. The existence in the spectra +of the other elements of a number of separate functions of~$n$ compels +us to assume the presence not of one but of a number of series of +stationary states, the energy of the $n$th~state of the $k$th~series apart +from an arbitrary additive constant being given by +\[ +E_{k}(n) = -h f_{k}(n). +\Tag{(15)} +\] +This complicated character of the ensemble of stationary states of +atoms of higher atomic number is exactly what was to be expected +\PageSep{30} +from the relation between the spectra calculated on the quantum +theory, and the decomposition of the motions of the atoms into +harmonic oscillations. From this point of view we may regard the +simple character of the stationary states of the hydrogen atom as +intimately connected with the simple periodic character of this +atom. Where the neutral atom contains more than one electron, we +find much more complicated motions with correspondingly complicated +harmonic components. We must therefore expect a more +complicated ensemble of stationary states, if we are still to have a +corresponding relation between the motions in the atom and the +spectrum. In the course of the lecture we shall trace this correspondence +in detail, and we shall be led to a simple explanation of +the apparent capriciousness in the occurrence of lines predicted by +the combination principle. + +The following figure gives a survey of the stationary states of +the sodium atom deduced from the series terms. +\Figure[Diagram of the series spectrum of sodium.]{}{30} + +The stationary states are represented by black dots whose distance +from the vertical line a---a is proportional to the numerical value +of the energy in the states. The arrows in the figure indicate the +transitions giving those lines of the sodium spectrum which appear +under the usual conditions of excitation. The arrangement of the +states in horizontal rows corresponds to the ordinary arrangement +of the ``spectral terms'' in the spectroscopic tables. Thus, the states +in the first row~($S$) correspond to the variable term in the ``sharp +series,'' the lines of which are emitted by transitions from these +states to the first state in the second row. The states in the second +\PageSep{31} +row~($P$) correspond to the variable term in the ``principal series'' +which is emitted by transitions from these states to the first state +in the $S$~row. The $D$~states correspond to the variable term in the +``diffuse series,'' which like the sharp series is emitted by transitions +to the first state in the $P$~row, and finally the $B$~states correspond +to the variable term in the ``Bergmann'' series (fundamental series), +in which transitions take place to the first state in the $D$~row. The +manner in which the various rows are arranged with reference to +one another will be used to illustrate the more detailed theory +which will be discussed later. The apparent capriciousness of the +combination principle, which I mentioned, consists in the fact that +under the usual conditions of excitation not all the lines belonging +to possible combinations of the terms of the sodium spectrum appear, +but only those indicated in the figure by arrows. + +The general question of the fixation of the stationary states of +an atom containing several electrons presents difficulties of a profound +character which are perhaps still far from completely solved. +It is possible, however, to obtain an immediate insight into the +stationary states involved in the emission of the series spectra by +considering the empirical laws which have been discovered about +the spectral terms. According to the well-known law discovered by +Rydberg for the spectra of elements emitted under the usual conditions +of excitation the functions~$f_{k}(n)$ appearing in formula~\Eq{(14)} +can be written in the form +\[ +f_{k}(n) = \frac{K}{n^{2}} \phi_{k}(n), +\Tag{(16)} +\] +where $\phi_{k}(n)$~represents a function which converges to unity for +large values of~$n$. $K$~is the same constant which appears in formula~\Eq{(5)} +for the spectrum of hydrogen. This result must evidently be +explained by supposing the atom to be electrically neutral in these +states and one electron to be moving round the nucleus in an orbit +the dimensions of which are very large in proportion to the distance +of the other electrons from the nucleus. We see, indeed, that in +this case the electric force acting on the outer electron will to a +first approximation be the same as that acting upon the electron +in the hydrogen atom, and the approximation will be the better +the larger the orbit. +\PageSep{32} + +On account of the limited time I shall not discuss how this +explanation of the universal appearance of Rydberg's constant in +the arc spectra is convincingly supported by the investigation of +the ``spark spectra.'' These are emitted by the elements under the +influence of very strong electrical discharges, and come from ionized +not neutral atoms. It is important, however, that I should indicate +briefly how the fundamental ideas of the theory and the assumption +that in the states corresponding to the spectra one electron moves +in an orbit around the others, are both supported by investigations +on selective absorption and the excitation of spectral lines by +bombardment by electrons. + +\Section{Absorption and excitation of radiation.}\Pagelabel{32} Just as we have +assumed that each emission of radiation is due to a transition from +a stationary state of higher to one of lower energy, so also we must +assume absorption of radiation by the atom to be due to a transition +in the opposite direction. For an element to absorb light corresponding +to a given line in its series spectrum, it is therefore +necessary for the atom of this element to be in that one of the two +states connected with the line possessing the smaller energy value. +If we now consider an element whose atoms in the gaseous state +do not combine into molecules, it will be necessary to assume that +under ordinary conditions nearly all the atoms exist in that stationary +state in which the value of the energy is a minimum. This state +I shall call the \emph{normal state}. We must therefore expect that the +absorption spectrum of a monatomic gas will contain only those +lines of the series spectrum, whose emission corresponds to transitions +to the normal state. This expectation is completely confirmed +by the spectra of the alkali metals. The absorption spectrum of +sodium vapour, for example, exhibits lines corresponding only to +the principal series, which as mentioned in the description of the +figure corresponds with transitions to the state of minimum energy. +Further confirmation of this view of the process of absorption is +given by experiments on \emph{resonance radiation}. Wood first showed +that sodium vapour subjected to light corresponding to the first +line of the principal series---the familiar yellow line---acquires the +ability of again emitting a radiation consisting only of the light of +this line. We can explain this by supposing the sodium atom to +\PageSep{33} +have been transferred from the normal state to the first state in +the second row. The fact that the resonance radiation does not +exhibit the same degree of polarization as the incident light is +in perfect agreement with our assumption that the radiation from +the excited vapour is not a resonance phenomenon in the sense of +the ordinary theory of radiation, but on the contrary depends on a +process which is not directly connected with the incident radiation. + +The phenomenon of the resonance radiation of the yellow sodium +line is, however, not quite so simple as I have indicated, since, as +you know, this line is really a doublet. This means that the variable +terms of the principal series are not simple but are represented by +two values slightly different from one another. According to our +picture of the origin of the sodium spectrum this means that the +$P$~states in the second row in the figure---as opposed to the $S$~states +in the first row---are not simple, but that for each place in this row +there are two stationary states. The energy values differ so little +from one another that it is impossible to represent them in the +figure as separate dots. The emission (and absorption) of the two +components of the yellow line are, therefore, connected with two +different processes. This was beautifully shown by some later researches +of Wood and Dunoyer. They found that if sodium vapour +is subjected to radiation from only one of the two components of +the yellow line, the resonance radiation, at least at low pressures, +consists only of this component. These experiments were later +continued by Strutt, and were extended to the case where the +exciting line corresponded to the second line in the principal series. +Strutt found that the resonance radiation consisted apparently only +to a small extent of light of the same frequency as the incident +light, while the greater part consisted of the familiar yellow line. +This result must appear very astonishing on the ordinary ideas of +resonance, since, as Strutt pointed out, no rational connection exists +between the frequencies of the first and second lines of the principal +series. It is however easily explained from our point of view. From +the figure it can be seen that when an atom has been transferred +into the second state in the second row, in addition to the direct +return to the normal state, there are still two other transitions +which may give rise to radiation, namely the transitions to the +second state in the first row and to the first state in the third row. +\PageSep{34} +The experiments seem to indicate that the second of these three +transitions is most probable, and I shall show later that there is +some theoretical justification for this conclusion. By this transition, +which results in the emission of an infra-red line which could not +be observed with the experimental arrangement, the atom is taken +to the second state of the first row, and from this state only +one transition is possible, which again gives an infra-red line. This +transition takes the atom to the first state in the second row, and +the subsequent transition to the normal state then gives rise to the +yellow line. Strutt discovered another equally surprising result, +that this yellow resonance radiation seemed to consist of both +components of the first line of the principal series, even when the +incident light consisted of only one component of the second line +of the principal series. This is in beautiful agreement with our +picture of the phenomenon. We must remember that the states in +the first row are simple, so when the atom has arrived in one of +these it has lost every possibility of later giving any indication +from which of the two states in the second row it originally came. + +Sodium vapour, in addition to the absorption corresponding to +the lines of the principal series, exhibits a \emph{selective absorption in a +continuous spectral region} beginning at the limit of this series and +extending into the ultra\Add{-}violet. This confirms in a striking manner +our assumption that the absorption of the lines of the principal +series of sodium results in final states of the atom in which one of +the electrons revolves in larger and larger orbits. For we must +assume that this continuous absorption corresponds to transitions +from the normal state to states in which the electron is in a position +to remove itself infinitely far from the nucleus. This phenomenon +exhibits a complete analogy with the \emph{photoelectric effect} from an +illuminated metal plate in which, by using light of a suitable +frequency, electrons of any velocity can be obtained. The frequency, +however, must always lie above a certain limit connected according +to Einstein's theory in a simple manner with the energy necessary +to bring an electron out of the metal. + +This view of the origin of the emission and absorption spectra +has been confirmed in a very interesting manner by experiments +on the \emph{excitation of spectral lines and production of ionization by +electron bombardment}. The chief advance in this field is due to the +\PageSep{35} +well-known experiments of Franck and Hertz. These investigators +obtained their first important results from their experiments on +mercury vapour, whose properties particularly facilitate such experiments. +On account of the great importance of the results, these +experiments have been extended to most gases and metals that can +be obtained in a gaseous state. With the aid of the figure I shall +briefly illustrate the results for the case of sodium vapour. It was +found that the electrons upon colliding with the atoms were thrown +back with undiminished velocity when their energy was less than +that required to transfer the atom from the normal state to the +next succeeding stationary state of higher energy value. In the +case of sodium vapour this means from the first state in the first +row to the first state in the second row. As soon, however, as the +energy of the electron reaches this critical value, a new type of +collision takes place, in which the electron loses all its kinetic +energy, while at the same time the vapour is excited and emits a +radiation corresponding to the yellow line. This is what would be +expected, if by the collision the atom was transferred from the +normal state to the first one in the second row. For some time it +was uncertain to what extent this explanation was correct, since +in the experiments on mercury vapour it was found that, together +with the occurrence of non-elastic impacts, ions were always formed +in the vapour. From our figure, however, we would expect ions +to be produced only when the kinetic energy of the electrons is +sufficiently great to bring the atom out of the normal state to the +common limit of the states. Later experiments, especially by Davis +and Goucher, have settled this point. It has been shown that ions +can only be directly produced by collisions when the kinetic energy +of the electrons corresponds to the limit of the series, and that the +ionization found at first was an indirect effect arising from the +photoelectric effect produced at the metal walls of the apparatus +by the radiation arising from the return of the mercury atoms to +the normal state. These experiments provide a direct and independent +proof of the reality of the distinctive stationary states, +whose existence we were led to infer from the series spectra. At +the same time we get a striking impression of the insufficiency of +the ordinary electrodynamical and mechanical conceptions for the +description of atomic processes, not only as regards the emission +\PageSep{36} +of radiation but also in such phenomena as the collision of free +electrons with atoms. + + +\Chapter{III.}{Development of the Quantum Theory +of Spectra} + +We see that it is possible by making use of a few simple ideas +to obtain a certain insight into the origin of the series spectra. +But when we attempt to penetrate more deeply, difficulties arise. +In fact, for systems which are not simply periodic it is not possible +to obtain sufficient information about the motions of these systems +in the stationary states from the numerical values of the energy +alone; more determining factors are required for the fixation of +the motion. We meet the same difficulties when we try to explain +in detail the characteristic effect of external forces upon the spectrum +of hydrogen. A foundation for further advances in this field has +been made in recent years through a development of the quantum +theory, which allows a fixation of the stationary states not only in +the case of simple periodic systems, but also for certain classes of +non-periodic systems. These are the \emph{conditionally periodic systems} +whose equations of motion can be solved by a ``separation of the +variables.'' If generalized coordinates are used the description of +the motion of these systems can be reduced to the consideration +of a number of generalized ``components of motion.'' Each of these +corresponds to the change of only one of the coordinates and may +therefore in a certain sense be regarded as ``independent.'' The +method for the fixation of the stationary states consists in fixing +the motion of each of these components by a condition, which can +be considered as a direct generalization of condition~\Eq{(1)} for a +Planck oscillator, so that the stationary states are in general +characterized by as many whole numbers as the number of the +degrees of freedom which the system possesses. A considerable +number of physicists have taken part in this development of the +quantum theory, including Planck himself. I also wish to mention +the important contribution made by Ehrenfest to this subject on +the limitations of the applicability of the laws of mechanics to +atomic processes. The decisive advance in the application of the +quantum theory to spectra, however, is due to Sommerfeld and his +followers. However, I shall not further discuss the systematic form +\PageSep{37} +in which these authors have presented their results. In a paper which +appeared some time ago in the Transactions of the Copenhagen +Academy, I have shown that the spectra, calculated with the aid +of this method for the fixation of the stationary states, exhibit a +correspondence with the spectra which should correspond to the +motion of the system similar to that which we have already considered +in the case of hydrogen. With the aid of this general +correspondence I shall try in the remainder of this lecture to +show how it is possible to present the theory of series spectra +and the effects produced by external fields of force upon these +spectra in a form which may be considered as the natural generalization +of the foregoing considerations. This form appears to me +to be especially suited for future work in the theory of spectra, +since it allows of an immediate insight into problems for which +the methods mentioned above fail on account of the complexity of +the motions in the atom. + +\Section{Effect of external forces on the hydrogen spectrum.} We +shall now proceed to investigate the effect of small perturbing +forces upon the spectrum of the simple system consisting of a single +electron revolving about a nucleus. For the sake of simplicity we +shall for the moment disregard the variation of the mass of the +electron with its velocity. The consideration of the small changes +in the motion due to this variation has been of great importance +in the development of Sommerfeld's theory which originated in the +explanation of the \emph{fine structure of the hydrogen lines}. This fine +structure is due to the fact, that taking into account the variation +of mass with velocity the orbit of the electron deviates a little +from a simple ellipse and is no longer exactly periodic. This deviation +from a Keplerian motion is, however, very small compared +with the perturbations due to the presence of external forces, such +as occur in experiments on the Zeeman and Stark effects. In atoms +of higher atomic number it is also negligible compared with the +disturbing effect of the inner electrons on the motion of the outer +electron. The neglect of the change in mass will therefore have no +important influence upon the explanation of the Zeeman and Stark +effects, or upon the explanation of the difference between the +hydrogen spectrum and the spectra of other elements. +\PageSep{38} + +We shall therefore as before consider the motion of the unperturbed +hydrogen atom as simply periodic and inquire in the +first place about the stationary states corresponding to this motion. +The energy in these states will then be determined by expression~\Eq{(7)} +which was derived from the spectrum of hydrogen. The energy of +the system being given, the major axis of the elliptical orbit of the +electron and its frequency of revolution are also determined. Substituting +in formulae \Eq{(7)} and~\Eq{(8)} the expression for~$K$ given in~\Eq{(12)}, +we obtain for the energy, major axis and frequency of revolution +in the $n$th~state of the unperturbed atom the expressions +\[ +\BreakMath{% +E_{n} = -W_{n} = -\frac{1}{n^{2}}\, \frac{2\pi^{2} e^{4} m}{h^{2}},\quad +2a_{n} = n^{2}\, \frac{h^{2}}{2\pi^{2} e^{2} m},\quad +\omega_{n} = \frac{1}{n^{3}}\, \frac{4\pi^{2} e^{4} m}{h^{3}}. +}{% +\begin{gathered} +E_{n} = -W_{n} = -\frac{1}{n^{2}}\, \frac{2\pi^{2} e^{4} m}{h^{2}}, \\ +2a_{n} = n^{2}\, \frac{h^{2}}{2\pi^{2} e^{2} m},\qquad +\omega_{n} = \frac{1}{n^{3}}\, \frac{4\pi^{2} e^{4} m}{h^{3}}. +\end{gathered} +} +\Tag{(17)} +\] + +We must further assume that in the stationary states of the +unperturbed system the form of the orbit is so far undetermined +that the \Chg{excentricity}{eccentricity} can vary continuously. This is not only immediately +indicated by the principle of correspondence,---since the +frequency of revolution is determined only by the energy and not +by the \Chg{excentricity}{eccentricity},---but also by the fact that the presence of any +small external forces will in general, in the course of time, produce +a finite change in the position as well as in the \Chg{excentricity}{eccentricity} of the +periodic orbit, while in the major axis it can produce only small +changes proportional to the intensity of the perturbing forces. + +In order to fix the stationary states of systems in the presence +of a given conservative external field of force, we shall have to +investigate, on the basis of the principle of correspondence, how +these forces affect the decomposition of the motion into harmonic +oscillations. Owing to the external forces the form and position of +the orbit will vary continuously. In the general case these changes +will be so complicated that it will not be possible to decompose the +perturbed motion into discrete harmonic oscillations. In such a +case we must expect that the perturbed system will not possess +any sharply separated stationary states. Although each emission +of radiation must be assumed to be monochromatic and to proceed +according to the general frequency condition we shall therefore +expect the final effect to be a broadening of the sharp spectral lines +of the unperturbed system. In certain cases, however, the perturbations +\PageSep{39} +will be of such a regular character that the perturbed system +can be decomposed into harmonic oscillations, although the ensemble +of these oscillations will naturally be of a more complicated kind +than in the unperturbed system. This happens, for example, when +the variations of the orbit with respect to time are periodic. In +this case harmonic oscillations will appear in the motion of the +system the frequencies of which are equal to whole multiples of the +period of the orbital perturbations, and in the spectrum to be +expected on the basis of the ordinary theory of radiation we would +expect components corresponding to these frequencies. According +to the principle of correspondence we are therefore immediately +led to the conclusion, that to each stationary state in the unperturbed +system there corresponds a number of stationary states in +the perturbed system in such a manner, that for a transition +between two of these states a radiation is emitted, whose frequency +stands in the same relationship to the periodic course of the +variations in the orbit, as the spectrum of a simple periodic system +does to its motion in the stationary states. + +\Section{The Stark effect.} An instructive example of the appearance of +periodic perturbations is obtained when hydrogen is subjected to +the effect of a homogeneous electric field. The \Chg{excentricity}{eccentricity} and +the position of the orbit vary continuously under the influence of +the field. During these changes, however, it is found that the +centre of the orbit remains in a plane perpendicular to the direction +of the electric force and that its motion in this plane is +simply periodic. When the centre has returned to its starting +point, the orbit will resume its original \Chg{excentricity}{eccentricity} and position, +and from this moment the entire cycle of orbits will be repeated. +In this case the determination of the energy of the stationary +states of the disturbed system is extremely simple, since it is found +that the period of the disturbance does not depend upon the +original configuration of the orbits nor therefore upon the position +of the plane in which the centre of the orbit moves, but only upon +the major axis and the frequency of revolution. From a simple +calculation it is found that the period a is given by the following +formula +\[ +\sigma = \frac{3eF}{8\pi^{2} ma\omega}, +\Tag{(18)} +\] +\PageSep{40} +where $F$~is the intensity of the external electric field. From +analogy with the fixation of the distinctive energy values of a +Planck oscillator we must therefore expect that the energy difference +between two different states, corresponding to the same stationary +state of the unperturbed system, will simply be equal to a whole +multiple of the product of $h$~by the period~$\sigma$ of the perturbations. +We are therefore immediately led to the following expression for +the energy of the stationary states of the perturbed system, +\[ +E = E_{n} + kh\sigma, +\Tag{(19)} +\] +where $E_{n}$~depends only upon the number~$n$ characterizing the +stationary state of the unperturbed system, while $k$~is a new whole +number which in this case may be either positive or negative. As +we shall see below, consideration of the relation between the energy +and the motion of the system shows that $k$~must be numerically +less than~$n$, if, as before, we place the quantity~$E_{n}$ equal to the +energy~$-W_{n}$ of the $n$th~stationary state of the undisturbed atom. +Substituting the values of $W_{n}$,~$\omega_{n}$ and~$a_{n}$ given by~\Eq{(17)} in formula~\Eq{(19)} +we get +\[ +E = -\frac{1}{n^{2}}\, \frac{2\pi^{2} e^{4} m}{h^{2}} + nk\, \frac{3h^{2} F}{8\pi^{2} em}. +\Tag{(20)} +\] +To find the effect of an electric field upon the lines of the hydrogen +spectrum, we use the frequency condition~\Eq{(4)} and obtain for the +frequency~$\nu$ of the radiation emitted by a transition between two +stationary states defined by the numbers $n'$,~$k'$ and $n''$,~$k''$ +\[ +\nu = \frac{2\pi^{2} e^{4} m}{h^{3}} \left(\frac{1}{(n'')^{2}} - \frac{1}{(n'')^{2}}\right) + + \frac{3h · F}{8\pi^{2} em} (n'k' - n''k''). +\Tag{(21)} +\] + +It is well known that this formula provides a complete explanation +of the Stark effect of the hydrogen lines. It corresponds +exactly with the one obtained by a different method by Epstein +and Schwarzschild. They used the fact that the hydrogen atom in +a homogeneous electric field is a conditionally periodic system +permitting a separation of variables by the use of parabolic coordinates. +The stationary states were fixed by applying quantum +conditions to each of these variables. + +We shall now consider more closely the correspondence between +the changes in the spectrum of hydrogen due to the presence of +\PageSep{41} +an electric field and the decomposition of the perturbed motion +of the atom into its harmonic components. Instead of the simple +decomposition into harmonic components corresponding to a simple +Kepler motion, the displacement~$\xi$ of the electron in a given +direction in space can be expressed in the present case by the +formula +\[ +\xi = \sum C_{\tau,\kappa} \cos 2\pi \bigl\{t(\tau\omega + \kappa\sigma) + c_{\tau,\kappa}\bigr\}, +\Tag{(22)} +\] +where $\omega$~is the average frequency of revolution in the perturbed +orbit and $\sigma$~is the period of the orbital perturbations, while $C_{\tau,\kappa}$~and +$c_{\tau,\kappa}$ are constants. The summation is to be extended over all integral +values for $\tau$~and~$\kappa$. + +If we now consider a transition between two stationary states +characterized by certain numbers $n'$,~$k'$ and $n''$,~$k''$, we find that in +the region where these numbers are large compared with their +differences $n' - n''$ and $k' - k''$, the frequency of the spectral line +which is emitted will be given approximately by the formula +\[ +\nu \sim (n' - n'')\omega + (k' - k'')\sigma. +\Tag{(23)} +\] +We see, therefore, that we have obtained a relation between the +spectrum and the motion of precisely the same character as in the +simple case of the unperturbed hydrogen atom. We have here a +similar correspondence between the harmonic component in the +motion, corresponding to definite values for $\tau$~and $\kappa$ in formula~\Eq{(22)}, +and the transition between two stationary states for which $n' - n'' = \tau$ +and $k' - k'' = \kappa$. + +A number of interesting results can be obtained from this +correspondence by considering the motion in more detail. Each +harmonic component in expression~\Eq{(22)} for which $\tau + \kappa$ is an even +number corresponds to a linear oscillation parallel to the direction +of the electric field, while each component for which $\tau + \kappa$ is odd +corresponds to an elliptical oscillation perpendicular to this direction. +The correspondence principle suggests at once that these +facts are connected with the \emph{characteristic polarization} observed in +the Stark effect. We would anticipate that a transition for which +$(n' - n'') + (k' - k'')$ is even would give rise to a component with an +electric vector parallel to the field, while a transition for which +$(n' - n'') + (k' - k'')$ is odd would correspond to a component with an +\PageSep{42} +electric vector perpendicular to the field. These results have been +fully confirmed by experiment and correspond to the empirical rule +of polarization, which Epstein proposed in his first paper on the +Stark effect. + +The applications of the correspondence principle that have so +far been described have been purely qualitative in character. It is +possible however to obtain a quantitative estimate of the relative +intensity of the various components of the Stark effect of hydrogen, +by correlating the numerical values of the coefficients~$C_{\tau,\kappa}$ in formula~\Eq{(22)} +with the probability of the corresponding transitions between +the stationary states. This problem has been treated in detail by +Kramers in a recently published dissertation. In this he gives a +thorough discussion of the application of the correspondence principle +to the question of the intensity of spectral lines. + +\Section{The Zeeman effect.} The problem of the effect of a homogeneous +magnetic field upon the hydrogen lines may be treated in an +entirely analogous manner. The effect on the motion of the hydrogen +atom consists simply of the superposition of a uniform rotation +upon the motion of the electron in the unperturbed atom. +The axis of rotation is parallel with the direction of the magnetic +force, while the frequency of revolution is given by the formula +\[ +\sigma = \frac{eH}{4\pi mc}, +\Tag{(24)} +\] +where $H$~is the intensity of the field and $c$~the velocity of light. + +Again we have a case where the perturbations are simply +periodic and where the period of the perturbations is independent +of the form and position of the orbit, and in the present case, even +of the major axis. Similar considerations apply therefore as in the +case of the Stark effect, and we must expect that the energy in the +stationary states will again be given by formula~\Eq{(19)}, if we substitute +for~$\sigma$ the value given in expression~\Eq{(24)}. This result is +also in complete agreement with that obtained by Sommerfeld and +Debye. The method they used involved the solution of the equations +of motion by the method of the separation of the variables. The +appropriate coordinates are polar ones about an axis parallel to +the field. + +If we try, however, to calculate directly the effect of the field by +\PageSep{43} +means of the frequency condition~\Eq{(4)}, we immediately meet with +an apparent disagreement which for some time was regarded as a +grave difficulty for the theory. As both Sommerfeld and Debye +have pointed out, lines are not observed corresponding to every +transition between the stationary states included in the formula. +We overcome this difficulty, however, as soon as we apply the +principle of correspondence. If we consider the harmonic components +of the motion we obtain a simple explanation both of the +non-occurrence of certain transitions and of the observed polarization. +In the magnetic field each elliptic harmonic component having +the frequency~$\tau\omega$ splits up into three harmonic components owing +to the uniform rotation of the orbit. Of these one is rectilinear +with frequency~$\tau\omega$ oscillating parallel to the magnetic field, and +two are circular with frequencies $\tau\omega + \sigma$ and $\tau\omega - \sigma$ oscillating in +opposite directions in a plane perpendicular to the direction of the +field. Consequently the motion represented by formula~\Eq{(22)} contains +no components for which $\kappa$~is numerically greater than~$1$, in contrast +to the Stark effect, where components corresponding to all values +of~$\kappa$ are present. Now formula~\Eq{(23)} again applies for large values +of $n$~and~$k$, and shows the asymptotic agreement between the +frequency of the radiation and the frequency of a harmonic component +in the motion. We arrive, therefore, at the conclusion that +transitions for which $k$~changes by more than unity \Chg{can not}{cannot} occur. +The argument is similar to that by which transitions between two +distinctive states of a Planck oscillator for which the values of~$n$ +in~\Eq{(1)} differ by more than unity are excluded. We must further +conclude that the various possible transitions consist of two types. +For the one type corresponding to the rectilinear component, $k$~remains +unchanged, and in the emitted radiation which possesses +the same frequency~$\nu_{0}$ as the original hydrogen line, the electric +vector will oscillate parallel with the field. For the second type, +corresponding to the circular components, $k$~will increase or decrease +by unity, and the radiation viewed in the direction of the field will +be circularly polarized and have frequencies $\nu_{0} + \sigma$ and $\nu_{0} - \sigma$ respectively. +These results agree with those of the familiar Lorentz +theory. The similarity in the two theories is remarkable, when we +recall the fundamental difference between the ideas of the quantum +theory and the ordinary theories of radiation. +\PageSep{44} + +\Section{Central perturbations.} An illustration based on similar considerations +which will throw light upon the spectra of other elements +consists in finding the effect of a small perturbing field of +force radially symmetrical with respect to the nucleus. In this case +neither the form of the orbit nor the position of its plane will +change with time, and the perturbing effect of the field will simply +consist of a uniform rotation of the major axis of the orbit. The +perturbations are periodic, so that we may assume that to each +energy value of a stationary state of the unperturbed system there +belongs a series of discrete energy values of the perturbed system, +characterized by different values of a whole number~$k$. The frequency~$\sigma$ +of the perturbations is equal to the frequency of rotation +of the major axis. For a given law of force for the perturbing +field we find that $\sigma$~depends both on the major axis and on the +\Chg{excentricity}{eccentricity}. The change in the energy of the stationary states, +therefore, will not be given by an expression as simple as the +second term in formula~\Eq{(19)}, but will be a function of~$k$, which is +different for different fields. It is possible, however, to characterize +by one and the same condition the motion in the stationary states +of a hydrogen atom which is perturbed by any central field. In +order to show this we must consider more closely the fixation of +the motion of a perturbed hydrogen atom. + +In the stationary states of the unperturbed hydrogen atom +only the major axis of the orbit is to be regarded as fixed, +while the \Chg{excentricity}{eccentricity} may assume any value. Since the change +in the energy of the atom due to the external field of force depends +upon the form and position of its orbit, the fixation of the +energy of the atom in the presence of such a field naturally +involves a closer determination of the orbit of the perturbed +system. + +Consider, for the sake of illustration, the change in the hydrogen +spectrum due to the presence of homogeneous electric and magnetic +fields which was described by equation~\Eq{(19)}. It is found that +this energy condition can be given a simple geometrical interpretation. +In the case of an electric field the distance from the +nucleus to the plane in which the centre of the orbit moves determines +the change in the energy of the system due to the presence +of the field. In the stationary states this distance is simply equal +\PageSep{45} +to $\dfrac{k}{n}$~times half the major axis of the orbit. In the case of a magnetic +field it is found that the quantity which determines the change +of energy of the system is the area of the projection of the orbit +upon a plane perpendicular to the magnetic force. In the various +stationary states this area is equal to $\dfrac{k}{n}$~times the area of a circle +whose radius is equal to half the major axis of the orbit. In the +case of a perturbing central force the correspondence between +the spectrum and the motion which is required by the quantum +theory leads now to the simple condition that in the stationary +states of the perturbed system the minor axis of the rotating orbit +is simply equal to $\dfrac{k}{n}$~times the major axis. This condition was first +derived by Sommerfeld from his general theory for the determination +of the stationary states of a central motion. It is easily shown +that this fixation of the value of the minor axis is equivalent to +the statement that the parameter~$2p$ of the elliptical orbit is given +by an expression of exactly the same form as that which gives the +major axis~$2a$ in the unperturbed atom. The only difference from +the expression for~$2a_{n}$ in~\Eq{(17)} is that $n$~is replaced by~$k$, so that +the value of the parameter in the stationary states of the perturbed +atom is given by +\[ +2p_{k} = k^{2}\, \frac{h^{2}}{2\pi^{2} e^{2} m}. +\Tag{(25)} +\] +The frequency of the radiation emitted by a transition between +two stationary states determined in this way for which $n'$~and~$n''$ are +large in proportion to their difference is given by an expression +which is the same as that in equation~\Eq{(23)}, if in this case $\omega$~is the +frequency of revolution of the electron in the slowly rotating orbit +and $\sigma$~represents the frequency of rotation of the major axis. + +Before proceeding further, it might be of interest to note that +this fixation of the stationary states of the hydrogen atom perturbed +by external electric and magnetic forces does not coincide in certain +respects with the theories of Sommerfeld, Epstein and Debye. +According to the theory of conditionally periodic systems the stationary +states for a system of three degrees of freedom will in general +be determined by three conditions, and therefore in these theories +\PageSep{46} +each state is characterized by three whole numbers. This would +mean that the stationary states of the perturbed hydrogen atom +corresponding to a certain stationary state of the unperturbed +hydrogen atom, fixed by one condition, should be subject to two +further conditions and should therefore be characterized by two +new whole numbers in addition to the number~$n$. But the perturbations +of the Keplerian motion are simply periodic and the +energy of the perturbed atom will therefore be fixed completely +by one additional condition. The introduction of a second condition +will add nothing further to the explanation of the phenomenon, +since with the appearance of new perturbing forces, even if +these are too small noticeably to affect the observed Zeeman and +Stark effects, the forms of motion characterized by such a condition +may be entirely changed. This is completely analogous to the +fact that the hydrogen spectrum as it is usually observed is not +noticeably affected by small forces, even when they are large enough +to produce a great change in the form and position of the orbit of +the electron. + +\Section{Relativity effect on hydrogen lines.} Before leaving the hydrogen +spectrum I shall consider briefly the effect of the variation of +the mass of the electron with its velocity. In the preceding sections +I have described how external fields of force split up the hydrogen +lines into several components, but it should be noticed that these +results are only accurate when the perturbations are large in comparison +with the small deviations from a pure Keplerian motion +due to the variation of the mass of the electron with its velocity. +When the variation of the mass is taken into account the motion +of the unperturbed atom will not be exactly periodic. Instead we +obtain a motion of precisely the same kind as that occurring in the +hydrogen atom perturbed by a small central field. According to +the correspondence principle an intimate connection is to be expected +between the frequency of revolution of the major axis of the +orbit and the difference of the frequencies of the fine structure +components, and the stationary states will be those orbits whose +parameters are given by expression~\Eq{(25)}. If we now consider the +effect of external forces upon the fine structure components of the +hydrogen lines it is necessary to keep in mind that this fixation of +\PageSep{47} +the stationary states only applies to the unperturbed hydrogen +atom, and that, as mentioned, the orbits in these states are in +general already strongly influenced by the presence of external +forces, which are small compared with those with which we are +concerned in experiments on the Stark and Zeeman effects. In +general the presence of such forces will lead to a great complexity +of perturbations, and the atom will no longer possess a group of +sharply defined stationary states. The fine structure components +of a given hydrogen line will therefore become diffuse and merged +together. There are, however, several important cases where this +does not happen on account of the simple character of the perturbations. +The simplest example is a hydrogen atom perturbed +by a central force acting from the nucleus. In this case it is evident +that the motion of the system will retain its centrally symmetrical +character, and that the perturbed motion will differ from the unperturbed +motion only in that the frequency of rotation of the major +axis will be different for different values of this axis and of the +parameter. This point is of importance in the theory of the +spectra of elements of higher atomic number, since, as we shall see, +the effect of the forces originating from the inner electrons may +to a first approximation be compared with that of a perturbing +central field. We \Chg{can not}{cannot} therefore expect these spectra to exhibit +a separate effect due to the variation of the mass of the electron +of the same kind as that found in the case of the hydrogen lines. +This variation will not give rise to a splitting up into separate +components but only to small displacements in the position of the +various lines. + +We obtain still another simple example in which the hydrogen +atom possesses sharp stationary states, although the change of mass +of the electron is considered, if we take an atom subject to a homogeneous +magnetic field. The effect of such a field will consist in +the superposition of a rotation of the entire system about an axis +through the nucleus and parallel with the magnetic force. It follows +immediately from this result according to the principle of correspondence +that each fine structure component must be expected +to split up into a normal Zeeman effect (Lorentz triplet). The +problem may also be solved by means of the theory of conditionally +periodic systems, since the equations of motion in the presence +\PageSep{48} +of a magnetic field, even when the change in the mass is considered, +will allow of a separation of the variables using polar +coordinates in space. This has been pointed out by Sommerfeld +and Debye. + +A more complicated case arises when the atom is exposed to a +homogeneous electric field which is not so strong that the effect +due to the change in the mass may be neglected. In this case there +is no system of coordinates by which the equations of motion can +be solved by separation of the variables, and the problem, therefore, +\Chg{can not}{cannot} be treated by the theory of the stationary states of conditionally +periodic systems. A closer investigation of the perturbations, +however, shows them to be of such a character that the motion +of the electrons may be decomposed into a number of separate harmonic +components. These fall into two groups for which the direction +of oscillation is either parallel with or perpendicular to the +field. According to the principle of correspondence, therefore, we +must expect that also in this case in the presence of the field each +hydrogen line will consist of a number of sharp, polarized components. +In fact by means of the principles I have described, it is +possible to give a unique fixation of the stationary states. The +problem of the effect of a homogeneous electric field upon the fine +structure components of the hydrogen lines has been treated in +detail from this point of view by Kramers in a paper which will +soon be published. In this paper it will be shown how it appears +possible to predict in detail the manner in which the fine structure +of the hydrogen lines gradually changes into the ordinary Stark +effect as the electric intensity increases. + +\Section{Theory of series spectra.} Let us now turn our attention once +more to the problem of the series spectra of elements of higher +atomic number. The general appearance of the Rydberg constant +in these spectra is to be explained by assuming that the atom is +neutral and that one electron revolves in an orbit the dimensions +of which are large in comparison with the distance of the inner electrons +from the nucleus. In a certain sense, therefore, the motion of +the outer electron may be compared with the motion of the electron +of the hydrogen atom perturbed by external forces, and the appearance +of the various series in the spectra of the other elements is +\PageSep{49} +from this point of view to be regarded as analogous to the splitting +up of the hydrogen lines into components on account of such forces. + +In his theory of the structure of series spectra of the type exhibited +by the alkali metals, Sommerfeld has made the assumption +that the orbit of the outer electron to a first approximation possesses +the same character as that produced by a simple perturbing +central field whose intensity diminishes rapidly with increasing +distance from the nucleus. He fixed the motion of the external +electron by means of his general theory for the fixation of the +stationary states of a central motion. The application of this +method depends on the possibility of separating the variables in +the equations of motion. In this manner Sommerfeld was able to +calculate a number of energy values which can be arranged in rows +just like the empirical spectral terms shown in the diagram of the +sodium spectrum (\PageRef[p.]{30}). The states grouped together by Sommerfeld +in the separate rows are exactly those which were characterized +by one and the same value of~$k$ in our investigation of the +hydrogen atom perturbed by a central force. The states in the +first row of the figure (row~$S$) correspond to the value $k = 1$, those +of the second row~($P$) correspond to $k = 2$, etc. The states corresponding +to one and the same value of~$n$ are connected by dotted +lines which are continued so that their vertical asymptotes correspond +to the energy value of the stationary states of the hydrogen +atom. The fact that for a constant~$n$ and increasing values of~$k$ +the energy values approach the corresponding values for the unperturbed +hydrogen atom is immediately evident from the theory +since the outer electron, for large values of the parameter of its +orbit, remains at a great distance from the inner system during the +whole revolution. The orbit will become almost elliptical and the +period of rotation of the major axis will be very large. It can be +seen, therefore, that the effect of the inner system on the energy +necessary to remove this electron from the atom must become less +for increasing values of~$k$. + +These beautiful results suggest the possibility of finding laws of +force for the perturbing central field which would account for the +spectra observed. Although Sommerfeld in this way has in fact +succeeded in deriving formulae for the spectral terms which vary +with~$n$ for a constant~$k$ in agreement with Rydberg's formulae, it +\PageSep{50} +has not been possible to explain the simultaneous variation with +both $k$~and~$n$ in any actual case. This is not surprising, since it is +to be anticipated that the effect of the inner electrons on the spectrum +could not be accounted for in such a simple manner. Further +consideration shows that it is necessary to consider not only the +forces which originate from the inner electrons but also to consider +the effect of the presence of the outer electron upon the motion of +the inner electrons. + +Before considering the series spectra of elements of low atomic +number I shall point out how the occurrence or non-occurrence of +certain transitions can be shown by the correspondence principle +to furnish convincing evidence in favour of Sommerfeld's assumption +about the orbit of the outer electron. For this purpose we +must describe the motion of the outer electron in terms of its harmonic +components. This is easily performed if we assume that the +presence of the inner electrons simply produces a uniform rotation +of the orbit of the outer electron in its plane. On account of this +rotation, the frequency of which we will denote by~$\sigma$, two circular +rotations with the periods $\tau\omega + \sigma$ and $\tau\omega - \sigma$ will appear in the +motion of the perturbed electron, instead of each of the harmonic +elliptical components with a period $\tau\omega$ in the unperturbed motion. +The decomposition of the perturbed motion into harmonic components +consequently will again be represented by a formula of the +type~\Eq{(22)}, in which only such terms appear for which $\kappa$~is equal +to $+1$ or~$-1$. Since the frequency of the emitted radiation in the +regions where $n$~and $k$ are large is again given by the asymptotic +formula~\Eq{(23)}, we at once deduce from the correspondence principle +that the only transitions which can take place are those for which +the values of~$k$ differ by unity. A glance at the figure for the sodium +spectrum shows that this agrees exactly with the experimental +results. This fact is all the more remarkable, since in Sommerfeld's +theory the arrangement of the energy values of the stationary +states in rows has no special relation to the possibility of transition +between these states. + +\Section{Correspondence principle and conservation of angular momentum.} +Besides these results the correspondence principle suggests +that the radiation emitted by the perturbed atom must +\PageSep{51} +exhibit circular polarization. On account of the indeterminateness +of the plane of the orbit, however, this polarization \Chg{can not}{cannot} be +directly observed. The assumption of such a polarization is a matter +of particular interest for the theory of radiation emission. On +account of the general correspondence between the spectrum of +an atom and the decomposition of its motion into harmonic +components, we are led to compare the radiation emitted during +the transition between two stationary states with the radiation +which would be emitted by a harmonically oscillating +electron on the basis of the classical electrodynamics. In particular +the radiation emitted according to the classical theory +by an electron revolving in a circular orbit possesses an angular +momentum and the energy~$\Delta E$ and the angular momentum~$\Delta P$ of +the radiation emitted during a certain time are connected by the +relation +\[ +\Delta E = 2\pi\omega · \Delta P. +\Tag{(26)} +\] + +Here $\omega$~represents the frequency of revolution of the electron, +and according to the classical theory this is equal to the frequency~$\nu$ +of the radiation. If we now assume that the total energy emitted +is equal to~$h\nu$ we obtain for the total angular momentum of the +radiation +\[ +\Delta P = \frac{h}{2\pi}. +\Tag{(27)} +\] + +It is extremely interesting to note that this expression is equal +to the change in the angular momentum which the atom suffers in +a transition where $k$~varies by unity. For in Sommerfeld's theory +the general condition for the fixation of the stationary states of a +central system, which in the special case of an approximately +Keplerian motion is equivalent to the relation~\Eq{(25)}, asserts that +the angular momentum of the system must be equal to a whole +multiple of~$\dfrac{h}{2\pi}$, a condition which may be written in our notation +\[ +P = k\, \frac{h}{2\pi}. +\Tag{(28)} +\] +We see, therefore, that this condition has obtained direct support +from a simple consideration of the conservation of angular momentum +during the emission of the radiation. I wish to emphasize +that this equation is to be regarded as a rational generalization of +\PageSep{52} +Planck's original statement about the distinctive states of a harmonic +oscillator. It may be of interest to recall that the possible +significance of the angular momentum in applications of the +quantum theory to atomic processes was first pointed out by +Nicholson on the basis of the fact that for a circular motion the +angular momentum is simply proportional to the ratio of the +kinetic energy to the frequency of revolution. + +In a previous paper which I presented to the Copenhagen +Academy I pointed out that these results confirm the conclusions +obtained by the application of the correspondence principle to +atomic systems possessing radial or axial symmetry. Rubinowicz +has independently indicated the conclusions which may be obtained +directly from a consideration of conservation of angular momentum +during the radiation process. In this way he has obtained several +of our results concerning the various types of possible transitions +and the polarization of the emitted radiation. Even for systems +possessing radial or axial symmetry, however, the conclusions which +we can draw by means of the correspondence principle are of a +more detailed character than can be obtained solely from a consideration +of the conservation of angular momentum. For example, +in the case of the hydrogen atom perturbed by a central force we +can only conclude that $k$~\Chg{can not}{cannot} change by more than unity, while +the correspondence principle requires that $k$~shall vary by unity +for every possible transition and that its value cannot remain unchanged. +Further, this principle enables us not only to exclude +certain transitions as being impossible---and can from this point of +view be considered as a ``selection principle''---but it also enables +us to draw conclusions about the relative probabilities of the various +possible types of transitions from the values of the amplitudes of +the harmonic components. In the present case, for example, the +fact that the amplitudes of those circular components which rotate +in the same sense as the electron are in general greater than the +amplitudes of those which rotate in the opposite sense leads us to +expect that lines corresponding to transitions for which $k$~decreases +by unity will in general possess greater intensity than lines during +the emission of which $k$~increases by unity. Simple considerations +like this, however, apply only to spectral lines corresponding to +transitions from one and the same stationary state. In other +\PageSep{53} +cases when we wish to estimate the relative intensities of two +spectral lines it is clearly necessary to take into consideration the +relative number of atoms which are present in each of the two +stationary states from which the transitions start. While the intensity +naturally \Chg{can not}{cannot} depend upon the number of atoms in the +final state, it is to be noticed, however, that in estimating the +probability of a transition between two stationary states it is necessary +to consider the character of the motion in the final as well as +in the initial state, since the values of the amplitudes of the components +of oscillation of both states are to be regarded as decisive +for the probability. + +To show how this method can be applied I shall return for a +moment to the problem which I mentioned in connection with +Strutt's experiment on the resonance radiation of sodium vapour. +This involved the discussion of the relative probability of the various +possible transitions which can start from that state corresponding +to the second term in the second row of the figure on \PageRef[p.]{30}. These +were transitions to the first and second states in the first row and +to the first state in the third row, and the results of experiment +indicate, as we saw, that the probability is greatest for the second +transitions. These transitions correspond to those harmonic components +having frequencies $2\omega + \sigma$, $\omega + \sigma$ and~$\sigma$, and it is seen +that only for the second transition do the amplitudes of the corresponding +harmonic component differ from zero in the initial as +well as in the final state. [In the next essay the reader will find +that the values of quantum number~$n$ assigned in \Fig{1} to the +various stationary states must be altered. While this correction +in no way influences the other conclusions in this essay it involves +that the reasoning in this passage \Chg{can not}{cannot} be maintained.] + +I have shown how the correspondence between the spectrum of +an element and the motion of the atom enables us to understand +the limitations in the direct application of the combination principle +in the prediction of spectral lines. The same ideas give an immediate +explanation of the interesting discovery made in recent years +by Stark and his collaborators, that certain \emph{new series of combination +lines} appear with considerable intensity when the radiating +atoms are subject to a strong external electric field. This phenomenon +is entirely analogous to the appearance of the so-called +\PageSep{54} +combination tones in acoustics. It is due to the fact that the +perturbation of the motion will not only consist in an effect upon +the components originally present, but in addition will give rise to +new components. The frequencies of these new components may be +$\tau\omega + \kappa\sigma$, where $\kappa$~is different from~$±1$. According to the correspondence +principle we must therefore expect that the electric field will +not only influence the lines appearing under ordinary circumstances, +but that it will also render possible new types of transitions which +give rise to the ``new'' combination lines observed. From an estimate +of the amplitudes of the particular components in the initial +and final states it has even been found possible to account for the +varying facility with which the new lines are brought up by the +external field. + +The general problem of the effect of an electric field on the spectra +of elements of higher atomic number differs essentially from the +simple Stark effect of the hydrogen lines, since we are here concerned +not with the perturbation of a purely periodic system, but +with the effect of the field on a periodic motion already subject to +a perturbation. The problem to a certain extent resembles the +effect of a weak electric force on the fine structure components of +the hydrogen atom. In much the same way the effect of an electric +field upon the series spectra of the elements may be treated directly +by investigating the perturbations of the external electron. A +continuation of my paper in the Transactions of the Copenhagen +Academy will soon appear in which I shall show how this method +enables us to understand the interesting observations Stark and +others have made in this field. + +\Section{The spectra of helium and lithium.} We see that it has been +possible to obtain a certain general insight into the origin of the +series spectra of a type like that of sodium. The difficulties encountered +in an attempt to give a detailed explanation of the +spectrum of a particular element, however, become very serious, +even when we consider the spectrum of helium whose neutral atom +contains only two electrons. The spectrum of this element has a +simple structure in that it consists of single lines or at any rate of +double lines whose components are very close together. We find, +however, that the lines fall into two groups each of which can be +\PageSep{55} +described by a formula of the type~\Eq{(14)}. These are usually called +the (ortho) helium and parhelium spectra. While the latter consists +of simple lines, the former possesses narrow doublets. The +discovery that helium, as opposed to the alkali metals, possesses +two complete spectra of the Rydberg type which do not exhibit any +mutual combinations was so surprising that at times there has been +a tendency to believe that helium consisted of two elements. This +way out of the difficulty is no longer open, since there is no room +for another element in this region of the periodic system, or more +correctly expressed, for an element possessing a new spectrum. The +existence of the two spectra can, however, be traced back to the fact +that in the stationary states corresponding to the series spectra we +have to do with a system possessing only one inner electron and in +consequence the motion of the inner system, in the absence of the +outer electron, will be simply periodic and therefore easily perturbed +by external forces. + +In order to illustrate this point we shall have to consider more +carefully the stationary states connected with the origin of a series +spectrum. We must assume that in these states one electron revolves +in an orbit outside the nucleus and the other electrons. We +might now suppose that in general a number of different groups of +such states might exist, each group corresponding to a different +stationary state of the inner system considered by itself. Further +consideration shows, however, that under the usual conditions of +excitation those groups have by far the greatest probability for which +the motion of the inner electrons corresponds to the ``normal'' state +of the inner system, \ie\ to that stationary state having the least +energy. Further the energy required to transfer the inner system +from its normal state to another stationary state is in general very +large compared with the energy which is necessary to transfer an +electron from the normal state of the neutral atom to a stationary +orbit of greater dimensions. Lastly the inner system is in general +capable of a permanent existence only in its normal state. Now, +the configuration of an atomic system in its stationary states and +also in the normal state will, in general, be completely determined. +We may therefore expect that the inner system under the influence +of the forces arising from the presence of the outer electron can in +the course of time suffer only small changes. For this reason we +\PageSep{56} +must assume that the influence of the inner system upon the motion +of the external electron will, in general, be of the same character +as the perturbations produced by a constant external field upon +the motion of the electron in the hydrogen atom. We must therefore +expect a spectrum consisting of an ensemble of spectral terms, +which in general form a connected group, even though in the +absence of external perturbing forces not every combination actually +occurs. The case of the helium spectrum, however, is quite different +since here the inner system contains only one electron the motion +of which in the absence of the external electron is simple periodic +provided the small changes due to the variation in the mass of the +electron with its velocity are neglected. For this reason the form of +the orbit in the stationary states of the inner system considered by +itself will not be determined. In other words, the stability of the +orbit is so slight, even if the variation in the mass is taken into +account, that small external forces are in a position to change the +\Chg{excentricity}{eccentricity} in the course of time to a finite extent. In this case, +therefore, it is possible to have several groups of stationary states, +for which the energy of the inner system is approximately the same +while the form of the orbit of the inner electron and its position +relative to the motion of the other electrons are so essentially +different, that no transitions between the states of different groups +can occur even in the presence of external forces. It can be seen +that these conclusions summarize the experimental observations +on the helium spectra. + +These\Pagelabel{56} considerations suggest an investigation of the nature of +the perturbations in the orbit of the inner electron of the helium +atom, due to the presence of the external electron. A discussion +of the helium spectrum from this point of view has recently been +given by Landé. The results of this work are of great interest particularly +in the demonstration of the large back effect on the outer +electron due to the perturbations of the inner orbit which themselves +arise from the presence of the outer electron. Nevertheless, it can +scarcely be regarded as a satisfactory explanation of the helium +spectrum. Apart from the serious objections which may be raised +against his calculation of the perturbations, difficulties arise if we +try to apply the correspondence principle to Landé's results in +order to account for the occurrence of two distinct spectra showing +\PageSep{57} +no mutual combinations. To explain this fact it seems necessary +to base the discussion on a more thorough investigation of the +mutual perturbations of the outer and the inner orbits. As a +result of these perturbations both electrons move in such an +extremely complicated way that the stationary states \Chg{can not}{cannot} be +fixed by the methods developed for conditionally periodic systems. +Dr~Kramers and I have in the last few years been engaged in such +an investigation, and in an address on atomic problems at the +meeting of the Dutch Congress of Natural and Medical Sciences +held in Leiden, April 1919, I gave a short communication of our +results. For various reasons we have up to the present time been +prevented from publishing, but in the very near future we hope to +give an account of these results and of the light which they seem +to throw upon the helium spectrum. + +The problem presented by the spectra of elements of higher +atomic number is simpler, since the inner system is better defined +in its normal state. On the other hand the difficulty of the mechanical +problem of course increases with the number of the particles in +the atom. We obtain an example of this in the case of lithium +with three electrons. The differences between the spectral terms +of the lithium spectrum and the corresponding spectral terms of +hydrogen are very small for the variable term of the principal series +($k = 2$) and for the diffuse series ($k = 3$), on the other hand it is very +considerable for the variable term of the sharp series ($k = 1$). This +is very different from what would be expected if it were possible to +describe the effect of the inner electron by a central force varying +in a simple manner with the distance. This must be because the +parameter of the orbit of the outer electron in the stationary states +corresponding to the terms of the sharp series is not much greater +than the linear dimensions of the orbits of the inner electrons. +According to the principle of correspondence the frequency of rotation +of the major axis of the orbit of the outer electron is to be regarded +as a measure of the deviation of the spectral terms from the corresponding +hydrogen terms. In order to calculate this frequency it +appears necessary to consider in detail the mutual effect of all three +electrons, at all events for that part of the orbit where the outer +electron is very close to the other two electrons. Even if we assumed +that we were fully acquainted with the normal state of the inner +\PageSep{58} +system in the absence of the outer electron---which would be +expected to be similar to the normal state of the neutral helium +atom---the exact calculation of this mechanical problem would +evidently form an exceedingly difficult task. + +\Section{Complex structure of series lines.} For the spectra of elements +of still higher atomic number the mechanical problem which has to +be solved in order to describe the motion in the stationary states +becomes still more difficult. This is indicated by the extraordinarily +complicated structure of many of the observed spectra. The fact that +the series spectra of the alkali metals, which possess the simplest +structure, consist of double lines whose separation increases with +the atomic number, indicates that here we have to do with systems +in which the motion of the outer electron possesses in general a +somewhat more complicated character than that of a simple central +motion. This gives rise to a more complicated ensemble of stationary +states. It would, however, appear that in the sodium atom the major +axis and the parameter of the stationary states corresponding to +each pair of spectral terms are given approximately by formulae +\Eq{(17)} and~\Eq{(25)}. This is indicated not only by the similar part played +by the two states in the experiments on the resonance radiation of +sodium vapour, but is also shown in a very instructive manner by +the peculiar effect of magnetic fields on the doublets. For small +fields each component splits up into a large number of sharp lines +instead of into the normal Lorentz triplet. With increasing field +strength Paschen and Back found that this \emph{anomalous Zeeman effect} +changed into the normal Lorentz triplet of a single line by a gradual +fusion of the components. + +This effect of a magnetic field upon the doublets of the alkali +spectrum is of interest in showing the intimate relation of the components +and confirms the reality of the simple explanation of the +general structure of the spectra of the alkali metals. If we may +again here rely upon the correspondence principle we have unambiguous +evidence that the effect of a magnetic field on the motion +of the electrons simply consists in the superposition of a uniform +rotation with a frequency given by equation~\Eq{(24)} as in the case of +the hydrogen atom. For if this were the case the correspondence +principle would indicate under all conditions a normal Zeeman effect +\PageSep{59} +for each component of the doublets. I want to emphasize that the +difference between the simple effect of a magnetic field, which the +theory predicts for the fine structure of components of the hydrogen +lines, and the observed effect on the alkali doublets is in no way to +be considered as a contradiction. The fine structure components +are not analogous to the individual doublet components, but each +single fine structure component corresponds to the ensemble of +components (doublet, triplet) which makes up one of the series lines +in Rydberg's scheme. The occurrence in strong fields of the effect +observed by Paschen and Back must therefore be regarded as a +strong support for the theoretical prediction of the effect of a magnetic +field on the fine structure components of the hydrogen lines. + +It does not appear necessary to assume the ``anomalous'' effect +of small fields on the doublet components to be due to a failure of +the ordinary electrodynamical laws for the description of the motion +of the outer electron, but rather to be connected with an effect of +the magnetic field on that intimate interaction between the motion +of the inner and outer electrons which is responsible for the occurrence +of the doublets. Such a view is probably not very different +from the ``coupling theory'' by which Voigt was able to account +formally for the details of the anomalous Zeeman effect. We might +even expect it to be possible to construct a theory of these effects +which would exhibit a formal analogy with the Voigt theory similar +to that between the quantum theory of the normal Zeeman effect and +the theory originally developed by Lorentz. Time unfortunately +does not permit me to enter further into this interesting problem, so +I must refer you to the continuation of my paper in the Transactions +of the Copenhagen Academy, which will contain a general discussion +of the origin of series spectra and of the effects of electric and +magnetic fields. + + +\Chapter{IV.}{Conclusion} + +In this lecture I have purposely not considered the question of +the structure of atoms and molecules although this is of course most +intimately connected with the kind of spectral theory I have developed. +We are encouraged to use results obtained from the spectra, +since even the simple theory of the hydrogen spectrum gives a +value for the major axis of the orbit of the electron in the normal +\PageSep{60} +state ($n = 1$) of the same order of magnitude as that derived from +the kinetic theory of gases. In my first paper on the subject I +attempted to sketch a theory of the structure of atoms and of +molecules of chemical compounds. This theory was based on a +simple generalization of the results for the stationary states of the +hydrogen atom. In several respects the theory was supported by +experiment, especially in the general way in which the properties +of the elements change with increasing atomic number, shown most +clearly by Moseley's results. I should like, however, to use this +occasion to state, that in view of the recent development of the +quantum theory, many of the special assumptions will certainly have +to be changed in detail. This has become clear from various sides +by the lack of agreement of the theory with experiment. It appears +no longer possible to justify the assumption that in the normal +states the electrons move in orbits of special geometrical simplicity, +like ``electronic rings.'' Considerations relating to the stability of +atoms and molecules against external influences and concerning the +possibility of the formation of an atom by successive addition of +the individual electrons compel us to claim, first that the configurations +of electrons are not only in mechanical equilibrium +but also possess a certain stability in the sense required by +ordinary mechanics, and secondly that the configurations employed +must be of such a nature that transitions to these from other +stationary states of the atom are possible. These requirements are +not in general fulfilled by such simple configurations as electronic +rings and they force us to look about for possibilities of more complicated +motions. It will not be possible here to consider further +these still open questions and I must content myself by referring +to the discussion in my forthcoming paper. In closing, however, +I should like to emphasize once more that in this lecture I have +only intended to bring out certain general points of view lying at +the basis of the spectral theory. In particular it was my intention +to show that, in spite of the fundamental differences between these +points of view and the ordinary conceptions of the phenomena of +radiation, it still appears possible on the basis of the general correspondence +between the spectrum and the motion in the atom to +employ these conceptions in a certain sense as guides in the investigation +of the spectra. +\PageSep{61} + + +\Essay{III}{The Structure of~the~Atom and the~Physical +and~Chemical~Properties of~the~Elements} +{Address delivered before a joint meeting of the Physical and Chemical +Societies in Copenhagen, October~18, 1921.} + +\Chapter{I.}{Preliminary} + +In an address which I delivered to you about a year ago I +described the main features of a theory of atomic structure which +I shall attempt to develop this evening. In the meantime this +theory has assumed more definite form, and in two recent letters +%[** TN: Footnote mark before punctuation in the original] +to \Title{Nature} I have given a somewhat further sketch of the development.\footnote + {\Title{Nature}, March~24, and October~13, 1921.} +The results which I am about to present to you are +of no final character; but I hope to be able to show you how this +view renders a correlation of the various properties of the elements +in such a way, that we avoid the difficulties which previously +appeared to stand in the way of a simple and consistent explanation. +Before proceeding, however, I must ask your forbearance if initially +I deal with matters already known to you, but in order to introduce +you to the subject it will first be necessary to give a brief +description of the most important results which have been obtained +in recent years in connection with the work on atomic structure. + +\Section{The nuclear atom.} The conception of atomic structure which +will form the basis of all the following remarks is the so-called +nuclear atom according to which an atom is assumed to consist of +a nucleus surrounded by a number of electrons whose distances +from one another and from the nucleus are very large compared +to the dimensions of the particles themselves. The nucleus +possesses almost the entire mass of the atom and has a positive +charge of such a magnitude that the number of electrons in a +neutral atom is equal to the number of the element in the periodic +system, the so-called \emph{atomic number}. This idea of the atom, which +is due principally to Rutherford's fundamental researches on radioactive +substances, exhibits extremely simple features, but just this +simplicity appears at first sight to present difficulties in explaining +the properties of the elements. When we treat this question on +\PageSep{62} +the basis of the ordinary mechanical and electrodynamical theories +it is impossible to find a starting point for an explanation of the +marked properties exhibited by the various elements, indeed not +even of their permanency. On the one hand the particles of the +atom apparently could not be at rest in a state of stable equilibrium, +and on the other hand we should have to expect that every motion +which might be present would give rise to the emission of electromagnetic +radiation which would not cease until all the energy of +the system had been emitted and all the electrons had fallen into +the nucleus. A method of escaping from these difficulties has now +been found in the application of ideas belonging to the quantum +theory, the basis of which was laid by Planck in his celebrated +work on the law of temperature radiation. This represented a +radical departure from previous conceptions since it was the first +instance in which the assumption of a discontinuity was employed +in the formulation of the general laws of nature. + +\Section{The postulates of the quantum theory.}\Pagelabel{62} The quantum theory +in the form in which it has been applied to the problems of atomic +structure rests upon two postulates which have a direct bearing +on the difficulties mentioned above. According to the first postulate +there are certain states in which the atom can exist without +emitting radiation, although the particles are supposed to have an +accelerated motion relative to one another. These \emph{stationary states} +are, in addition, supposed to possess a peculiar kind of stability, so +that it is impossible either to add energy to or remove energy from +the atom except by a process involving a transition of the atom +into another of these states. According to the second postulate +each emission of radiation from the atom resulting from such a +transition always consists of a train of purely harmonic waves. +The frequency of these waves does not depend directly upon the +motion of the atom, but is determined by a \emph{frequency relation}, +according to which the frequency multiplied by the universal constant +introduced by Planck is equal to the total energy emitted +during the process. For a transition between two stationary states +for which the values of the energy of the atom before and after the +emission of radiation are $E'$~and $E''$ respectively, we have therefore +\[ +h\nu = E' - E'', +\Tag{(1)} +\] +\PageSep{63} +where $h$~is Planck's constant and $\nu$~is the frequency of the emitted +radiation. Time does not permit me to give a systematic survey +of the quantum theory, the recent development of which has gone +hand in hand with its applications to atomic structure. I shall +therefore immediately proceed to the consideration of those applications +of the theory which are of direct importance in connection +with our subject. + +\Section{Hydrogen atom.} We shall commence by considering the +simplest atom conceivable, namely, an atom consisting of a nucleus +and one electron. If the charge on the nucleus corresponds to that +of a single electron and the system consequently is neutral we have +a hydrogen atom. Those developments of the quantum theory which +have made possible its application to atomic structure started with +the interpretation of the well-known simple spectrum emitted by +hydrogen. This spectrum consists of a series of lines, the frequencies +of which are given by the extremely simple Balmer formula +\[ +\nu = K\left(\frac{1}{(n'')^{2}} - \frac{1}{(n')^{2}}\right), +\Tag{(2)} +\] +where $n''$~and $n'$ are integers. According to the quantum theory +we shall now assume that the atom possesses a series of stationary +states characterized by a series of integers, and it can be seen how +the frequencies given by formula~\Eq{(2)} may be derived from the +frequency relation if it is assumed that a hydrogen line is connected +with a radiation emitted during a transition between two +of these states corresponding to the numbers $n'$~and~$n''$, and if the +energy in the $n$th~state apart from an arbitrary additive constant +is supposed to be given by the formula +\[ +E_{n} = -\frac{Kh}{n^{2}}. +\Tag{(3)} +\] +The negative sign is used because the energy of the atom is +measured most simply by the work required to remove the electron +to infinite distance from the nucleus, and we shall assume that the +numerical value of the expression on the right-hand side of formula~\Eq{(3)} +is just equal to this work. + +As regards the closer description of the stationary states we find +that the electron will very nearly describe an ellipse with the +nucleus in the focus. The major axis of this ellipse is connected +\PageSep{64} +with the energy of the atom in a simple way, and corresponding to +the energy values of the stationary states given by formula~\Eq{(3)} +there are a series of values for the major axis~$2a$ of the orbit of the +electron given by the formula +\[ +2a_{n} = \frac{n^{2} e^{2}}{hK}, +\Tag{(4)} +\] +where $e$~is the numerical value of the charge of the electron and +the nucleus. + +On the whole we may say that the spectrum of hydrogen shows +us the \emph{formation of the hydrogen atom}, since the stationary states +may be regarded as different stages of a process by which the electron +under the emission of radiation is bound in orbits of smaller +and smaller dimensions corresponding to states with decreasing +values of~$n$. It will be seen that this view has certain characteristic +features in common with the binding process of an electron +to the nucleus if this were to take place according to the ordinary +electrodynamics, but that our view differs from it in just such a +way that it is possible to account for the observed properties of +hydrogen. In particular it is seen that the final result of the +binding process leads to a quite definite stationary state of the +atom, namely that state for which $n = 1$. This state which corresponds +to the minimum energy of the atom will be called the +\emph{normal state} of the atom. It may be stated here that the values of +the energy of the atom and the major axis of the orbit of the +electron which are found if we put $n = 1$ in formulae \Eq{(3)} and~\Eq{(4)} +are of the same order of magnitude as the values of the firmness +of binding of electrons and of the dimensions of the atoms which +have been obtained from experiments on the electrical and mechanical +properties of gases. A more accurate check of formulae +\Eq{(3)} and~\Eq{(4)} can however not be obtained from such a comparison, +because in such experiments hydrogen is not present in the form +of simple atoms but as molecules. + +The formal basis of the quantum theory consists not only of the +frequency relation, but also of conditions which permit the determination +of the stationary states of atomic systems. The latter +conditions, like that assumed for the frequency, may be regarded as +natural generalizations of that assumption regarding the interaction +between simple electrodynamic systems and a surrounding field of +\PageSep{65} +electromagnetic radiation which forms the basis of Planck's theory +of temperature radiation. I shall not here go further into the +nature of these conditions but only mention that by their means +the stationary states are characterized by a number of integers, +the so-called \emph{quantum numbers}. For a purely periodic motion like +that assumed in the case of the hydrogen atom only a single +quantum number is necessary for the determination of the stationary +states. This number determines the energy of the atom and also +the major axis of the orbit of the electron, but not its \Chg{excentricity}{eccentricity}. +The energy in the various stationary states, if the small influence +of the motion of the nucleus is neglected, is given by the following +formula: +\[ +E_{n} = -\frac{2\pi^{2} N^{2} e^{4} m}{n^{2} h^{2}}, +\Tag{(5)} +\] +where $e$~and $m$ are respectively the charge and the mass of the +electron, and where for the sake of subsequent applications the +charge on the nucleus has been designated by~$Ne$. + +For the atom of hydrogen $N = 1$, and a comparison with +equation~\Eq{(3)} leads to the following theoretical expression for~$K$ in +formula~\Eq{(2)}, namely +\[ +K = \frac{2\pi^{2} e^{4} m}{h^{3}}. +\Tag{(6)} +\] +This agrees with the empirical value of the constant for the spectrum +of hydrogen within the limit of accuracy with which the various +quantities can be determined. + +\Section{Hydrogen spectrum and X-ray spectra.} If in the above +formula we put $N = 2$ which corresponds to an atom consisting of +an electron revolving around a nucleus with a double charge, we +get values for the energies in the stationary states, which are four +times larger than the energies in the corresponding states of the +hydrogen atom, and we obtain the following formula for the +spectrum which would be emitted by such an atom: +\[ +\nu = 4K \left(\frac{1}{(n'')^{2}} - \frac{1}{(n')^{2}}\right). +\Tag{(7)} +\] +This formula represents certain lines which have been known for +some time and which had been attributed to hydrogen on account +of the great similarity between formulae \Eq{(2)} and~\Eq{(7)} since it had +\PageSep{66} +never been anticipated that two different substances could exhibit +properties so closely resembling each other. According to the theory +we may, however, expect that the emission of the spectrum given by~\Eq{(7)} +corresponds to the \emph{first stage of the formation of the helium atom}, +\ie\ to the binding of a first electron by the doubly charged nucleus +of this atom. This interpretation has been found to agree with +more recent information. For instance it has been possible to +obtain this spectrum from pure helium. I have dwelt on this point +in order to show how this intimate connection between the properties +of two elements, which at first sight might appear quite +surprising, is to be regarded as an immediate expression of the +characteristic simple structure of the nuclear atom. A short time +after the elucidation of this question, new evidence of extraordinary +interest was obtained of such a similarity between the properties of +the elements. I refer to Moseley's fundamental researches on the +X-ray spectra of the elements. Moseley found that these spectra +varied in an extremely simple manner from one element to the +next in the periodic system. It is well known that the lines of +the X-ray spectra may be divided into groups corresponding to the +different characteristic absorption regions for X-rays discovered by +Barkla. As regards the $K$~group which contains the most penetrating +X-rays, Moseley found that the strongest line for all elements +investigated could be represented by a formula which with +a small simplification can be written +\[ +\nu = N^{2} K \left(\frac{1}{1^{2}} - \frac{1}{2^{2}}\right). +\Tag{(8)} +\] +$K$~is the same constant as in the hydrogen spectrum, and $N$~the +atomic number. The great significance of this discovery lies in +the fact that it would seem firmly to establish the view that this +atomic number is equal to the number of electrons in the atom. +This assumption had already been used as a basis for work on +atomic structure and was first stated by van~den Broek. While +the significance of this aspect of Moseley's discovery was at once +clear to all, it has on the other hand been more difficult to understand +the very great similarity between the spectrum of hydrogen +and the X-ray spectra. This similarity is shown, not only by the +lines of the $K$~group, but also by groups of less penetrating X-rays. +\PageSep{67} +Thus Moseley found for all the elements he investigated that the +frequencies of the strongest line in the $L$~group may be represented +by a formula which with a simplification similar to that employed +in formula~\Eq{(8)} can be written +\[ +\nu = N^{2} K \left(\frac{1}{2^{2}} - \frac{1}{3^{2}}\right). +\Tag{(9)} +\] +Here again we obtain an expression for the frequency which corresponds +to a line in the spectrum which would be emitted by the +\emph{binding of an electron to a nucleus, whose charge is~$Ne$}. + +\Section{The fine structure of the hydrogen lines.} This similarity between +the structure of the X-ray spectra and the hydrogen spectrum +was still further extended in a very interesting manner by Sommerfeld's +important theory of the fine structure of the hydrogen lines. +The calculation given above of the energy in the stationary states +of the hydrogen system, where each state is characterized by a +single quantum number, rests upon the assumption that the orbit +of the electron in the atom is simply periodic. This is, however, +only approximately true. It is found that if the change in the mass +of the electron due to its velocity is taken into consideration the +orbit of the electron no longer remains a simple ellipse, but its +motion may be described as a \emph{central motion} obtained by superposing +a slow and uniform rotation upon a simple periodic motion in a +very nearly elliptical orbit. For a central motion of this kind the +stationary states are characterized by \emph{two quantum numbers}. In the +case under consideration one of these may be so chosen that to a +very close approximation it will determine the energy of the atom +in the same manner as the quantum number previously used +determined the energy in the case of a simple elliptical orbit. This +quantum number which will always be denoted by~$n$ will therefore +be called the ``principal quantum number.'' Besides this condition, +which to a very close approximation determines the major axis in the +rotating and almost elliptical orbit, a second condition will be imposed +upon the stationary states of a central orbit, namely that the angular +momentum of the electron about the centre shall be equal to a whole +multiple of Planck's constant divided by~$2\pi$. The whole number, which +occurs as a factor in this expression, may be regarded as the second +quantum number and will be denoted by~$k$. The latter condition fixes +\PageSep{68} +the \Chg{excentricity}{eccentricity} of the rotating orbit which in the case of a simple +periodic orbit was undetermined. It should be mentioned that the +possible importance of the angular momentum in the quantum theory +was pointed out by Nicholson before the application of this theory to +the spectrum of hydrogen, and that a determination of the stationary +states for the hydrogen atom similar to that employed by Sommerfeld +was proposed almost simultaneously by Wilson, although the +latter did not succeed in giving a physical application to his results. + +The simplest description of the form of the rotating nearly +elliptical electronic orbit in the hydrogen atom is obtained by +considering the chord which passes through the focus and is +perpendicular to the major axis, the so-called ``parameter.'' The +length~$2p$ of this parameter is given to a very close approximation +by an expression of exactly the same form as the expression for the +major axis, except that $k$~takes the place of~$n$. Using the same +notation as before we have therefore +\[ +2a = n^{2}\, \frac{h^{2}}{2\pi^{2} N e^{2} m},\quad +2p = k^{2}\, \frac{h^{2}}{2\pi^{2} N e^{2} m}. +\Tag{(10)} +\] +For each of the stationary states which had previously been denoted +by a given value of~$n$, we obtain therefore a set of stationary states +corresponding to values of~$k$ from $1$ to~$n$. Instead of the simple +formula~\Eq{(5)} Sommerfeld found a more complicated expression for +the energy in the stationary states which depends on~$k$ as well as~$n$. +Taking the variation of the mass of the electron with velocity +into account and neglecting terms of higher order of magnitude he +obtained +\[ +E_{n,k} = -\frac{2\pi^{2} N^{2} e^{4} m}{n^{2} h^{2}} + \left[1 + \frac{4\pi^{2} N^{2} e^{4}}{h^{2} c^{2}}\left(-\frac{3}{4n^{2}} + \frac{1}{nk}\right)\right], +\Tag{(11)} +\] +where $c$~is the velocity of light. + +Corresponding to each of the energy values for the stationary +states of the hydrogen atom given by the simple formula~\Eq{(5)} we +obtain $n$~values differing only very little from one another, since +the second term within the bracket is very small. With the aid of +the general frequency relation~\Eq{(1)} we therefore obtain a number of +components with nearly coincident frequencies instead of each +hydrogen line given by the simple formula~\Eq{(2)}. Sommerfeld has +now shown that this calculation actually agrees with measurements +\PageSep{69} +of the fine structure. This agreement applies not only to the fine +structure of the hydrogen lines which is very difficult to measure +on account of the extreme proximity of the components, but it is +also possible to account in detail for the fine structure of the helium +lines given by formula~\Eq{(7)} which has been very carefully investigated +by Paschen. Sommerfeld in connection with this theory +also pointed out that formula~\Eq{(11)} could be applied to the X-ray +spectra. Thus he showed that in the $K$~and $L$ groups pairs of lines +appeared the differences of whose frequencies could be determined +by the expression~\Eq{(11)} for the energy in the stationary states which +correspond to the binding of a single electron by a nucleus of +charge~$Ne$. + +\Section{Periodic table.} In spite of the great formal similarity between +the X-ray spectra and the hydrogen spectrum indicated by these +results a far-reaching difference must be assumed to exist between +the processes which give rise to the appearance of these two types +of spectra. While the emission of the hydrogen spectrum, like the +emission of the ordinary optical spectra of other elements, may be +assumed to be connected with the binding of an electron by an +atom, observations on the appearance and absorption of X-ray +spectra clearly indicate that these spectra are connected with a +process which may be described as a \emph{reorganization of the electronic +arrangement} after a disturbance within the atom due to the effect +of external agencies. We should therefore expect that the appearance +of the X-ray spectra would depend not only upon the direct +interaction between a single electron and the nucleus, but also on +the manner in which the electrons are arranged in the completely +formed atom. + +The peculiar manner in which the properties of the elements +vary with the atomic number, as expressed in the periodic system, +provides a guide of great value in considering this latter problem. +A simple survey of this system is given in \Fig{1}. The number preceding +each element indicates the atomic number, and the elements +within the various vertical columns form the different ``periods'' of +the system. The lines, which connect pairs of elements in successive +columns, indicate homologous properties of such elements. Compared +with usual representations of the periodic system, this method, +\PageSep{70} +proposed more than twenty years ago by Julius Thomsen, of indicating +the periodic variations in the properties of the elements is +more suited for comparison with theories of atomic constitution. +The meaning of the frames round certain sequences of elements +within the later periods of the table will be explained later. They +refer to certain characteristic features of the theory of atomic +constitution. +\Figure{1}{70} + +In an explanation of the periodic system it is natural to assume +a division of the electrons in the atom into distinct groups +in such a manner that the grouping of the elements in the system +is attributed to the gradual formation of the groups of electrons +in the atoms as the atomic number increases. Such a grouping +\PageSep{71} +of the electrons in the atom has formed a prominent part of all +more detailed views of atomic structure ever since J.~J. Thomson's +famous attempt to explain the periodic system on the basis +of an investigation of the stability of various electronic configurations. +Although Thomson's assumption regarding the distribution +of the positive electricity in the atom is not consistent with more +recent experimental evidence, nevertheless his work has exerted +great influence upon the later development of the atomic theory on +account of the many original ideas which it contained. + +With the aid of the information concerning the binding of +electrons by the nucleus obtained from the theory of the hydrogen +spectrum I attempted in the same paper in which this theory was +set forth to sketch in broad outlines a picture of the structure of +the nucleus atom. In this it was assumed that each electron in its +normal state moved in a manner analogous to the motion in +the last stages of the binding of a single electron by a nucleus. +As in Thomson's theory, it was assumed that the electrons moved +in circular orbits and that the electrons in each separate group +during this motion occupied positions with reference to one another +corresponding to the vertices of plane regular polygons. Such an +arrangement is frequently described as a distribution of the electrons +in ``rings.'' By means of these assumptions it was possible to +account for the orders of magnitude of the dimensions of the atoms +as well as the firmness with which the electrons were bound by the +atom, a measure of which may be obtained by means of experiments +on the excitation of the various types of spectra. It was not +possible, however, in this way to arrive at a detailed explanation +of the characteristic properties of the elements even after it had +become apparent from the results of Moseley and the work of +Sommerfeld and others that this simple picture ought to be extended +to include orbits in the fully formed atom characterized by +higher quantum numbers corresponding to previous stages in the +formation of the hydrogen atom. This point has been especially +emphasized by Vegard. + +The difficulty of arriving at a satisfactory picture of the atom is +intimately connected with the difficulty of accounting for the pronounced +``stability'' which the properties of the elements demand. +As I emphasized when considering the formation of the hydrogen +\PageSep{72} +atom, the postulates of the quantum theory aim directly at this +point, but the results obtained in this way for an atom containing +a single electron do not permit of a direct elucidation of problems +like that of the distribution in groups of the electrons in an atom +containing several electrons. If we imagine that the electrons in +the groups of the atom are orientated relatively to one another at any +moment, like the vertices of regular polygons, and rotating in either +circles or ellipses, the postulates do not give sufficient information to +determine the difference in the stability of electronic configurations +with different numbers of electrons in the groups. + +The peculiar character of stability of the atomic structure, demanded +by the properties of the elements, is brought out in an +interesting way by Kossel in two important papers. In the first +paper he shows that a more detailed explanation of the origin of +the high frequency spectra can be obtained on the basis of the +group structure of the atom. He assumes that a line in the X-ray +spectrum is due to a process which may be described as follows: an +electron is removed from the atom by some external action after +which an electron in one of the other groups takes its place; this +exchange of place may occur in as many ways as there are groups +of more loosely bound electrons. This view of the origin of the +characteristic X-rays afforded a simple explanation of the peculiar +absorption phenomena observed. It has also led to the prediction +of certain simple relations between the frequencies of the X-ray +lines from one and the same element and has proved to be a suitable +basis for the classification of the complete spectrum. However it has +not been possible to develop a theory which reconciles in a satisfactory +way Sommerfeld's work on the fine structure of the X-ray +lines with Kossel's general scheme. As we shall see later the +adoption of a new point of view when considering the stability of +the atom renders it possible to bring the different results in a natural +way in connection with one another. + +In his second paper Kossel investigates the possibilities for an +explanation of the periodic system on the basis of the atomic theory. +Without entering further into the problem of the causes of the +division of the electrons into groups, or the reasons for the different +stability of the various electronic configurations, he points out in +connection with ideas which had already played a part in Thomson's +\PageSep{73} +theory, how the periodic system affords evidence of a periodic appearance +of especially stable configurations of electrons. These configurations +appear in the neutral atoms of elements occupying the +final position in each period in \Fig{1}, and the stability in question is +assumed in order to explain not only the inactive chemical properties +of these elements but also the characteristic active properties of the +immediately preceding or succeeding elements. If we consider for +instance an inactive gas like argon, the atomic number of which is~$18$, +we must assume that the $18$~electrons in the atom are arranged in +an exceedingly regular configuration possessing a very marked +stability. The pronounced electronegative character of the preceding +element, chlorine, may then be explained by supposing the neutral +atom which contains only $17$~electrons to possess a tendency to +capture an additional electron. This gives rise to a negative chlorine +ion with a configuration of $18$~electrons similar to that occurring +in the neutral argon atom. On the other hand the marked electropositive +character of potassium may be explained by supposing +one of the $19$~electrons in the neutral atom to be as it were superfluous, +and that this electron therefore is easily lost; the rest of the +atom forming a positive ion of potassium having a constitution similar +to that of the argon atom. In a corresponding manner it is possible +to account for the electronegative and electropositive character of +elements like sulphur and calcium, whose atomic numbers are $16$ and~$20$. +In contrast to chlorine and potassium these elements are divalent, +and the stable configuration of $18$~electrons is formed by the addition +of two electrons to the sulphur atom and by the loss of two electrons +from the calcium atom. Developing these ideas Kossel has succeeded +not only in giving interesting explanations of a large number of +chemical facts, but has also been led to certain general conclusions +about the grouping of the electrons in elements belonging to the +first periods of the periodic system, which in certain respects are +in conformity with the results to be discussed in the following +paragraphs. Kossel's\Pagelabel{73} work was later continued in an interesting +manner by Ladenburg with special reference to the grouping of the +electrons in atoms of elements belonging to the later periods of the +periodic table. It will be seen that Ladenburg's conclusions also +exhibit points of similarity with the results which we shall discuss +later. +\PageSep{74} + +\Section{Recent atomic models.} Up to the present time it has not been +possible to obtain a satisfactory account based upon a consistent application +of the quantum theory to the nuclear atom of the ultimate +cause of the pronounced stability of certain arrangements of electrons. +Nevertheless it has been apparent for some time that the solution +should be sought for by investigating the possibilities of a \emph{spatial +distribution of the electronic orbits} in the atom instead of limiting +the investigation to configurations in which all electrons belonging +to a particular group move in the same plane as was assumed for +simplicity in my first papers on the structure of the atom. The +necessity of assuming a spatial distribution of the configurations +of electrons has been drawn attention to by various writers. Born +and Landé, in connection with their investigations of the structure +and properties of crystals, have pointed out that the assumption of +spatial configurations appears necessary for an explanation of these +properties. Landé has pursued this question still further, and as +will be mentioned later has proposed a number of different ``spatial +atomic models'' in which the electrons in each separate group of +the atom at each moment form configurations possessing regular +polyhedral symmetry. These models constitute in certain respects +a distinct advance, although they have not led to decisive results +on questions of the stability of atomic structure. + +The importance of spatial electronic configurations has, in addition, +been pointed out by Lewis and Langmuir in connection with their +atomic models. Thus Lewis, who in several respects independently +came to the same conclusions as Kossel, suggested that the number~$8$ +characterizing the first groups of the periodic system might indicate +a constitution of the outer atomic groups where the electrons +within each group formed a configuration like the corners of a cube. +He emphasized how a configuration of this kind leads to instructive +models of the molecular structure of chemical combinations. It is +to be remarked, however, that such a ``static'' model of electronic +configuration will not be possible if we assume the forces within +the atom to be due exclusively to the electric charges of the +particles. Langmuir, who has attempted to develop Lewis' conceptions +still further and to account not only for the occurrence of +the first octaves, but also for the longer periods of the periodic +system, supposes therefore the structure of the atoms to be governed +\PageSep{75} +by forces whose nature is unknown to us. He conceives the atom +to possess a ``cellular structure,'' so that each electron is in advance +assigned a place in a cell and these cells are arranged in shells in +such a manner, that the various shells from the nucleus of the atom +outward contain exactly the same number of places as the periods +in the periodic system proceeding in the direction of increasing +atomic number. Langmuir's work has attracted much attention +among chemists, since it has to some extent thrown light on the +conceptions with which empirical chemical science is concerned. +On his theory the explanation of the properties of the various +elements is based on a number of postulates about the structure of +the atoms formulated for that purpose. Such a descriptive theory +is sharply differentiated from one where an attempt is made to +explain the specific properties of the elements with the aid of +general laws applying to the interaction between the particles in +each atom. The principal task of this lecture will consist in an +attempt to show that an advance along these lines appears by no +means hopeless, but on the contrary that with the aid of a consistent +application of the postulates of the quantum theory it +actually appears possible to obtain an insight into the structure +and stability of the atom. + + +\Chapter{II.}{Series Spectra and the Capture of Electrons +by\protect~Atoms} + +We attack the problem of atomic constitution by asking the +question: ``How may an atom be formed by the successive capture +and binding of the electrons one by one in the field of force surrounding +the nucleus?'' + +Before attempting to answer this question it will first be +necessary to consider in more detail what the quantum theory +teaches us about the general character of the binding process. We +have already seen how the hydrogen spectrum gives us definite +information about the course of this process of binding the electron +by the nucleus. In considering the formation of the atoms of other +elements we have also in their spectra sources for the elucidation +of the formation processes, but the direct information obtained in +this way is not so complete as in the case of the hydrogen atom. +For an element of atomic number~$N$ the process of formation may +\PageSep{76} +be regarded as occurring in $N$~stages, corresponding with the successive +binding of $N$~electrons in the field of the nucleus. A spectrum +must be assumed to correspond to each of these binding processes; +but only for the first two elements, hydrogen and helium, do we +possess a detailed knowledge of these spectra. For other elements +of higher atomic number, where several spectra will be connected +with the formation of the atom, we are at present acquainted with +only two types, called the ``arc'' and ``spark'' spectra respectively, +according to the experimental conditions of excitation. Although +these spectra show a much more complicated structure than the +hydrogen spectrum, given by formula~\Eq{(2)} and the helium spectrum +given by formula~\Eq{(7)}, nevertheless in many cases it has been +possible to find simple laws for the frequencies exhibiting a close +analogy with the laws expressed by these formulae. + +\Section{Arc and spark spectra.} If for the sake of simplicity we disregard +the complex structure shown by the lines of most spectra +(occurrence of doublets, triplets etc.), the frequency of the lines of +many arc spectra can be represented to a close approximation by +the Rydberg formula +\[ +\nu = \frac{K}{(n'' + \alpha_{k''})^{2}} - \frac{K}{(n' + \alpha_{k'})^{2}}, +\Tag{(12)} +\] +where $n'$~and $n''$ are integral numbers, $K$~the same constant as in +the hydrogen spectrum, while $\alpha_{k'}$~and $\alpha_{k''}$ are two constants belonging +to a set characteristic of the element. A spectrum with a +structure of this kind is, like the hydrogen spectrum, called a series +spectrum, since the lines can be arranged into series in which the +frequencies converge to definite limiting values. These series are +for example represented by formula~\Eq{(12)} if, using two definite +constants for $\alpha_{k''}$~and~$\alpha_{k'}$, $n''$~remains unaltered, while $n'$~assumes a +series of successive, gradually increasing integral values. + +Formula~\Eq{(12)} applies only approximately, but it is always found +that the frequencies of the spectral lines can be written, as in +formulae \Eq{(2)} and~\Eq{(12)}, as a difference of two functions of integral +numbers. Thus the latter formula applies accurately, if the +quantities~$\alpha_{k}$ are not considered as constants, but as representatives +of a set of series of numbers~$\alpha_{k}(n)$ characteristic of the element, +whose values for increasing~$n$ within each series quickly approach +\PageSep{77} +a constant limiting value. The fact that the frequencies of the +spectra always appear as the difference of two terms, the so-called +``spectral terms,'' from the combinations of which the complete +spectrum is formed, has been pointed out by Ritz, who with the +establishment of the combination principle has greatly advanced +the study of the spectra. The quantum theory offers an immediate +interpretation of this principle, since, according to the frequency +relation we are led to consider the lines as due to transitions +between stationary states of the atom, just as in the hydrogen +spectrum, only in the spectra of the other elements we have to do +not with a single series of stationary states, but with a set of such +series. From formula~\Eq{(12)} we thus obtain for an arc spectrum---if +we temporarily disregard the structure of the individual lines---information +about an ensemble of stationary states, for which the +energy of the atom in the $n$th~state of the $k$th~series is given by +\[ +E_{k}(n) = -\frac{Kh}{(n + \alpha_{k})^{2}} +\Tag{(13)} +\] +very similar to the simple formula~\Eq{(3)} for the energy in the stationary +states of the hydrogen atom. + +As regards the spark spectra, the structure of which has been +cleared up mainly by Fowler's investigations, it has been possible +in the case of many elements to express the frequencies approximately +by means of a formula of exactly the same type as~\Eq{(12)}, +only with the difference that~$K$, just as in the helium spectrum +given by formula~\Eq{(7)}, is replaced by a constant, which is four times +as large. For the spark spectra, therefore, the energy values in the +corresponding stationary states of the atom will be given by an +expression of the same type as~\Eq{(13)}, only with the difference that +$K$~is replaced by~$4K$. + +This remarkable similarity between the structure of these types +of spectra and the simple spectra given by \Eq{(2)}~and~\Eq{(7)} is explained +simply by assuming the arc spectra to be connected with the \emph{last +stage in the formation of the neutral atom} consisting in the capture +and binding of the $N$th~electron. On the other hand the spark +spectra are connected with the \emph{last stage but one in the formation +of the atom}, namely the binding of the $(N - 1)$th~electron. In these +cases the field of force in which the electron moves will be much +\PageSep{78} +the same as that surrounding the nucleus of a hydrogen or helium +atom respectively, at least in the earlier stages of the binding +process, where during the greater part of its revolution it moves +at a distance from the nucleus which is large in proportion to the +dimensions of the orbits of the electrons previously bound. From +analogy with formula~\Eq{(3)} giving the stationary states of the +hydrogen atom, we shall therefore assume that the numerical value +of the expression on the right-hand side of~\Eq{(13)} will be equal to the +work required to remove the last captured electron from the atom, +the binding of which gives rise to the arc spectrum of the element. + +\Section{Series diagram.} While the origin of the arc and spark spectra +was to this extent immediately interpreted on the basis of the +original simple theory of the hydrogen spectrum, it was Sommerfeld's +theory of the fine structure of the hydrogen lines which first gave +us a clear insight into the characteristic difference between the +hydrogen spectrum and the spark spectrum of helium on the one +hand, and the arc and spark spectra of other elements on the other. +When we consider the binding not of the first but of the subsequent +electrons in the atom, the orbit of the electron under consideration---at +any rate in the latter stages of the binding process where the +electron last bound comes into intimate interaction with those +previously bound---will no longer be to a near approximation a +closed ellipse, but on the contrary will to a first approximation be a +central orbit of the same type as in the hydrogen atom, when we +take into account the change with velocity in the mass of the +electron. This motion, as we have seen, may be resolved into a +plane periodic motion upon which a uniform rotation is superposed +in the plane of the orbit; only the superposed rotation will in this +case be comparatively much more rapid and the deviation of the +periodic orbit from an ellipse much greater than in the case of the +hydrogen atom. For an orbit of this type the stationary states, just +as in the theory of the fine structure, will be determined by two +quantum numbers which we shall denote by $n$~and~$k$, connected in +a very simple manner with the kinematic properties of the orbit. +For brevity I shall only mention that while the quantum number~$k$ +is connected with the value of the constant angular momentum +of the electron about the centre in the simple manner previously +\PageSep{79} +indicated, the determination of the principal quantum number~$n$ +requires an investigation of the whole course of the orbit and for +an arbitrary central orbit will not be related in a simple way to +the dimensions of the rotating periodic orbit if this deviates essentially +from a Keplerian ellipse. +\Figure{2}{79} + +These results are represented in \Fig{2} which is a reproduction +of an illustration I have used on a previous occasion +(see Essay~II, \PageRef{30}), and which gives a survey of the origin +of the sodium spectrum. The black dots represent the stationary +states corresponding to the various series of spectral terms, +shown on the right by the letters $S$,~$P$,~$D$ and~$B$. These letters +correspond to the usual notations employed in spectroscopic +literature and indicate the nature of the series (sharp series, +principal series, diffuse series, etc.)\ obtained by combinations of +the corresponding spectral terms. The distances of the separate +points from the vertical line at the right of the figure are proportional +to the numerical value of the energy of the atom given +by equation~\Eq{(13)}. The oblique, black arrows indicate finally the +transitions between the stationary states giving rise to the +appearance of the lines in the commonly observed sodium +spectrum. The values of $n$~and $k$ attached to the various states +indicate the quantum numbers, which, according to Sommerfeld's +theory, from a preliminary consideration might be regarded as +characterizing the orbit of the outer electron. For the sake of +convenience the states which were regarded as corresponding to +the same value of~$n$ are connected by means of dotted lines, and these +are so drawn that their vertical asymptotes correspond to the +\PageSep{80} +terms in the hydrogen spectrum which belong to the same value +of the principal quantum number. The course of the curves illustrates +how the deviation from the hydrogen terms may be expected +to decrease with increasing values of~$k$, corresponding to states, +where the minimum distance between the electron in its revolution +and the nucleus constantly increases. + +It should be noted that even though the theory represents the +principal features of the structure of the series spectra it has not +yet been possible to give a detailed account of the spectrum of any +element by a closer investigation of the electronic orbits which may +occur in a simple field of force possessing central symmetry. As +I have mentioned already the lines of most spectra show a complex +structure. In the sodium spectrum for instance the lines of the +principal series are doublets indicating that to each $P$-term not +one stationary state, but two such states correspond with slightly +different values of the energy. This difference is so little that +it would not be recognizable in a diagram on the same scale as +\Fig{2}. The appearance of these doublets is undoubtedly due to +the small deviations from central symmetry of the field of force +originating from the inner system in consequence of which the +general type of motion of the external electron will possess a +more complicated character than that of a simple central motion. +As a result the stationary states must be characterized by more +than two quantum numbers, in the same way that the occurrence +of deviations of the orbit of the electron in the hydrogen atom from +a simple periodic orbit requires that the stationary states of this +atom shall be characterized by more than one quantum number. +Now the rules of the quantum theory lead to the introduction of +a third quantum number through the condition that the resultant +angular momentum of the atom, multiplied by~$2\pi$, is equal to an +entire multiple of Planck's constant. This determines the orientation +of the orbit of the outer electron relative to the axis of the +inner system. + +In this way Sommerfeld, Landé and others have shown that it +is possible not only to account in a formal way for the complex +structure of the lines of the series spectra, but also to obtain a +promising interpretation of the complicated effect of external +magnetic fields on this structure. We shall not enter here on these +\PageSep{81} +problems but shall confine ourselves to the problem of the fixation +of the two quantum numbers $n$~and~$k$, which to a first approximation +describe the orbit of the outer electron in the stationary +states, and whose determination is a matter of prime importance +in the following discussion of the formation of the atom. In +the determination of these numbers we at once encounter difficulties +of a profound nature, which---as we shall see---are intimately +connected with the question of the remarkable stability of atomic +structure. I shall here only remark that the values of the quantum +number~$n$, given in the figure, undoubtedly \Chg{can not}{cannot} be retained, +neither for the~$S$ nor the $P$~series. On the other hand, so far as +the values employed for the quantum number~$k$ are concerned, it +may be stated with certainty, that the interpretation of the properties +of the orbits, which they indicate, is correct. A starting +point for the investigation of this question has been obtained from +considerations of an entirely different kind from those previously +mentioned, which have made it possible to establish a close connection +between the motion in the atom and the appearance of +spectral lines. + +\Section{Correspondence principle.} So far as the principles of the +quantum theory are concerned, the point which has been emphasized +hitherto is the radical departure of these principles from our +usual conceptions of mechanical and electrodynamical phenomena. +As I have attempted to show in recent years, it appears +possible, however, to adopt a point of view which suggests that the +quantum theory may, nevertheless, be regarded as a rational +generalization of our ordinary conceptions. As may be seen from +the postulates of the quantum theory, and particularly the frequency +relation, a direct connection between the spectra and the motion +of the kind required by the classical dynamics is excluded, but at +the same time the form of these postulates leads us to another +relation of a remarkable nature. Let us consider an electrodynamic +system and inquire into the nature of the radiation which would +result from the motion of the system on the basis of the ordinary +conceptions. We imagine the motion to be decomposed into purely +harmonic oscillations, and the radiation is assumed to consist of +the simultaneous emission of series of electromagnetic waves +\PageSep{82} +possessing the same frequency as these harmonic components and +intensities which depend upon the amplitudes of the components. +An investigation of the formal basis of the quantum theory shows +us now, that it is possible to trace the question of the origin of the +radiation processes which accompany the various transitions back +to an investigation of the various harmonic components, which +appear in the motion of the atom. The possibility, that a particular +transition shall occur, may be regarded as being due to the +presence of a definitely assignable ``corresponding'' component in +the motion. This principle of correspondence at the same time +throws light upon a question mentioned several times previously, +namely the relation between the number of quantum numbers, +which must be used to describe the stationary states of an atom, +and the types to which the orbits of the electrons belong. The +classification of these types can be based very simply on a decomposition +of the motion into its harmonic components. Time does +not permit me to consider this question any further, and I shall +confine myself to a statement of some simple conclusions, which +the correspondence principle permits us to draw concerning the +occurrence of transitions between various pairs of stationary states. +These conclusions are of decisive importance in the subsequent +argument. + +The simplest example of such a conclusion is obtained by +considering an atomic system, which contains a particle describing +a \emph{purely periodic orbit}, and where the stationary states are characterized +by a single quantum number~$n$. In this case the motion +can according to Fourier's theorem be decomposed into a simple +series of harmonic oscillations whose frequency may be written~$\tau\omega$, +where $\tau$~is a whole number, and $\omega$~is the frequency of revolution +in the orbit. It can now be shown that a transition between two +stationary states, for which the values of the quantum number are +respectively equal to $n'$~and~$n''$, will correspond to a harmonic +component, for which $\tau = n' - n''$. This throws at once light upon +the remarkable difference which exists between the possibilities +of transitions between the stationary states of a hydrogen atom +on the one hand and of a simple system consisting of an electric +particle capable of executing simple harmonic oscillations about a +position of equilibrium on the other. For the latter system, which +\PageSep{83} +is frequently called a Planck oscillator, the energy in the stationary +states is determined by the familiar formula $E = nh\omega$, and with the +aid of the frequency relation we obtain therefore for the radiation +which will be emitted during a transition between two stationary +states $\nu = (n' - n'') \omega$. Now, an important assumption, which is not +only essential in Planck's theory of temperature radiation, but +which also appears necessary to account for the molecular absorption +in the infra-red region of radiation, states that a harmonic oscillator +will only emit and absorb radiation, for which the frequency~$\nu$ is +equal to the frequency of oscillation~$\omega$ of the oscillator. We are +therefore compelled to assume that in the case of the oscillator +transitions can occur only between stationary states which are +characterized by quantum numbers differing by only one unit, +while in the hydrogen spectrum represented by formula~\Eq{(2)} all +possible transitions could take place between the stationary states +given by formula~\Eq{(5)}. From the point of view of the principle of +correspondence it is seen, however, that this apparent difficulty is +explained by the occurrence in the motion of the hydrogen atom, +as opposed to the motion of the oscillator, of harmonic components +corresponding to values of~$\tau$, which are different from~$1$; or using +a terminology well known from acoustics, there appear overtones +in the motion of the hydrogen atom. + +Another simple example of the application of the correspondence +principle is afforded by a \emph{central motion}, to the investigation of +which the explanation of the series spectra in the first approximation +may be reduced. Referring once more to the figure of the +sodium spectrum, we see that the black arrows, which correspond +to the spectral lines appearing under the usual conditions of +excitation, only connect pairs of points in consecutive rows. Now +it is found that this remarkable limitation of the occurrence of +combinations between spectral terms may quite naturally be +explained by an investigation of the harmonic components into +which a central motion can be resolved. It can readily be shown +that such a motion can be decomposed into two series of harmonic +components, whose frequencies can be expressed by $\tau\omega + \sigma$ and +$\tau\omega - \sigma$ respectively, where $\tau$~is a whole number, $\omega$~the frequency of +revolution in the rotating periodic orbit and $\sigma$~the frequency of the +superposed rotation. These components correspond with transitions +\PageSep{84} +where the principal number~$n$ decreases by $\tau$~units, while the +quantum number~$k$ decreases or increases, respectively, by one +unit, corresponding exactly with the transitions indicated by the +black arrows in the figure. This may be considered as a very +important result, because we may say, that the quantum theory, +which for the first time has offered a simple interpretation of the +fundamental principle of combination of spectral lines has at the +same time removed the mystery which has hitherto adhered +to the application of this principle on account of the apparent +capriciousness of the appearance of predicted combination lines. +Especially attention may be drawn to the simple interpretation +which the quantum theory offers of the appearance observed by +Stark and his collaborators of certain new series of lines, which do +not appear under ordinary circumstances, but which are excited +when the emitting atoms are subject to intense external electric +fields. In fact, on the correspondence principle this is immediately +explained from an examination of the perturbations in the motion +of the outer electron which give rise to the appearance in this +motion---besides the harmonic components already present in a +simple central orbit---of a number of constituent harmonic vibrations +of new type and of amplitudes proportional to the intensity +of the external forces. + +It may be of interest to note that an investigation of the +limitation of the possibility of transitions between stationary +states, based upon a simple consideration of conservation of angular +momentum during the process of radiation, does not, contrary to +what has previously been supposed (compare Essay~II, \PageRef{62}), +suffice to throw light on the remarkably simple structure of series +spectra illustrated by the figure. As mentioned above we must +assume that the ``complexity'' of the spectral terms, corresponding +to given values of $n$~and~$k$, which we witness in the fine +structure of the spectral lines, may be ascribed to states, corresponding +to different values of this angular momentum, in +which the plane of the electronic orbit is orientated in a different +manner, relative to the configuration of the previously bound +electrons in the atom. Considerations of conservation of angular +momentum can, in connection with the series spectra, therefore only +contribute to an understanding of the limitation of the possibilities +\PageSep{85} +of combination observed in the peculiar laws applying to the +number of components in the complex structure of the lines. So +far as the last question is concerned, such considerations offer a +direct support for the consequences of the correspondence principle. + + +\Chapter{III.}{Formation of Atoms and the Periodic Table} + +A correspondence has been shown to exist between the motion +of the electron last captured and the occurrence of transitions +between the stationary states corresponding to the various stages +of the binding process. This fact gives a point of departure for a +choice between the numerous possibilities which present themselves +when considering the formation of the atoms by the successive +capture and binding of the electrons. Among the processes which +are conceivable and which according to the quantum theory might +occur in the atom we shall reject those whose occurrence \Chg{can not}{cannot} be +regarded as consistent with a correspondence of the required nature. + +\Section{First Period. Hydrogen---Helium.} It will not be necessary to +concern ourselves long with the question of the constitution of the +hydrogen atom. From what has been said previously we may assume +that the final result of the process of \emph{binding of the first electron} in +any atom will be a stationary state, where the energy of the atom +is given by~\Eq{(5)}, if we put $n = 1$, or more precisely by formula~\Eq{(11)}, +if we put $n = 1$ and $k = 1$. The orbit of the electron will be a circle +whose radius will be given by formulae~\Eq{(10)}, if $n$~and $k$ are each +put equal to~$1$. Such an orbit will be called a $1$-quantum orbit, +and in general an orbit for which the principal quantum number +has a given value~$n$ will be called an $n$-quantum orbit. Where it +is necessary to differentiate between orbits corresponding to various +values of the quantum number~$k$, a central orbit, characterized by +given values of the quantum numbers $n$~and~$k$, will be referred to +as an $n_{k}$~orbit. + +In the question of the constitution of the helium atom we meet +the much more complicated problem of the \emph{binding of the second +electron}. Information about this binding process may, however, be +obtained from the arc spectrum of helium. This spectrum, as +opposed to most other simple spectra, consists of two complete +systems of lines with frequencies given by formulae of the type~\Eq{(12)}. +\PageSep{86} +On this account helium was at first assumed to be a mixture +of two different gases, ``orthohelium'' and ``parhelium,'' but now we +know that the two spectra simply mean that the binding of the second +electron can occur in two different ways. A theoretical explanation of +the main features of the helium spectrum has recently been attempted +in an interesting paper by Landé. He supposes the emission of the +orthohelium spectrum to be due to transitions between stationary +states where both electrons move in the same plane and revolve +in the same sense. The parhelium spectrum, on the other hand, is +ascribed by him to stationary states where the planes of the orbits +form an angle with each other. Dr~Kramers and I have made a +closer investigation of the interaction between the two orbits in +the different stationary states. The results of our investigation +which was begun several years before the appearance of Landé's +work have not yet been published. Without going into details +I may say, that even though our results in several respects differ +materially from those of Landé (compare Essay~II, \PageRef{56}), we agree +with his general conclusions concerning the origin of the orthohelium +and parhelium spectra. + +The final result of the binding of the second electron is intimately +related to the origin of the two helium spectra. Important +information on this point has been obtained recently by Franck +and his co-workers. As is well known he has thrown light upon +many features of the structure of the atom and of the origin +of spectra by observing the effect of bombarding atoms by +electrons of various velocities. A short time ago these experiments +showed that the impact of electrons could bring helium into a +``metastable'' state from which the atom cannot return to its +normal state by means of a simple transition accompanied by the +emission of radiation, but only by means of a process analogous to +a chemical reaction involving interaction with atoms of other +elements. This result is closely connected with the fact that the +binding of the second electron can occur in two different ways, as +is shown by the occurrence of two distinct spectra. Thus it is +evident from Franck's experiments that the normal state of the +atom is the last stage in the binding process involving the emission +of the parhelium spectrum by which the electron last captured as +well as the one first captured will be bound in a $1_{1}$~orbit. The +\PageSep{87} +metastable state, on the contrary, is the final stage of the process +giving the orthohelium spectrum. In this case the second electron, +as opposed to the first, will move in a $2_{1}$~orbit. This corresponds to +a firmness of binding which is about six times less than for the +electron in the normal state of the atom. + +If we now consider somewhat more closely this apparently +surprising result, it is found that a clear grasp of it may be obtained +from the point of view of correspondence. It can be shown that +the coherent class of motions to which the orthohelium orbits +belong does not contain a $1_{1}$~orbit. If on the whole we would claim +the existence of a state where the two electrons move in $1_{1}$~orbits +in the same plane, and if in addition it is claimed that the motion +should possess the periodic properties necessary for the definition +of stationary states, then there seems that no possibility is afforded +other than the assumption that the two electrons move around the +nucleus in one and the same orbit, in such a manner that at each +moment they are situated at the ends of a diameter. This extremely +simple ring-configuration might be expected to correspond to +the firmest possible binding of the electrons in the atom, and it +was on this account proposed as a model for the helium atom in +my first paper on atomic structure. If, however, we inquire about +the possibility of a transition from one of the orthohelium states +to a configuration of this type we meet conditions which are very +different from those which apply to transitions between two of +the orthohelium orbits. In fact, the occurrence of each of these +transitions is due to the existence of well-defined corresponding +constituent harmonic vibration in the central orbits which the outer +electron describes in the class of motions to which the stationary +states belong. The transition we have to discuss, on the other +hand, is one by which the last captured electron is transferred from +a state in which it is moving ``outside'' the other to a state in which +it moves round the nucleus on equal terms with the other electron. +Now it is impossible to find a series of simple intermediate forms +for the motion of those two electrons in which the orbit of the last +captured electron exhibits a sufficient similarity to a central motion +that for this transition there could be a correspondence of the +necessary kind. It is therefore evident, that where the two electrons +move in the same plane, the electron captured last \Chg{can not}{cannot} be +\PageSep{88} +bound firmer than in a $2_{1}$~orbit. If, on the other hand, we consider +the binding process which accompanies the emission of the parhelium +spectrum and where the electrons in the stationary states move in +orbits whose planes form angles with one another we meet essentially +different conditions. A corresponding intimate change in the +interaction between the electron last captured and the one previously +bound is not required here for the two electrons in the atom to +become equivalent. We may therefore imagine the last stage of +the binding process to take place in a manner similar to those +stages corresponding to transitions between orbits characterized by +greater values of $n$~and~$k$. + +In the \emph{normal state of the helium atom} the two electrons must +be assumed to move in equivalent $1_{1}$~orbits. As a first approximation +these may be described as two circular orbits, whose planes make +an angle of~$120°$ with one another, in agreement with the conditions +which the angular momentum of an atom according to the quantum +theory must satisfy. On account of the interaction between the +two electrons these planes at the same time turn slowly around +the fixed impulse axis of the atom. Starting from a distinctly +different point of view Kemble has recently suggested a similar +model for the helium atom. He has at the same time directed +attention to a possible type of motion of very marked symmetry +in which the electrons during their entire revolution assume +symmetrical positions with reference to a fixed axis. Kemble has +not, however, investigated this motion further. Previous to the +appearance of this paper Kramers had commenced a closer investigation +of precisely this type of motion in order to find out to what +extent it was possible from such a calculation to account for the +firmness with which the electrons are bound in the helium atom, +that is to account for the ionization potential. Early measurements +of this potential had given values corresponding approximately to +that which would result from the ring-configuration already mentioned. +This requires $17/8$~as much work to remove a single +electron as is necessary to remove an electron from the hydrogen +atom in its normal state. As the theoretical value for the latter +amount of work---which for the sake of simplicity will be represented +by~$W$---corresponds to an ionization potential of $13.53$~volts, +the ionization potential of helium would be expected to be $28.8$~volts. +\PageSep{89} +Recent and more accurate determinations, however, have +given a value for the ionization potential of helium which is considerably +lower and lies in the neighbourhood of $25$~volts. This +showed therefore the untenability of the ring-configuration quite +independently of any other considerations. A careful investigation of +the spatial atomic configuration requires elaborate calculation, and +Kramers has not yet obtained final results. With the approximation +to which they have been so far completed the calculations point to +the possibility of an agreement with the experimental results. The +final result may be awaited with great interest, since it offers in +the simplest case imaginable a test of the principles by which we +are attempting to determine stationary states of atoms containing +more than one electron. + +Hydrogen and helium, as seen in the survey of the periodic +system given in \Fig{1}, together form the first period in the system +of elements, since helium is the first of the inactive gases. The great +difference in the chemical properties of hydrogen and helium is +closely related to the great difference in the nature of the binding +of the electron. This is directly indicated by the spectra and +ionization potentials. While helium possesses the highest known +ionization potential of all the elements, the binding of the electron +in the hydrogen atom is sufficiently loose to account for the tendency +of hydrogen to form positive ions in aqueous solutions and chemical +combinations. Further consideration of this particular question +requires, however, a comparison between the nature and firmness +of the electronic configurations of other atoms, and it can therefore +not be discussed at the moment. + +\Section{Second Period. Lithium---Neon.} When considering the atomic +structure of elements which contain more than two electrons in the +neutral atom, we shall assume first of all that what has previously +been said about the formation of the helium atom will in the main +features also apply to the capture and binding of the first two +electrons. These electrons may, therefore, in the normal state of +the atom be regarded as moving in equivalent orbits characterized +by the quantum symbol~$1_{1}$. We obtain direct information about +the \emph{binding of the third electron} from the spectrum of lithium. +This spectrum shows the existence of a number of series of +\PageSep{90} +stationary states, where the firmness with which the last captured +electron is bound is very nearly the same as in the stationary states +of the hydrogen atom. These states correspond to orbits where $k$~is +greater than or equal to~$2$, and where the last captured electron +moves entirely outside the region where the first two electrons +move. But in addition this spectrum gives us information about a +series of states corresponding to $k = 1$ in which the energy differs +essentially from the corresponding stationary states of the hydrogen +atom. In these states the last captured electron, even if it remains +at a considerable distance from the nucleus during the greater part +of its revolution, will at certain moments during the revolution +approach to a distance from the nucleus which is of the same order +of magnitude as the dimensions of the orbits of the previously +bound electrons. On this account the electrons will be bound with +a firmness which is considerably greater than that with which the +electrons are bound in the stationary states of the hydrogen atom +corresponding to the same value of~$n$. + +Now as regards the lithium spectrum as well as the other alkali +spectra we are so fortunate (see \PageRef{32}) as to possess definite evidence +about the normal state of the atom from experiments on selective +absorption. In fact these experiments tell us that the first member +of the sequence of $S$-terms corresponds to this state. This term +corresponds to a strength of binding which is only a little more than +a third of that of the hydrogen atom. We must therefore conclude +that the outer electron in the normal state of the lithium atom +moves in a $2_{1}$~orbit, just as the outer electron in the metastable +state of the helium atom. The reason why the binding of the +outer electron \Chg{can not}{cannot} proceed to an orbit characterized by a smaller +value for the total quantum number may also be considered as +analogous in the two cases. In fact, a transition by which the third +electron in the lithium atom was ultimately bound in a $1_{1}$~orbit +would lead to a state in the atom in which the electron would play +an equivalent part with the two electrons previously bound. Such +a process would be of a type entirely different from the transitions +between the stationary states connected with the emission of the +lithium spectrum, and would, contrary to these, not exhibit a +correspondence with a harmonic component in the motion of the +atom. +\PageSep{91} + +We obtain, therefore, a picture of the formation and structure of +the lithium atom which offers a natural explanation of the great +difference of the chemical properties of lithium from those of helium +and hydrogen. This difference is at once explained by the fact that +the firmness by which the last captured electron is bound in its +$2_{1}$~orbit in the lithium atom is only about a third of that with which +the electron in the hydrogen atom is held, and almost five times +smaller than the firmness of the binding of the electrons in the +helium atom. + +What has been said here applies not alone to the formation of +the lithium atom, but may also be assumed to apply to the binding +of the third electron in every atom, so that in contrast to the first +two electrons which move in $1_{1}$~orbits this may be assumed to move +in a $2_{1}$~orbit. As regards the \emph{binding of the fourth, fifth and sixth +electrons} in the atom, we do not possess a similar guide as no simple +series spectra are known of beryllium, boron and carbon. Although +conclusions of the same degree of certainty \Chg{can not}{cannot} be reached it +seems possible, however, to arrive at results consistent with general +physical and chemical evidence by proceeding by means of considerations +of the same kind as those applied to the binding of the +first three electrons. In fact, we shall assume that the fourth, fifth +and sixth electrons will be bound in $2_{1}$~orbits. The reason why the +binding of a first electron in an orbit of this type will not prevent the +capture of the others in two quanta orbits may be ascribed to the fact +that $2_{1}$~orbits are not circular but very \Chg{excentric}{eccentric}; For example, the +$3$rd~electron cannot keep the remaining electrons away from the inner +system in the same way in which the first two electrons bound in +the lithium atom prevent the third from being bound in a +$1$-quantum orbit. Thus we shall expect that the $4$th, $5$th and $6$th +electrons in a similar way to the $3$rd will at certain moments of +their revolution enter into the region where the first two +bound electrons move. We must not imagine, however, that these +visits into the inner system take place at the same time, but +that the four electrons visit the nucleus separately at equal +intervals of time. In earlier work on atomic structure it was supposed +that the electrons in the various groups in the atom moved +in separate regions within the atom and that at each moment the +electrons within each separate group were arranged in configurations +\PageSep{92} +possessing symmetry like that of a regular polygon or polyhedron. +Among other things this involved that the electrons in each group +were supposed to be at the point of the orbit nearest the nucleus +at the same time. A structure of this kind may be described as one +where the motions of the electrons within the groups are coupled +together in a manner which is largely independent of the interaction +between the various groups. On the contrary, the characteristic +feature of a structure like that I have suggested is the \emph{intimate +coupling between the motions of the electrons in the various groups} +characterized by different quantum numbers, as well as the \emph{greater +independence in the mode of binding within one and the same group +of electrons} the orbits of which are characterized by the same +quantum number. In emphasizing this last feature I have two +points in mind. Firstly the smaller effect of the presence of previously +bound electrons on the firmness of binding of succeeding +electrons in the same group. Secondly the way in which the motions +of the electrons within the group reflect the independence both of +the processes by which the group can be formed and by which it +can be reorganized by change of position of the different electrons +in the atom after a disturbance by external forces. The last point +will be considered more closely when we deal with the origin and +nature of the X-ray spectra; for the present we shall continue the +consideration of the structure of the atom to which we are led by +the investigation of the processes connected with the successive +capture of the electrons. + +The preceding considerations enable us to understand the fact +that the two elements beryllium and boron immediately succeeding +lithium can appear electropositively with $2$~and $3$~valencies respectively +in combination with other substances. For like the third +electron in the lithium atom, the last captured electrons in these +elements will be much more lightly bound than the first two +electrons. At the same time we understand why the electropositive +character of these elements is less marked than in the case of +lithium, since the electrons in the $2$-quanta orbits will be much +more firmly bound on account of the stronger field in which they +are moving. New conditions arise, however, in the case of the +next element, carbon, as this element in its typical chemical combinations +\Chg{can not}{cannot} be supposed to occur as an ion, but rather as a +\PageSep{93} +neutral atom. This must be assumed to be due not only to the great +firmness in the binding of the electrons but also to be an essential +consequence of the symmetrical configuration of the electrons. + +With the binding of the $4$th, $5$th and $6$th electrons in $2_{1}$~orbits, +the spatial symmetry of the regular configuration of the orbits +must be regarded as steadily increasing, until with the binding of +the $6$th electron the orbits of the four last bound electrons may be +expected to form an exceptionally symmetrical configuration in +which the normals to the planes of the orbits occupy positions +relative to one another nearly the same as the lines from the centre +to the vertices of a regular tetrahedron. Such a configuration +of groups of $2$-quanta orbits in the carbon atom seems capable +of furnishing a suitable foundation for explaining the structure of +organic compounds. I shall not discuss this question any further, +for it would require a thorough study of the interaction between +the motions of the electrons in the atoms forming the molecule. +I might mention, however, that the types of molecular models to +which we are led are very different from the molecular models +which were suggested in my first papers. In these the chemical +``valence bonds'' were represented by ``electron rings'' of the same +type as those which were assumed to compose the groups of +electrons within the individual atoms. It is nevertheless possible +to give a general explanation of the chemical properties of the +elements without touching on those matters at all. This is largely +due to the fact that the structures of combinations of atoms of the +same element and of many organic compounds do not have the +same significance for our purpose as those molecular structures in +which the individual atoms occur as electrically charged ions. The +latter kind of compounds, to which the greater number of simple +inorganic compounds belong, is frequently called ``heteropolar'' and +possesses a far more typical character than the first compounds +which are called ``homoeopolar,'' and whose properties to quite a +different degree exhibit the individual peculiarities of the elements. +My main purpose will therefore be to consider the fitness which +the configurations of the electrons in the various atoms offer for +the formation of ions. + +Before leaving the carbon atom I should mention, that a model +of this atom in which the orbits of the four most lightly bound +\PageSep{94} +electrons possess a pronounced tetrahedric symmetry had already +been suggested by Landé. In order to agree with the measurements +of the size of the atoms he also assumed that these electrons moved +in $2_{1}$~orbits. There is, however, this difference between Landé's +view and that given here, that while Landé deduced the characteristic +properties of the carbon atom solely from an investigation of +the simplest form of motion which four electrons can execute +employing spatial symmetry, our view originates from a consideration +of the stability of the whole atom. For our assumptions about +the orbits of the electrons are based directly on an investigation of +the interaction between these electrons and the first two bound +electrons. The result is that our model of the carbon atom has +dynamic properties which are essentially different from the properties +of Landé's model. + +In order to account for the properties of \emph{the elements in the second +half of the second period} it will first of all be necessary to show +why the configuration of ten electrons occurring in the neutral atom +of neon possesses such a remarkable degree of stability. Previously +it has been assumed that the properties of this configuration were +due to the interaction between eight electrons which moved in +equivalent orbits outside the nucleus and an inner group of two +electrons like that in the helium atom. It will be seen, however, +that the solution must be sought in an entirely different direction. +It \Chg{can not}{cannot} be expected that \emph{the $7$th electron} will be bound in a $2_{1}$~orbit +equivalent to the orbits of the four preceding electrons. The occurrence +of five such orbits would so definitely destroy the symmetry +in the interaction of these electrons that it is inconceivable that a +process resulting in the accession of a fifth electron to this group +would be in agreement with the correspondence principle. On the +contrary it will be necessary to assume that the four electrons in +their exceptionally symmetrical orbital configuration will keep out +later captured electrons with the result that these electrons will be +bound in orbits of other types. + +The orbits which come into consideration for the $7$th electron in +the nitrogen atom and the $7$th, $8$th, $9$th and $10$th electrons in the +atoms of the immediately following elements will be circular orbits +of the type~$2_{2}$. The diameters of these orbits are considerably larger +than those of the $l_{1}$~orbits of the first two electrons; on the other +\PageSep{95} +hand the outermost part of the \Chg{excentric}{eccentric} $2_{1}$~orbits will extend some +distance beyond these circular $2_{2}$~orbits. I shall not here discuss the +capture and binding of these electrons. This requires a further investigation +of the interaction between the motions of the electrons +in the two types of $2$-quanta orbits. I shall simply mention, that +in the atom of neon in which we will assume that there are four +electrons in $2_{2}$~orbits the planes of these orbits must be regarded not +only as occupying a position relative to one another characterized +by a high degree of spatial symmetry, but also as possessing a +configuration harmonizing with the four elliptical $2_{1}$~orbits. An +interaction of this kind in which the orbital planes do not +coincide can be attained only if the configurations in both subgroups +exhibit a systematic deviation from tetrahedral symmetry. +This will have the result that the electron groups with $2$-quanta +orbits in the neon atom will have only a single axis of symmetry +which must be supposed to coincide with the axis of symmetry of +the innermost group of two electrons. + +Before leaving the description of the elements within the second +period it may be pointed out that the above considerations offer a +basis for interpreting that tendency of the neutral atoms of oxygen +and fluorine for capturing further electrons which is responsible for +the marked electronegative character of these elements. In fact, +this tendency may be ascribed to the fact that the orbits of +the last captured electrons will find their place within the region, +in which the previously captured electrons move in $2_{1}$~orbits. This +suggests an explanation of the great difference between the properties +of the elements in the latter half of the second period of the +periodic system and those of the elements in the first half, in whose +atoms there is only a single type of $2$-quanta orbits. + +\Section{Third Period. Sodium---Argon.} We shall now consider the +structure of atoms of elements in the third period of the periodic +system. This brings us immediately 'to the question of \emph{the binding +of the $11$th electron} in the atom. Here we meet conditions which +in some respects are analogous to those connected with the binding +of the $7$th electron. The same type of argument that applied to +the carbon atom shows that the symmetry of the configuration in +the neon atom would be essentially, if not entirely, destroyed by +\PageSep{96} +the addition of another electron in an orbit of the same type as +that in which the last captured electrons were bound. Just as in +the case of the $3$rd~and $7$th electrons we may therefore expect to +meet a new type of orbit for the 11th electron in the atom, and the +orbits which present themselves this time are the $3_{1}$~orbits. An +electron in such an orbit will for the greater part of the time remain +outside the orbits of the first ten electrons. But at certain moments +during the revolution it will penetrate not only into the region of +the $2$-quanta orbits, but like the $2_{1}$~orbits it will penetrate to +distances from the nucleus which are smaller than the radii of +the $1$-quantum orbits of the two electrons first bound. This fact, +which has a most important bearing on the stability of the atom, +leads to a peculiar result as regards the binding of the $11$th electron. +In the sodium atom this electron will move in a field which so far +as the outer part of the orbit is concerned deviates only very little +from that surrounding the nucleus in the hydrogen atom, but the +dimensions of this part of the orbit will, nevertheless, be essentially +different from the dimensions of the corresponding part of a $3_{1}$~orbit +in the hydrogen atom. This arises from the fact, that even +though the electron only enters the inner configuration of the first +ten electrons for short intervals during its revolution, this part of +the orbit will nevertheless exert an essential influence upon the +determination of the principal quantum number. This is directly +related to the fact that the motion of the electron in the first part +of the orbit deviates only a little from the motion which each of +the previously bound electrons in $2_{1}$~orbits executes during a complete +revolution. The uncertainty which has prevailed in the +determination of the quantum numbers for the stationary states +corresponding to a spectrum like that of sodium is connected with +this. This question has been discussed by several physicists. From +a comparison of the spectral terms of the various alkali metals, +Roschdestwensky has drawn the conclusion that the normal state +does not, as we might be inclined to expect a~priori, correspond to +a $1_{1}$~orbit as shown in \Fig{2} on \PageRef{79}, but that this state corresponds +to a $2_{1}$~orbit. Schrödinger has arrived at a similar result +in an attempt to account for the great difference between the +$S$~terms and the terms in the $P$~and $D$ series of the alkali spectra. +He assumes that the ``outer'' electron in the states corresponding +\PageSep{97} +to the $S$~terms---in contrast to those corresponding to the $P$~and +$D$ terms---penetrates partly into the region of the orbits of the +inner electrons during the course of its revolution. These investigations +contain without doubt important hints, but in reality the +conditions must be very different for the different alkali spectra. +Instead of a $2_{1}$~orbit as in lithium we must thus assume for +the spectrum of sodium not only that the first spectral term in +the $S$~series corresponds to a $3_{1}$~orbit, but also, as a more detailed +consideration shows, that the first term in the $P$~series corresponds +not to a $2_{2}$~orbit as indicated in \Fig{2}, but to a $3_{2}$~orbit. If the +numbers in this figure were correct, it would require among other +things that the $P$~terms should be smaller than the hydrogen terms +\Figure{3}{97} +corresponding to the same principal quantum number. This would +mean that the average effect of the inner electrons could be described +as a repulsion greater than would occur if their total electrical charge +were united in the nucleus. This, however, \Chg{can not}{cannot} be expected from +our view of atomic structure. The fact that the last captured electron, +at any rate for low values of~$k$, revolves partly inside the orbits of the +previously bound electrons will on the contrary involve that the +presence of these electrons will give rise to a virtual repulsion +which is considerably smaller than that which would be due to +their combined charges. Instead of the curves drawn between +points in \Fig{2} which represent stationary states corresponding +to the same value of the principal quantum number running from +right to left, we obtain curves which run from left to right, as +is indicated in \Fig{3}. The stationary states are labelled with +\PageSep{98} +quantum numbers corresponding to the structure I have described. +According to the view underlying \Fig{2} the sodium spectrum +might be described simply as a distorted hydrogen spectrum, +whereas according to \Fig{3} there is not only distortion but also +complete disappearance of certain terms of low quantum numbers. +It may be stated, that this view not only appears to offer an explanation +of the magnitude of the terms, but that the complexity +of the terms in the $P$~and $D$ series finds a natural explanation in +the deviation of the configuration of the ten electrons first bound +from a purely central symmetry. This lack of symmetry has its +origin in the configuration of the two innermost electrons and +``transmits'' itself to the outer parts of the atomic structure, since +the $2_{1}$~orbits penetrate partly into the region of these electrons. + +This view of the sodium spectrum provides at the same time an +immediate explanation of the pronounced electropositive properties +of sodium, since the last bound electron in the sodium atom is still +more loosely bound than the last captured electron in the lithium +atom. In this connection it might be mentioned that the increase +in atomic volume with increasing atomic number in the family of +the alkali metals finds a simple explanation in the successively +looser binding of the valency electrons. In his work on the X-ray +spectra Sommerfeld at an earlier period regarded this increase in +the atomic volumes as supporting the assumption that the principal +quantum number of the orbit of the valency electrons increases by +unity as we pass from one metal to the next in the family. His +later investigations on the series spectra have led him, however, +definitely to abandon this assumption. At first sight it might also +appear to entail a far greater increase in the atomic volume than +that actually observed. A simple explanation of this fact is however +afforded by realizing that the orbit of the electron will run +partly inside the region of the inner orbit and that therefore the +``effective'' quantum number which corresponds to the outer almost +elliptical loop will be much smaller than the principal quantum +number, by which the whole central orbit is described. It may +be mentioned that Vegard in his investigations on the X-ray spectra +has also proposed the assumption of successively increasing quantum +numbers for the electronic orbits in the various groups of the atom, +reckoned from the nucleus outward. He has introduced assumptions +\PageSep{99} +about the relations between the numbers of electrons in the various +groups of the atom and the lengths of the periods in the periodic +system which exhibit certain formal similarities with the results +presented here. But Vegard's considerations do not offer points of +departure for a further consideration of the evolution and stability +of the groups, and consequently no basis for a detailed interpretation +of the properties of the elements. + +When we consider the elements following sodium in the third +period of the periodic system we meet in \emph{the binding of the $12$th, +$13$th and $14$th electrons} conditions which are analogous to those +we met in the binding of the $4$th, $5$th and $6$th electrons. In the +elements of the third periods, however, we possess a far more +detailed knowledge of the series spectra. Too little is known +about the beryllium spectrum to draw conclusions about the +binding of the fourth electron, but we may infer directly from the +well-known arc spectrum of magnesium that the $12$th electron +in the atom of this element is bound in a $3_{1}$~orbit. As regards +the binding of the $13$th electron we meet in aluminium an +absorption spectrum different in structure to that of the alkali +metals. In fact here not the lines of the principal series but the +lines of the sharp and diffuse series are absorption lines. Consequently +it is the first member of the $P$~terms and not of the $S$~terms +which corresponds to the normal state of the aluminium +atom, and we must assume that the $13$th electron is bound in +a $3_{2}$~orbit. This, however, would hardly seem to be a general +property of the binding of the $13$th electron in atoms, but rather +to arise from the special conditions for the binding of the last +electron in an atom, where already there are two other electrons +bound as loosely as the valency electron of aluminium. At the +present state of the theory it seems best to assume that in the +silicon atom the four last captured electrons will move in $3_{1}$~orbits +forming a configuration possessing symmetrical properties +similar to the outer configuration of the four electrons in $2_{1}$~orbits +in carbon. Like what we assumed for the latter configuration we +shall expect that the configuration of the $3_{1}$~orbits occurring for the +first time in silicon possesses such a completion, that the addition +of a further electron in a $3_{1}$~orbit to the atom of the following elements +is impossible, and that \emph{the $15$th electron} in the elements of +\PageSep{100} +higher atomic number will be bound in a new type of orbit. In this +case, however, the orbits with which we meet will not be circular, +as in the capture of the $7$th electron, but will be rotating \Chg{excentric}{eccentric} +orbits of the type~$3_{2}$. This is very closely related to the fact, mentioned +above, that the non-circular orbits will correspond to a +firmer binding than the circular orbits having the same value for +the principal quantum number, since the electrons will at certain +moments penetrate much farther into the interior of the atom. +Even though a $3_{2}$~orbit will not penetrate into the innermost configuration +of $1_{1}$~orbits, it will penetrate to distances from the nucleus +which are considerably less than the radii of the circular $2_{}2$~orbits. +In the case of the $16$th, $17$th and $18$th electrons the conditions are +similar to those for the $15$th. So for argon we may expect a configuration +in which the ten innermost electrons move in orbits of +the same type as in the neon atom while the last eight electrons will +form a configuration of four $3_{1}$~orbits and four $3_{2}$~orbits, whose +symmetrical properties must be regarded as closely corresponding +to the configuration of $2$-quanta orbits in the neon atom. At the +same time, as this picture suggests a qualitative explanation of the +similarity of the chemical properties of the elements in the latter +part of the second and third periods, it also opens up the possibility +of a natural explanation of the conspicuous difference from a +quantitative aspect. + +\Section{Fourth Period. Potassium---Krypton.} In the fourth period +we meet at first elements which resemble chemically those at the +beginning of the two previous periods. This is also what we should +expect. We must thus assume that \emph{the $19$th electron} is bound in +a new type of orbit, and a closer consideration shows that this will +be a $4_{1}$~orbit. The points which were emphasized in connection +with the binding of the last electron in the sodium atom will be +even more marked here on account of the larger quantum number +by which the orbits of the inner electrons are characterized. In +fact, in the potassium atom the $4_{1}$~orbit of the $19$th electron will, +as far as inner loops are concerned, coincide closely with the shape +of a $3_{1}$~orbit. On this account, therefore, the dimensions of the +outer part of the orbit will not only deviate greatly from the +dimensions of a $4_{1}$~orbit in the hydrogen atom, but will coincide +\PageSep{101} +closely with a hydrogen orbit of the type~$2_{1}$, the dimensions of +which are about four times smaller than the $4_{1}$~hydrogen orbit. +This result allows an immediate explanation of the main features of +the chemical properties and the spectrum of potassium. Corresponding +results apply to calcium, in the neutral atom of which +there will be two valency electrons in equivalent $4_{1}$~orbits. + +After calcium the properties of the elements in the fourth period +of the periodic system deviate, however, more and more from the +corresponding elements in the previous periods, until in the family +of the iron metals we meet elements whose properties are essentially +different. Proceeding to still higher atomic numbers we again +meet different conditions. Thus we find in the latter part of the +fourth period a series of elements whose chemical properties approach +more and more to the properties of the elements at the end +of the preceding periods, until finally with atomic number~$36$ we +again meet one of the inactive gases, namely krypton. This is +exactly what we should expect. The formation and stability of the +atoms of the elements in the first three periods require that each +of the first $18$ electrons in the atom shall be bound in each succeeding +element in an orbit of the same principal quantum number +as that possessed by the particular electron, when it first appeared. +It is readily seen that this is no longer the case for the $19$th +electron. With increasing nuclear charge and the consequent +decrease in the difference between the fields of force inside and +outside the region of the orbits of the first $18$ bound electrons, the +dimensions of those parts of a $4_{1}$~orbit which fall outside will +approach more and more to the dimensions of a $4$-quantum orbit +calculated on the assumption that the interaction between the +electrons in the atom may be neglected. \emph{With increasing atomic +number a point will therefore be reached where a $3_{3}$~orbit will correspond +to a firmer binding of the $19$th electron than a $4_{1}$~orbit}, and +this occurs as early as at the beginning of the fourth period. This +cannot only be anticipated from a simple calculation but is confirmed +in a striking way from an examination of the series spectra. While +the spectrum of potassium indicates that the $4_{1}$~orbit corresponds +to a binding which is more than twice as firm as in a $3_{3}$~orbit +corresponding to the first spectral term in the $D$~series, the conditions +are entirely different as soon as calcium is reached. We +\PageSep{102} +shall not consider the arc spectrum which is emitted during the +capture of the $20$th electron but the spark spectrum which corresponds +to the capture and binding of the $19$th electron. While the +spark spectrum of magnesium exhibits great similarity with the +sodium spectrum as regards the values of the spectral terms in the +various series---apart from the fact that the constant appearing in +formula~\Eq{(12)} is four times as large as the Rydberg constant---we +meet in the spark spectrum of calcium the remarkable condition +\Figure{4}{102} +that the first term of the $D$~series is larger than the first term of +the $P$~series and is only a little smaller than the first term of the +$S$~series, which may be regarded as corresponding to the binding +of the $19$th electron in the normal state of the calcium atom. +These facts are shown in \Fig[figure]{4} which gives a survey of the +stationary states corresponding to the arc spectra of sodium and +potassium. As in figures \FigNum{2} and~\FigNum{3} of the sodium spectrum, we +have disregarded the complexity of the spectral terms, and the +numbers characterizing the stationary states are simply the quantum +\PageSep{103} +numbers $n$~and~$k$. For the sake of comparison the scale in which the +energy of the different states is indicated is chosen four times as +small for the spark spectra as for the arc spectra. Consequently +the vertical lines indicated with various values of~$n$ correspond for +the arc spectra to the spectral terms of hydrogen, for the spark +spectra to the terms of the helium spectrum given by formula~\Eq{(7)}. +Comparing the change in the relative firmness in the binding of +the $19$th electron in a $4_{1}$~and $3_{3}$~orbit for potassium and calcium we +see that we must be prepared already for the next element, +scandium, to find that the $3_{3}$~orbit will correspond to a stronger +binding of this electron than a $4_{1}$~orbit. On the other hand it +follows from previous remarks that the binding will be much lighter +than for the first $18$ electrons which agrees that in chemical combinations +scandium appears electropositively with three valencies. + +If we proceed to the following elements, a still larger number of +$3_{3}$~orbits will occur in the normal state of these atoms, since the +number of such electron orbits will depend upon the firmness of +their binding compared to the firmness with which an electron is +bound in a $4_{1}$~orbit, in which type of orbit at least the last captured +electron in the atom may be assumed to move. We therefore meet +conditions which are essentially different from those which we have +considered in connection with the previous periods, so that here +we have to do with \emph{the successive development of one of the inner +groups of electrons in the atom}, in this case with groups of electrons +in $3$-quanta orbits. Only when the development of this group has +been completed may we expect to find once more a corresponding +change in the properties of the elements with increasing atomic +number such as we find in the preceding periods. The properties +of the elements in the latter part of the fourth period show +immediately that the group, when completed, will possess $18$~electrons. +Thus in krypton, for example, we may expect besides +the groups of $1$,~$2$ and $3$-quanta orbits a markedly symmetrical +configuration of $8$~electrons in $4$-quanta orbits consisting of four $4_{1}$~orbits +and four $4_{2}$~orbits. + +The question now arises: In which way will the gradual formation +of the group of electrons having $3$-quanta orbits take place? +From analogy with the constitution of the groups of electrons with +$2$-quanta orbits we might at first sight be inclined to suppose that +\PageSep{104} +the complete group of $3$-quanta orbits would consist of three subgroups +of four electrons each in orbits of the types $3_{1}$,~$3_{2}$ and~$3_{3}$ +respectively, so that the total number of electrons would be $12$ +instead of~$18$. Further consideration shows, however, that such an +expectation would not be justified. The stability of the configuration +of eight electrons with $2$-quanta orbits occurring in neon must +be ascribed not only to the symmetrical configuration of the electronic +orbits in the two subgroups of $2_{1}$~and $2_{2}$ orbits respectively, +but fully as much to the possibility of bringing the orbits inside these +subgroups into harmonic relation with one another. The situation +is different, however, for the groups of electrons with $3$-quanta +orbits. Three subgroups of four orbits each \Chg{can not}{cannot} in this case be +expected to come into interaction with one another in a correspondingly +simple manner. On the contrary we must assume that +the presence of electrons in $3_{3}$~orbits will diminish the harmony of +the orbits within the first two $3$-quanta subgroups, at any rate +when a point is reached where the $19$th electron is no longer, as +was the case with scandium, bound considerably more lightly than +the previously bound electrons in $3$-quanta orbits, but has been +drawn so far into the atom that it revolves within essentially +the same region of the atom where these electrons move. We +shall now assume that this decrease in the harmony will so to +say ``open'' the previously ``closed'' configuration of electrons +in orbits of these types. As regards the final result, the number~$18$ +indicates that after the group is finally formed there will +be three subgroups containing six electrons each. Even if it has +not at present been possible to follow in detail the various +steps in the formation of the group this result is nevertheless +confirmed in an interesting manner by the fact that it is possible +to arrange three configurations having six electrons each in a simple +manner relative to one another. The configuration of the subgroups +does not exhibit a tetrahedral symmetry like the groups of $2$-quanta +orbits in carbon, but a symmetry which, so far as the relative +orientation of the normals to the planes of the orbits is concerned, +may be described as trigonal. + +In spite of the great difference in the properties of the elements +of this period, compared with those of the preceding period, the +completion of the group of $18$~electrons in $3$-quanta orbits in the +\PageSep{105} +fourth period may to a certain extent be said to have the same +characteristic results as the completion of the group of $2$-quanta +orbits in the second period. As we have seen, this determined not +only the properties of neon as an inactive gas, but in addition the +electronegative properties of the preceding elements and the +electropositive properties of the elements which follow. The fact +that there is no inactive gas possessing an outer group of $18$~electrons +is very easily accounted for by the much larger dimensions +which a $3_{3}$~orbit has in comparison with a $2_{2}$~orbit revolving in the +same field of force. On this account a complete $3$-quanta group +\Chg{can not}{cannot} occur as the outermost group in a neutral atom, but only +in positively charged ions. The characteristic decrease in valency +which we meet in copper, shown by the appearance of the singly +charged cuprous ions, indicates the same tendency towards the +completion of a symmetrical configuration of electrons that we +found in the marked electronegative character of an element like +fluorine. Direct evidence that a complete group of $3$-quanta orbits +is present in the cuprous ion is given by the spectrum of copper +which, in contrast to the extremely complicated spectra of the +preceding elements resulting from the unsymmetrical character of +the inner system, possesses a simple structure very much like that +of the sodium spectrum. This may no doubt be ascribed to a +simple symmetrical structure present in the cuprous ion similar to +that in the sodium ion, although the great difference in the constitution +of the outer group of electrons in these ions is shown +both by the considerable difference in the values of the spectral +terms and in the separation of the doublets in the $P$~terms of the +two spectra. The occurrence of the cupric compounds shows, however, +that the firmness of binding in the group of $3$-quanta orbits +in the copper atom is not as great as the firmness with which the +electrons are bound in the group of $2$-quanta orbits in the sodium +atom. Zinc, which is always divalent, is the first element in which +the groups of the electrons are so firmly bound that they \Chg{can not}{cannot} +be removed by ordinary chemical processes. + +The picture I have given of the formation and structure of the +atoms of the elements in the fourth period gives an explanation of +the chemical and spectral properties. In addition it is supported +by evidence of a different nature to that which we have hitherto +\PageSep{106} +used. It is a familiar fact, that the elements in the fourth period +differ markedly from the elements in the preceding periods +partly in their \emph{magnetic properties} and partly in the \emph{characteristic +colours} of their compounds. Paramagnetism and colours do occur +in elements belonging to the foregoing periods, but not in simple +compounds where the atoms considered enter as ions. Many +elements of the fourth period, on the contrary, exhibit paramagnetic +properties and characteristic colours even in dissociated +aqueous solutions. The importance of this has been emphasized +by Ladenburg in his attempt to explain the properties of the +elements in the long periods of the periodic system (see \PageRef{73}). +Langmuir in order to account for the difference between the fourth +period and the preceding periods simply assumed that the atom, +in addition to the layers of cells containing $8$~electrons each, possesses +an outer layer of cells with room for $18$~electrons which is completely +filled for the first time in the case of krypton. Ladenburg, +on the other hand, assumes that for some reason or other an +intermediate layer is developed between the inner electronic +configuration in the atom appearing already in argon, and the +external group of valency electrons. This layer commences with +scandium and is completed exactly at the end of the family of iron +metals. In support of this assumption Ladenburg not only mentions +the chemical properties of the elements in the fourth period, but +also refers to the paramagnetism and colours which occur exactly +in the elements, where this intermediate layer should be in +development. It is seen that Ladenburg's ideas exhibit certain +formal similarities with the interpretation I have given above of +the appearance of the fourth period, and it is interesting to note that +our view, based on a direct investigation of the conditions for the +formation of the atoms, enables us to understand the relation +emphasized by Ladenburg. + +Our ordinary electrodynamic conceptions are probably insufficient +to form a basis for an explanation of atomic magnetism. This is +hardly to be wondered at when we remember that they have not +proved adequate to account for the phenomena of radiation which +are connected with the intimate interaction between the electric +and magnetic forces arising from the motion of the electrons. In +whatever way these difficulties may be solved it seems simplest to +\PageSep{107} +assume that the occurrence of magnetism, such as we meet in the +elements of the fourth period, results from a lack of symmetry in +the internal structure of the atom, thus preventing the magnetic +forces arising from the motion of the electrons from forming a +system of closed lines of force running wholly within the atom. +While it has been assumed that the ions of the elements in the +previous periods, whether positively or negatively charged, contain +configurations of marked symmetrical character, we must, however, +be prepared to encounter a definite lack of symmetry in the +electronic configurations in ions of those elements within the fourth +period which contain a group of electrons in $3$-quanta orbits in the +transition stage between symmetrical configurations of $8$~and $18$ +electrons respectively. As pointed out by Kossel, the experimental +results exhibit an extreme simplicity, the magnetic moment of the +ions depending only on the number of electrons in the ion. Ferric +ions, for example, exhibit the same atomic magnetism as manganous +ions, while manganic ions exhibit the same atomic magnetism as +chromous ions. It is in beautiful agreement with what we have +assumed about the structure of the atoms of copper and zinc, that +the magnetism disappears with those ions containing $28$~electrons +which, as I stated, must be assumed to contain a complete group +of $3$-quanta orbits. On the whole a consideration of the magnetic +properties of the elements within the fourth period gives us a vivid +impression of how a wound in the otherwise symmetrical inner +structure is first developed and then healed as we pass from element +to element. It is to be hoped that a further investigation of the +magnetic properties will give us a clue to the way in which the +group of electrons in $3$-quanta orbits is developed step by step. + +Also the colours of the ions directly support our view of atomic +structure. According to the postulates of the quantum theory +absorption as well as emission of radiation is regarded as taking +place during transitions between stationary states. The occurrence +of colours, that is to say the absorption of light in the visible region +of the spectrum, is evidence of transitions involving energy changes +of the same order of magnitude as those giving the usual optical +spectra of the elements. In contrast to the ions of the elements of +the preceding periods where all the electrons are assumed to be very +firmly bound, the occurrence of such processes in the fourth period +\PageSep{108} +is exactly what we should expect. For the development and completion +of the electronic groups with $3$-quanta orbits will proceed, +so to say, in competition with the binding of electrons in orbits of +higher quanta, since the binding of electrons in $3$-quanta orbits +occurs when the electrons in these orbits are bound more firmly +than electrons in $4_{1}$~orbits. The development of the group will +therefore proceed to the point where we may say there is equilibrium +between the two kinds of orbits. This condition may be +assumed to be intimately connected not only with the colour of the +ions, but also with the tendency of the elements to form ions with +different valencies. This is in contrast to the elements of the first +periods where the charge of the ions in aqueous solutions is always +the same for one and the same element. + +\Section{Fifth Period. Rubidium---Xenon.} The structure of the atoms +in the remaining periods may be followed up in complete analogy +with what has already been said. Thus we shall assume that the +$37$th and $38$th electrons in the elements of the fifth period are +bound in $5_{1}$~orbits. This is supported by the measurements of the +arc spectrum of rubidium and the spark spectrum of strontium. +The latter spectrum indicates at the same time that $4_{3}$~orbits will +soon appear, and therefore in this period, which like the $4$th +contains $18$~elements, we must assume that we are witnessing a +\emph{further stage in the development of the electronic group of $4$-quanta +orbits}. The first stage in the formation of this group may be said +to have been attained in krypton with the appearance of a symmetrical +configuration of eight electrons consisting of two subgroups +each of four electrons in $4_{1}$~and $4_{2}$~orbits. A second preliminary +completion must be regarded as having been reached with the +appearance of a symmetrical configuration of $18$~electrons in the +case of silver, consisting of three subgroups with six electrons each +in orbits of the types $4_{1}$,~$4_{2}$ and~$4_{3}$. Everything that has been said +about the successive formation of the group of electrons with $3$-quanta +orbits applies unchanged to this stage in the transformation +of the group with $4$-quanta orbits. For in no case have we made +use of the absolute values of the quantum numbers nor of assumptions +concerning the form of the orbits but only of the number of +possible types of orbits which might come into consideration. At +\PageSep{109} +the same time it may be of interest to mention that the properties +of these elements compared with those of the foregoing period +nevertheless show a difference corresponding exactly to what would +be expected from the difference in the types of orbits. For instance, +the divergencies from the characteristic valency conditions of the +elements in the second and third periods appear later in the fifth +period than for elements in the fourth period. While an element +like titanium in the fourth period already shows a marked tendency +to occur with various valencies, on the other hand an element like +zirconium is still quadri-valent like carbon in the second period +and silicon in the third. A simple investigation of the kinematic +properties of the orbits of the electrons shows in fact that an +electron in an \Chg{excentric}{eccentric} $4_{3}$~orbit of an element in the fifth +period will be considerably more loosely bound than an electron in +a circular $3_{3}$~orbit of the corresponding element in the fourth +period, while electrons which are bound in \Chg{excentric}{eccentric} orbits of the +types $5_{1}$~and $4_{1}$ respectively will correspond to a binding of about +the same firmness. + +At the end of the fifth period we may assume that xenon, the +atomic number of which is~$54$, has a structure which in addition to +the two $1$-quantum, eight $2$-quanta, eighteen $3$-quanta and eighteen +$4$-quanta orbits already mentioned contains a symmetrical +configuration of eight electrons in $5$-quanta orbits consisting of two +subgroups with four electrons each in $5_{1}$~and $5_{2}$ orbits respectively. + +\Section{Sixth Period. Caesium---Niton.} If we now consider the atoms +of elements of still higher atomic number, we must first of all +assume that the $55$th and $56$th electrons in the atoms of caesium +and barium are bound in $6_{1}$~orbits. This is confirmed by the spectra +of these elements. It is clear, however, that we must be prepared +shortly to meet entirely new conditions. With increasing nuclear +charge we shall have to expect not only that an electron in a $5_{3}$~orbit +will be bound more firmly than in a $6_{1}$~orbit, but we must also +expect that a moment will arrive when during the formation of the +atom a $4_{4}$~orbit will represent a firmer binding of the electron than +an orbit of $5$~or $6$-quanta, in much the same way as in the elements +of the fourth period a new stage in the development of the $3$-quanta +group was started when a point was reached where for the first +\PageSep{110} +time the $19$th electron was bound in a $3_{3}$~orbit instead of in a $4_{1}$~orbit. +We shall thus expect in the sixth period to meet with a new +stage in the development of the group with $4$-quanta orbits. Once +this point has been reached we must be prepared to find with increasing +atomic number a number of elements following one another, +which as in the family of the iron metals have very nearly the same +properties. The similarity will, however, be still more pronounced, +since in this case we are concerned with the successive transformation +of a configuration of electrons which lies deeper in the interior +of the atom. You will have already guessed that what I have in view +is a simple explanation of the occurrence of the \emph{family of rare earths} +at the beginning of the sixth period. As in the case of the transformation +and completion of the group of $3$-quanta orbits in the fourth +period and the partial completion of groups of $4$-quanta orbits in +the fifth period, we may immediately deduce from the length of the +sixth period the number of electrons, namely~$32$, which are finally +contained in the $4$-quanta group of orbits. Analogous to what +applied to the group of $3$-quanta orbits it is probable that, when +the group is completed, it will contain eight electrons in each of the +four subgroups. Even though it has not yet been possible to follow +the development of the group step by step, we can even here give +some theoretical evidence in favour of the occurrence of a symmetrical +configuration of exactly this number of electrons. I shall +simply mention that it is not possible without coincidence of the +planes of the orbits to arrive at an interaction between four subgroups +of six electrons each in a configuration of simple trigonal +symmetry, which is equally simple as that shown by three subgroups. +The difficulties which we meet make it probable that a harmonic +interaction can be attained precisely by four groups each containing +eight electrons the orbital configurations of which exhibit axial +symmetry. + +Just as in the case of the family of the iron metals in the fourth +period, the proposed explanation of the occurrence of the family of +rare earths in the sixth period is supported in an interesting +manner by an investigation of the magnetic properties of these +elements. In spite of the great chemical similarity the members +of this family exhibit very different magnetic properties, so that +while some of them exhibit but very little magnetism others exhibit +\PageSep{111} +a greater magnetic moment per atom than any other element which +has been investigated. It is also possible to give a simple interpretation +of the peculiar colours exhibited by the compounds of these +elements in much the same way as in the case of the family of iron +metals in the fourth period. The idea that the appearance of the +group of the rare earths is connected with the development of inner +groups in the atom is not in itself new and has for instance been +considered by Vegard in connection with his work on X-ray spectra. +The new feature of the present considerations lies, however, in the +emphasis laid on the peculiar way in which the relative strength of +the binding for two orbits of the same principal quantum number +but of different shapes varies with the nuclear charge and with the +number of electrons previously bound. Due to this fact the presence +of a group like that of the rare earths in the sixth period may be +considered as a direct consequence of the theory and might actually +have been predicted on a quantum theory, adapted to the explanation +of the properties of the elements within the preceding periods +in the way I have shown. + +Besides \emph{the final development of the group of $4$-quanta orbits} we +observe in the sixth period in the family of the platinum metals \emph{the +second stage in the development of the group of $5$-quanta orbits}. +Also in the radioactive, chemically inactive gas niton, which completes +this period, we observe the first preliminary step in the +development of a group of electrons with $6$-quanta orbits. In the +atom of this element, in addition to the groups of electrons of two +$1$-quantum, eight $2$-quanta, eighteen $3$-quanta, thirty-two $4$-quanta +and eighteen $5$-quanta orbits respectively, there is also an outer +symmetrical configuration of eight electrons in $6$-quanta orbits, +which we shall assume to consist of two subgroups with four electrons +each in $6_{1}$~and $6_{2}$ orbits respectively. + +\Section{Seventh Period.} In the seventh and last period of the periodic +system we may expect the appearance of $7$-quanta orbits in the +normal state of the atom. Thus in the neutral atom of radium in +addition to the electronic structure of niton there will be two +electrons in $7_{1}$~orbits which will penetrate during their revolution +not only into the region of the orbits of electrons possessing lower +values for the principal quantum number, but even to distances +\PageSep{112} +from the nucleus which are less than the radii of the orbits of the +innermost $1$-quantum orbits. The properties of the elements in the +seventh period are very similar to the properties of the elements in the +fifth period. Thus, in contrast to the conditions in the sixth period, +there are no elements whose properties resemble one another like +those of the rare earths. In exact analogy with what has already +been said about the relations between the properties of the elements +in the fourth and fifth periods this may be very simply explained by +the fact that an \Chg{excentric}{eccentric} $5_{4}$~orbit will correspond to a considerably +looser binding of an electron in the atom of an element of the +seventh period than the binding of an electron in a circular $4_{4}$~orbit +in the corresponding element of the sixth period, while there will be +a much smaller difference in the firmness of the binding of these +electrons in orbits of the types $7_{1}$~and $6_{1}$ respectively. + +It is well known that the seventh period is not complete, for no atom +has been found having an atomic number greater than~$92$. This is +probably connected with the fact that the last elements in the +system are radioactive and that nuclei of atoms with a total charge +greater than~$92$ will not be sufficiently stable to exist under conditions +where the elements can be observed. It is tempting to +sketch a picture of the atoms formed by the capture and binding +of electrons around nuclei having higher charges, and thus to +obtain some idea of the properties which the corresponding hypothetical +elements might be expected to exhibit. I shall not develop +this matter further, however, since the general results we should +get will be evident to you from the views I have developed to +explain the properties of the elements actually observed. A survey +of these results is given in the following table, which gives a symbolical +representation of the atomic structure of the inactive gases +which complete the first six periods in the periodic system. In +order to emphasize the progressive change the table includes the +probable arrangement of electrons in the next atom which would +possess properties like the inactive gases. + +The view of atomic constitution underlying this table, which +involves configurations of electrons moving with large velocities +between each other, so that the electrons in the ``outer'' groups +penetrate into the region of the orbits of the electrons of the ``inner'' +groups, is of course completely different from such statical models +\PageSep{113} +of the atom as are proposed by Langmuir. But quite apart from this +it will be seen that the arrangement of the electronic groups in +the atom, to which we have been lead by tracing the way in which +each single electron has been bound, is essentially different from +the arrangement of the groups in Langmuir's theory. In order to +explain the properties of the elements of the sixth period Langmuir +assumes for instance that, in addition to the inner layers of cells +containing $2$,~$8$, $8$, $18$ and $18$ electrons respectively, which are +employed to account for the properties of the elements in the +earlier periods, the atom also possesses a layer of cells with room +for $32$~electrons which is just completed in the case of niton. + +\Figure{}{113} + +In this connection it may be of interest to mention a recent +paper by Bury, to which my attention was first drawn after the +deliverance of this address, and which contains an interesting +survey of the chemical properties of the elements based on similar +conceptions of atomic structure as those applied by Lewis and +Langmuir. From purely chemical considerations Bury arrives at +conclusions which as regards the arrangement and completion of +the groups in the main coincide with those of the present theory, +the outlines of which were given in my letters to Nature mentioned +in the introduction. + +\Section{Survey of the periodic table.} The results given in this address +are also illustrated by means of the representation of the periodic +system given in \Fig{1}. In this figure the frames are meant to +indicate such elements in which one of the ``inner'' groups is +in a stage of development. Thus there will be found in the +\PageSep{114} +fourth and fifth periods a single frame indicating the final completion +of the electronic group with $3$-quanta orbits, and the +last stage but one in the development of the group with $4$-quanta +orbits respectively. In the sixth period it has been necessary to +introduce two frames, of which the inner one indicates the last +stage of the evolution of the group with $4$-quanta orbits, giving rise +to the rare earths. This occurs at a place in the periodic system +where the third stage in the development of an electronic group +with $5$-quanta orbits, indicated by the outer frame, has already +begun. In this connection it will be seen that the inner frame +encloses a smaller number of elements than is usually attributed +to the family of the rare earths. At the end of this group an +uncertainty exists, due to the fact that no element of atomic +number~$72$ is known with certainty. However, as indicated in +\Fig{1}, we must conclude from the theory that the group with +$4$-quanta orbits is finally completed in lutetium~($71$). This element +therefore ought to be the last in the sequence of consecutive +elements with similar properties in the first half of the sixth period, +and at the place~$72$ an element must be expected which in its +chemical and physical properties is homologous with zirconium and +thorium. This, which is already indited on Julius Thomsen's old +table, has also been pointed out by Bury. [Quite recently Dauvillier +has in an investigation of the X-ray spectrum excited in preparations +containing rare earths, observed certain faint lines which he ascribes +to an element of atomic number~$72$. This element is identified by +him as the element celtium, belonging to the family of rare earths, +the existence of which had previously been suspected by Urbain. +Quite apart from the difficulties which this result, if correct, might +entail for atomic theories, it would, since the rare earths according +to chemical view possess three valencies, imply a rise in positive +valency of two units when passing from the element~$72$ to the +next element~$73$, tantalum. This would mean an exception from +the otherwise general rule, that the valency never increases by +more than one unit when passing from one element to the next in +the periodic table\Add{.}] In the case of the incomplete seventh period +the full drawn frame indicates the third stage in the development +of the electronic group with $6$-quanta orbits, which must begin in +actinium. The dotted frame indicates the last stage but one in +\PageSep{115} +the development of the group with $5$-quanta orbits, which hitherto +has not been observed, but which ought to begin shortly after +uranium, if it has not already begun in this element. + +With reference to the homology of the elements the exceptional +position of the elements enclosed by frames in \Fig{1} is further +emphasized by taking care that, in spite of the large similarity +many elements exhibit, no connecting lines are drawn between +two elements which occupy different positions in the system with +respect to framing. In fact, the large chemical similarity between, +for instance, aluminium and scandium, both of which are trivalent +and pronounced electropositive elements, is directly or indirectly +emphasized in the current representations of the periodic table. +While this procedure is justified by the analogous structure of the +trivalent ions of these elements, our more detailed ideas of atomic +structure suggest, however, marked differences in the physical +properties of aluminium and scandium, originating in the essentially +different character of the way in which the last three electrons +in the neutral atom are bound. This fact gives probably a direct +explanation of the marked difference existing between the spectra +of aluminium and scandium. Even if the spectrum of scandium is +not yet sufficiently cleared up, this difference seems to be of a much +more fundamental character than for instance the difference between +the arc spectra of sodium and copper, which apart from the large +difference in the absolute values of the spectral terms possess a +completely analogous structure, as previously mentioned in this +essay. On the whole we must expect that the spectra of elements +in the later periods lying inside a frame will show new features +compared with the spectra of the elements in the first three periods. +This expectation seems supported by recent work on the spectrum +of manganese by Catalan, which appeared just before the printing +of this essay. + +Before I leave the interpretation of the chemical properties by +means of this atomic model I should like to remind you once again +of the fundamental principles which we have used. The whole +theory has evolved from an investigation of the way in which +electrons can be captured by an atom. The formation of an atom +was held to consist in the successive binding of electrons, this +binding resulting in radiation according to the quantum theory. +\PageSep{116} +According to the fundamental postulates of the theory this binding +takes place in stages by transitions between stationary states +accompanied by emission of radiation. For the problem of the +stability of the atom the essential problem is at what stage such a +process comes to an end. As regards this point the postulates give +no direct information, but here the correspondence principle is +brought in. Even though it has been possible to penetrate considerably +further at many points than the time has permitted me +to indicate to you, still it has not yet been possible to follow in +detail all stages in the formation of the atoms. We cannot say, for +instance, that the above table of the atomic constitution of the +inert gases may in every detail be considered as the unambiguous +result of applying the correspondence principle. On the other hand +it appears that our considerations already place the empirical data +in a light which scarcely permits of an essentially different interpretation +of the properties of the elements based upon the postulates of +the quantum theory. This applies not only to the series spectra +and the close relationship of these to the chemical properties of the +elements, but also to the X-ray spectra, the consideration of which +leads us into an investigation of interatomic processes of an entirely +different character. As we have already mentioned, it is necessary +to assume that the emission of the latter spectra is connected with +processes which may be described as a reorganization of the completely +formed atom after a disturbance produced in the interior +of the atom by the action of external forces. + + +\Chapter{IV.}{Reorganization of Atoms and X-Ray Spectra} + +As in the case of the series spectra it has also been possible to represent +the frequency of each line in the X-ray spectrum of an element +as the difference of two of a set of spectral terms. We shall therefore +assume that each X-ray line is due to a transition between +two stationary states of the atom. The values of the atomic energy +corresponding to these states are frequently referred to as the +``energy levels'' of the X-ray spectra. The great difference between +the origin of the X-ray and the series spectra is clearly seen, however, +in the difference of the laws applying to the absorption of +radiation in the X-ray and the optical regions of the spectra. The +absorption by non-excited atoms in the latter case is connected +\PageSep{117} +with those lines in the series spectrum which correspond to combinations +of the various spectral terms with the largest of these +terms. As has been shown, especially by the investigations of +Wagner and de~Broglie, the absorption in the X-ray region, on +the other hand, is connected not with the X-ray lines but with +certain spectral regions commencing at the so-called ``absorption +edges.'' The frequencies of these edges agree very closely with the +spectral terms used to account for the X-ray lines. We shall now +see how the conception of atomic structure developed in the preceding +pages offers a simple interpretation of these facts. Let us +consider the following question: What changes in the state of the +atom can be produced by the absorption of radiation, and which +processes of emission can be initiated by such changes? + +\Section{Absorption and emission of X-rays and correspondence +principle.} The possibility of producing a change at all in the +motion of an electron in the interior of an atom by means of radiation +must in the first place be regarded as intimately connected +with the character of the interaction between the electrons within +the separate groups. In contrast to the forms of motion where at +every moment the position of the electrons exhibits polygonal or +polyhedral symmetry, the conception of this interaction evolved from +a consideration of the possible formation of atoms by successive +binding of electrons has such a character that the harmonic components +in the motion of an electron are in general represented in +the resulting electric moment of the atom. As a result of this it +will be possible to release a single electron from the interaction +with the other electrons in the same group by a process which +possesses the necessary analogy with an absorption process on +the ordinary electrodynamic view claimed by the correspondence +principle. The points of view on which we based the interpretation +of the development and completion of the groups during the +formation of an atom imply, on the other hand, that just as no +additional electron can be taken up into a previously completed +group in the atom by a change involving emission of radiation, +similarly it will not be possible for a new electron to be added +to such a group, when the state of the atom is changed by +absorption of radiation. This means that an electron which belongs +\PageSep{118} +to one of the inner groups of the atom, as a consequence of an +absorption process---besides the case where it leaves the atom +completely---can only go over either to an incompleted group, or +to an orbit where the electron during the greater part of its revolution +moves at a distance from the nucleus large compared to the +distance of the other electrons. On account of the peculiar conditions +of stability which control the occurrence of incomplete groups in +the interior of the atom, the energy which is necessary to bring +about a transition to such a group will in general differ very little +from that required to remove the particular electron completely +from the atom. We must therefore assume that the energy levels +corresponding to the absorption edges indicate to a first approximation +the amount of work that is required to remove an electron +in one of the inner groups completely from the atom. The +correspondence principle also provides a basis for understanding +the experimental evidence about the appearance of the emission +lines of the X-ray spectra due to transitions between the stationary +states corresponding to these energy levels. Thus the nature of the +interaction between the electrons in the groups of the atom implies +that each electron in the atom is so to say prepared, independently +of the other electrons in the same group, to seize any opportunity +which is offered to become more firmly bound by being taken up +into a group of electrons with orbits corresponding to smaller values +of the principal quantum number. It is evident, however, that on +the basis of our views of atomic structure, such an opportunity is +always at hand as soon as an electron has been removed from one +of these groups. + +At the same time that our view of the atom leads to a natural +conception of the phenomena of emission and absorption of X-rays, +agreeing closely with that by which Kossel has attempted to give +a formal explanation of the experimental observations, it also suggests +a simple explanation of those quantitative relations holding for the +frequencies of the lines which have been discovered by Moseley and +Sommerfeld. These researches brought to light a remarkable and +far-reaching similarity between the Röntgen spectrum of a given +element and the spectrum which would be expected to appear upon +the binding of a single electron by the nucleus. This similarity we +immediately understand if we recall that in the normal state of the +\PageSep{119} +atom there are electrons moving in orbits which, with certain +limitations, correspond to all stages of such a binding process and +that, when an electron is removed from its original place in the +atom, processes may be started within the atom which will correspond +to all transitions between these stages permitted by the +correspondence principle. This brings us at once out of those +difficulties which apparently arise, when one attempts to account +for the origin of the X-ray spectra by means of an atomic structure, +suited to explain the periodic system. This difficulty has been felt +to such an extent that it has led Sommerfeld for example in his +recent work to assume that the configurations of the electrons in +the various atoms of one and the same element may be different +even under usual conditions. Since, in contrast to our ideas, he +supposed all electrons in the principal groups of the atom to move +in equivalent orbits, he is compelled to assume that these groups +are different in the different atoms, corresponding to different +possible types of orbital shapes. Such an assumption, however, seems +inconsistent with an interpretation of the definite character of the +physical and chemical properties of the elements, and stands in marked +contradiction with the points of view about the stability of the atoms +which form the basis of the view of atomic structure here proposed. + +\Section{X-ray spectra and atomic structure.} In this connection it is +of interest to emphasize that the group distribution of the electrons +in the atom, on which we have based both the explanation of the +periodic system and the classification of the lines in the X-ray +spectra, shows itself in an entirely different manner in these two +phenomena. While the characteristic change of the chemical +properties with atomic number is due to the gradual development +and completion of the groups of the loosest bound electrons, the +characteristic absence of almost every trace of a periodic change in +the X-ray spectra is due to two causes. Firstly the electronic +configuration of the completed groups is repeated unchanged for +increasing atomic number, and secondly the gradual way in which +the incompleted groups are developed implies that a type of orbit, +from the moment when it for the first time appears in the normal +state of the neutral atom, always will occur in this state and will +correspond to a steadily increasing firmness of binding. The development +\PageSep{120} +of the groups in the atom with increasing atomic number, +which governs the chemical properties of the elements shows itself +in the X-ray spectra mainly in the appearance of new lines. Swinne +has already referred to a connection of this kind between the periodic +system and the X-ray spectra in connection with Kossel's theory. +We can only expect a closer connection between the X-ray phenomena +and the chemical properties of the elements, when the conditions +on the surface of the atom are concerned. In agreement +with what has been brought to light by investigations on absorption +of X-rays in elements of lower atomic number, such as have +been performed in recent years in the physical laboratory at Lund, +we understand immediately that the position and eventual structure +of the absorption edges will to a certain degree depend upon +the physical and chemical conditions under which the element +investigated exists, while such a dependence does not appear in +the characteristic emission lines. + +If we attempt to obtain a more detailed explanation of the +experimental observations, we meet the question of the influence +of the presence of the other electrons in the atom upon the firmness +of the binding of an electron in a given type of orbit. This influence +will, as we at once see, be least for the inner parts of the atom, +where for each electron the attraction of the nucleus is large in +proportion to the repulsion of the other electrons. It should also +be recalled, that while the relative influence of the presence of the +other electrons upon the firmness of the binding will decrease with +increasing charge of the nucleus, the effect of the variation in the +mass of the electron with the velocity upon the firmness of the +binding will increase strongly. This may be seen from Sommerfeld's +formula~\Eq{(11)}. While we obtain a fairly good agreement for the +levels corresponding to the removal of one of the innermost electrons +in the atom by using the simple formula~\Eq{(11)}, it is, however, already +necessary to take the influence of the other electrons into consideration +in making an approximate calculation of the levels corresponding +to a removal of an electron from one of the outer groups in the +atom. Just this circumstance offers us, however, a possibility of +obtaining information about the configurations of the electrons in +the interior of the atoms from the X-ray spectra. Numerous +investigations have been directed at this question both by +\PageSep{121} +Sommerfeld and his pupils and by Debye, Vegard and others. It +may also be remarked that de~Broglie and Dauvillier in a recent +paper have thought it possible to find support in the experimental +material for certain assumptions about the numbers of electrons in +the groups of the atom to which Dauvillier had been led by considerations +about the periodic system similar to those proposed by +Langmuir and Ladenburg. In calculations made in connection with +these investigations it is assumed that the electrons in the various +groups move in separate concentric regions of the atom, so that +the effect of the presence of the electrons in inner groups upon the +motion of the electrons in outer groups as a first approximation +may be expected to consist in a simple screening of the nucleus. +On our view, however, the conditions are essentially different, since +for the calculation of the firmness of the binding of the electrons +it is necessary to take into consideration that the electrons in the +more lightly bound groups in general during a certain fraction of +their revolution will penetrate into the region of the orbits of +electrons in the more firmly bound groups. On account of this +fact, many examples of which we saw in the series spectra, we \Chg{can not}{cannot} +expect to give an account of the firmness of the binding of the +separate electrons, simply by means of a ``screening correction'' +consisting in the subtraction of a constant quantity from the value +for~$N$ in such formulae as \Eq{(5)} and~\Eq{(11)}. Furthermore in the calculation +of the work corresponding to the energy levels we must take +account not only of the interaction between, the electrons in the +normal state of the atom, but also of the changes in the configuration +and interaction of the remaining electrons, which establish +themselves automatically without emission of radiation during the +removal of the electron. Even though such calculations have not +yet been made very accurately, a preliminary investigation has +already shown that it is possible approximately to account for the +experimental results. + +\Section{Classification of X-ray spectra.} Independently of a definite +view of atomic structure it has been possible by means of a formal +application of Kossel's and Sommerfeld's theories to disentangle +the large amount of experimental material on X-ray spectra. This +material is drawn mainly from the accurate measurements of +\PageSep{122} +Siegbahn and his collaborators. From this disentanglement of the +experimental observations, in which besides Sommerfeld and his +students especially Smekal and Coster have taken part, we have +obtained a nearly complete classification of the energy levels corresponding +to the X-ray spectra. These levels are formally referred +to types of orbits characterized by two quantum numbers $n$ and~$k$, +and certain definite rules for the possibilities of combination +between the various levels have also been found. In this way a +number of results of great interest for the further elucidation of +the origin of the X-ray spectra have been attained. First it has +not only been possible to find levels, which within certain limits +correspond to all possible pairs of numbers for $n$ and~$k$, but it has +been found that in general to each such pair more than one level +must be assigned. This result, which at first may appear very +surprising, upon further consideration can be given a simple +interpretation. We must remember that the levels depend not +only upon the constitution of the atom in the normal state, but +also upon the configurations which appear after the removal +of one of the inner electrons and which in contrast to the normal +state do not possess a uniquely completed character. If we thus +consider a process in which one of the electrons in a group +(subgroup) is removed we must be prepared to find that after the +process the orbits of the remaining electrons in this group may be +orientated in more than one way in relation to one another, and +still fulfil the conditions required of the stationary states by the +quantum theory. Such a view of the ``complexity'' of the levels, as +further consideration shows, just accounts for the manner in which +the energy difference of the two levels varies with the atomic +number. Without attempting to develop a more detailed picture +of atomic structure, Smekal has already discussed the possibility +of accounting for the multiplicity of levels. Besides referring to +the possibility that the separate electrons in the principal groups +do not move in equivalent orbits, Smekal suggests the introduction +of three quantum numbers for the description of the various groups, +but does not further indicate to what extent these quantum +numbers shall be regarded as characterizing a complexity in the +structure of the groups in the normal state itself or on the +contrary characterizing the incompleted groups which appear +when an electron is removed. +\PageSep{123} + +It will be seen that the complexity of the X-ray levels exhibits a +close analogy with the explanation of the complexity of the terms +of the series spectra. There exists, however, this difference between +the complex structure of the X-ray spectra and the complex +structure of the lines in the series spectra, that in the X-ray +spectra there occur not only combinations between spectral terms, +for which $k$~varies by unity, but also between terms corresponding +to the same value of~$k$. This may be assumed to be +due to the fact, that in the X-ray spectra in contrast to the series +\Figure{5}{123} +spectra we have to do with transitions between stationary states +where, both before and after the transition, the electron concerned +takes part in an intimate interaction with other electrons in orbits +with the same principal quantum number. Even though this +interaction may be assumed to be of such a nature that the +harmonic components which would appear in the motion of an +electron in the absence of the others will in general also appear +in the resulting moment of the atom, we must expect that the +interaction between the electrons will give rise to the appearance +in this moment of new types of harmonic components. +\PageSep{124} + +It may be of interest to insert here a few words about a new +paper of Coster which appeared after this address was given, +and in which he has succeeded in obtaining an extended and +detailed connection between the X-ray spectra and the ideas +of atomic structure given in this essay. The classification mentioned +above was based on measurements of the spectra of the +heaviest elements, and the results in their complete form, which +were principally due to independent work of Coster and Wentzel, +may be represented by the diagram in \Fig{5}, which refers to +elements in the neighbourhood of niton. The vertical arrows +\Figure{6}{124} +represent the observed lines arising from combinations between +the different energy levels which are represented by horizontal lines. +In each group the levels are arranged in the same succession as +their energy values, but their distances do not give a quantitative +picture of the actual energy-differences, since this would require a +much larger figure. The numbers~$n_{k}$ attached to the different levels +indicate the type of the corresponding orbit. The letters $a$ and~$b$ +refer to the rules of combination which I mentioned. According +to these rules the possibility of combination is limited (1)~by the +exclusion of combinations, for which $k$~changes by more than one +unit, (2)~by the condition that only combinations between an $a$- +and a $b$-level can take place. The latter rule was given in this +\PageSep{125} +form by Coster; Wentzel formulated it in a somewhat different +way by the formal introduction of a third quantum number. In +his new paper Coster has established a similar classification for the +lighter elements. For the elements in the neighbourhood of xenon +and krypton he has obtained results illustrated by the diagrams +given in \Fig{6}. Just as in \Fig{5} the levels correspond exactly to +those types of orbits which, as seen from the table on \PageRef[page]{113}, +according to the theory will be present in the atoms of these elements. +In xenon several of the levels present in niton have disappeared, +and in krypton still more levels have fallen away. Coster +has also investigated in which elements these particular levels +appear for the last time, when passing from higher to lower atomic +number. His results concerning this point confirm in detail the +predictions of the theory. Further he proves that the change in +the firmness of binding of the electrons in the outer groups in +the elements of the family of the rare earths shows a dependence +on the atomic number which strongly supports the assumption that +in these elements a completion of an inner group of $4$-quanta +orbits takes place. For details the reader is referred to Coster's +paper in the \Title{Philosophical Magazine}. Another important contribution +to our systematic knowledge of the X-ray spectra is +contained in a recent paper by Wentzel. He shows that various +lines, which find no place in the classification hitherto considered, +can be ascribed in a natural manner to processes of reorganization, +initiated by the removal of more than one electron from the +atom; these lines are therefore in a certain sense analogous to +the enhanced lines in the optical spectra. + +\Chapter{}{Conclusion} + +Before bringing this address to a close I wish once more to +emphasize the complete analogy in the application of the +quantum theory to the stability of the atom, used in explaining +two so different phenomena as the periodic system and X-ray +spectra. This point is of the greatest importance in judging the +reality of the theory, since the justification for employing considerations, +relating to the formation of atoms by successive capture +of electrons, as a guiding principle for the investigation of atomic +\PageSep{126} +structure might appear doubtful if such considerations could not +be brought into natural agreement with views on the reorganization +of the atom after a disturbance in the normal electronic +arrangement. Even though a certain inner consistency in this +view of atomic structure will be recognized, it is, however, hardly +necessary for me to emphasize the incomplete character of the +theory, not only as regards the elaboration of details, but also so +far as the foundation of the general points of view is concerned. +There seems, however, to be no other way of advance in atomic +problems than that which hitherto has been followed, namely to let +the work in these two directions go hand in hand. + +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% +\PGLicense +\begin{PGtext} +End of the Project Gutenberg EBook of The Theory of Spectra and Atomic +Constitution, by Niels (Niels Henrik David) Bohr + +*** END OF THIS PROJECT GUTENBERG EBOOK THEORY OF SPECTRA *** + +***** This file should be named 47464-pdf.pdf or 47464-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/4/7/4/6/47464/ + +Produced by Andrew D. 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You may copy it, give it away or re-use it under the terms of +% the Project Gutenberg License included with this eBook or online at % +% www.gutenberg.org. If you are not located in the United States, you'll have +% to check the laws of the country where you are located before using this ebook. +% % +% % +% % +% Title: The Theory of Spectra and Atomic Constitution % +% Three Essays % +% % +% Author: Niels (Niels Henrik David) Bohr % +% % +% Release Date: November 26, 2014 [EBook #47464] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK THEORY OF SPECTRA *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{47464} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Latin-1 text encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. 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+\begin{minipage}{\textwidth} +\small +\begin{PGtext} +The Project Gutenberg EBook of The Theory of Spectra and Atomic +Constitution, by Niels (Niels Henrik David) Bohr + +This eBook is for the use of anyone anywhere in the United States and most +other parts of the world at no cost and with almost no restrictions +whatsoever. You may copy it, give it away or re-use it under the terms of +the Project Gutenberg License included with this eBook or online at +www.gutenberg.org. If you are not located in the United States, you'll have +to check the laws of the country where you are located before using this ebook. + + + +Title: The Theory of Spectra and Atomic Constitution + Three Essays + +Author: Niels (Niels Henrik David) Bohr + +Release Date: November 26, 2014 [EBook #47464] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK THEORY OF SPECTRA *** +\end{PGtext} +\end{minipage} +\end{center} +\newpage +%% Credits and transcriber's note %% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. 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OF +CANADA, Ltd. +TOKYO : MARUZEN-KABUSHIKI-KAISHA + +ALL RIGHTS RESERVED +\fi +%[** TN: End of omitted half-title] +\PageSep{iii} +\newpage +\begin{center} +\Huge\bfseries +The Theory of Spectra \\ +and \\ +Atomic Constitution +\bigskip + +\large\normalfont +THREE ESSAYS \\ +BY \\ +\Large +NIELS BOHR +\medskip + +\normalsize +Professor of Theoretical Physics in~the~University~of~Copenhagen +\vfill + +\Large +CAMBRIDGE \\ +AT THE UNIVERSITY PRESS \\ +1922 +\end{center} +\newpage +\PageSep{iv} +\ifthenelse{\boolean{ForPrinting}}{% Publisher's verso +\begin{center} +\null\vfill +\footnotesize +PRINTED IN GREAT BRITAIN \\ +AT THE CAMBRIDGE UNIVERSITY PRESS +\end{center} +}{}% Omit for screen-formatted version +\PageSep{v} + +\FrontMatter + +\Preface + +\First{The} three essays which here appear in English all deal with +the application of the quantum theory to problems of atomic +structure, and refer to the different stages in the development of +this theory. + +The first essay ``On the spectrum of hydrogen'' is a translation of +a Danish address given before the Physical Society of Copenhagen +on the 20th~of December 1913, and printed in \Title{Fysisk Tidsskrift}, +\No{xii.}\ p.~97, 1914. Although this address was delivered at a time +when the formal development of the quantum theory was only at +its beginning, the reader will find the general trend of thought +very similar to that expressed in the later addresses, which +form the other two essays. As emphasized at several points the +theory does not attempt an ``explanation'' in the usual sense of +this word, but only the establishment of a connection between facts +which in the present state of science are unexplained, that is to +say the usual physical conceptions do not offer sufficient basis for +a detailed description. + +The second essay ``On the series spectra of the elements'' is a +translation of a German address given before the Physical Society +of Berlin on the 27th~of April 1920, and printed in \Title{Zeitschrift für +Physik}, \No{vi.}\ p.~423, 1920. This address falls into two main parts. +The considerations in the first part are closely related to the contents +of the first essay; especially no use is made of the new +formal conceptions established through the later development of +the quantum theory. The second part contains a survey of the +results reached by this development. An attempt is made to +elucidate the problems by means of a general principle which postulates +a formal correspondence between the fundamentally different +conceptions of the classical electrodynamics and those of the +quantum theory. The first germ of this correspondence principle +may be found in the first essay in the deduction of the expression +for the constant of the hydrogen spectrum in terms of +Planck's constant and of the quantities which in Rutherford's +\PageSep{vi} +atomic model are necessary for the description of the hydrogen +atom. + +The third essay ``The structure of the atom and the physical +and chemical properties of the elements'' is based on a Danish +address, given before a joint meeting of the Physical and Chemical +Societies of Copenhagen on the 18th~of October 1921, and printed +in \Title{Fysisk Tidsskrift}, \No{xix.}\ p.~153, 1921. While the first two essays +form verbal translations of the respective addresses, this essay +differs from the Danish original in certain minor points. Besides +the addition of a few new figures with explanatory text, certain +passages dealing with problems discussed in the second essay are +left out, and some remarks about recent contributions to the +subject are inserted. Where such insertions have been introduced +will clearly appear from the text. This essay is divided into +four parts. The first two parts contain a survey of previous results +concerning atomic problems and a short account of the theoretical +ideas of the quantum theory. In the following parts it is shown +how these ideas lead to a view of atomic constitution which seems +to offer an explanation of the observed physical and chemical +properties of the elements, and especially to bring the characteristic +features of the periodic table into close connection with the +interpretation of the optical and high frequency spectra of the +elements. + +For the convenience of the reader all three essays are subdivided +into smaller paragraphs, each with a headline. Conforming to the +character of the essays there is, however, no question of anything +like a full account or even a proportionate treatment of the subject +stated in these headlines, the principal object being to emphasize +certain general views in a freer form than is usual in scientific +treatises or text books. For the same reason no detailed references +to the literature are given, although an attempt is made to mention +the main contributions to the development of the subject. As +regards further information the reader in the case of the second +essay is referred to a larger treatise ``On the quantum theory of +line spectra,'' two parts of which have appeared in the Transactions of +the Copenhagen Academy (\Title{D.\ Kgl.\ Danske Vidensk.\ Selsk.\ Skrifter}, +8.\ Række, \No{iv.}~1, I~and~II, 1918),\footnote + {See \href{http://www.gutenberg.org/ebooks/47167}{www.gutenberg.org/ebooks/47167}.---\textit{Trans.}} +where full references to the literature +may be found. The proposed continuation of this treatise, mentioned +\PageSep{vii} +at several places in the second essay, has for various reasons been +delayed, but in the near future the work will be completed by the +publication of a third part. It is my intention to deal more fully +with the problems discussed in the third essay by a larger systematic +account of the application of the quantum theory to atomic +problems, which is under preparation. + +As mentioned both in the beginning and at the end of the +third essay, the considerations which it contains are clearly still +incomplete in character. This holds not only as regards the +elaboration of details, but also as regards the development of the +theoretical ideas. It may be useful once more to emphasize, +that---although the word ``explanation'' has been used more +liberally than for instance in the first essay---we are not concerned +with a description of the phenomena, based on a well-defined +physical picture. It may rather be said that hitherto every +progress in the problem of atomic structure has tended to emphasize +the well-known ``mysteries'' of the quantum theory more and more. +I hope the exposition in these essays is sufficiently clear, nevertheless, +to give the reader an impression of the peculiar charm +which the study of atomic physics possesses just on this account. + +I wish to express my best thanks to Dr~A.~D. Udden, University +of Pennsylvania, who has undertaken the translation of the +original addresses into English, and to Mr~C.~D. Ellis, Trinity +College, Cambridge, who has looked through the manuscript and +suggested many valuable improvements in the exposition of the +subject. +\Signature{N. BOHR.}{Copenhagen,}{May}{1922.} +\PageSep{viii} + +\TableofContents + +\iffalse +%[** TN: Original ToC text (not manually verified)] +CONTENTS + +ESSAY I +ON THE SPECTRUM OF HYDROGEN + +PAGE + +Empirical Spectral Laws 1 +Laws of Temperature Radiation 4 +The Nuclear Theory of the Atom 7 +Quantum Theory of Spectra 10 +Hydrogen Spectrum 12 +The Pickering Lines 15 +Other Spectra 18 + + +ESSAY II +ON THE SERIES SPECTRA OF THE ELEMENTS + +I. Introduction .20 + +II. General Principles of the Quantum Theory of Spectra . 23 +Hydrogen Spectrum 24 +The Correspondence Principle 27 +General Spectral Laws 29 +Absorption and Excitation of Radiation 32 + +III. Development of the Quantum Theory of Spectra . . 36 +Effect of External Forces on the Hydrogen Spectrum . . 37 +The Stark Effect 39 +The Zeeman Effect 42 +Central Perturbations . 44 +Relativity Effect on Hydrogen Lines 46 +Theory of Series Spectra 48 +Correspondence Principle and Conservation of Angular Momentum 50 +The Spectra of Helium and Lithium 54 +Complex Structure of Series Lines 58 + +IV. Conclusion 59 + +\PageSep{ix} +CONTENTS + +ESSAY III + +THE STRUCTURE OF THE ATOM AND THE PHYSICAL +AND CHEMICAL PROPERTIES OF THE ELEMENTS + +PAGE + +I. Preliminary 61 +The Nuclear Atom 61 +The Postulates of the Quantum Theory 62 +Hydrogen Atom 63 +Hydrogen Spectrum and X-ray Spectra 65 +The Fine Structure of the Hydrogen Lines .... 67 +Periodic Table 69 +Recent Atomic Models 74 + +II. Series Spectra and the Capture of Electrons by Atoms . 75 +Arc and Spark Spectra 76 +Series Diagram 78 +Correspondence Principle 81 + +III. Formation of Atoms and the Periodic Table ... 85 +First Period. Hydrogen---Helium 85 +Second Period. Lithium---Neon 89 +Third Period. Sodium---Argon 95 +Fourth Period. Potassium---Krypton 100 +Fifth Period. Rubidium--- Xenon 108 +Sixth Period. Caesium---Niton 109 +Seventh Period 111 +Survey of the Periodic Table 113 + +IV. Reorganization of Atoms and X-ray Spectra . . .116 +Absorption and Emission of X-rays and Correspondence Principle 117 +X-ray Spectra and Atomic Structure 119 +Classification of X-ray Spectra 121 +Conclusion 125 +\fi +%[** TN: End of original ToC text] +\PageSep{1} +\MainMatter + +\Essay{I}{On the Spectrum of Hydrogen} + {Address delivered before the Physical Society in Copenhagen, Dec.~20, 1913.} + +\Section{Empirical spectral laws.} Hydrogen possesses not only the +smallest atomic weight of all the elements, but it also occupies a +peculiar position both with regard to its physical and its chemical +properties. One of the points where this becomes particularly apparent +is the hydrogen line spectrum. + +The spectrum of hydrogen observed in an ordinary Geissler tube +consists of a series of lines, the strongest of which lies at the red +end of the spectrum, while the others extend out into the ultra\Add{-}violet, +the distance between the various lines, as well as their intensities, +constantly decreasing. In the ultra\Add{-}violet the series converges +to a limit. + +Balmer, as we know, discovered (1885) that it was possible to +represent the wave lengths of these lines very accurately by the +simple law +\[ +\frac{1}{\lambda_{n}} = R \left(\frac{1}{4} - \frac{1}{n^{2}}\right), +\Tag{(1)} +\] +where $R$~is a constant and $n$~is a whole number. The wave lengths +of the five strongest hydrogen lines, corresponding to $n = 3$, $4$,~$5$, $6$,~$7$, +measured in air at ordinary pressure and temperature, and the +values of these wave lengths multiplied by $\left(\dfrac{1}{4} - \dfrac{1}{n^{2}}\right)$ are given in +the following table:\Pagelabel{1} +\[ +%[** TN: Original uses a period for multiplication and a center dot as a decimal point] +\begin{array}{*{2}{c<{\qquad\qquad}}c} +n & \lambda · 10^{8} & \lambda · \left(\dfrac{1}{4} - \dfrac{1}{n^{2}}\right) · 10^{10} \\ +3 & 6563.04 & 91153.3 \\ +4 & 4861.49 & 91152.9 \\ +5 & 4340.66 & 91153.9 \\ +6 & 4101.85 & 91152.2 \\ +7 & 3970.25 & 91153.7 \\ +\end{array} +\] +The table shows that the product is nearly constant, while the deviations +are not greater than might be ascribed to experimental errors. + +As you already know, Balmer's discovery of the law relating to +the hydrogen spectrum led to the discovery of laws applying to +the spectra of other elements. The most important work in this +\PageSep{2} +connection was done by Rydberg (1890) and Ritz (1908). Rydberg +pointed out that the spectra of many elements contain series of +lines whose wave lengths are given approximately by the formula +\[ +\frac{1}{\lambda_{n}} = A - \frac{R}{(n + \alpha)^{2}}, +\] +where $A$~and~$\alpha$ are constants having different values for the various +series, while $R$~is a universal constant equal to the constant in the +spectrum of hydrogen. If the wave lengths are measured in vacuo +Rydberg calculated the value of~$R$ to be~$109675$. In the spectra of +many elements, as opposed to the simple spectrum of hydrogen, there +are several series of lines whose wave lengths are to a close approximation +given by Rydberg's formula if different values are assigned to +the constants $A$~and~$\alpha$. Rydberg showed, however, in his earliest +work, that certain relations existed between the constants in the +various series of the spectrum of one and the same element. These +relations were later very successfully generalized by Ritz through +the establishment of the ``combination principle.'' According to +this principle, the wave lengths of the various lines in the spectrum +of an element may be expressed by the formula +\[ +\frac{1}{\lambda} = F_{r}(n_{1}) - F_{s}(n_{2}). +\Tag{(2)} +\] +In this formula $n_{1}$~and~$n_{2}$ are whole numbers, and $F_{1}(n)$, $F_{2}(n)$,~\dots\ is +a series of functions of~$n$, which may be written approximately +\[ +F_{r}(n) = \frac{R}{(n + \alpha_{r})^{2}}, +\] +where $R$~is Rydberg's universal constant and $\alpha_{r}$ is a constant which +is different for the different functions. A particular spectral line will, +according to this principle, correspond to each combination of $n_{1}$~and~$n_{2}$, +as well as to the functions $F_{1}$, $F_{2}$,~\dots. The establishment of +this principle led therefore to the prediction of a great number of +lines which were not included in the spectral formulae previously +considered, and in a large number of cases the calculations were +found to be in close agreement with the experimental observations. +In the case of hydrogen Ritz assumed that formula~\Eq{(1)} was a special +case of the general formula +\[ +\frac{1}{\lambda} = R\left(\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}\right), +\Tag{(3)} +\] +\PageSep{3} +and therefore predicted among other things a series of lines in the +infra\Add{-}red given by the formula +\[ +\frac{1}{\lambda} = R\left(\frac{1}{9} - \frac{1}{n^{2}}\right). +\] +In 1909 Paschen succeeded in observing the first two lines of this +series corresponding to $n = 4$ and $n = 5$. + +The part played by hydrogen in the development of our +knowledge of the spectral laws is not solely due to its ordinary +simple spectrum, but it can also be traced in other less direct +ways. At a time when Rydberg's laws were still in want of +further confirmation Pickering (1897) found in the spectrum of a +star a series of lines whose wave lengths showed a very simple relation +to the ordinary hydrogen spectrum, since to a very close +approximation they could be expressed by the formula +\[ +\frac{1}{\lambda} = R\left(\frac{1}{4} - \frac{1}{(n + \frac{1}{2})^{2}}\right). +\] +Rydberg considered these lines to represent a new series of lines +in the spectrum of hydrogen, and predicted according to his theory +the existence of still another series of hydrogen lines the wave +lengths of which would be given by +\[ +\frac{1}{\lambda} = R\left(\frac{1}{(\frac{3}{2})^{2}} - \frac{1}{n^{2}}\right). +\] +By examining earlier observations it was actually found that a line +had been observed in the spectrum of certain stars which coincided +closely with the first line in this series (corresponding to $n = 2$); +from analogy with other spectra it was also to be expected that this +would be the strongest line. This was regarded as a great triumph +for Rydberg's theory and tended to remove all doubt that the new +spectrum was actually due to hydrogen. Rydberg's view has therefore +been generally accepted by physicists up to the present moment. +Recently however the question has been reopened and Fowler +(1912) has succeeded in observing the Pickering lines in ordinary +laboratory experiments. We shall return to this question again +later. + +The discovery of these beautiful and simple laws concerning the +line spectra of the elements has naturally resulted in many attempts +at a theoretical explanation. Such attempts are very alluring +\PageSep{4} +because the simplicity of the spectral laws and the exceptional accuracy +with which they apply appear to promise that the correct explanation +will be very simple and will give valuable information +about the properties of matter. I should like to consider some of +these theories somewhat more closely, several of which are extremely +interesting and have been developed with the greatest keenness +and ingenuity, but unfortunately space does not permit me to do +so here. I shall have to limit myself to the statement that not +one of the theories so far proposed appears to offer a satisfactory or +even a plausible way of explaining the laws of the line spectra. +Considering our deficient knowledge of the laws which determine +the processes inside atoms it is scarcely possible to give an explanation +of the kind attempted in these theories. The inadequacy of +our ordinary theoretical conceptions has become especially apparent +from the important results which have been obtained in recent years +from the theoretical and experimental study of the laws of temperature +radiation. You will therefore understand that I shall not +attempt to propose an explanation of the spectral laws; on the +contrary I shall try to indicate a way in which it appears possible +to bring the spectral laws into close connection with other properties +of the elements, which appear to be equally inexplicable on +the basis of the present state of the science. In these considerations +I shall employ the results obtained from the study of temperature +radiation as well as the view of atomic structure which has been +reached by the study of the radioactive elements. + +\Section{Laws of temperature radiation.} I shall commence by mentioning +the conclusions which have been drawn from experimental +and theoretical work on temperature radiation. + +Let us consider an enclosure surrounded by bodies which are in +temperature equilibrium. In this space there will be a certain +amount of energy contained in the rays emitted by the surrounding +substances and crossing each other in every direction. By making +the assumption that the temperature equilibrium will not be disturbed +by the mutual radiation of the various bodies Kirchhoff +(1860) showed that the amount of energy per unit volume as well +as the distribution of this energy among the various wave lengths +is independent of the form and size of the space and of the nature +\PageSep{5} +of the surrounding bodies and depends only on the temperature. +Kirchhoff's result has been confirmed by experiment, and the +amount of energy and its distribution among the various wave +lengths and the manner in which it depends on the temperature +are now fairly well known from a great amount of experimental +work; or, as it is usually expressed, we have a fairly +accurate experimental knowledge of the ``laws of temperature +radiation.'' + +Kirchhoff's considerations were only capable of predicting the +existence of a law of temperature radiation, and many physicists +have subsequently attempted to find a more thorough explanation +of the experimental results. You will perceive that the electromagnetic +theory of light together with the electron theory suggests +a method of solving this problem. According to the electron theory +of matter a body consists of a system of electrons. By making +certain definite assumptions concerning the forces acting on the +electrons it is possible to calculate their motion and consequently +the energy radiated from the body per second in the form of +electromagnetic oscillations of various wave lengths. In a similar +manner the absorption of rays of a given wave length by a substance +can be determined by calculating the effect of electromagnetic +oscillations upon the motion of the electrons. Having investigated +the emission and absorption of a body at all temperatures, and for +rays of all wave lengths, it is possible, as Kirchhoff has shown, to +determine immediately the laws of temperature radiation. Since +the result is to be independent of the nature of the body we are +justified in expecting an agreement with experiment, even though +very special assumptions are made about the forces acting upon +the electrons of the hypothetical substance. This naturally +simplifies the problem considerably, but it is nevertheless sufficiently +difficult and it is remarkable that it has been possible +to make any advance at all in this direction. As is well known +this has been done by Lorentz (1903). He calculated the +emissive as well as the absorptive power of a metal for long +wave lengths, using the same assumptions about the motions +of the electrons in the metal that Drude (1900) employed in +his calculation of the ratio of the electrical and thermal conductivities. +Subsequently, by calculating the ratio of the emissive +\PageSep{6} +to the absorptive power, Lorentz really obtained an expression +for the law of temperature radiation which for long wave lengths +agrees remarkably well with experimental facts. In spite of this +beautiful and promising result, it has nevertheless become apparent +that the electromagnetic theory is incapable of explaining the law +of temperature radiation. For, it is possible to show, that, if the +investigation is not confined to oscillations of long wave lengths, +as in Lorentz's work, but is also extended to oscillations corresponding +to small wave lengths, results are obtained which are +contrary to experiment. This is especially evident from Jeans' +investigations (1905) in which he employed a very interesting +statistical method first proposed by Lord Rayleigh. + +We are therefore compelled to assume, that the classical electrodynamics +does not agree with reality, or expressed more carefully, +that it \Chg{can not}{cannot} be employed in calculating the absorption and +emission of radiation by atoms. Fortunately, the law of temperature +radiation has also successfully indicated the direction in which the +necessary changes in the electrodynamics are to be sought. Even +before the appearance of the papers by Lorentz and Jeans, Planck +(1900) had derived theoretically a formula for the black body radiation +which was in good agreement with the results of experiment. +Planck did not limit himself exclusively to the classical electrodynamics, +but introduced the further assumption that a system of +oscillating electrical particles (elementary resonators) will neither +radiate nor absorb energy continuously, as required by the ordinary +electrodynamics, but on the contrary will radiate and absorb discontinuously. +The energy contained within the system at any +moment is always equal to a whole multiple of the so-called +quantum of energy the magnitude of which is equal to~$h\nu$, where +$h$~is Planck's constant and $\nu$~is the frequency of oscillation of the +system per second. In formal respects Planck's theory leaves much +to be desired; in certain calculations the ordinary electrodynamics +is used, while in others assumptions distinctly at variance with it +are introduced without any attempt being made to show that it +is possible to give a consistent explanation of the procedure used. +Planck's theory would hardly have acquired general recognition +merely on the ground of its agreement with experiments on black +body radiation, but, as you know, the theory has also contributed +\PageSep{7} +quite remarkably to the elucidation of many different physical +phenomena, such as specific heats, photoelectric effect, X-rays and +the absorption of heat rays by gases. These explanations involve +more than the qualitative assumption of a discontinuous transformation +of energy, for with the aid of Planck's constant~$h$ it +seems to be possible, at least approximately, to account for a great +number of phenomena about which nothing could be said previously. +It is therefore hardly too early to express the opinion that, whatever +the final explanation will be, the discovery of ``energy quanta'' +must be considered as one of the most important results arrived at +in physics, and must be taken into consideration in investigations +of the properties of atoms and particularly in connection with any +explanation of the spectral laws in which such phenomena as +the emission and absorption of electromagnetic radiation are +concerned. + +\Section{The nuclear theory of the atom.} We shall now consider the +second part of the foundation on which we shall build, namely the +conclusions arrived at from experiments with the rays emitted by +radioactive substances. I have previously here in the Physical +Society had the opportunity of speaking of the scattering of $\alpha$~rays +in passing through thin plates, and to mention how Rutherford +(1911) has proposed a theory for the structure of the atom in +order to explain the remarkable and unexpected results of these +experiments. I shall, therefore, only remind you that the characteristic +feature of Rutherford's theory is the assumption of the +existence of a positively charged nucleus inside the atom. A number +of electrons are supposed to revolve in closed orbits around the +nucleus, the number of these electrons being sufficient to neutralize +the positive charge of the nucleus. The dimensions of the nucleus +are supposed to be very small in comparison with the dimensions +of the orbits of the electrons, and almost the entire mass of the +atom is supposed to be concentrated in the nucleus. + +According to Rutherford's calculation the positive charge of the +nucleus corresponds to a number of electrons equal to about half +the atomic weight. This number coincides approximately with the +number of the particular element in the periodic system and it is +therefore natural to assume that the number of electrons in the +\PageSep{8} +atom is exactly equal to this number. This hypothesis, which was +first stated by van~den Broek (1912), opens the possibility of +obtaining a simple explanation of the periodic system. This assumption +is strongly confirmed by experiments on the elements +of small atomic weight. In the first place, it is evident that according +to Rutherford's theory the $\alpha$~particle is the same as the +nucleus of a helium atom. Since the $\alpha$~particle has a double positive +charge it follows immediately that a neutral helium atom contains +two electrons. Further the concordant results obtained from calculations +based on experiments as different as the diffuse scattering +of X-rays and the decrease in velocity of $\alpha$~rays in passing +through matter render the conclusion extremely likely that a +hydrogen atom contains only a single electron. This agrees most +beautifully with the fact that J.~J. Thomson in his well-known +experiments on rays of positive electricity has never observed a +hydrogen atom with more than a single positive charge, while all +other elements investigated may have several charges. + +Let us now assume that a hydrogen atom simply consists of an +electron revolving around a nucleus of equal and opposite charge, +and of a mass which is very large in comparison with that of the +electron. It is evident that this assumption may explain the peculiar +position already referred to which hydrogen occupies among the +elements, but it appears at the outset completely hopeless to attempt +to explain anything at all of the special properties of hydrogen, +still less its line spectrum, on the basis of considerations relating +to such a simple system. + +Let us assume for the sake of brevity that the mass of the nucleus +is infinitely large in proportion to that of the electron, and that the +velocity of the electron is very small in comparison with that of +light. If we now temporarily disregard the energy radiation, which, +according to the ordinary electrodynamics, will accompany the accelerated +motion of the electron, the latter in accordance with +Kepler's first law will describe an ellipse with the nucleus in one +of the foci. Denoting the frequency of revolution by~$\omega$, and the +major axis of the ellipse by~$2a$ we find that +\[ +\omega^{2} = \frac{2W^{3}}{\pi^{2} e^{4} m},\quad +2a = \frac{e^{2}}{W}, +\Tag{(4)} +\] +\PageSep{9} +where $e$~is the charge of the electron and $m$~its mass, while $W$~is +the work which must be added to the system in order to remove +the electron to an infinite distance from the nucleus. + +These expressions are extremely simple and they show that the +magnitude of the frequency of revolution as well as the length of +the major axis depend only on~$W$, and are independent of the +\Chg{excentricity}{eccentricity} of the orbit. By varying~$W$ we may obtain all possible +values for $\omega$~and~$2a$. This condition shows, however, that it is not +possible to employ the above formulae directly in calculating the +orbit of the electron in a hydrogen atom. For this it will be necessary +to assume that the orbit of the electron \Chg{can not}{cannot} take on all values, +and in any event, the line spectrum clearly indicates that the +oscillations of the electron cannot vary continuously between wide +limits. The impossibility of making any progress with a simple +system like the one considered here might have been foretold from +a consideration of the dimensions involved; for with the aid of $e$~and +$m$~alone it is impossible to obtain a quantity which can be +interpreted as a diameter of an atom or as a frequency. + +If we attempt to account for the radiation of energy in the manner +required by the ordinary electrodynamics it will only make matters +worse. As a result of the radiation of energy~$W$ would continually +increase, and the above expressions~\Eq{(4)} show that at the same time +the frequency of revolution of the system would increase, and the +dimensions of the orbit decrease. This process would not stop until +the particles had approached so closely to one another that they no +longer attracted each other. The quantity of energy which would +be radiated away before this happened would be very great. If we +were to treat these particles as geometrical points this energy would +be infinitely great, and with the dimensions of the electrons as +calculated from their mass (about $10^{-13}$~cm.), and of the nucleus as +calculated by Rutherford (about $10^{-12}$~cm.), this energy would be +many times greater than the energy changes with which we are +familiar in ordinary atomic processes. + +It can be seen that it is impossible to employ Rutherford's atomic +model so long as we confine ourselves exclusively to the ordinary +electrodynamics. But this is nothing more than might have been +expected. As I have mentioned we may consider it to be an +established fact that it is impossible to obtain a satisfactory +\PageSep{10} +explanation of the experiments on temperature radiation with the +aid of electrodynamics, no matter what atomic model be employed. +The fact that the deficiencies of the atomic model we are +considering stand out so plainly is therefore perhaps no serious +drawback; even though the defects of other atomic models are +much better concealed they must nevertheless be present and will +be just as serious. + +\Section{Quantum theory of spectra.} Let us now try to overcome these +difficulties by applying Planck's theory to the problem. + +It is readily seen that there can be no question of a direct application +of Planck's theory. This theory is concerned with the emission +and absorption of energy in a system of electrical particles, which +oscillate with a given frequency per second, dependent only on the +nature of the system and independent of the amount of energy +contained in the system. In a system consisting of an electron and +a nucleus the period of oscillation corresponds to the period of +revolution of the electron. But the formula~\Eq{(4)} for~$\omega$ shows that the +frequency of revolution depends upon~$W$, \ie\ on the energy of the +system. Still the fact that we \Chg{can not}{cannot} immediately apply Planck's +theory to our problem is not as serious as it might seem to be, for +in assuming Planck's theory we have manifestly acknowledged the +inadequacy of the ordinary electrodynamics and have definitely +parted with the coherent group of ideas on which the latter theory +is based. In fact in taking such a step we \Chg{can not}{cannot} expect that all +cases of disagreement between the theoretical conceptions hitherto +employed and experiment will be removed by the use of Planck's +assumption regarding the quantum of the energy momentarily +present in an oscillating system. We stand here almost entirely on +virgin ground, and upon introducing new assumptions we need only +take care not to get into contradiction with experiment. Time will +have to show to what extent this can be avoided; but the safest +way is, of course, to make as few assumptions as possible. + +With this in mind let us first examine the experiments on +temperature radiation. The subject of direct observation is the +distribution of radiant energy over oscillations of the various wave +lengths. Even though we may assume that this energy comes from +systems of oscillating particles, we know little or nothing about +\PageSep{11} +these systems. No one has ever seen a Planck's resonator, nor +indeed even measured its frequency of oscillation; we can observe +only the period of oscillation of the radiation which is emitted. It +is therefore very convenient that it is possible to show that to +obtain the laws of temperature radiation it is not necessary to +make any assumptions about the systems which emit the radiation +except that the amount of energy emitted each time shall be equal +to~$h\nu$, where $h$~is Planck's constant and $\nu$~is the frequency of the +radiation. Indeed, it is possible to derive Planck's law of radiation +from this assumption alone, as shown by Debye, who employed a +method which is a combination of that of Planck and of Jeans. +Before considering any further the nature of the oscillating systems +let us see whether it is possible to bring this assumption about the +emission of radiation into agreement with the spectral laws. + +If the spectrum of some element contains a spectral line corresponding +to the frequency~$\nu$ it will be assumed that one of the +atoms of the element (or some other elementary system) can emit +an amount of energy~$h\nu$. Denoting the energy of the atom before +and after the emission of the radiation by $E_{1}$ and~$E_{2}$ we have +\[ +h\nu = E_{1} - E_{2} \text{ or } +\nu = \frac{E_{1}}{h} - \frac{E_{2}}{h}. +\Tag{(5)} +\] + +During the emission of the radiation the system may be regarded +as passing from one state to another; in order to introduce a name +for these states, we shall call them ``stationary'' states, simply +indicating thereby that they form some kind of waiting places +between which occurs the emission of the energy corresponding to +the various spectral lines. As previously mentioned the spectrum +of an element consists of a series of lines whose wave lengths may +be expressed by the formula~\Eq{(2)}. By comparing this expression +with the relation given above it is seen that---since $\nu = \dfrac{c}{\lambda}$, where $c$~is +the velocity of light---each of the spectral lines may be regarded +as being emitted by the transition of a system between two stationary +states in which the energy apart from an additive arbitrary +constant is given by $-ch F_{r}(n_{1})$ and $-ch F_{s}(n_{2})$ respectively. Using +this interpretation the combination principle asserts that a series +of stationary states exists for the given system, and that it can +\PageSep{12} +pass from one to any other of these states with the emission of +a monochromatic radiation. We see, therefore, that with a simple +extension of our first assumption it is possible to give a formal +explanation of the most general law of line spectra. + +\Section{Hydrogen spectrum.} This result encourages us to make an +attempt to obtain a clear conception of the stationary states which +have so far only been regarded as formal. With this end in view, +we naturally turn to the spectrum of hydrogen. The formula +applying to this spectrum is given by the expression +\[ +\frac{1}{\lambda} = \frac{R}{n_{1}^{2}} - \frac{R}{n_{2}^{2}}. +\] +According to our assumption this spectrum is produced by transitions +between a series of stationary states of a system, concerning +which we can for the present only say that the energy of the system +in the $n$th~state, apart from an additive constant, is given by +$-\dfrac{Rhc}{n^{2}}$. Let us now try to find a connection between this and the +model of the hydrogen atom. We assume that in the calculation +of the frequency of revolution of the electron in the stationary states +of the atom it will be possible to employ the above formula for~$\omega$. +It is quite natural to make this assumption; since, in trying to +form a reasonable conception of the stationary states, there is, for +the present at least, no other means available besides the ordinary +mechanics. + +Corresponding to the $n$th~stationary state in formula~\Eq{(4)} for~$\omega$, +let us by way of experiment put $W = \dfrac{Rhc}{n^{2}}$. This gives us +\[ +\omega_{n}^{2} = \frac{2}{\pi^{2}}\, \frac{R^{3} h^{3} c^{3}}{e^{4} mn^{6}}. +\Tag{(6)} +\] + +The radiation of light corresponding to a particular spectral line +is according to our assumption emitted by a transition between +two stationary states, corresponding to two different frequencies of +revolution, and we are not justified in expecting any simple relation +between these frequencies of revolution of the electron and +the frequency of the emitted radiation. You understand, of course, +that I am by no means trying to give what might ordinarily be +described as an explanation; nothing has been said here about +\PageSep{13} +how or why the radiation is emitted. On one point, however, we +may expect a connection with the ordinary conceptions; namely, +that it will be possible to calculate the emission of slow electromagnetic +oscillations on the basis of the classical electrodynamics. +This assumption is very strongly supported by the result of +Lorentz's calculations which have already been described. From +the formula for~$\omega$ it is seen that the frequency of revolution decreases +as $n$~increases, and that the expression~$\dfrac{\omega_{n}}{\omega_{n+1}}$ approaches the +value~$1$. + +According to what has been said above, the frequency of the +radiation corresponding to the transition between the $(n + 1)$th +and the $n$th~stationary state is given by +\[ +\nu = Rc \left(\frac{1}{n^{2}} - \frac{1}{(n + 1)^{2}}\right). +\] +If $n$~is very large this expression is approximately equal to +\[ +\nu = 2Rc/n^{3}. +\] +In order to obtain a connection with the ordinary electrodynamics +let us now place this frequency equal to the frequency of revolution, +that is +\[ +\omega_{n} = 2Rc/n^{3}. +\] +Introducing this value of~$\omega_{n}$ in~\Eq{(6)} we see that $n$~disappears from +the equation, and further that the equation will be satisfied only if +\[ +R = \frac{2\pi^{2} e^{4} m}{ch^{3}}. +\Tag{(7)} +\] +The constant~$R$ is very accurately known, and is, as I have said +before, equal to~$109675$. By introducing the most recent values +for $e$,~$m$ and~$h$ the expression on the right-hand side of the equation +becomes equal to $1.09 · 10^{5}$. The agreement is as good as +could be expected, considering the uncertainty in the experimental +determination of the constants $e$,~$m$ and~$h$. The agreement between +our calculations and the classical electrodynamics is, therefore, +fully as good as we are justified in expecting. + +We \Chg{can not}{cannot} expect to obtain a corresponding explanation of the +frequency values of the other stationary states. Certain simple +formal relations apply, however, to all the stationary states. By +introducing the expression, which has been found for~$R$, we +get for the $n$th~state $W_{n} = \frac{1}{2}nh\omega_{n}$. This equation is entirely +\PageSep{14} +analogous to Planck's assumption concerning the energy of a +resonator. $W$~in our system is readily shown to be equal to the +average value of the kinetic energy of the electron during a +single revolution. The energy of a resonator was shown by Planck +you may remember to be always equal to~$nh\nu$. Further the average +value of the kinetic energy of Planck's resonator is equal to its +potential energy, so that the average value of the kinetic energy +of the resonator, according to Planck, is equal to~$\frac{1}{2}nh\omega$. This +analogy suggests another manner of presenting the theory, and it +was just in this way that I was originally led into these considerations. +When we consider how differently the equation is +employed here and in Planck's theory it appears to me misleading +to use this analogy as a foundation, and in the account I have +given I have tried to free myself as much as possible from it. + +Let us continue with the elucidation of the calculations, and in +the expression for~$2a$ introduce the value of~$W$ which corresponds +to the $n$th~stationary state. This gives us +\[ +2a = n^{2} · \frac{e^{2}}{chR} + = n^{2} · \frac{h^{2}}{2\pi^{2} me^{2}} + = n^{2} · 1.1 · 10^{-8}. +\Tag{(8)} +\] + +It is seen that for small values of~$n$, we obtain values for the +major axis of the orbit of the electron which are of the same +order of magnitude as the values of the diameters of the atoms +calculated from the kinetic theory of gases. For large values of~$n$, +$2a$~becomes very large in proportion to the calculated dimensions +of the atoms. This, however, does not necessarily disagree with +experiment. Under ordinary circumstances a hydrogen atom will +probably exist only in the state corresponding to $n = 1$. For this +state $W$~will have its greatest value and, consequently, the atom +will have emitted the largest amount of energy possible; this will +therefore represent the most stable state of the atom from which +the system \Chg{can not}{cannot} be transferred except by adding energy to it +from without. The large values for~$2a$ corresponding to large~$n$ need +not, therefore, be contrary to experiment; indeed, we may in these +large values seek an explanation of the fact, that in the laboratory +it has hitherto not been possible to observe the hydrogen lines +corresponding to large values of~$n$ in Balmer's formula, while they +have been observed in the spectra of certain stars. In order that +the large orbits of the electrons may not be disturbed by electrical +\PageSep{15} +forces from the neighbouring atoms the pressure will have to be +very low, so low, indeed, that it is impossible to obtain sufficient +light from a Geissler tube of ordinary dimensions. In the stars, +however, we may assume that we have to do with hydrogen which +is exceedingly attenuated and distributed throughout an enormously +large region of space. + +\Section{The Pickering lines.} You have probably noticed that we have +not mentioned at all the spectrum found in certain stars which +according to the opinion then current was assigned to hydrogen, +and together with the ordinary hydrogen spectrum was considered +by Rydberg to form a connected system of lines completely +analogous to the spectra of other elements. You have probably +also perceived that difficulties would arise in interpreting this +spectrum by means of the assumptions which have been employed. +If such an attempt were to be made it would be necessary to give +up the simple considerations which lead to the expression~\Eq{(7)} for +the constant~$R$. We shall see, however, that it appears possible to +explain the occurrence of this spectrum in another way. Let us +suppose that it is not due to hydrogen, but to some other simple +system consisting of a single electron revolving about a nucleus +with an electrical charge~$Ne$. The expression for~$\omega$ becomes then +\[ +\omega^{2} = \frac{2}{\pi^{2}}\, \frac{W^{3}}{N^{2} e^{4} m}. +\] +Repeating the same calculations as before only in the inverse +order we find, that this system will emit a line spectrum given by +the expression +\[ +\frac{1}{\lambda} + = \frac{2\pi^{2} N^{2} e^{4} m}{ch^{3}}\left(\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}\right) + = R\raisebox{-4pt}{$\Biggl($}\frac{1}{\left(\dfrac{n_{1}}{N}\right)^{2}} - \frac{1}{\left(\dfrac{n_{2}}{N}\right)^{2}}\raisebox{-4pt}{$\Biggr)$}. +\Tag{(9)} +\] + +By comparing this formula with the formula for Pickering's and +Rydberg's series, we see that the observed lines can be explained +on the basis of the theory, if it be assumed that the spectrum is +due to an electron revolving about a nucleus with a charge~$2e$, or +according to Rutherford's theory around the nucleus of a helium +atom. The fact that the spectrum in question is not observed in +an ordinary helium tube, but only in stars, may be accounted for +\PageSep{16} +by the high degree of ionization which is required for the production +of this spectrum; a neutral helium atom contains of course +two electrons while the system under consideration contains +only one. + +These conclusions appear to be supported by experiment. +Fowler, as I have mentioned, has recently succeeded in observing +Pickering's and Rydberg's lines in a laboratory experiment. By +passing a very heavy current through a mixture of hydrogen and +helium Fowler observed not only these lines but also a new series +of lines. This new series was of the same general type, the wave +length being given approximately by +\[ +\frac{1}{\lambda} + = R\left(\frac{1}{(\frac{3}{2})^{2}} - \frac{1}{(n + \frac{1}{2})^{2}}\right). +\] +Fowler interpreted all the observed lines as the hydrogen spectrum +sought for. With the observation of the latter series of lines, +however, the basis of the analogy between the hypothetical +hydrogen spectrum and the other spectra disappeared, and thereby +also the foundation upon which Rydberg had founded his conclusions; +on the contrary it is seen, that the occurrence of the lines +was exactly what was to be expected on our view. + +In the following table the first column contains the wave +lengths measured by Fowler, while the second contains the limiting +values of the experimental errors given by him; in the third +column we find the products of the wave lengths by the quantity +$\left(\dfrac{1}{n_{1}^{2}} - \dfrac{1}{n_{2}^{2}}\right) \Add{·} 10^{10}$; the values employed for $n_{1}$~and~$n_{2}$ are enclosed in +parentheses in the last column. +\begin{table}[hbt] +\Pagelabel{16} +\[ +\begin{array}{l*{2}{>{\qquad}l}l} +\ColHead{\lambda · 10^{8}} & +\ColHead{\text{Limit of error}} & +\ColHead{\lambda · \left(\dfrac{1}{n_{1}^{2}} - \dfrac{1}{n_{2}^{2}}\right) · 10^{10}} & \\ +4685.98 & 0.01 & 22779.1 & (3 : 4) \\ +3203.30 & 0.05 & 22779.0 & (3 : 5) \\ +2733.34 & 0.05 & 22777.8 & (3 : 6) \\ +2511.31 & 0.05 & 22778.3 & (3 : 7) \\ +2385.47 & 0.05 & 22777.9 & (3 : 8) \\ +2306.20 & 0.10 & 22777.3 & (3 : 9) \\ +2252.88 & 0.10 & 22779.1 & (3 : 10) \\ +5410.5 & 1.0 & 22774 & (4 : 7) \\ +4541.3 & 0.25 & 22777 & (4 : 9) \\ +4200.3 & 0.5 & 22781 & (4 : 11) \\ +\end{array} +\] +\end{table} +\PageSep{17} + +The values of the products are seen to be very nearly equal, +while the deviations are of the same order of magnitude as the +limits of experimental error. The value of the product +\[ +\lambda \left(\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}\right) +\] +should for this spectrum, according to the formula~\Eq{(9)}, be exactly +$\frac{1}{4}$~of the corresponding product for the hydrogen spectrum. From +the tables on pages \PageNum{1} and~\PageNum{16} we find for these products $91153$ +and $22779$, and dividing the former by the latter we get $4.0016$. +This value is very nearly equal to~$4$; the deviation is, however, +much greater than can be accounted for in any way by the errors +of the experiments. It has been easy, however, to find a theoretical +explanation of this point. In all the foregoing calculations +we have assumed that the mass of the nucleus is infinitely great +compared to that of the electron. This is of course not the +case, even though it holds to a very close approximation; for a +hydrogen atom the ratio of the mass of the nucleus to that of the +electron will be about $1850$ and for a helium atom four times as +great. + +If we consider a system consisting of an electron revolving about +a nucleus with a charge~$Ne$ and a mass~$M$, we find the following +expression for the frequency of revolution of the system: +\[ +\omega^{2} = \frac{2}{\pi^{2}}\, \frac{W^{3} (M + m)}{N^{2} e^{4} Mm}. +\] + +From this formula we find in a manner quite similar to that +previously employed that the system will emit a line spectrum, +the wave lengths of which are given by the formula +\[ +\frac{1}{\lambda} + = \frac{2\pi^{2} N^{2} e^{4} mM}{ch^{3} (M + m)} + \left(\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}\right). +\Tag{(10)} +\] + +If with the aid of this formula we try to find the ratio of the +product for the hydrogen spectrum, to that of the hypothetical +helium spectrum we get the value $4.00163$ which is in complete +agreement with the preceding value calculated from the experimental +observations. + +I must further mention that Evans has made some experiments +to determine whether the spectrum in question is due to hydrogen +or helium. He succeeded in observing one of the lines in very +\PageSep{18} +pure helium; there was, at any rate, not enough hydrogen present +to enable the hydrogen lines to be observed. Since in any event +Fowler does not seem to consider such evidence as conclusive it is +to be hoped that these experiments will be continued. There is, +however, also another possibility of deciding this question. As is +evident from the formula~\Eq{(10)}, the helium spectrum under consideration +should contain, besides the lines observed by Fowler, a +series of lines lying close to the ordinary hydrogen lines. These +lines may be obtained by putting $n_{1} = 4$, $n_{2} = 6$, $8$, $10$,~etc. Even +if these lines were present, it would be extremely difficult to +observe them on account of their position with regard to the +hydrogen lines, but should they be observed this would probably +also settle the question of the origin of the spectrum, since no +reason would seem to be left to assume the spectrum to be due to +hydrogen. + +\Section{Other spectra.} For the spectra of other elements the problem +becomes more complicated, since the atoms contain a larger +number of electrons. It has not yet been possible on the basis of +this theory to explain any other spectra besides those which I +have already mentioned. On the other hand it ought to be +mentioned that the general laws applying to the spectra are very +simply interpreted on the basis of our assumptions. So far as the +combination principle is concerned its explanation is obvious. In +the method we have employed our point of departure was largely +determined by this particular principle. But a simple explanation +can be also given of the other general law, namely, the occurrence +of Rydberg's constant in all spectral formulae. Let us assume +that the spectra under consideration, like the spectrum of hydrogen, +are emitted by a neutral system, and that they are produced by +the binding of an electron previously removed from the system. +If such an electron revolves about the nucleus in an orbit which +is large in proportion to that of the other electrons it will be +subjected to forces much the same as the electron in a hydrogen +atom, since the inner electrons individually will approximately +neutralize the effect of a part of the positive charge of the nucleus. +We may therefore assume that for this system there will exist a +series of stationary states in which the motion of the outermost +\PageSep{19} +electron is approximately the same as in the stationary states of a +hydrogen atom. I shall not discuss these matters any further, +but shall only mention that they lead to the conclusion that +Rydberg's constant is not exactly the same for all elements. +The expression for this constant will in fact contain the factor +$\dfrac{M}{M + m}$, where $M$~is the mass of the nucleus. The correction is +exceedingly small for elements of large atomic weight, but for +hydrogen it is, from the point of view of spectrum analysis, very +considerable. If the procedure employed leads to correct results, it +is not therefore permissible to calculate Rydberg's constant directly +from the hydrogen spectrum; the value of the universal constant +should according to the theory be~$109735$ and not~$109675$. + +I shall not tire you any further with more details; I hope to +return to these questions here in the Physical Society, and to +show how, on the basis of the underlying ideas, it is possible +to develop a theory for the structure of atoms and molecules. +Before closing I only wish to say that I hope I have expressed +myself sufficiently clearly so that you have appreciated the extent +to which these considerations conflict with the admirably coherent +group of conceptions which have been rightly termed the classical +theory of electrodynamics. On the other hand, by emphasizing +this conflict, I have tried to convey to you the impression that it +may be also possible in the course of time to discover a certain +coherence in the new ideas. +\PageSep{20} + + +\Essay{II}{On the Series Spectra of the Elements} + {Address delivered before the Physical Society in Berlin, April~27, 1920.} + +\Chapter{I.}{Introduction} + +The subject on which I have the honour to speak here, at the +kind invitation of the Council of your society, is very extensive and +it would be impossible in a single address to give a comprehensive +survey of even the most important results obtained in the theory +of spectra. In what follows I shall try merely to emphasize some +points of view which seem to me important when considering the +present state of the theory of spectra and the possibilities of its +development in the near future. I regret in this connection not to +have time to describe the history of the development of spectral +theories, although this would be of interest for our purpose. No +difficulty, however, in understanding this lecture need be experienced +on this account, since the points of view underlying previous +attempts to explain the spectra differ fundamentally from those upon +which the following considerations rest. This difference exists both +in the development of our ideas about the structure of the atom +and in the manner in which these ideas are used in explaining the +spectra. + +We shall assume, according to Rutherford's theory, that an atom +consists of a positively charged nucleus with a number of electrons +revolving about it. Although the nucleus is assumed to be very +small in proportion to the size of the whole atom, it will contain +nearly the entire mass of the atom. I shall not state the reasons +which led to the establishment of this nuclear theory of the atom, +nor describe the very strong support which this theory has received +from very different sources. I shall mention only that result +which lends such charm and simplicity to the modern development +of the atomic theory. I refer to the idea that the number of electrons +in a neutral atom is exactly equal to the number, giving the +position of the element in the periodic table, the so-called ``atomic +number.'' This assumption, which was first proposed by van~den +Broek, immediately suggests the possibility ultimately of deriving +\PageSep{21} +the explanation of the physical and chemical properties of the +elements from their atomic numbers. If, however, an explanation +of this kind is attempted on the basis of the classical laws of +mechanics and electrodynamics, insurmountable difficulties are encountered. +These difficulties become especially apparent when we +consider the spectra of the elements. In fact, the difficulties are +here so obvious that it would be a waste of time to discuss them in +detail. It is evident that systems like the nuclear atom, if based +upon the usual mechanical and electrodynamical conceptions, +would not even possess sufficient stability to give a spectrum consisting +of sharp lines. + +In this lecture I shall use the ideas of the quantum theory. It +will not be necessary, particularly here in Berlin, to consider in +detail how Planck's fundamental work on temperature radiation +has given rise to this theory, according to which the laws governing +atomic processes exhibit a definite element of discontinuity. I shall +mention only Planck's chief result about the properties of an exceedingly +simple kind of atomic system, the Planck ``oscillator.'' +This consists of an electrically charged particle which can execute +harmonic oscillations about its position of equilibrium with a frequency +independent of the amplitude. By studying the statistical +equilibrium of a number of such systems in a field of radiation +Planck was led to the conclusion that the emission and absorption +of radiation take place in such a manner, that, so far as a statistical +equilibrium is concerned only certain distinctive states of the +oscillator are to be taken into consideration. In these states the +energy of the system is equal to a whole multiple of a so-called +``energy quantum,'' which was found to be proportional to the frequency +of the oscillator. The particular energy values are therefore +given by the well-known formula +\[ +E_{n} = nh\omega, +\Tag{(1)} +\] +where $n$~is a whole number, $\omega$~the frequency of vibration of the +oscillator, and $h$~is Planck's constant. + +If we attempt to use this result to explain the spectra of the +elements, however, we encounter difficulties, because the motion of +the particles in the atom, in spite of its simple structure, is in general +exceedingly complicated compared with the motion of a Planck +\PageSep{22} +oscillator. The question then arises, how Planck's result ought to +be generalized in order to make its application possible. Different +points of view immediately suggest themselves. Thus we might +regard this equation as a relation expressing certain characteristic +properties of the distinctive motions of an atomic system and try +to obtain the general form of these properties. On the other hand, +we may also regard equation~\Eq{(1)} as a statement about a property +of the process of radiation and inquire into the general laws which +control this process. + +In Planck's theory it is taken for granted that the frequency of +the radiation emitted and absorbed by the oscillator is equal to its +own frequency, an assumption which may be written +\[ +\nu \equiv \omega, +\Tag{(2)} +\] +if in order to make a sharp distinction between the frequency of +the emitted radiation and the frequency of the particles in the atoms, +we here and in the following denote the former by~$\nu$ and the latter +by~$\omega$. We see, therefore, that Planck's result may be interpreted to +mean, that the oscillator can emit and absorb radiation only in +``radiation quanta'' of magnitude +\[ +\Delta E = h\nu. +\Tag{(3)} +\] +It is well known that ideas of this kind led Einstein to a theory +of the photoelectric effect. This is of great importance, since it +represents the first instance in which the quantum theory was +applied to a phenomenon of non-statistical character. I shall not +here discuss the familiar difficulties to which the ``hypothesis of +light quanta'' leads in connection with the phenomena of interference, +for the explanation of which the classical theory of radiation +has shown itself to be so remarkably suited. Above all I shall not +consider the problem of the nature of radiation, I shall only attempt +to show how it has been possible in a purely formal manner to +develop a spectral theory, the essential elements of which may be +considered as a simultaneous rational development of the two ways +of interpreting Planck's result. +\PageSep{23} + + +\Chapter{II.}{General Principles of the Quantum Theory +of\protect~Spectra} + +In order to explain the appearance of line spectra we are compelled +to assume that the emission of radiation by an atomic system +takes place in such a manner that it is not possible to follow the +emission in detail by means of the usual conceptions. Indeed, these +do not even offer us the means of calculating the frequency of the +emitted radiation. We shall see, however, that it is possible to give +a very simple explanation of the general empirical laws for the +frequencies of the spectral lines, if for each emission of radiation +by the atom we assume the fundamental law to hold, that during +the entire period of the emission the radiation possesses one and +the same frequency~$\nu$, connected with the total energy emitted by +the \emph{frequency relation} +\[ +h\nu = E' - E''. +\Tag{(4)} +\] +Here $E'$~and $E''$ represent the energy of the system before and +after the emission. + +If this law is assumed, the spectra do not give us information +about the motion of the particles in the atom, as is supposed in the +usual theory of radiation, but only a knowledge of the energy +changes in the various processes which can occur in the atom. +From this point of view the spectra show the existence of certain, +definite energy values corresponding to certain distinctive states +of the atoms. These states will be called the \emph{stationary states} of +the atoms, since we shall assume that the atom can remain a finite +time in each state, and can leave this state only by a process of +transition to another stationary state. Notwithstanding the fundamental +departure from the ordinary mechanical and electrodynamical +conceptions, we shall see, however, that it is possible to give a +rational interpretation of the evidence provided by the spectra on +the basis of these ideas. + +Although we must assume that the ordinary mechanics \Chg{can not}{cannot} +be used to describe the transitions between the stationary states, +nevertheless, it has been found possible to develop a consistent +theory on the assumption that the motion in these states can be +described by the use of the ordinary mechanics. Moreover, although +the process of radiation \Chg{can not}{cannot} be described on the basis of the +\PageSep{24} +ordinary theory of electrodynamics, according to which the nature +of the radiation emitted by an atom is directly related to the harmonic +components occurring in the motion of the system, there is +found, nevertheless, to exist a far-reaching \emph{correspondence} between +the various types of possible transitions between the stationary +states on the one hand and the various harmonic components of the +motion on the other hand. This correspondence is of such a nature, +that the present theory of spectra is in a certain sense to be regarded +as a rational generalization of the ordinary theory of radiation. + +\Section{Hydrogen spectrum.} In order that the principal points may +stand out as clearly as possible I shall, before considering the more +complicated types of series spectra, first consider the simplest spectrum, +namely, the series spectrum of hydrogen. This spectrum +consists of a number of lines whose frequencies are given with great +exactness by Balmer's formula +\[ +\nu = \frac{K}{(n'')^{2}} - \frac{K}{(n')^{2}}, +\Tag{(5)} +\] +where $K$~is a constant, and $n'$~and $n''$ are whole numbers. If we put +$n'' = 2$ and give to~$n'$ the values $3$,~$4$,~etc., we get the well-known +Balmer series of hydrogen. If we put $n'' = 1$ or $n'' = 3$ we obtain +respectively the ultra-violet and infra-red series. We shall assume +the hydrogen atom simply to consist of a positively charged nucleus +with a single electron revolving about it. For the sake of simplicity +we shall suppose the mass of the nucleus to be infinite in comparison +with the mass of the electron, and further we shall disregard the +small variations in the motion due to the change in mass of the +electron with its velocity. With these simplifications the electron +will describe a closed elliptical orbit with the nucleus at one of the +foci. The frequency of revolution~$\omega$ and the major axis~$2a$ of the +orbit will be connected with the energy of the system by the following +equations: +\[ +\omega = \sqrt{\frac{2W^{3}}{\pi^{2} e^{4} m}},\quad +2a = \frac{e^{2}}{W}. +\Tag{(6)} +\] +Here $e$~is the charge of the electron and $m$~its mass, while $W$~is the +work required to remove the electron to infinity. + +The simplicity of these formulae suggests the possibility of using +them in an attempt to explain the spectrum of hydrogen. This, +\PageSep{25} +however, is not possible so long as we use the classical theory of +radiation. It would not even be possible to understand how hydrogen +could emit a spectrum consisting of sharp lines; for since $\omega$~varies +with~$W$, the frequency of the emitted radiation would vary continuously +during the emission. We can avoid these difficulties if +we use the ideas of the quantum theory. If for each line we form +the product~$h\nu$ by multiplying both sides of~\Eq{(5)} by~$h$, then, since +the right-hand side of the resulting relation may be written as +the difference of two simple expressions, we are led by comparison +with formula~\Eq{(4)} to the assumption that the separate lines of the +spectrum will be emitted by transitions between two stationary +states, forming members of an infinite series of states, in which the +energy in the $n$th~state apart from an arbitrary additive constant is +determined by the expression +\[ +E_{n} = -\frac{Kh}{n^{2}}. +\Tag{(7)} +\] +The negative sign has been chosen because the energy of the atom +will be most simply characterized by the work~$W$ required to remove +the electron completely from the atom. If we now substitute $\dfrac{Kh}{n^{2}}$ +for~$W$ in formula~\Eq{(6)}, we obtain the following expression for the frequency +and the major axis in the $n$th~stationary state: +\[ +\omega_{n} = \frac{1}{n^{3}} \sqrt{\frac{2h^{3} K^{3}}{\pi^{2} e^{4} m}},\quad +2a_{n} = \frac{n^{2} e^{2}}{hK}. +\Tag{(8)} +\] +A comparison between the motions determined by these equations +and the distinctive states of a Planck resonator may be shown to +offer a theoretical determination of the constant~$K$. Instead of +doing this I shall show how the value of~$K$ can be found by a simple +comparison of the spectrum emitted with the motion in the stationary +states, a comparison which at the same time will lead us to the +principle of correspondence. + +We have assumed that each hydrogen line is the result of a +transition between two stationary states of the atom corresponding +to different values of~$n$. Equations~\Eq{(8)} show that the frequency of +revolution and the major axis of the orbit can be entirely different +in the two states, since, as the energy decreases, the major axis of +the orbit becomes smaller and the frequency of revolution increases. +\PageSep{26} +In general, therefore, it will be impossible to obtain a relation between +the frequency of revolution of the electrons and the frequency +of the radiation as in the ordinary theory of radiation. If, however, +we consider the ratio of the frequencies of revolution in two stationary +states corresponding to given values of $n'$~and~$n''$, we see that this +ratio approaches unity as $n'$~and $n''$ gradually increase, if at the +same time the difference $n' - n''$ remains unchanged. By considering +transitions corresponding to large values of $n'$~and~$n''$ we may +therefore hope to establish a certain connection with the ordinary +theory. For the frequency of the radiation emitted by a transition, +we get according to~\Eq{(5)} +\[ +\nu = \frac{K}{(n'')^{2}} - \frac{K}{(n')^{2}} + = (n' - n'') K\, \frac{n' + n''}{(n')^{2} (n'')^{2}}. +\Tag{(9)} +\] +If now the numbers $n'$~and $n''$ are large in proportion to their difference, +we see that by equations~\Eq{(8)} this expression may be written +approximately, +\[ +\nu \sim (n' - n'') \omega \sqrt{\frac{2\pi^{2} e^{4} m}{Kh^{3}}}, +\Tag{(10)} +\] +where $\omega$~represents the frequency of revolution in the one or the +other of the two stationary states. Since $n' - n''$ is a whole number, +we see that the first part of this expression, \ie\ $(n' - n'')\omega$, is the +same as the frequency of one of the harmonic components into +which the elliptical motion may be decomposed. This involves the +well-known result that for a system of particles having a periodic +motion of frequency~$\omega$, the displacement~$\xi$ of the particles in a given +direction in space may be represented as a function of the time by +a trigonometric series of the form +\[ +\xi = \sum C_{\tau} \cos 2\pi(\tau\omega t + c_{\tau}), +\Tag{(11)} +\] +where the summation is to be extended over all positive integral +values of~$\tau$. + +We see, therefore, that the frequency of the radiation emitted +by a transition between two stationary states, for which the numbers +$n'$~and $n''$ are large in proportion to their difference, will coincide +with the frequency of one of the components of the radiation, which +according to the ordinary ideas of radiation would be expected from +the motion of the atom in these states, provided the last factor on +\PageSep{27} +the right-hand side of equation~\Eq{(10)} is equal to~$1$. This condition, +which is identical to the condition +\[ +K = \frac{2\pi^{2} e^{4} m}{h^{3}}, +\Tag{(12)} +\] +is in fact fulfilled, if we give to~$K$ its value as found from measurements +on the hydrogen spectrum, and if for $e$,~$m$ and~$h$ we use the +values obtained directly from experiment. This agreement clearly +gives us a \emph{connection between the spectrum and the atomic model of +hydrogen}, which is as close as could reasonably be expected considering +the fundamental difference between the ideas of the quantum +theory and of the ordinary theory of radiation. + +\Section{The correspondence principle.} Let us now consider somewhat +more closely this relation between the spectra one would expect on +the basis of the quantum theory, and on the ordinary theory of +radiation. The frequencies of the spectral lines calculated according +to both methods agree completely in the region where the stationary +states deviate only little from one another. We must not forget, +however, that the mechanism of emission in both cases is different. +The different frequencies corresponding to the various harmonic +components of the motion are emitted simultaneously according to +the ordinary theory of radiation and with a relative intensity depending +directly upon the ratio of the amplitudes of these oscillations. +But according to the quantum theory the various spectral +lines are emitted by entirely distinct processes, consisting of transitions +from one stationary state to various adjacent states, so that +the radiation corresponding to the $\tau$th~``harmonic'' will be emitted +by a transition for which $n' - n'' = \tau$. The relative intensity +with which each particular line is emitted depends consequently +upon the relative probability of the occurrence of the different +transitions. + +This correspondence between the frequencies determined by the +two methods must have a deeper significance and we are led to +anticipate that it will also apply to the intensities. This is equivalent +to the statement that, when the quantum numbers are large, +the relative probability of a particular transition is connected in a +simple manner with the amplitude of the corresponding harmonic +component in the motion. +\PageSep{28} + +This peculiar relation suggests a \emph{general law for the occurrence +of transitions between stationary states}. Thus we shall assume that +even when the quantum numbers are small the possibility of +transition between two stationary states is connected with the +presence of a certain harmonic component in the motion of the +system. If the numbers $n'$~and $n''$ are not large in proportion to +their difference, the numerical value of the amplitudes of these +components in the two stationary states may be entirely different. +We must be prepared to find, therefore, that the exact connection +between the probability of a transition and the amplitude of the +corresponding harmonic component in the motion is in general +complicated like the connection between the frequency of the radiation +and that of the component. From this point of view, for +example, the green line~$H_{\beta}$ of the hydrogen spectrum which corresponds +to a transition from the fourth to the second stationary +state may be considered in a certain sense to be an ``octave'' of the +red line~$H_{\alpha}$, corresponding to a transition from the third to the +second state, even though the frequency of the first line is by no +means twice as great as that of the latter. In fact, the transition +giving rise to~$H_{\beta}$ may be regarded as due to the presence of a harmonic +oscillation in the motion of the atom, which is an octave +higher than the oscillation giving rise to the emission of~$H_{\alpha}$. + +Before considering other spectra, where numerous opportunities +will be found to use this point of view, I shall briefly mention an +interesting application to the Planck oscillator. If from \Eq{(1)}~and \Eq{(4)} +we calculate the frequency, which would correspond to a transition +between two particular states of such an oscillator, we find +\[ +\nu = (n' - n'')\omega, +\Tag{(13)} +\] +where $n'$~and $n''$ are the numbers characterizing the states. It was +an essential assumption in Planck's theory that the frequency of +the radiation emitted and absorbed by the oscillator is always equal +to~$\omega$. We see that this assumption is equivalent to the assertion +that transitions occur only between two successive stationary states +in sharp contrast to the hydrogen atom. According to our view, +however, this was exactly what might have been expected, for we +must assume that the essential difference between the oscillator +and the hydrogen atom is that the motion of the oscillator is simple +\PageSep{29} +harmonic. We can see that it is possible to develop a formal theory +of radiation, in which the spectrum of hydrogen and the simple +spectrum of a Planck oscillator appear completely analogous. This +theory can only be formulated by one and the same condition for +a system as simple as the oscillator. In general this condition +breaks up into two parts, one concerning the fixation of the stationary +states, and the other relating to the frequency of the radiation +emitted by a transition between these states. + +\Section{General spectral laws.} Although the series spectra of the +elements of higher atomic number have a more complicated structure +than the hydrogen spectrum, simple laws have been discovered +showing a remarkable analogy to the Balmer formula. Rydberg +and Ritz showed that the frequencies in the series spectra of many +elements can be expressed by a formula of the type +\[ +\nu = f_{k''}(n'') - f_{k'}(n'), +\Tag{(14)} +\] +where $n'$~and $n''$ are two whole numbers and $f_{k'}$~and $f_{k''}$ are two +functions belonging to a series of functions characteristic of the +element. These functions vary in a simple manner with~$n$ and in +particular converge to zero for increasing values of~$n$. The various +series of lines are obtained from this formula by allowing the first +term~$f_{k''}(n'')$ to remain constant, while a series of consecutive whole +numbers are substituted for~$n'$ in the second term~$f_{k'}(n')$. According +to the Ritz \emph{combination principle} the entire spectrum may then +be obtained by forming every possible combination of two values +among all the quantities~$f_{k}(n)$. + +The fact that the frequency of each line of the spectrum may be +written as the difference of two simple expressions depending upon +whole numbers suggests at once that the terms on the right-hand +side multiplied by~$h$ may be placed equal to the energy in the +various stationary states of the atom. The existence in the spectra +of the other elements of a number of separate functions of~$n$ compels +us to assume the presence not of one but of a number of series of +stationary states, the energy of the $n$th~state of the $k$th~series apart +from an arbitrary additive constant being given by +\[ +E_{k}(n) = -h f_{k}(n). +\Tag{(15)} +\] +This complicated character of the ensemble of stationary states of +atoms of higher atomic number is exactly what was to be expected +\PageSep{30} +from the relation between the spectra calculated on the quantum +theory, and the decomposition of the motions of the atoms into +harmonic oscillations. From this point of view we may regard the +simple character of the stationary states of the hydrogen atom as +intimately connected with the simple periodic character of this +atom. Where the neutral atom contains more than one electron, we +find much more complicated motions with correspondingly complicated +harmonic components. We must therefore expect a more +complicated ensemble of stationary states, if we are still to have a +corresponding relation between the motions in the atom and the +spectrum. In the course of the lecture we shall trace this correspondence +in detail, and we shall be led to a simple explanation of +the apparent capriciousness in the occurrence of lines predicted by +the combination principle. + +The following figure gives a survey of the stationary states of +the sodium atom deduced from the series terms. +\Figure[Diagram of the series spectrum of sodium.]{}{30} + +The stationary states are represented by black dots whose distance +from the vertical line a---a is proportional to the numerical value +of the energy in the states. The arrows in the figure indicate the +transitions giving those lines of the sodium spectrum which appear +under the usual conditions of excitation. The arrangement of the +states in horizontal rows corresponds to the ordinary arrangement +of the ``spectral terms'' in the spectroscopic tables. Thus, the states +in the first row~($S$) correspond to the variable term in the ``sharp +series,'' the lines of which are emitted by transitions from these +states to the first state in the second row. The states in the second +\PageSep{31} +row~($P$) correspond to the variable term in the ``principal series'' +which is emitted by transitions from these states to the first state +in the $S$~row. The $D$~states correspond to the variable term in the +``diffuse series,'' which like the sharp series is emitted by transitions +to the first state in the $P$~row, and finally the $B$~states correspond +to the variable term in the ``Bergmann'' series (fundamental series), +in which transitions take place to the first state in the $D$~row. The +manner in which the various rows are arranged with reference to +one another will be used to illustrate the more detailed theory +which will be discussed later. The apparent capriciousness of the +combination principle, which I mentioned, consists in the fact that +under the usual conditions of excitation not all the lines belonging +to possible combinations of the terms of the sodium spectrum appear, +but only those indicated in the figure by arrows. + +The general question of the fixation of the stationary states of +an atom containing several electrons presents difficulties of a profound +character which are perhaps still far from completely solved. +It is possible, however, to obtain an immediate insight into the +stationary states involved in the emission of the series spectra by +considering the empirical laws which have been discovered about +the spectral terms. According to the well-known law discovered by +Rydberg for the spectra of elements emitted under the usual conditions +of excitation the functions~$f_{k}(n)$ appearing in formula~\Eq{(14)} +can be written in the form +\[ +f_{k}(n) = \frac{K}{n^{2}} \phi_{k}(n), +\Tag{(16)} +\] +where $\phi_{k}(n)$~represents a function which converges to unity for +large values of~$n$. $K$~is the same constant which appears in formula~\Eq{(5)} +for the spectrum of hydrogen. This result must evidently be +explained by supposing the atom to be electrically neutral in these +states and one electron to be moving round the nucleus in an orbit +the dimensions of which are very large in proportion to the distance +of the other electrons from the nucleus. We see, indeed, that in +this case the electric force acting on the outer electron will to a +first approximation be the same as that acting upon the electron +in the hydrogen atom, and the approximation will be the better +the larger the orbit. +\PageSep{32} + +On account of the limited time I shall not discuss how this +explanation of the universal appearance of Rydberg's constant in +the arc spectra is convincingly supported by the investigation of +the ``spark spectra.'' These are emitted by the elements under the +influence of very strong electrical discharges, and come from ionized +not neutral atoms. It is important, however, that I should indicate +briefly how the fundamental ideas of the theory and the assumption +that in the states corresponding to the spectra one electron moves +in an orbit around the others, are both supported by investigations +on selective absorption and the excitation of spectral lines by +bombardment by electrons. + +\Section{Absorption and excitation of radiation.}\Pagelabel{32} Just as we have +assumed that each emission of radiation is due to a transition from +a stationary state of higher to one of lower energy, so also we must +assume absorption of radiation by the atom to be due to a transition +in the opposite direction. For an element to absorb light corresponding +to a given line in its series spectrum, it is therefore +necessary for the atom of this element to be in that one of the two +states connected with the line possessing the smaller energy value. +If we now consider an element whose atoms in the gaseous state +do not combine into molecules, it will be necessary to assume that +under ordinary conditions nearly all the atoms exist in that stationary +state in which the value of the energy is a minimum. This state +I shall call the \emph{normal state}. We must therefore expect that the +absorption spectrum of a monatomic gas will contain only those +lines of the series spectrum, whose emission corresponds to transitions +to the normal state. This expectation is completely confirmed +by the spectra of the alkali metals. The absorption spectrum of +sodium vapour, for example, exhibits lines corresponding only to +the principal series, which as mentioned in the description of the +figure corresponds with transitions to the state of minimum energy. +Further confirmation of this view of the process of absorption is +given by experiments on \emph{resonance radiation}. Wood first showed +that sodium vapour subjected to light corresponding to the first +line of the principal series---the familiar yellow line---acquires the +ability of again emitting a radiation consisting only of the light of +this line. We can explain this by supposing the sodium atom to +\PageSep{33} +have been transferred from the normal state to the first state in +the second row. The fact that the resonance radiation does not +exhibit the same degree of polarization as the incident light is +in perfect agreement with our assumption that the radiation from +the excited vapour is not a resonance phenomenon in the sense of +the ordinary theory of radiation, but on the contrary depends on a +process which is not directly connected with the incident radiation. + +The phenomenon of the resonance radiation of the yellow sodium +line is, however, not quite so simple as I have indicated, since, as +you know, this line is really a doublet. This means that the variable +terms of the principal series are not simple but are represented by +two values slightly different from one another. According to our +picture of the origin of the sodium spectrum this means that the +$P$~states in the second row in the figure---as opposed to the $S$~states +in the first row---are not simple, but that for each place in this row +there are two stationary states. The energy values differ so little +from one another that it is impossible to represent them in the +figure as separate dots. The emission (and absorption) of the two +components of the yellow line are, therefore, connected with two +different processes. This was beautifully shown by some later researches +of Wood and Dunoyer. They found that if sodium vapour +is subjected to radiation from only one of the two components of +the yellow line, the resonance radiation, at least at low pressures, +consists only of this component. These experiments were later +continued by Strutt, and were extended to the case where the +exciting line corresponded to the second line in the principal series. +Strutt found that the resonance radiation consisted apparently only +to a small extent of light of the same frequency as the incident +light, while the greater part consisted of the familiar yellow line. +This result must appear very astonishing on the ordinary ideas of +resonance, since, as Strutt pointed out, no rational connection exists +between the frequencies of the first and second lines of the principal +series. It is however easily explained from our point of view. From +the figure it can be seen that when an atom has been transferred +into the second state in the second row, in addition to the direct +return to the normal state, there are still two other transitions +which may give rise to radiation, namely the transitions to the +second state in the first row and to the first state in the third row. +\PageSep{34} +The experiments seem to indicate that the second of these three +transitions is most probable, and I shall show later that there is +some theoretical justification for this conclusion. By this transition, +which results in the emission of an infra-red line which could not +be observed with the experimental arrangement, the atom is taken +to the second state of the first row, and from this state only +one transition is possible, which again gives an infra-red line. This +transition takes the atom to the first state in the second row, and +the subsequent transition to the normal state then gives rise to the +yellow line. Strutt discovered another equally surprising result, +that this yellow resonance radiation seemed to consist of both +components of the first line of the principal series, even when the +incident light consisted of only one component of the second line +of the principal series. This is in beautiful agreement with our +picture of the phenomenon. We must remember that the states in +the first row are simple, so when the atom has arrived in one of +these it has lost every possibility of later giving any indication +from which of the two states in the second row it originally came. + +Sodium vapour, in addition to the absorption corresponding to +the lines of the principal series, exhibits a \emph{selective absorption in a +continuous spectral region} beginning at the limit of this series and +extending into the ultra\Add{-}violet. This confirms in a striking manner +our assumption that the absorption of the lines of the principal +series of sodium results in final states of the atom in which one of +the electrons revolves in larger and larger orbits. For we must +assume that this continuous absorption corresponds to transitions +from the normal state to states in which the electron is in a position +to remove itself infinitely far from the nucleus. This phenomenon +exhibits a complete analogy with the \emph{photoelectric effect} from an +illuminated metal plate in which, by using light of a suitable +frequency, electrons of any velocity can be obtained. The frequency, +however, must always lie above a certain limit connected according +to Einstein's theory in a simple manner with the energy necessary +to bring an electron out of the metal. + +This view of the origin of the emission and absorption spectra +has been confirmed in a very interesting manner by experiments +on the \emph{excitation of spectral lines and production of ionization by +electron bombardment}. The chief advance in this field is due to the +\PageSep{35} +well-known experiments of Franck and Hertz. These investigators +obtained their first important results from their experiments on +mercury vapour, whose properties particularly facilitate such experiments. +On account of the great importance of the results, these +experiments have been extended to most gases and metals that can +be obtained in a gaseous state. With the aid of the figure I shall +briefly illustrate the results for the case of sodium vapour. It was +found that the electrons upon colliding with the atoms were thrown +back with undiminished velocity when their energy was less than +that required to transfer the atom from the normal state to the +next succeeding stationary state of higher energy value. In the +case of sodium vapour this means from the first state in the first +row to the first state in the second row. As soon, however, as the +energy of the electron reaches this critical value, a new type of +collision takes place, in which the electron loses all its kinetic +energy, while at the same time the vapour is excited and emits a +radiation corresponding to the yellow line. This is what would be +expected, if by the collision the atom was transferred from the +normal state to the first one in the second row. For some time it +was uncertain to what extent this explanation was correct, since +in the experiments on mercury vapour it was found that, together +with the occurrence of non-elastic impacts, ions were always formed +in the vapour. From our figure, however, we would expect ions +to be produced only when the kinetic energy of the electrons is +sufficiently great to bring the atom out of the normal state to the +common limit of the states. Later experiments, especially by Davis +and Goucher, have settled this point. It has been shown that ions +can only be directly produced by collisions when the kinetic energy +of the electrons corresponds to the limit of the series, and that the +ionization found at first was an indirect effect arising from the +photoelectric effect produced at the metal walls of the apparatus +by the radiation arising from the return of the mercury atoms to +the normal state. These experiments provide a direct and independent +proof of the reality of the distinctive stationary states, +whose existence we were led to infer from the series spectra. At +the same time we get a striking impression of the insufficiency of +the ordinary electrodynamical and mechanical conceptions for the +description of atomic processes, not only as regards the emission +\PageSep{36} +of radiation but also in such phenomena as the collision of free +electrons with atoms. + + +\Chapter{III.}{Development of the Quantum Theory +of Spectra} + +We see that it is possible by making use of a few simple ideas +to obtain a certain insight into the origin of the series spectra. +But when we attempt to penetrate more deeply, difficulties arise. +In fact, for systems which are not simply periodic it is not possible +to obtain sufficient information about the motions of these systems +in the stationary states from the numerical values of the energy +alone; more determining factors are required for the fixation of +the motion. We meet the same difficulties when we try to explain +in detail the characteristic effect of external forces upon the spectrum +of hydrogen. A foundation for further advances in this field has +been made in recent years through a development of the quantum +theory, which allows a fixation of the stationary states not only in +the case of simple periodic systems, but also for certain classes of +non-periodic systems. These are the \emph{conditionally periodic systems} +whose equations of motion can be solved by a ``separation of the +variables.'' If generalized coordinates are used the description of +the motion of these systems can be reduced to the consideration +of a number of generalized ``components of motion.'' Each of these +corresponds to the change of only one of the coordinates and may +therefore in a certain sense be regarded as ``independent.'' The +method for the fixation of the stationary states consists in fixing +the motion of each of these components by a condition, which can +be considered as a direct generalization of condition~\Eq{(1)} for a +Planck oscillator, so that the stationary states are in general +characterized by as many whole numbers as the number of the +degrees of freedom which the system possesses. A considerable +number of physicists have taken part in this development of the +quantum theory, including Planck himself. I also wish to mention +the important contribution made by Ehrenfest to this subject on +the limitations of the applicability of the laws of mechanics to +atomic processes. The decisive advance in the application of the +quantum theory to spectra, however, is due to Sommerfeld and his +followers. However, I shall not further discuss the systematic form +\PageSep{37} +in which these authors have presented their results. In a paper which +appeared some time ago in the Transactions of the Copenhagen +Academy, I have shown that the spectra, calculated with the aid +of this method for the fixation of the stationary states, exhibit a +correspondence with the spectra which should correspond to the +motion of the system similar to that which we have already considered +in the case of hydrogen. With the aid of this general +correspondence I shall try in the remainder of this lecture to +show how it is possible to present the theory of series spectra +and the effects produced by external fields of force upon these +spectra in a form which may be considered as the natural generalization +of the foregoing considerations. This form appears to me +to be especially suited for future work in the theory of spectra, +since it allows of an immediate insight into problems for which +the methods mentioned above fail on account of the complexity of +the motions in the atom. + +\Section{Effect of external forces on the hydrogen spectrum.} We +shall now proceed to investigate the effect of small perturbing +forces upon the spectrum of the simple system consisting of a single +electron revolving about a nucleus. For the sake of simplicity we +shall for the moment disregard the variation of the mass of the +electron with its velocity. The consideration of the small changes +in the motion due to this variation has been of great importance +in the development of Sommerfeld's theory which originated in the +explanation of the \emph{fine structure of the hydrogen lines}. This fine +structure is due to the fact, that taking into account the variation +of mass with velocity the orbit of the electron deviates a little +from a simple ellipse and is no longer exactly periodic. This deviation +from a Keplerian motion is, however, very small compared +with the perturbations due to the presence of external forces, such +as occur in experiments on the Zeeman and Stark effects. In atoms +of higher atomic number it is also negligible compared with the +disturbing effect of the inner electrons on the motion of the outer +electron. The neglect of the change in mass will therefore have no +important influence upon the explanation of the Zeeman and Stark +effects, or upon the explanation of the difference between the +hydrogen spectrum and the spectra of other elements. +\PageSep{38} + +We shall therefore as before consider the motion of the unperturbed +hydrogen atom as simply periodic and inquire in the +first place about the stationary states corresponding to this motion. +The energy in these states will then be determined by expression~\Eq{(7)} +which was derived from the spectrum of hydrogen. The energy of +the system being given, the major axis of the elliptical orbit of the +electron and its frequency of revolution are also determined. Substituting +in formulae \Eq{(7)} and~\Eq{(8)} the expression for~$K$ given in~\Eq{(12)}, +we obtain for the energy, major axis and frequency of revolution +in the $n$th~state of the unperturbed atom the expressions +\[ +\BreakMath{% +E_{n} = -W_{n} = -\frac{1}{n^{2}}\, \frac{2\pi^{2} e^{4} m}{h^{2}},\quad +2a_{n} = n^{2}\, \frac{h^{2}}{2\pi^{2} e^{2} m},\quad +\omega_{n} = \frac{1}{n^{3}}\, \frac{4\pi^{2} e^{4} m}{h^{3}}. +}{% +\begin{gathered} +E_{n} = -W_{n} = -\frac{1}{n^{2}}\, \frac{2\pi^{2} e^{4} m}{h^{2}}, \\ +2a_{n} = n^{2}\, \frac{h^{2}}{2\pi^{2} e^{2} m},\qquad +\omega_{n} = \frac{1}{n^{3}}\, \frac{4\pi^{2} e^{4} m}{h^{3}}. +\end{gathered} +} +\Tag{(17)} +\] + +We must further assume that in the stationary states of the +unperturbed system the form of the orbit is so far undetermined +that the \Chg{excentricity}{eccentricity} can vary continuously. This is not only immediately +indicated by the principle of correspondence,---since the +frequency of revolution is determined only by the energy and not +by the \Chg{excentricity}{eccentricity},---but also by the fact that the presence of any +small external forces will in general, in the course of time, produce +a finite change in the position as well as in the \Chg{excentricity}{eccentricity} of the +periodic orbit, while in the major axis it can produce only small +changes proportional to the intensity of the perturbing forces. + +In order to fix the stationary states of systems in the presence +of a given conservative external field of force, we shall have to +investigate, on the basis of the principle of correspondence, how +these forces affect the decomposition of the motion into harmonic +oscillations. Owing to the external forces the form and position of +the orbit will vary continuously. In the general case these changes +will be so complicated that it will not be possible to decompose the +perturbed motion into discrete harmonic oscillations. In such a +case we must expect that the perturbed system will not possess +any sharply separated stationary states. Although each emission +of radiation must be assumed to be monochromatic and to proceed +according to the general frequency condition we shall therefore +expect the final effect to be a broadening of the sharp spectral lines +of the unperturbed system. In certain cases, however, the perturbations +\PageSep{39} +will be of such a regular character that the perturbed system +can be decomposed into harmonic oscillations, although the ensemble +of these oscillations will naturally be of a more complicated kind +than in the unperturbed system. This happens, for example, when +the variations of the orbit with respect to time are periodic. In +this case harmonic oscillations will appear in the motion of the +system the frequencies of which are equal to whole multiples of the +period of the orbital perturbations, and in the spectrum to be +expected on the basis of the ordinary theory of radiation we would +expect components corresponding to these frequencies. According +to the principle of correspondence we are therefore immediately +led to the conclusion, that to each stationary state in the unperturbed +system there corresponds a number of stationary states in +the perturbed system in such a manner, that for a transition +between two of these states a radiation is emitted, whose frequency +stands in the same relationship to the periodic course of the +variations in the orbit, as the spectrum of a simple periodic system +does to its motion in the stationary states. + +\Section{The Stark effect.} An instructive example of the appearance of +periodic perturbations is obtained when hydrogen is subjected to +the effect of a homogeneous electric field. The \Chg{excentricity}{eccentricity} and +the position of the orbit vary continuously under the influence of +the field. During these changes, however, it is found that the +centre of the orbit remains in a plane perpendicular to the direction +of the electric force and that its motion in this plane is +simply periodic. When the centre has returned to its starting +point, the orbit will resume its original \Chg{excentricity}{eccentricity} and position, +and from this moment the entire cycle of orbits will be repeated. +In this case the determination of the energy of the stationary +states of the disturbed system is extremely simple, since it is found +that the period of the disturbance does not depend upon the +original configuration of the orbits nor therefore upon the position +of the plane in which the centre of the orbit moves, but only upon +the major axis and the frequency of revolution. From a simple +calculation it is found that the period a is given by the following +formula +\[ +\sigma = \frac{3eF}{8\pi^{2} ma\omega}, +\Tag{(18)} +\] +\PageSep{40} +where $F$~is the intensity of the external electric field. From +analogy with the fixation of the distinctive energy values of a +Planck oscillator we must therefore expect that the energy difference +between two different states, corresponding to the same stationary +state of the unperturbed system, will simply be equal to a whole +multiple of the product of $h$~by the period~$\sigma$ of the perturbations. +We are therefore immediately led to the following expression for +the energy of the stationary states of the perturbed system, +\[ +E = E_{n} + kh\sigma, +\Tag{(19)} +\] +where $E_{n}$~depends only upon the number~$n$ characterizing the +stationary state of the unperturbed system, while $k$~is a new whole +number which in this case may be either positive or negative. As +we shall see below, consideration of the relation between the energy +and the motion of the system shows that $k$~must be numerically +less than~$n$, if, as before, we place the quantity~$E_{n}$ equal to the +energy~$-W_{n}$ of the $n$th~stationary state of the undisturbed atom. +Substituting the values of $W_{n}$,~$\omega_{n}$ and~$a_{n}$ given by~\Eq{(17)} in formula~\Eq{(19)} +we get +\[ +E = -\frac{1}{n^{2}}\, \frac{2\pi^{2} e^{4} m}{h^{2}} + nk\, \frac{3h^{2} F}{8\pi^{2} em}. +\Tag{(20)} +\] +To find the effect of an electric field upon the lines of the hydrogen +spectrum, we use the frequency condition~\Eq{(4)} and obtain for the +frequency~$\nu$ of the radiation emitted by a transition between two +stationary states defined by the numbers $n'$,~$k'$ and $n''$,~$k''$ +\[ +\nu = \frac{2\pi^{2} e^{4} m}{h^{3}} \left(\frac{1}{(n'')^{2}} - \frac{1}{(n'')^{2}}\right) + + \frac{3h · F}{8\pi^{2} em} (n'k' - n''k''). +\Tag{(21)} +\] + +It is well known that this formula provides a complete explanation +of the Stark effect of the hydrogen lines. It corresponds +exactly with the one obtained by a different method by Epstein +and Schwarzschild. They used the fact that the hydrogen atom in +a homogeneous electric field is a conditionally periodic system +permitting a separation of variables by the use of parabolic coordinates. +The stationary states were fixed by applying quantum +conditions to each of these variables. + +We shall now consider more closely the correspondence between +the changes in the spectrum of hydrogen due to the presence of +\PageSep{41} +an electric field and the decomposition of the perturbed motion +of the atom into its harmonic components. Instead of the simple +decomposition into harmonic components corresponding to a simple +Kepler motion, the displacement~$\xi$ of the electron in a given +direction in space can be expressed in the present case by the +formula +\[ +\xi = \sum C_{\tau,\kappa} \cos 2\pi \bigl\{t(\tau\omega + \kappa\sigma) + c_{\tau,\kappa}\bigr\}, +\Tag{(22)} +\] +where $\omega$~is the average frequency of revolution in the perturbed +orbit and $\sigma$~is the period of the orbital perturbations, while $C_{\tau,\kappa}$~and +$c_{\tau,\kappa}$ are constants. The summation is to be extended over all integral +values for $\tau$~and~$\kappa$. + +If we now consider a transition between two stationary states +characterized by certain numbers $n'$,~$k'$ and $n''$,~$k''$, we find that in +the region where these numbers are large compared with their +differences $n' - n''$ and $k' - k''$, the frequency of the spectral line +which is emitted will be given approximately by the formula +\[ +\nu \sim (n' - n'')\omega + (k' - k'')\sigma. +\Tag{(23)} +\] +We see, therefore, that we have obtained a relation between the +spectrum and the motion of precisely the same character as in the +simple case of the unperturbed hydrogen atom. We have here a +similar correspondence between the harmonic component in the +motion, corresponding to definite values for $\tau$~and $\kappa$ in formula~\Eq{(22)}, +and the transition between two stationary states for which $n' - n'' = \tau$ +and $k' - k'' = \kappa$. + +A number of interesting results can be obtained from this +correspondence by considering the motion in more detail. Each +harmonic component in expression~\Eq{(22)} for which $\tau + \kappa$ is an even +number corresponds to a linear oscillation parallel to the direction +of the electric field, while each component for which $\tau + \kappa$ is odd +corresponds to an elliptical oscillation perpendicular to this direction. +The correspondence principle suggests at once that these +facts are connected with the \emph{characteristic polarization} observed in +the Stark effect. We would anticipate that a transition for which +$(n' - n'') + (k' - k'')$ is even would give rise to a component with an +electric vector parallel to the field, while a transition for which +$(n' - n'') + (k' - k'')$ is odd would correspond to a component with an +\PageSep{42} +electric vector perpendicular to the field. These results have been +fully confirmed by experiment and correspond to the empirical rule +of polarization, which Epstein proposed in his first paper on the +Stark effect. + +The applications of the correspondence principle that have so +far been described have been purely qualitative in character. It is +possible however to obtain a quantitative estimate of the relative +intensity of the various components of the Stark effect of hydrogen, +by correlating the numerical values of the coefficients~$C_{\tau,\kappa}$ in formula~\Eq{(22)} +with the probability of the corresponding transitions between +the stationary states. This problem has been treated in detail by +Kramers in a recently published dissertation. In this he gives a +thorough discussion of the application of the correspondence principle +to the question of the intensity of spectral lines. + +\Section{The Zeeman effect.} The problem of the effect of a homogeneous +magnetic field upon the hydrogen lines may be treated in an +entirely analogous manner. The effect on the motion of the hydrogen +atom consists simply of the superposition of a uniform rotation +upon the motion of the electron in the unperturbed atom. +The axis of rotation is parallel with the direction of the magnetic +force, while the frequency of revolution is given by the formula +\[ +\sigma = \frac{eH}{4\pi mc}, +\Tag{(24)} +\] +where $H$~is the intensity of the field and $c$~the velocity of light. + +Again we have a case where the perturbations are simply +periodic and where the period of the perturbations is independent +of the form and position of the orbit, and in the present case, even +of the major axis. Similar considerations apply therefore as in the +case of the Stark effect, and we must expect that the energy in the +stationary states will again be given by formula~\Eq{(19)}, if we substitute +for~$\sigma$ the value given in expression~\Eq{(24)}. This result is +also in complete agreement with that obtained by Sommerfeld and +Debye. The method they used involved the solution of the equations +of motion by the method of the separation of the variables. The +appropriate coordinates are polar ones about an axis parallel to +the field. + +If we try, however, to calculate directly the effect of the field by +\PageSep{43} +means of the frequency condition~\Eq{(4)}, we immediately meet with +an apparent disagreement which for some time was regarded as a +grave difficulty for the theory. As both Sommerfeld and Debye +have pointed out, lines are not observed corresponding to every +transition between the stationary states included in the formula. +We overcome this difficulty, however, as soon as we apply the +principle of correspondence. If we consider the harmonic components +of the motion we obtain a simple explanation both of the +non-occurrence of certain transitions and of the observed polarization. +In the magnetic field each elliptic harmonic component having +the frequency~$\tau\omega$ splits up into three harmonic components owing +to the uniform rotation of the orbit. Of these one is rectilinear +with frequency~$\tau\omega$ oscillating parallel to the magnetic field, and +two are circular with frequencies $\tau\omega + \sigma$ and $\tau\omega - \sigma$ oscillating in +opposite directions in a plane perpendicular to the direction of the +field. Consequently the motion represented by formula~\Eq{(22)} contains +no components for which $\kappa$~is numerically greater than~$1$, in contrast +to the Stark effect, where components corresponding to all values +of~$\kappa$ are present. Now formula~\Eq{(23)} again applies for large values +of $n$~and~$k$, and shows the asymptotic agreement between the +frequency of the radiation and the frequency of a harmonic component +in the motion. We arrive, therefore, at the conclusion that +transitions for which $k$~changes by more than unity \Chg{can not}{cannot} occur. +The argument is similar to that by which transitions between two +distinctive states of a Planck oscillator for which the values of~$n$ +in~\Eq{(1)} differ by more than unity are excluded. We must further +conclude that the various possible transitions consist of two types. +For the one type corresponding to the rectilinear component, $k$~remains +unchanged, and in the emitted radiation which possesses +the same frequency~$\nu_{0}$ as the original hydrogen line, the electric +vector will oscillate parallel with the field. For the second type, +corresponding to the circular components, $k$~will increase or decrease +by unity, and the radiation viewed in the direction of the field will +be circularly polarized and have frequencies $\nu_{0} + \sigma$ and $\nu_{0} - \sigma$ respectively. +These results agree with those of the familiar Lorentz +theory. The similarity in the two theories is remarkable, when we +recall the fundamental difference between the ideas of the quantum +theory and the ordinary theories of radiation. +\PageSep{44} + +\Section{Central perturbations.} An illustration based on similar considerations +which will throw light upon the spectra of other elements +consists in finding the effect of a small perturbing field of +force radially symmetrical with respect to the nucleus. In this case +neither the form of the orbit nor the position of its plane will +change with time, and the perturbing effect of the field will simply +consist of a uniform rotation of the major axis of the orbit. The +perturbations are periodic, so that we may assume that to each +energy value of a stationary state of the unperturbed system there +belongs a series of discrete energy values of the perturbed system, +characterized by different values of a whole number~$k$. The frequency~$\sigma$ +of the perturbations is equal to the frequency of rotation +of the major axis. For a given law of force for the perturbing +field we find that $\sigma$~depends both on the major axis and on the +\Chg{excentricity}{eccentricity}. The change in the energy of the stationary states, +therefore, will not be given by an expression as simple as the +second term in formula~\Eq{(19)}, but will be a function of~$k$, which is +different for different fields. It is possible, however, to characterize +by one and the same condition the motion in the stationary states +of a hydrogen atom which is perturbed by any central field. In +order to show this we must consider more closely the fixation of +the motion of a perturbed hydrogen atom. + +In the stationary states of the unperturbed hydrogen atom +only the major axis of the orbit is to be regarded as fixed, +while the \Chg{excentricity}{eccentricity} may assume any value. Since the change +in the energy of the atom due to the external field of force depends +upon the form and position of its orbit, the fixation of the +energy of the atom in the presence of such a field naturally +involves a closer determination of the orbit of the perturbed +system. + +Consider, for the sake of illustration, the change in the hydrogen +spectrum due to the presence of homogeneous electric and magnetic +fields which was described by equation~\Eq{(19)}. It is found that +this energy condition can be given a simple geometrical interpretation. +In the case of an electric field the distance from the +nucleus to the plane in which the centre of the orbit moves determines +the change in the energy of the system due to the presence +of the field. In the stationary states this distance is simply equal +\PageSep{45} +to $\dfrac{k}{n}$~times half the major axis of the orbit. In the case of a magnetic +field it is found that the quantity which determines the change +of energy of the system is the area of the projection of the orbit +upon a plane perpendicular to the magnetic force. In the various +stationary states this area is equal to $\dfrac{k}{n}$~times the area of a circle +whose radius is equal to half the major axis of the orbit. In the +case of a perturbing central force the correspondence between +the spectrum and the motion which is required by the quantum +theory leads now to the simple condition that in the stationary +states of the perturbed system the minor axis of the rotating orbit +is simply equal to $\dfrac{k}{n}$~times the major axis. This condition was first +derived by Sommerfeld from his general theory for the determination +of the stationary states of a central motion. It is easily shown +that this fixation of the value of the minor axis is equivalent to +the statement that the parameter~$2p$ of the elliptical orbit is given +by an expression of exactly the same form as that which gives the +major axis~$2a$ in the unperturbed atom. The only difference from +the expression for~$2a_{n}$ in~\Eq{(17)} is that $n$~is replaced by~$k$, so that +the value of the parameter in the stationary states of the perturbed +atom is given by +\[ +2p_{k} = k^{2}\, \frac{h^{2}}{2\pi^{2} e^{2} m}. +\Tag{(25)} +\] +The frequency of the radiation emitted by a transition between +two stationary states determined in this way for which $n'$~and~$n''$ are +large in proportion to their difference is given by an expression +which is the same as that in equation~\Eq{(23)}, if in this case $\omega$~is the +frequency of revolution of the electron in the slowly rotating orbit +and $\sigma$~represents the frequency of rotation of the major axis. + +Before proceeding further, it might be of interest to note that +this fixation of the stationary states of the hydrogen atom perturbed +by external electric and magnetic forces does not coincide in certain +respects with the theories of Sommerfeld, Epstein and Debye. +According to the theory of conditionally periodic systems the stationary +states for a system of three degrees of freedom will in general +be determined by three conditions, and therefore in these theories +\PageSep{46} +each state is characterized by three whole numbers. This would +mean that the stationary states of the perturbed hydrogen atom +corresponding to a certain stationary state of the unperturbed +hydrogen atom, fixed by one condition, should be subject to two +further conditions and should therefore be characterized by two +new whole numbers in addition to the number~$n$. But the perturbations +of the Keplerian motion are simply periodic and the +energy of the perturbed atom will therefore be fixed completely +by one additional condition. The introduction of a second condition +will add nothing further to the explanation of the phenomenon, +since with the appearance of new perturbing forces, even if +these are too small noticeably to affect the observed Zeeman and +Stark effects, the forms of motion characterized by such a condition +may be entirely changed. This is completely analogous to the +fact that the hydrogen spectrum as it is usually observed is not +noticeably affected by small forces, even when they are large enough +to produce a great change in the form and position of the orbit of +the electron. + +\Section{Relativity effect on hydrogen lines.} Before leaving the hydrogen +spectrum I shall consider briefly the effect of the variation of +the mass of the electron with its velocity. In the preceding sections +I have described how external fields of force split up the hydrogen +lines into several components, but it should be noticed that these +results are only accurate when the perturbations are large in comparison +with the small deviations from a pure Keplerian motion +due to the variation of the mass of the electron with its velocity. +When the variation of the mass is taken into account the motion +of the unperturbed atom will not be exactly periodic. Instead we +obtain a motion of precisely the same kind as that occurring in the +hydrogen atom perturbed by a small central field. According to +the correspondence principle an intimate connection is to be expected +between the frequency of revolution of the major axis of the +orbit and the difference of the frequencies of the fine structure +components, and the stationary states will be those orbits whose +parameters are given by expression~\Eq{(25)}. If we now consider the +effect of external forces upon the fine structure components of the +hydrogen lines it is necessary to keep in mind that this fixation of +\PageSep{47} +the stationary states only applies to the unperturbed hydrogen +atom, and that, as mentioned, the orbits in these states are in +general already strongly influenced by the presence of external +forces, which are small compared with those with which we are +concerned in experiments on the Stark and Zeeman effects. In +general the presence of such forces will lead to a great complexity +of perturbations, and the atom will no longer possess a group of +sharply defined stationary states. The fine structure components +of a given hydrogen line will therefore become diffuse and merged +together. There are, however, several important cases where this +does not happen on account of the simple character of the perturbations. +The simplest example is a hydrogen atom perturbed +by a central force acting from the nucleus. In this case it is evident +that the motion of the system will retain its centrally symmetrical +character, and that the perturbed motion will differ from the unperturbed +motion only in that the frequency of rotation of the major +axis will be different for different values of this axis and of the +parameter. This point is of importance in the theory of the +spectra of elements of higher atomic number, since, as we shall see, +the effect of the forces originating from the inner electrons may +to a first approximation be compared with that of a perturbing +central field. We \Chg{can not}{cannot} therefore expect these spectra to exhibit +a separate effect due to the variation of the mass of the electron +of the same kind as that found in the case of the hydrogen lines. +This variation will not give rise to a splitting up into separate +components but only to small displacements in the position of the +various lines. + +We obtain still another simple example in which the hydrogen +atom possesses sharp stationary states, although the change of mass +of the electron is considered, if we take an atom subject to a homogeneous +magnetic field. The effect of such a field will consist in +the superposition of a rotation of the entire system about an axis +through the nucleus and parallel with the magnetic force. It follows +immediately from this result according to the principle of correspondence +that each fine structure component must be expected +to split up into a normal Zeeman effect (Lorentz triplet). The +problem may also be solved by means of the theory of conditionally +periodic systems, since the equations of motion in the presence +\PageSep{48} +of a magnetic field, even when the change in the mass is considered, +will allow of a separation of the variables using polar +coordinates in space. This has been pointed out by Sommerfeld +and Debye. + +A more complicated case arises when the atom is exposed to a +homogeneous electric field which is not so strong that the effect +due to the change in the mass may be neglected. In this case there +is no system of coordinates by which the equations of motion can +be solved by separation of the variables, and the problem, therefore, +\Chg{can not}{cannot} be treated by the theory of the stationary states of conditionally +periodic systems. A closer investigation of the perturbations, +however, shows them to be of such a character that the motion +of the electrons may be decomposed into a number of separate harmonic +components. These fall into two groups for which the direction +of oscillation is either parallel with or perpendicular to the +field. According to the principle of correspondence, therefore, we +must expect that also in this case in the presence of the field each +hydrogen line will consist of a number of sharp, polarized components. +In fact by means of the principles I have described, it is +possible to give a unique fixation of the stationary states. The +problem of the effect of a homogeneous electric field upon the fine +structure components of the hydrogen lines has been treated in +detail from this point of view by Kramers in a paper which will +soon be published. In this paper it will be shown how it appears +possible to predict in detail the manner in which the fine structure +of the hydrogen lines gradually changes into the ordinary Stark +effect as the electric intensity increases. + +\Section{Theory of series spectra.} Let us now turn our attention once +more to the problem of the series spectra of elements of higher +atomic number. The general appearance of the Rydberg constant +in these spectra is to be explained by assuming that the atom is +neutral and that one electron revolves in an orbit the dimensions +of which are large in comparison with the distance of the inner electrons +from the nucleus. In a certain sense, therefore, the motion of +the outer electron may be compared with the motion of the electron +of the hydrogen atom perturbed by external forces, and the appearance +of the various series in the spectra of the other elements is +\PageSep{49} +from this point of view to be regarded as analogous to the splitting +up of the hydrogen lines into components on account of such forces. + +In his theory of the structure of series spectra of the type exhibited +by the alkali metals, Sommerfeld has made the assumption +that the orbit of the outer electron to a first approximation possesses +the same character as that produced by a simple perturbing +central field whose intensity diminishes rapidly with increasing +distance from the nucleus. He fixed the motion of the external +electron by means of his general theory for the fixation of the +stationary states of a central motion. The application of this +method depends on the possibility of separating the variables in +the equations of motion. In this manner Sommerfeld was able to +calculate a number of energy values which can be arranged in rows +just like the empirical spectral terms shown in the diagram of the +sodium spectrum (\PageRef[p.]{30}). The states grouped together by Sommerfeld +in the separate rows are exactly those which were characterized +by one and the same value of~$k$ in our investigation of the +hydrogen atom perturbed by a central force. The states in the +first row of the figure (row~$S$) correspond to the value $k = 1$, those +of the second row~($P$) correspond to $k = 2$, etc. The states corresponding +to one and the same value of~$n$ are connected by dotted +lines which are continued so that their vertical asymptotes correspond +to the energy value of the stationary states of the hydrogen +atom. The fact that for a constant~$n$ and increasing values of~$k$ +the energy values approach the corresponding values for the unperturbed +hydrogen atom is immediately evident from the theory +since the outer electron, for large values of the parameter of its +orbit, remains at a great distance from the inner system during the +whole revolution. The orbit will become almost elliptical and the +period of rotation of the major axis will be very large. It can be +seen, therefore, that the effect of the inner system on the energy +necessary to remove this electron from the atom must become less +for increasing values of~$k$. + +These beautiful results suggest the possibility of finding laws of +force for the perturbing central field which would account for the +spectra observed. Although Sommerfeld in this way has in fact +succeeded in deriving formulae for the spectral terms which vary +with~$n$ for a constant~$k$ in agreement with Rydberg's formulae, it +\PageSep{50} +has not been possible to explain the simultaneous variation with +both $k$~and~$n$ in any actual case. This is not surprising, since it is +to be anticipated that the effect of the inner electrons on the spectrum +could not be accounted for in such a simple manner. Further +consideration shows that it is necessary to consider not only the +forces which originate from the inner electrons but also to consider +the effect of the presence of the outer electron upon the motion of +the inner electrons. + +Before considering the series spectra of elements of low atomic +number I shall point out how the occurrence or non-occurrence of +certain transitions can be shown by the correspondence principle +to furnish convincing evidence in favour of Sommerfeld's assumption +about the orbit of the outer electron. For this purpose we +must describe the motion of the outer electron in terms of its harmonic +components. This is easily performed if we assume that the +presence of the inner electrons simply produces a uniform rotation +of the orbit of the outer electron in its plane. On account of this +rotation, the frequency of which we will denote by~$\sigma$, two circular +rotations with the periods $\tau\omega + \sigma$ and $\tau\omega - \sigma$ will appear in the +motion of the perturbed electron, instead of each of the harmonic +elliptical components with a period $\tau\omega$ in the unperturbed motion. +The decomposition of the perturbed motion into harmonic components +consequently will again be represented by a formula of the +type~\Eq{(22)}, in which only such terms appear for which $\kappa$~is equal +to $+1$ or~$-1$. Since the frequency of the emitted radiation in the +regions where $n$~and $k$ are large is again given by the asymptotic +formula~\Eq{(23)}, we at once deduce from the correspondence principle +that the only transitions which can take place are those for which +the values of~$k$ differ by unity. A glance at the figure for the sodium +spectrum shows that this agrees exactly with the experimental +results. This fact is all the more remarkable, since in Sommerfeld's +theory the arrangement of the energy values of the stationary +states in rows has no special relation to the possibility of transition +between these states. + +\Section{Correspondence principle and conservation of angular momentum.} +Besides these results the correspondence principle suggests +that the radiation emitted by the perturbed atom must +\PageSep{51} +exhibit circular polarization. On account of the indeterminateness +of the plane of the orbit, however, this polarization \Chg{can not}{cannot} be +directly observed. The assumption of such a polarization is a matter +of particular interest for the theory of radiation emission. On +account of the general correspondence between the spectrum of +an atom and the decomposition of its motion into harmonic +components, we are led to compare the radiation emitted during +the transition between two stationary states with the radiation +which would be emitted by a harmonically oscillating +electron on the basis of the classical electrodynamics. In particular +the radiation emitted according to the classical theory +by an electron revolving in a circular orbit possesses an angular +momentum and the energy~$\Delta E$ and the angular momentum~$\Delta P$ of +the radiation emitted during a certain time are connected by the +relation +\[ +\Delta E = 2\pi\omega · \Delta P. +\Tag{(26)} +\] + +Here $\omega$~represents the frequency of revolution of the electron, +and according to the classical theory this is equal to the frequency~$\nu$ +of the radiation. If we now assume that the total energy emitted +is equal to~$h\nu$ we obtain for the total angular momentum of the +radiation +\[ +\Delta P = \frac{h}{2\pi}. +\Tag{(27)} +\] + +It is extremely interesting to note that this expression is equal +to the change in the angular momentum which the atom suffers in +a transition where $k$~varies by unity. For in Sommerfeld's theory +the general condition for the fixation of the stationary states of a +central system, which in the special case of an approximately +Keplerian motion is equivalent to the relation~\Eq{(25)}, asserts that +the angular momentum of the system must be equal to a whole +multiple of~$\dfrac{h}{2\pi}$, a condition which may be written in our notation +\[ +P = k\, \frac{h}{2\pi}. +\Tag{(28)} +\] +We see, therefore, that this condition has obtained direct support +from a simple consideration of the conservation of angular momentum +during the emission of the radiation. I wish to emphasize +that this equation is to be regarded as a rational generalization of +\PageSep{52} +Planck's original statement about the distinctive states of a harmonic +oscillator. It may be of interest to recall that the possible +significance of the angular momentum in applications of the +quantum theory to atomic processes was first pointed out by +Nicholson on the basis of the fact that for a circular motion the +angular momentum is simply proportional to the ratio of the +kinetic energy to the frequency of revolution. + +In a previous paper which I presented to the Copenhagen +Academy I pointed out that these results confirm the conclusions +obtained by the application of the correspondence principle to +atomic systems possessing radial or axial symmetry. Rubinowicz +has independently indicated the conclusions which may be obtained +directly from a consideration of conservation of angular momentum +during the radiation process. In this way he has obtained several +of our results concerning the various types of possible transitions +and the polarization of the emitted radiation. Even for systems +possessing radial or axial symmetry, however, the conclusions which +we can draw by means of the correspondence principle are of a +more detailed character than can be obtained solely from a consideration +of the conservation of angular momentum. For example, +in the case of the hydrogen atom perturbed by a central force we +can only conclude that $k$~\Chg{can not}{cannot} change by more than unity, while +the correspondence principle requires that $k$~shall vary by unity +for every possible transition and that its value cannot remain unchanged. +Further, this principle enables us not only to exclude +certain transitions as being impossible---and can from this point of +view be considered as a ``selection principle''---but it also enables +us to draw conclusions about the relative probabilities of the various +possible types of transitions from the values of the amplitudes of +the harmonic components. In the present case, for example, the +fact that the amplitudes of those circular components which rotate +in the same sense as the electron are in general greater than the +amplitudes of those which rotate in the opposite sense leads us to +expect that lines corresponding to transitions for which $k$~decreases +by unity will in general possess greater intensity than lines during +the emission of which $k$~increases by unity. Simple considerations +like this, however, apply only to spectral lines corresponding to +transitions from one and the same stationary state. In other +\PageSep{53} +cases when we wish to estimate the relative intensities of two +spectral lines it is clearly necessary to take into consideration the +relative number of atoms which are present in each of the two +stationary states from which the transitions start. While the intensity +naturally \Chg{can not}{cannot} depend upon the number of atoms in the +final state, it is to be noticed, however, that in estimating the +probability of a transition between two stationary states it is necessary +to consider the character of the motion in the final as well as +in the initial state, since the values of the amplitudes of the components +of oscillation of both states are to be regarded as decisive +for the probability. + +To show how this method can be applied I shall return for a +moment to the problem which I mentioned in connection with +Strutt's experiment on the resonance radiation of sodium vapour. +This involved the discussion of the relative probability of the various +possible transitions which can start from that state corresponding +to the second term in the second row of the figure on \PageRef[p.]{30}. These +were transitions to the first and second states in the first row and +to the first state in the third row, and the results of experiment +indicate, as we saw, that the probability is greatest for the second +transitions. These transitions correspond to those harmonic components +having frequencies $2\omega + \sigma$, $\omega + \sigma$ and~$\sigma$, and it is seen +that only for the second transition do the amplitudes of the corresponding +harmonic component differ from zero in the initial as +well as in the final state. [In the next essay the reader will find +that the values of quantum number~$n$ assigned in \Fig{1} to the +various stationary states must be altered. While this correction +in no way influences the other conclusions in this essay it involves +that the reasoning in this passage \Chg{can not}{cannot} be maintained.] + +I have shown how the correspondence between the spectrum of +an element and the motion of the atom enables us to understand +the limitations in the direct application of the combination principle +in the prediction of spectral lines. The same ideas give an immediate +explanation of the interesting discovery made in recent years +by Stark and his collaborators, that certain \emph{new series of combination +lines} appear with considerable intensity when the radiating +atoms are subject to a strong external electric field. This phenomenon +is entirely analogous to the appearance of the so-called +\PageSep{54} +combination tones in acoustics. It is due to the fact that the +perturbation of the motion will not only consist in an effect upon +the components originally present, but in addition will give rise to +new components. The frequencies of these new components may be +$\tau\omega + \kappa\sigma$, where $\kappa$~is different from~$±1$. According to the correspondence +principle we must therefore expect that the electric field will +not only influence the lines appearing under ordinary circumstances, +but that it will also render possible new types of transitions which +give rise to the ``new'' combination lines observed. From an estimate +of the amplitudes of the particular components in the initial +and final states it has even been found possible to account for the +varying facility with which the new lines are brought up by the +external field. + +The general problem of the effect of an electric field on the spectra +of elements of higher atomic number differs essentially from the +simple Stark effect of the hydrogen lines, since we are here concerned +not with the perturbation of a purely periodic system, but +with the effect of the field on a periodic motion already subject to +a perturbation. The problem to a certain extent resembles the +effect of a weak electric force on the fine structure components of +the hydrogen atom. In much the same way the effect of an electric +field upon the series spectra of the elements may be treated directly +by investigating the perturbations of the external electron. A +continuation of my paper in the Transactions of the Copenhagen +Academy will soon appear in which I shall show how this method +enables us to understand the interesting observations Stark and +others have made in this field. + +\Section{The spectra of helium and lithium.} We see that it has been +possible to obtain a certain general insight into the origin of the +series spectra of a type like that of sodium. The difficulties encountered +in an attempt to give a detailed explanation of the +spectrum of a particular element, however, become very serious, +even when we consider the spectrum of helium whose neutral atom +contains only two electrons. The spectrum of this element has a +simple structure in that it consists of single lines or at any rate of +double lines whose components are very close together. We find, +however, that the lines fall into two groups each of which can be +\PageSep{55} +described by a formula of the type~\Eq{(14)}. These are usually called +the (ortho) helium and parhelium spectra. While the latter consists +of simple lines, the former possesses narrow doublets. The +discovery that helium, as opposed to the alkali metals, possesses +two complete spectra of the Rydberg type which do not exhibit any +mutual combinations was so surprising that at times there has been +a tendency to believe that helium consisted of two elements. This +way out of the difficulty is no longer open, since there is no room +for another element in this region of the periodic system, or more +correctly expressed, for an element possessing a new spectrum. The +existence of the two spectra can, however, be traced back to the fact +that in the stationary states corresponding to the series spectra we +have to do with a system possessing only one inner electron and in +consequence the motion of the inner system, in the absence of the +outer electron, will be simply periodic and therefore easily perturbed +by external forces. + +In order to illustrate this point we shall have to consider more +carefully the stationary states connected with the origin of a series +spectrum. We must assume that in these states one electron revolves +in an orbit outside the nucleus and the other electrons. We +might now suppose that in general a number of different groups of +such states might exist, each group corresponding to a different +stationary state of the inner system considered by itself. Further +consideration shows, however, that under the usual conditions of +excitation those groups have by far the greatest probability for which +the motion of the inner electrons corresponds to the ``normal'' state +of the inner system, \ie\ to that stationary state having the least +energy. Further the energy required to transfer the inner system +from its normal state to another stationary state is in general very +large compared with the energy which is necessary to transfer an +electron from the normal state of the neutral atom to a stationary +orbit of greater dimensions. Lastly the inner system is in general +capable of a permanent existence only in its normal state. Now, +the configuration of an atomic system in its stationary states and +also in the normal state will, in general, be completely determined. +We may therefore expect that the inner system under the influence +of the forces arising from the presence of the outer electron can in +the course of time suffer only small changes. For this reason we +\PageSep{56} +must assume that the influence of the inner system upon the motion +of the external electron will, in general, be of the same character +as the perturbations produced by a constant external field upon +the motion of the electron in the hydrogen atom. We must therefore +expect a spectrum consisting of an ensemble of spectral terms, +which in general form a connected group, even though in the +absence of external perturbing forces not every combination actually +occurs. The case of the helium spectrum, however, is quite different +since here the inner system contains only one electron the motion +of which in the absence of the external electron is simple periodic +provided the small changes due to the variation in the mass of the +electron with its velocity are neglected. For this reason the form of +the orbit in the stationary states of the inner system considered by +itself will not be determined. In other words, the stability of the +orbit is so slight, even if the variation in the mass is taken into +account, that small external forces are in a position to change the +\Chg{excentricity}{eccentricity} in the course of time to a finite extent. In this case, +therefore, it is possible to have several groups of stationary states, +for which the energy of the inner system is approximately the same +while the form of the orbit of the inner electron and its position +relative to the motion of the other electrons are so essentially +different, that no transitions between the states of different groups +can occur even in the presence of external forces. It can be seen +that these conclusions summarize the experimental observations +on the helium spectra. + +These\Pagelabel{56} considerations suggest an investigation of the nature of +the perturbations in the orbit of the inner electron of the helium +atom, due to the presence of the external electron. A discussion +of the helium spectrum from this point of view has recently been +given by Landé. The results of this work are of great interest particularly +in the demonstration of the large back effect on the outer +electron due to the perturbations of the inner orbit which themselves +arise from the presence of the outer electron. Nevertheless, it can +scarcely be regarded as a satisfactory explanation of the helium +spectrum. Apart from the serious objections which may be raised +against his calculation of the perturbations, difficulties arise if we +try to apply the correspondence principle to Landé's results in +order to account for the occurrence of two distinct spectra showing +\PageSep{57} +no mutual combinations. To explain this fact it seems necessary +to base the discussion on a more thorough investigation of the +mutual perturbations of the outer and the inner orbits. As a +result of these perturbations both electrons move in such an +extremely complicated way that the stationary states \Chg{can not}{cannot} be +fixed by the methods developed for conditionally periodic systems. +Dr~Kramers and I have in the last few years been engaged in such +an investigation, and in an address on atomic problems at the +meeting of the Dutch Congress of Natural and Medical Sciences +held in Leiden, April 1919, I gave a short communication of our +results. For various reasons we have up to the present time been +prevented from publishing, but in the very near future we hope to +give an account of these results and of the light which they seem +to throw upon the helium spectrum. + +The problem presented by the spectra of elements of higher +atomic number is simpler, since the inner system is better defined +in its normal state. On the other hand the difficulty of the mechanical +problem of course increases with the number of the particles in +the atom. We obtain an example of this in the case of lithium +with three electrons. The differences between the spectral terms +of the lithium spectrum and the corresponding spectral terms of +hydrogen are very small for the variable term of the principal series +($k = 2$) and for the diffuse series ($k = 3$), on the other hand it is very +considerable for the variable term of the sharp series ($k = 1$). This +is very different from what would be expected if it were possible to +describe the effect of the inner electron by a central force varying +in a simple manner with the distance. This must be because the +parameter of the orbit of the outer electron in the stationary states +corresponding to the terms of the sharp series is not much greater +than the linear dimensions of the orbits of the inner electrons. +According to the principle of correspondence the frequency of rotation +of the major axis of the orbit of the outer electron is to be regarded +as a measure of the deviation of the spectral terms from the corresponding +hydrogen terms. In order to calculate this frequency it +appears necessary to consider in detail the mutual effect of all three +electrons, at all events for that part of the orbit where the outer +electron is very close to the other two electrons. Even if we assumed +that we were fully acquainted with the normal state of the inner +\PageSep{58} +system in the absence of the outer electron---which would be +expected to be similar to the normal state of the neutral helium +atom---the exact calculation of this mechanical problem would +evidently form an exceedingly difficult task. + +\Section{Complex structure of series lines.} For the spectra of elements +of still higher atomic number the mechanical problem which has to +be solved in order to describe the motion in the stationary states +becomes still more difficult. This is indicated by the extraordinarily +complicated structure of many of the observed spectra. The fact that +the series spectra of the alkali metals, which possess the simplest +structure, consist of double lines whose separation increases with +the atomic number, indicates that here we have to do with systems +in which the motion of the outer electron possesses in general a +somewhat more complicated character than that of a simple central +motion. This gives rise to a more complicated ensemble of stationary +states. It would, however, appear that in the sodium atom the major +axis and the parameter of the stationary states corresponding to +each pair of spectral terms are given approximately by formulae +\Eq{(17)} and~\Eq{(25)}. This is indicated not only by the similar part played +by the two states in the experiments on the resonance radiation of +sodium vapour, but is also shown in a very instructive manner by +the peculiar effect of magnetic fields on the doublets. For small +fields each component splits up into a large number of sharp lines +instead of into the normal Lorentz triplet. With increasing field +strength Paschen and Back found that this \emph{anomalous Zeeman effect} +changed into the normal Lorentz triplet of a single line by a gradual +fusion of the components. + +This effect of a magnetic field upon the doublets of the alkali +spectrum is of interest in showing the intimate relation of the components +and confirms the reality of the simple explanation of the +general structure of the spectra of the alkali metals. If we may +again here rely upon the correspondence principle we have unambiguous +evidence that the effect of a magnetic field on the motion +of the electrons simply consists in the superposition of a uniform +rotation with a frequency given by equation~\Eq{(24)} as in the case of +the hydrogen atom. For if this were the case the correspondence +principle would indicate under all conditions a normal Zeeman effect +\PageSep{59} +for each component of the doublets. I want to emphasize that the +difference between the simple effect of a magnetic field, which the +theory predicts for the fine structure of components of the hydrogen +lines, and the observed effect on the alkali doublets is in no way to +be considered as a contradiction. The fine structure components +are not analogous to the individual doublet components, but each +single fine structure component corresponds to the ensemble of +components (doublet, triplet) which makes up one of the series lines +in Rydberg's scheme. The occurrence in strong fields of the effect +observed by Paschen and Back must therefore be regarded as a +strong support for the theoretical prediction of the effect of a magnetic +field on the fine structure components of the hydrogen lines. + +It does not appear necessary to assume the ``anomalous'' effect +of small fields on the doublet components to be due to a failure of +the ordinary electrodynamical laws for the description of the motion +of the outer electron, but rather to be connected with an effect of +the magnetic field on that intimate interaction between the motion +of the inner and outer electrons which is responsible for the occurrence +of the doublets. Such a view is probably not very different +from the ``coupling theory'' by which Voigt was able to account +formally for the details of the anomalous Zeeman effect. We might +even expect it to be possible to construct a theory of these effects +which would exhibit a formal analogy with the Voigt theory similar +to that between the quantum theory of the normal Zeeman effect and +the theory originally developed by Lorentz. Time unfortunately +does not permit me to enter further into this interesting problem, so +I must refer you to the continuation of my paper in the Transactions +of the Copenhagen Academy, which will contain a general discussion +of the origin of series spectra and of the effects of electric and +magnetic fields. + + +\Chapter{IV.}{Conclusion} + +In this lecture I have purposely not considered the question of +the structure of atoms and molecules although this is of course most +intimately connected with the kind of spectral theory I have developed. +We are encouraged to use results obtained from the spectra, +since even the simple theory of the hydrogen spectrum gives a +value for the major axis of the orbit of the electron in the normal +\PageSep{60} +state ($n = 1$) of the same order of magnitude as that derived from +the kinetic theory of gases. In my first paper on the subject I +attempted to sketch a theory of the structure of atoms and of +molecules of chemical compounds. This theory was based on a +simple generalization of the results for the stationary states of the +hydrogen atom. In several respects the theory was supported by +experiment, especially in the general way in which the properties +of the elements change with increasing atomic number, shown most +clearly by Moseley's results. I should like, however, to use this +occasion to state, that in view of the recent development of the +quantum theory, many of the special assumptions will certainly have +to be changed in detail. This has become clear from various sides +by the lack of agreement of the theory with experiment. It appears +no longer possible to justify the assumption that in the normal +states the electrons move in orbits of special geometrical simplicity, +like ``electronic rings.'' Considerations relating to the stability of +atoms and molecules against external influences and concerning the +possibility of the formation of an atom by successive addition of +the individual electrons compel us to claim, first that the configurations +of electrons are not only in mechanical equilibrium +but also possess a certain stability in the sense required by +ordinary mechanics, and secondly that the configurations employed +must be of such a nature that transitions to these from other +stationary states of the atom are possible. These requirements are +not in general fulfilled by such simple configurations as electronic +rings and they force us to look about for possibilities of more complicated +motions. It will not be possible here to consider further +these still open questions and I must content myself by referring +to the discussion in my forthcoming paper. In closing, however, +I should like to emphasize once more that in this lecture I have +only intended to bring out certain general points of view lying at +the basis of the spectral theory. In particular it was my intention +to show that, in spite of the fundamental differences between these +points of view and the ordinary conceptions of the phenomena of +radiation, it still appears possible on the basis of the general correspondence +between the spectrum and the motion in the atom to +employ these conceptions in a certain sense as guides in the investigation +of the spectra. +\PageSep{61} + + +\Essay{III}{The Structure of~the~Atom and the~Physical +and~Chemical~Properties of~the~Elements} +{Address delivered before a joint meeting of the Physical and Chemical +Societies in Copenhagen, October~18, 1921.} + +\Chapter{I.}{Preliminary} + +In an address which I delivered to you about a year ago I +described the main features of a theory of atomic structure which +I shall attempt to develop this evening. In the meantime this +theory has assumed more definite form, and in two recent letters +%[** TN: Footnote mark before punctuation in the original] +to \Title{Nature} I have given a somewhat further sketch of the development.\footnote + {\Title{Nature}, March~24, and October~13, 1921.} +The results which I am about to present to you are +of no final character; but I hope to be able to show you how this +view renders a correlation of the various properties of the elements +in such a way, that we avoid the difficulties which previously +appeared to stand in the way of a simple and consistent explanation. +Before proceeding, however, I must ask your forbearance if initially +I deal with matters already known to you, but in order to introduce +you to the subject it will first be necessary to give a brief +description of the most important results which have been obtained +in recent years in connection with the work on atomic structure. + +\Section{The nuclear atom.} The conception of atomic structure which +will form the basis of all the following remarks is the so-called +nuclear atom according to which an atom is assumed to consist of +a nucleus surrounded by a number of electrons whose distances +from one another and from the nucleus are very large compared +to the dimensions of the particles themselves. The nucleus +possesses almost the entire mass of the atom and has a positive +charge of such a magnitude that the number of electrons in a +neutral atom is equal to the number of the element in the periodic +system, the so-called \emph{atomic number}. This idea of the atom, which +is due principally to Rutherford's fundamental researches on radioactive +substances, exhibits extremely simple features, but just this +simplicity appears at first sight to present difficulties in explaining +the properties of the elements. When we treat this question on +\PageSep{62} +the basis of the ordinary mechanical and electrodynamical theories +it is impossible to find a starting point for an explanation of the +marked properties exhibited by the various elements, indeed not +even of their permanency. On the one hand the particles of the +atom apparently could not be at rest in a state of stable equilibrium, +and on the other hand we should have to expect that every motion +which might be present would give rise to the emission of electromagnetic +radiation which would not cease until all the energy of +the system had been emitted and all the electrons had fallen into +the nucleus. A method of escaping from these difficulties has now +been found in the application of ideas belonging to the quantum +theory, the basis of which was laid by Planck in his celebrated +work on the law of temperature radiation. This represented a +radical departure from previous conceptions since it was the first +instance in which the assumption of a discontinuity was employed +in the formulation of the general laws of nature. + +\Section{The postulates of the quantum theory.}\Pagelabel{62} The quantum theory +in the form in which it has been applied to the problems of atomic +structure rests upon two postulates which have a direct bearing +on the difficulties mentioned above. According to the first postulate +there are certain states in which the atom can exist without +emitting radiation, although the particles are supposed to have an +accelerated motion relative to one another. These \emph{stationary states} +are, in addition, supposed to possess a peculiar kind of stability, so +that it is impossible either to add energy to or remove energy from +the atom except by a process involving a transition of the atom +into another of these states. According to the second postulate +each emission of radiation from the atom resulting from such a +transition always consists of a train of purely harmonic waves. +The frequency of these waves does not depend directly upon the +motion of the atom, but is determined by a \emph{frequency relation}, +according to which the frequency multiplied by the universal constant +introduced by Planck is equal to the total energy emitted +during the process. For a transition between two stationary states +for which the values of the energy of the atom before and after the +emission of radiation are $E'$~and $E''$ respectively, we have therefore +\[ +h\nu = E' - E'', +\Tag{(1)} +\] +\PageSep{63} +where $h$~is Planck's constant and $\nu$~is the frequency of the emitted +radiation. Time does not permit me to give a systematic survey +of the quantum theory, the recent development of which has gone +hand in hand with its applications to atomic structure. I shall +therefore immediately proceed to the consideration of those applications +of the theory which are of direct importance in connection +with our subject. + +\Section{Hydrogen atom.} We shall commence by considering the +simplest atom conceivable, namely, an atom consisting of a nucleus +and one electron. If the charge on the nucleus corresponds to that +of a single electron and the system consequently is neutral we have +a hydrogen atom. Those developments of the quantum theory which +have made possible its application to atomic structure started with +the interpretation of the well-known simple spectrum emitted by +hydrogen. This spectrum consists of a series of lines, the frequencies +of which are given by the extremely simple Balmer formula +\[ +\nu = K\left(\frac{1}{(n'')^{2}} - \frac{1}{(n')^{2}}\right), +\Tag{(2)} +\] +where $n''$~and $n'$ are integers. According to the quantum theory +we shall now assume that the atom possesses a series of stationary +states characterized by a series of integers, and it can be seen how +the frequencies given by formula~\Eq{(2)} may be derived from the +frequency relation if it is assumed that a hydrogen line is connected +with a radiation emitted during a transition between two +of these states corresponding to the numbers $n'$~and~$n''$, and if the +energy in the $n$th~state apart from an arbitrary additive constant +is supposed to be given by the formula +\[ +E_{n} = -\frac{Kh}{n^{2}}. +\Tag{(3)} +\] +The negative sign is used because the energy of the atom is +measured most simply by the work required to remove the electron +to infinite distance from the nucleus, and we shall assume that the +numerical value of the expression on the right-hand side of formula~\Eq{(3)} +is just equal to this work. + +As regards the closer description of the stationary states we find +that the electron will very nearly describe an ellipse with the +nucleus in the focus. The major axis of this ellipse is connected +\PageSep{64} +with the energy of the atom in a simple way, and corresponding to +the energy values of the stationary states given by formula~\Eq{(3)} +there are a series of values for the major axis~$2a$ of the orbit of the +electron given by the formula +\[ +2a_{n} = \frac{n^{2} e^{2}}{hK}, +\Tag{(4)} +\] +where $e$~is the numerical value of the charge of the electron and +the nucleus. + +On the whole we may say that the spectrum of hydrogen shows +us the \emph{formation of the hydrogen atom}, since the stationary states +may be regarded as different stages of a process by which the electron +under the emission of radiation is bound in orbits of smaller +and smaller dimensions corresponding to states with decreasing +values of~$n$. It will be seen that this view has certain characteristic +features in common with the binding process of an electron +to the nucleus if this were to take place according to the ordinary +electrodynamics, but that our view differs from it in just such a +way that it is possible to account for the observed properties of +hydrogen. In particular it is seen that the final result of the +binding process leads to a quite definite stationary state of the +atom, namely that state for which $n = 1$. This state which corresponds +to the minimum energy of the atom will be called the +\emph{normal state} of the atom. It may be stated here that the values of +the energy of the atom and the major axis of the orbit of the +electron which are found if we put $n = 1$ in formulae \Eq{(3)} and~\Eq{(4)} +are of the same order of magnitude as the values of the firmness +of binding of electrons and of the dimensions of the atoms which +have been obtained from experiments on the electrical and mechanical +properties of gases. A more accurate check of formulae +\Eq{(3)} and~\Eq{(4)} can however not be obtained from such a comparison, +because in such experiments hydrogen is not present in the form +of simple atoms but as molecules. + +The formal basis of the quantum theory consists not only of the +frequency relation, but also of conditions which permit the determination +of the stationary states of atomic systems. The latter +conditions, like that assumed for the frequency, may be regarded as +natural generalizations of that assumption regarding the interaction +between simple electrodynamic systems and a surrounding field of +\PageSep{65} +electromagnetic radiation which forms the basis of Planck's theory +of temperature radiation. I shall not here go further into the +nature of these conditions but only mention that by their means +the stationary states are characterized by a number of integers, +the so-called \emph{quantum numbers}. For a purely periodic motion like +that assumed in the case of the hydrogen atom only a single +quantum number is necessary for the determination of the stationary +states. This number determines the energy of the atom and also +the major axis of the orbit of the electron, but not its \Chg{excentricity}{eccentricity}. +The energy in the various stationary states, if the small influence +of the motion of the nucleus is neglected, is given by the following +formula: +\[ +E_{n} = -\frac{2\pi^{2} N^{2} e^{4} m}{n^{2} h^{2}}, +\Tag{(5)} +\] +where $e$~and $m$ are respectively the charge and the mass of the +electron, and where for the sake of subsequent applications the +charge on the nucleus has been designated by~$Ne$. + +For the atom of hydrogen $N = 1$, and a comparison with +equation~\Eq{(3)} leads to the following theoretical expression for~$K$ in +formula~\Eq{(2)}, namely +\[ +K = \frac{2\pi^{2} e^{4} m}{h^{3}}. +\Tag{(6)} +\] +This agrees with the empirical value of the constant for the spectrum +of hydrogen within the limit of accuracy with which the various +quantities can be determined. + +\Section{Hydrogen spectrum and X-ray spectra.} If in the above +formula we put $N = 2$ which corresponds to an atom consisting of +an electron revolving around a nucleus with a double charge, we +get values for the energies in the stationary states, which are four +times larger than the energies in the corresponding states of the +hydrogen atom, and we obtain the following formula for the +spectrum which would be emitted by such an atom: +\[ +\nu = 4K \left(\frac{1}{(n'')^{2}} - \frac{1}{(n')^{2}}\right). +\Tag{(7)} +\] +This formula represents certain lines which have been known for +some time and which had been attributed to hydrogen on account +of the great similarity between formulae \Eq{(2)} and~\Eq{(7)} since it had +\PageSep{66} +never been anticipated that two different substances could exhibit +properties so closely resembling each other. According to the theory +we may, however, expect that the emission of the spectrum given by~\Eq{(7)} +corresponds to the \emph{first stage of the formation of the helium atom}, +\ie\ to the binding of a first electron by the doubly charged nucleus +of this atom. This interpretation has been found to agree with +more recent information. For instance it has been possible to +obtain this spectrum from pure helium. I have dwelt on this point +in order to show how this intimate connection between the properties +of two elements, which at first sight might appear quite +surprising, is to be regarded as an immediate expression of the +characteristic simple structure of the nuclear atom. A short time +after the elucidation of this question, new evidence of extraordinary +interest was obtained of such a similarity between the properties of +the elements. I refer to Moseley's fundamental researches on the +X-ray spectra of the elements. Moseley found that these spectra +varied in an extremely simple manner from one element to the +next in the periodic system. It is well known that the lines of +the X-ray spectra may be divided into groups corresponding to the +different characteristic absorption regions for X-rays discovered by +Barkla. As regards the $K$~group which contains the most penetrating +X-rays, Moseley found that the strongest line for all elements +investigated could be represented by a formula which with +a small simplification can be written +\[ +\nu = N^{2} K \left(\frac{1}{1^{2}} - \frac{1}{2^{2}}\right). +\Tag{(8)} +\] +$K$~is the same constant as in the hydrogen spectrum, and $N$~the +atomic number. The great significance of this discovery lies in +the fact that it would seem firmly to establish the view that this +atomic number is equal to the number of electrons in the atom. +This assumption had already been used as a basis for work on +atomic structure and was first stated by van~den Broek. While +the significance of this aspect of Moseley's discovery was at once +clear to all, it has on the other hand been more difficult to understand +the very great similarity between the spectrum of hydrogen +and the X-ray spectra. This similarity is shown, not only by the +lines of the $K$~group, but also by groups of less penetrating X-rays. +\PageSep{67} +Thus Moseley found for all the elements he investigated that the +frequencies of the strongest line in the $L$~group may be represented +by a formula which with a simplification similar to that employed +in formula~\Eq{(8)} can be written +\[ +\nu = N^{2} K \left(\frac{1}{2^{2}} - \frac{1}{3^{2}}\right). +\Tag{(9)} +\] +Here again we obtain an expression for the frequency which corresponds +to a line in the spectrum which would be emitted by the +\emph{binding of an electron to a nucleus, whose charge is~$Ne$}. + +\Section{The fine structure of the hydrogen lines.} This similarity between +the structure of the X-ray spectra and the hydrogen spectrum +was still further extended in a very interesting manner by Sommerfeld's +important theory of the fine structure of the hydrogen lines. +The calculation given above of the energy in the stationary states +of the hydrogen system, where each state is characterized by a +single quantum number, rests upon the assumption that the orbit +of the electron in the atom is simply periodic. This is, however, +only approximately true. It is found that if the change in the mass +of the electron due to its velocity is taken into consideration the +orbit of the electron no longer remains a simple ellipse, but its +motion may be described as a \emph{central motion} obtained by superposing +a slow and uniform rotation upon a simple periodic motion in a +very nearly elliptical orbit. For a central motion of this kind the +stationary states are characterized by \emph{two quantum numbers}. In the +case under consideration one of these may be so chosen that to a +very close approximation it will determine the energy of the atom +in the same manner as the quantum number previously used +determined the energy in the case of a simple elliptical orbit. This +quantum number which will always be denoted by~$n$ will therefore +be called the ``principal quantum number.'' Besides this condition, +which to a very close approximation determines the major axis in the +rotating and almost elliptical orbit, a second condition will be imposed +upon the stationary states of a central orbit, namely that the angular +momentum of the electron about the centre shall be equal to a whole +multiple of Planck's constant divided by~$2\pi$. The whole number, which +occurs as a factor in this expression, may be regarded as the second +quantum number and will be denoted by~$k$. The latter condition fixes +\PageSep{68} +the \Chg{excentricity}{eccentricity} of the rotating orbit which in the case of a simple +periodic orbit was undetermined. It should be mentioned that the +possible importance of the angular momentum in the quantum theory +was pointed out by Nicholson before the application of this theory to +the spectrum of hydrogen, and that a determination of the stationary +states for the hydrogen atom similar to that employed by Sommerfeld +was proposed almost simultaneously by Wilson, although the +latter did not succeed in giving a physical application to his results. + +The simplest description of the form of the rotating nearly +elliptical electronic orbit in the hydrogen atom is obtained by +considering the chord which passes through the focus and is +perpendicular to the major axis, the so-called ``parameter.'' The +length~$2p$ of this parameter is given to a very close approximation +by an expression of exactly the same form as the expression for the +major axis, except that $k$~takes the place of~$n$. Using the same +notation as before we have therefore +\[ +2a = n^{2}\, \frac{h^{2}}{2\pi^{2} N e^{2} m},\quad +2p = k^{2}\, \frac{h^{2}}{2\pi^{2} N e^{2} m}. +\Tag{(10)} +\] +For each of the stationary states which had previously been denoted +by a given value of~$n$, we obtain therefore a set of stationary states +corresponding to values of~$k$ from $1$ to~$n$. Instead of the simple +formula~\Eq{(5)} Sommerfeld found a more complicated expression for +the energy in the stationary states which depends on~$k$ as well as~$n$. +Taking the variation of the mass of the electron with velocity +into account and neglecting terms of higher order of magnitude he +obtained +\[ +E_{n,k} = -\frac{2\pi^{2} N^{2} e^{4} m}{n^{2} h^{2}} + \left[1 + \frac{4\pi^{2} N^{2} e^{4}}{h^{2} c^{2}}\left(-\frac{3}{4n^{2}} + \frac{1}{nk}\right)\right], +\Tag{(11)} +\] +where $c$~is the velocity of light. + +Corresponding to each of the energy values for the stationary +states of the hydrogen atom given by the simple formula~\Eq{(5)} we +obtain $n$~values differing only very little from one another, since +the second term within the bracket is very small. With the aid of +the general frequency relation~\Eq{(1)} we therefore obtain a number of +components with nearly coincident frequencies instead of each +hydrogen line given by the simple formula~\Eq{(2)}. Sommerfeld has +now shown that this calculation actually agrees with measurements +\PageSep{69} +of the fine structure. This agreement applies not only to the fine +structure of the hydrogen lines which is very difficult to measure +on account of the extreme proximity of the components, but it is +also possible to account in detail for the fine structure of the helium +lines given by formula~\Eq{(7)} which has been very carefully investigated +by Paschen. Sommerfeld in connection with this theory +also pointed out that formula~\Eq{(11)} could be applied to the X-ray +spectra. Thus he showed that in the $K$~and $L$ groups pairs of lines +appeared the differences of whose frequencies could be determined +by the expression~\Eq{(11)} for the energy in the stationary states which +correspond to the binding of a single electron by a nucleus of +charge~$Ne$. + +\Section{Periodic table.} In spite of the great formal similarity between +the X-ray spectra and the hydrogen spectrum indicated by these +results a far-reaching difference must be assumed to exist between +the processes which give rise to the appearance of these two types +of spectra. While the emission of the hydrogen spectrum, like the +emission of the ordinary optical spectra of other elements, may be +assumed to be connected with the binding of an electron by an +atom, observations on the appearance and absorption of X-ray +spectra clearly indicate that these spectra are connected with a +process which may be described as a \emph{reorganization of the electronic +arrangement} after a disturbance within the atom due to the effect +of external agencies. We should therefore expect that the appearance +of the X-ray spectra would depend not only upon the direct +interaction between a single electron and the nucleus, but also on +the manner in which the electrons are arranged in the completely +formed atom. + +The peculiar manner in which the properties of the elements +vary with the atomic number, as expressed in the periodic system, +provides a guide of great value in considering this latter problem. +A simple survey of this system is given in \Fig{1}. The number preceding +each element indicates the atomic number, and the elements +within the various vertical columns form the different ``periods'' of +the system. The lines, which connect pairs of elements in successive +columns, indicate homologous properties of such elements. Compared +with usual representations of the periodic system, this method, +\PageSep{70} +proposed more than twenty years ago by Julius Thomsen, of indicating +the periodic variations in the properties of the elements is +more suited for comparison with theories of atomic constitution. +The meaning of the frames round certain sequences of elements +within the later periods of the table will be explained later. They +refer to certain characteristic features of the theory of atomic +constitution. +\Figure{1}{70} + +In an explanation of the periodic system it is natural to assume +a division of the electrons in the atom into distinct groups +in such a manner that the grouping of the elements in the system +is attributed to the gradual formation of the groups of electrons +in the atoms as the atomic number increases. Such a grouping +\PageSep{71} +of the electrons in the atom has formed a prominent part of all +more detailed views of atomic structure ever since J.~J. Thomson's +famous attempt to explain the periodic system on the basis +of an investigation of the stability of various electronic configurations. +Although Thomson's assumption regarding the distribution +of the positive electricity in the atom is not consistent with more +recent experimental evidence, nevertheless his work has exerted +great influence upon the later development of the atomic theory on +account of the many original ideas which it contained. + +With the aid of the information concerning the binding of +electrons by the nucleus obtained from the theory of the hydrogen +spectrum I attempted in the same paper in which this theory was +set forth to sketch in broad outlines a picture of the structure of +the nucleus atom. In this it was assumed that each electron in its +normal state moved in a manner analogous to the motion in +the last stages of the binding of a single electron by a nucleus. +As in Thomson's theory, it was assumed that the electrons moved +in circular orbits and that the electrons in each separate group +during this motion occupied positions with reference to one another +corresponding to the vertices of plane regular polygons. Such an +arrangement is frequently described as a distribution of the electrons +in ``rings.'' By means of these assumptions it was possible to +account for the orders of magnitude of the dimensions of the atoms +as well as the firmness with which the electrons were bound by the +atom, a measure of which may be obtained by means of experiments +on the excitation of the various types of spectra. It was not +possible, however, in this way to arrive at a detailed explanation +of the characteristic properties of the elements even after it had +become apparent from the results of Moseley and the work of +Sommerfeld and others that this simple picture ought to be extended +to include orbits in the fully formed atom characterized by +higher quantum numbers corresponding to previous stages in the +formation of the hydrogen atom. This point has been especially +emphasized by Vegard. + +The difficulty of arriving at a satisfactory picture of the atom is +intimately connected with the difficulty of accounting for the pronounced +``stability'' which the properties of the elements demand. +As I emphasized when considering the formation of the hydrogen +\PageSep{72} +atom, the postulates of the quantum theory aim directly at this +point, but the results obtained in this way for an atom containing +a single electron do not permit of a direct elucidation of problems +like that of the distribution in groups of the electrons in an atom +containing several electrons. If we imagine that the electrons in +the groups of the atom are orientated relatively to one another at any +moment, like the vertices of regular polygons, and rotating in either +circles or ellipses, the postulates do not give sufficient information to +determine the difference in the stability of electronic configurations +with different numbers of electrons in the groups. + +The peculiar character of stability of the atomic structure, demanded +by the properties of the elements, is brought out in an +interesting way by Kossel in two important papers. In the first +paper he shows that a more detailed explanation of the origin of +the high frequency spectra can be obtained on the basis of the +group structure of the atom. He assumes that a line in the X-ray +spectrum is due to a process which may be described as follows: an +electron is removed from the atom by some external action after +which an electron in one of the other groups takes its place; this +exchange of place may occur in as many ways as there are groups +of more loosely bound electrons. This view of the origin of the +characteristic X-rays afforded a simple explanation of the peculiar +absorption phenomena observed. It has also led to the prediction +of certain simple relations between the frequencies of the X-ray +lines from one and the same element and has proved to be a suitable +basis for the classification of the complete spectrum. However it has +not been possible to develop a theory which reconciles in a satisfactory +way Sommerfeld's work on the fine structure of the X-ray +lines with Kossel's general scheme. As we shall see later the +adoption of a new point of view when considering the stability of +the atom renders it possible to bring the different results in a natural +way in connection with one another. + +In his second paper Kossel investigates the possibilities for an +explanation of the periodic system on the basis of the atomic theory. +Without entering further into the problem of the causes of the +division of the electrons into groups, or the reasons for the different +stability of the various electronic configurations, he points out in +connection with ideas which had already played a part in Thomson's +\PageSep{73} +theory, how the periodic system affords evidence of a periodic appearance +of especially stable configurations of electrons. These configurations +appear in the neutral atoms of elements occupying the +final position in each period in \Fig{1}, and the stability in question is +assumed in order to explain not only the inactive chemical properties +of these elements but also the characteristic active properties of the +immediately preceding or succeeding elements. If we consider for +instance an inactive gas like argon, the atomic number of which is~$18$, +we must assume that the $18$~electrons in the atom are arranged in +an exceedingly regular configuration possessing a very marked +stability. The pronounced electronegative character of the preceding +element, chlorine, may then be explained by supposing the neutral +atom which contains only $17$~electrons to possess a tendency to +capture an additional electron. This gives rise to a negative chlorine +ion with a configuration of $18$~electrons similar to that occurring +in the neutral argon atom. On the other hand the marked electropositive +character of potassium may be explained by supposing +one of the $19$~electrons in the neutral atom to be as it were superfluous, +and that this electron therefore is easily lost; the rest of the +atom forming a positive ion of potassium having a constitution similar +to that of the argon atom. In a corresponding manner it is possible +to account for the electronegative and electropositive character of +elements like sulphur and calcium, whose atomic numbers are $16$ and~$20$. +In contrast to chlorine and potassium these elements are divalent, +and the stable configuration of $18$~electrons is formed by the addition +of two electrons to the sulphur atom and by the loss of two electrons +from the calcium atom. Developing these ideas Kossel has succeeded +not only in giving interesting explanations of a large number of +chemical facts, but has also been led to certain general conclusions +about the grouping of the electrons in elements belonging to the +first periods of the periodic system, which in certain respects are +in conformity with the results to be discussed in the following +paragraphs. Kossel's\Pagelabel{73} work was later continued in an interesting +manner by Ladenburg with special reference to the grouping of the +electrons in atoms of elements belonging to the later periods of the +periodic table. It will be seen that Ladenburg's conclusions also +exhibit points of similarity with the results which we shall discuss +later. +\PageSep{74} + +\Section{Recent atomic models.} Up to the present time it has not been +possible to obtain a satisfactory account based upon a consistent application +of the quantum theory to the nuclear atom of the ultimate +cause of the pronounced stability of certain arrangements of electrons. +Nevertheless it has been apparent for some time that the solution +should be sought for by investigating the possibilities of a \emph{spatial +distribution of the electronic orbits} in the atom instead of limiting +the investigation to configurations in which all electrons belonging +to a particular group move in the same plane as was assumed for +simplicity in my first papers on the structure of the atom. The +necessity of assuming a spatial distribution of the configurations +of electrons has been drawn attention to by various writers. Born +and Landé, in connection with their investigations of the structure +and properties of crystals, have pointed out that the assumption of +spatial configurations appears necessary for an explanation of these +properties. Landé has pursued this question still further, and as +will be mentioned later has proposed a number of different ``spatial +atomic models'' in which the electrons in each separate group of +the atom at each moment form configurations possessing regular +polyhedral symmetry. These models constitute in certain respects +a distinct advance, although they have not led to decisive results +on questions of the stability of atomic structure. + +The importance of spatial electronic configurations has, in addition, +been pointed out by Lewis and Langmuir in connection with their +atomic models. Thus Lewis, who in several respects independently +came to the same conclusions as Kossel, suggested that the number~$8$ +characterizing the first groups of the periodic system might indicate +a constitution of the outer atomic groups where the electrons +within each group formed a configuration like the corners of a cube. +He emphasized how a configuration of this kind leads to instructive +models of the molecular structure of chemical combinations. It is +to be remarked, however, that such a ``static'' model of electronic +configuration will not be possible if we assume the forces within +the atom to be due exclusively to the electric charges of the +particles. Langmuir, who has attempted to develop Lewis' conceptions +still further and to account not only for the occurrence of +the first octaves, but also for the longer periods of the periodic +system, supposes therefore the structure of the atoms to be governed +\PageSep{75} +by forces whose nature is unknown to us. He conceives the atom +to possess a ``cellular structure,'' so that each electron is in advance +assigned a place in a cell and these cells are arranged in shells in +such a manner, that the various shells from the nucleus of the atom +outward contain exactly the same number of places as the periods +in the periodic system proceeding in the direction of increasing +atomic number. Langmuir's work has attracted much attention +among chemists, since it has to some extent thrown light on the +conceptions with which empirical chemical science is concerned. +On his theory the explanation of the properties of the various +elements is based on a number of postulates about the structure of +the atoms formulated for that purpose. Such a descriptive theory +is sharply differentiated from one where an attempt is made to +explain the specific properties of the elements with the aid of +general laws applying to the interaction between the particles in +each atom. The principal task of this lecture will consist in an +attempt to show that an advance along these lines appears by no +means hopeless, but on the contrary that with the aid of a consistent +application of the postulates of the quantum theory it +actually appears possible to obtain an insight into the structure +and stability of the atom. + + +\Chapter{II.}{Series Spectra and the Capture of Electrons +by\protect~Atoms} + +We attack the problem of atomic constitution by asking the +question: ``How may an atom be formed by the successive capture +and binding of the electrons one by one in the field of force surrounding +the nucleus?'' + +Before attempting to answer this question it will first be +necessary to consider in more detail what the quantum theory +teaches us about the general character of the binding process. We +have already seen how the hydrogen spectrum gives us definite +information about the course of this process of binding the electron +by the nucleus. In considering the formation of the atoms of other +elements we have also in their spectra sources for the elucidation +of the formation processes, but the direct information obtained in +this way is not so complete as in the case of the hydrogen atom. +For an element of atomic number~$N$ the process of formation may +\PageSep{76} +be regarded as occurring in $N$~stages, corresponding with the successive +binding of $N$~electrons in the field of the nucleus. A spectrum +must be assumed to correspond to each of these binding processes; +but only for the first two elements, hydrogen and helium, do we +possess a detailed knowledge of these spectra. For other elements +of higher atomic number, where several spectra will be connected +with the formation of the atom, we are at present acquainted with +only two types, called the ``arc'' and ``spark'' spectra respectively, +according to the experimental conditions of excitation. Although +these spectra show a much more complicated structure than the +hydrogen spectrum, given by formula~\Eq{(2)} and the helium spectrum +given by formula~\Eq{(7)}, nevertheless in many cases it has been +possible to find simple laws for the frequencies exhibiting a close +analogy with the laws expressed by these formulae. + +\Section{Arc and spark spectra.} If for the sake of simplicity we disregard +the complex structure shown by the lines of most spectra +(occurrence of doublets, triplets etc.), the frequency of the lines of +many arc spectra can be represented to a close approximation by +the Rydberg formula +\[ +\nu = \frac{K}{(n'' + \alpha_{k''})^{2}} - \frac{K}{(n' + \alpha_{k'})^{2}}, +\Tag{(12)} +\] +where $n'$~and $n''$ are integral numbers, $K$~the same constant as in +the hydrogen spectrum, while $\alpha_{k'}$~and $\alpha_{k''}$ are two constants belonging +to a set characteristic of the element. A spectrum with a +structure of this kind is, like the hydrogen spectrum, called a series +spectrum, since the lines can be arranged into series in which the +frequencies converge to definite limiting values. These series are +for example represented by formula~\Eq{(12)} if, using two definite +constants for $\alpha_{k''}$~and~$\alpha_{k'}$, $n''$~remains unaltered, while $n'$~assumes a +series of successive, gradually increasing integral values. + +Formula~\Eq{(12)} applies only approximately, but it is always found +that the frequencies of the spectral lines can be written, as in +formulae \Eq{(2)} and~\Eq{(12)}, as a difference of two functions of integral +numbers. Thus the latter formula applies accurately, if the +quantities~$\alpha_{k}$ are not considered as constants, but as representatives +of a set of series of numbers~$\alpha_{k}(n)$ characteristic of the element, +whose values for increasing~$n$ within each series quickly approach +\PageSep{77} +a constant limiting value. The fact that the frequencies of the +spectra always appear as the difference of two terms, the so-called +``spectral terms,'' from the combinations of which the complete +spectrum is formed, has been pointed out by Ritz, who with the +establishment of the combination principle has greatly advanced +the study of the spectra. The quantum theory offers an immediate +interpretation of this principle, since, according to the frequency +relation we are led to consider the lines as due to transitions +between stationary states of the atom, just as in the hydrogen +spectrum, only in the spectra of the other elements we have to do +not with a single series of stationary states, but with a set of such +series. From formula~\Eq{(12)} we thus obtain for an arc spectrum---if +we temporarily disregard the structure of the individual lines---information +about an ensemble of stationary states, for which the +energy of the atom in the $n$th~state of the $k$th~series is given by +\[ +E_{k}(n) = -\frac{Kh}{(n + \alpha_{k})^{2}} +\Tag{(13)} +\] +very similar to the simple formula~\Eq{(3)} for the energy in the stationary +states of the hydrogen atom. + +As regards the spark spectra, the structure of which has been +cleared up mainly by Fowler's investigations, it has been possible +in the case of many elements to express the frequencies approximately +by means of a formula of exactly the same type as~\Eq{(12)}, +only with the difference that~$K$, just as in the helium spectrum +given by formula~\Eq{(7)}, is replaced by a constant, which is four times +as large. For the spark spectra, therefore, the energy values in the +corresponding stationary states of the atom will be given by an +expression of the same type as~\Eq{(13)}, only with the difference that +$K$~is replaced by~$4K$. + +This remarkable similarity between the structure of these types +of spectra and the simple spectra given by \Eq{(2)}~and~\Eq{(7)} is explained +simply by assuming the arc spectra to be connected with the \emph{last +stage in the formation of the neutral atom} consisting in the capture +and binding of the $N$th~electron. On the other hand the spark +spectra are connected with the \emph{last stage but one in the formation +of the atom}, namely the binding of the $(N - 1)$th~electron. In these +cases the field of force in which the electron moves will be much +\PageSep{78} +the same as that surrounding the nucleus of a hydrogen or helium +atom respectively, at least in the earlier stages of the binding +process, where during the greater part of its revolution it moves +at a distance from the nucleus which is large in proportion to the +dimensions of the orbits of the electrons previously bound. From +analogy with formula~\Eq{(3)} giving the stationary states of the +hydrogen atom, we shall therefore assume that the numerical value +of the expression on the right-hand side of~\Eq{(13)} will be equal to the +work required to remove the last captured electron from the atom, +the binding of which gives rise to the arc spectrum of the element. + +\Section{Series diagram.} While the origin of the arc and spark spectra +was to this extent immediately interpreted on the basis of the +original simple theory of the hydrogen spectrum, it was Sommerfeld's +theory of the fine structure of the hydrogen lines which first gave +us a clear insight into the characteristic difference between the +hydrogen spectrum and the spark spectrum of helium on the one +hand, and the arc and spark spectra of other elements on the other. +When we consider the binding not of the first but of the subsequent +electrons in the atom, the orbit of the electron under consideration---at +any rate in the latter stages of the binding process where the +electron last bound comes into intimate interaction with those +previously bound---will no longer be to a near approximation a +closed ellipse, but on the contrary will to a first approximation be a +central orbit of the same type as in the hydrogen atom, when we +take into account the change with velocity in the mass of the +electron. This motion, as we have seen, may be resolved into a +plane periodic motion upon which a uniform rotation is superposed +in the plane of the orbit; only the superposed rotation will in this +case be comparatively much more rapid and the deviation of the +periodic orbit from an ellipse much greater than in the case of the +hydrogen atom. For an orbit of this type the stationary states, just +as in the theory of the fine structure, will be determined by two +quantum numbers which we shall denote by $n$~and~$k$, connected in +a very simple manner with the kinematic properties of the orbit. +For brevity I shall only mention that while the quantum number~$k$ +is connected with the value of the constant angular momentum +of the electron about the centre in the simple manner previously +\PageSep{79} +indicated, the determination of the principal quantum number~$n$ +requires an investigation of the whole course of the orbit and for +an arbitrary central orbit will not be related in a simple way to +the dimensions of the rotating periodic orbit if this deviates essentially +from a Keplerian ellipse. +\Figure{2}{79} + +These results are represented in \Fig{2} which is a reproduction +of an illustration I have used on a previous occasion +(see Essay~II, \PageRef{30}), and which gives a survey of the origin +of the sodium spectrum. The black dots represent the stationary +states corresponding to the various series of spectral terms, +shown on the right by the letters $S$,~$P$,~$D$ and~$B$. These letters +correspond to the usual notations employed in spectroscopic +literature and indicate the nature of the series (sharp series, +principal series, diffuse series, etc.)\ obtained by combinations of +the corresponding spectral terms. The distances of the separate +points from the vertical line at the right of the figure are proportional +to the numerical value of the energy of the atom given +by equation~\Eq{(13)}. The oblique, black arrows indicate finally the +transitions between the stationary states giving rise to the +appearance of the lines in the commonly observed sodium +spectrum. The values of $n$~and $k$ attached to the various states +indicate the quantum numbers, which, according to Sommerfeld's +theory, from a preliminary consideration might be regarded as +characterizing the orbit of the outer electron. For the sake of +convenience the states which were regarded as corresponding to +the same value of~$n$ are connected by means of dotted lines, and these +are so drawn that their vertical asymptotes correspond to the +\PageSep{80} +terms in the hydrogen spectrum which belong to the same value +of the principal quantum number. The course of the curves illustrates +how the deviation from the hydrogen terms may be expected +to decrease with increasing values of~$k$, corresponding to states, +where the minimum distance between the electron in its revolution +and the nucleus constantly increases. + +It should be noted that even though the theory represents the +principal features of the structure of the series spectra it has not +yet been possible to give a detailed account of the spectrum of any +element by a closer investigation of the electronic orbits which may +occur in a simple field of force possessing central symmetry. As +I have mentioned already the lines of most spectra show a complex +structure. In the sodium spectrum for instance the lines of the +principal series are doublets indicating that to each $P$-term not +one stationary state, but two such states correspond with slightly +different values of the energy. This difference is so little that +it would not be recognizable in a diagram on the same scale as +\Fig{2}. The appearance of these doublets is undoubtedly due to +the small deviations from central symmetry of the field of force +originating from the inner system in consequence of which the +general type of motion of the external electron will possess a +more complicated character than that of a simple central motion. +As a result the stationary states must be characterized by more +than two quantum numbers, in the same way that the occurrence +of deviations of the orbit of the electron in the hydrogen atom from +a simple periodic orbit requires that the stationary states of this +atom shall be characterized by more than one quantum number. +Now the rules of the quantum theory lead to the introduction of +a third quantum number through the condition that the resultant +angular momentum of the atom, multiplied by~$2\pi$, is equal to an +entire multiple of Planck's constant. This determines the orientation +of the orbit of the outer electron relative to the axis of the +inner system. + +In this way Sommerfeld, Landé and others have shown that it +is possible not only to account in a formal way for the complex +structure of the lines of the series spectra, but also to obtain a +promising interpretation of the complicated effect of external +magnetic fields on this structure. We shall not enter here on these +\PageSep{81} +problems but shall confine ourselves to the problem of the fixation +of the two quantum numbers $n$~and~$k$, which to a first approximation +describe the orbit of the outer electron in the stationary +states, and whose determination is a matter of prime importance +in the following discussion of the formation of the atom. In +the determination of these numbers we at once encounter difficulties +of a profound nature, which---as we shall see---are intimately +connected with the question of the remarkable stability of atomic +structure. I shall here only remark that the values of the quantum +number~$n$, given in the figure, undoubtedly \Chg{can not}{cannot} be retained, +neither for the~$S$ nor the $P$~series. On the other hand, so far as +the values employed for the quantum number~$k$ are concerned, it +may be stated with certainty, that the interpretation of the properties +of the orbits, which they indicate, is correct. A starting +point for the investigation of this question has been obtained from +considerations of an entirely different kind from those previously +mentioned, which have made it possible to establish a close connection +between the motion in the atom and the appearance of +spectral lines. + +\Section{Correspondence principle.} So far as the principles of the +quantum theory are concerned, the point which has been emphasized +hitherto is the radical departure of these principles from our +usual conceptions of mechanical and electrodynamical phenomena. +As I have attempted to show in recent years, it appears +possible, however, to adopt a point of view which suggests that the +quantum theory may, nevertheless, be regarded as a rational +generalization of our ordinary conceptions. As may be seen from +the postulates of the quantum theory, and particularly the frequency +relation, a direct connection between the spectra and the motion +of the kind required by the classical dynamics is excluded, but at +the same time the form of these postulates leads us to another +relation of a remarkable nature. Let us consider an electrodynamic +system and inquire into the nature of the radiation which would +result from the motion of the system on the basis of the ordinary +conceptions. We imagine the motion to be decomposed into purely +harmonic oscillations, and the radiation is assumed to consist of +the simultaneous emission of series of electromagnetic waves +\PageSep{82} +possessing the same frequency as these harmonic components and +intensities which depend upon the amplitudes of the components. +An investigation of the formal basis of the quantum theory shows +us now, that it is possible to trace the question of the origin of the +radiation processes which accompany the various transitions back +to an investigation of the various harmonic components, which +appear in the motion of the atom. The possibility, that a particular +transition shall occur, may be regarded as being due to the +presence of a definitely assignable ``corresponding'' component in +the motion. This principle of correspondence at the same time +throws light upon a question mentioned several times previously, +namely the relation between the number of quantum numbers, +which must be used to describe the stationary states of an atom, +and the types to which the orbits of the electrons belong. The +classification of these types can be based very simply on a decomposition +of the motion into its harmonic components. Time does +not permit me to consider this question any further, and I shall +confine myself to a statement of some simple conclusions, which +the correspondence principle permits us to draw concerning the +occurrence of transitions between various pairs of stationary states. +These conclusions are of decisive importance in the subsequent +argument. + +The simplest example of such a conclusion is obtained by +considering an atomic system, which contains a particle describing +a \emph{purely periodic orbit}, and where the stationary states are characterized +by a single quantum number~$n$. In this case the motion +can according to Fourier's theorem be decomposed into a simple +series of harmonic oscillations whose frequency may be written~$\tau\omega$, +where $\tau$~is a whole number, and $\omega$~is the frequency of revolution +in the orbit. It can now be shown that a transition between two +stationary states, for which the values of the quantum number are +respectively equal to $n'$~and~$n''$, will correspond to a harmonic +component, for which $\tau = n' - n''$. This throws at once light upon +the remarkable difference which exists between the possibilities +of transitions between the stationary states of a hydrogen atom +on the one hand and of a simple system consisting of an electric +particle capable of executing simple harmonic oscillations about a +position of equilibrium on the other. For the latter system, which +\PageSep{83} +is frequently called a Planck oscillator, the energy in the stationary +states is determined by the familiar formula $E = nh\omega$, and with the +aid of the frequency relation we obtain therefore for the radiation +which will be emitted during a transition between two stationary +states $\nu = (n' - n'') \omega$. Now, an important assumption, which is not +only essential in Planck's theory of temperature radiation, but +which also appears necessary to account for the molecular absorption +in the infra-red region of radiation, states that a harmonic oscillator +will only emit and absorb radiation, for which the frequency~$\nu$ is +equal to the frequency of oscillation~$\omega$ of the oscillator. We are +therefore compelled to assume that in the case of the oscillator +transitions can occur only between stationary states which are +characterized by quantum numbers differing by only one unit, +while in the hydrogen spectrum represented by formula~\Eq{(2)} all +possible transitions could take place between the stationary states +given by formula~\Eq{(5)}. From the point of view of the principle of +correspondence it is seen, however, that this apparent difficulty is +explained by the occurrence in the motion of the hydrogen atom, +as opposed to the motion of the oscillator, of harmonic components +corresponding to values of~$\tau$, which are different from~$1$; or using +a terminology well known from acoustics, there appear overtones +in the motion of the hydrogen atom. + +Another simple example of the application of the correspondence +principle is afforded by a \emph{central motion}, to the investigation of +which the explanation of the series spectra in the first approximation +may be reduced. Referring once more to the figure of the +sodium spectrum, we see that the black arrows, which correspond +to the spectral lines appearing under the usual conditions of +excitation, only connect pairs of points in consecutive rows. Now +it is found that this remarkable limitation of the occurrence of +combinations between spectral terms may quite naturally be +explained by an investigation of the harmonic components into +which a central motion can be resolved. It can readily be shown +that such a motion can be decomposed into two series of harmonic +components, whose frequencies can be expressed by $\tau\omega + \sigma$ and +$\tau\omega - \sigma$ respectively, where $\tau$~is a whole number, $\omega$~the frequency of +revolution in the rotating periodic orbit and $\sigma$~the frequency of the +superposed rotation. These components correspond with transitions +\PageSep{84} +where the principal number~$n$ decreases by $\tau$~units, while the +quantum number~$k$ decreases or increases, respectively, by one +unit, corresponding exactly with the transitions indicated by the +black arrows in the figure. This may be considered as a very +important result, because we may say, that the quantum theory, +which for the first time has offered a simple interpretation of the +fundamental principle of combination of spectral lines has at the +same time removed the mystery which has hitherto adhered +to the application of this principle on account of the apparent +capriciousness of the appearance of predicted combination lines. +Especially attention may be drawn to the simple interpretation +which the quantum theory offers of the appearance observed by +Stark and his collaborators of certain new series of lines, which do +not appear under ordinary circumstances, but which are excited +when the emitting atoms are subject to intense external electric +fields. In fact, on the correspondence principle this is immediately +explained from an examination of the perturbations in the motion +of the outer electron which give rise to the appearance in this +motion---besides the harmonic components already present in a +simple central orbit---of a number of constituent harmonic vibrations +of new type and of amplitudes proportional to the intensity +of the external forces. + +It may be of interest to note that an investigation of the +limitation of the possibility of transitions between stationary +states, based upon a simple consideration of conservation of angular +momentum during the process of radiation, does not, contrary to +what has previously been supposed (compare Essay~II, \PageRef{62}), +suffice to throw light on the remarkably simple structure of series +spectra illustrated by the figure. As mentioned above we must +assume that the ``complexity'' of the spectral terms, corresponding +to given values of $n$~and~$k$, which we witness in the fine +structure of the spectral lines, may be ascribed to states, corresponding +to different values of this angular momentum, in +which the plane of the electronic orbit is orientated in a different +manner, relative to the configuration of the previously bound +electrons in the atom. Considerations of conservation of angular +momentum can, in connection with the series spectra, therefore only +contribute to an understanding of the limitation of the possibilities +\PageSep{85} +of combination observed in the peculiar laws applying to the +number of components in the complex structure of the lines. So +far as the last question is concerned, such considerations offer a +direct support for the consequences of the correspondence principle. + + +\Chapter{III.}{Formation of Atoms and the Periodic Table} + +A correspondence has been shown to exist between the motion +of the electron last captured and the occurrence of transitions +between the stationary states corresponding to the various stages +of the binding process. This fact gives a point of departure for a +choice between the numerous possibilities which present themselves +when considering the formation of the atoms by the successive +capture and binding of the electrons. Among the processes which +are conceivable and which according to the quantum theory might +occur in the atom we shall reject those whose occurrence \Chg{can not}{cannot} be +regarded as consistent with a correspondence of the required nature. + +\Section{First Period. Hydrogen---Helium.} It will not be necessary to +concern ourselves long with the question of the constitution of the +hydrogen atom. From what has been said previously we may assume +that the final result of the process of \emph{binding of the first electron} in +any atom will be a stationary state, where the energy of the atom +is given by~\Eq{(5)}, if we put $n = 1$, or more precisely by formula~\Eq{(11)}, +if we put $n = 1$ and $k = 1$. The orbit of the electron will be a circle +whose radius will be given by formulae~\Eq{(10)}, if $n$~and $k$ are each +put equal to~$1$. Such an orbit will be called a $1$-quantum orbit, +and in general an orbit for which the principal quantum number +has a given value~$n$ will be called an $n$-quantum orbit. Where it +is necessary to differentiate between orbits corresponding to various +values of the quantum number~$k$, a central orbit, characterized by +given values of the quantum numbers $n$~and~$k$, will be referred to +as an $n_{k}$~orbit. + +In the question of the constitution of the helium atom we meet +the much more complicated problem of the \emph{binding of the second +electron}. Information about this binding process may, however, be +obtained from the arc spectrum of helium. This spectrum, as +opposed to most other simple spectra, consists of two complete +systems of lines with frequencies given by formulae of the type~\Eq{(12)}. +\PageSep{86} +On this account helium was at first assumed to be a mixture +of two different gases, ``orthohelium'' and ``parhelium,'' but now we +know that the two spectra simply mean that the binding of the second +electron can occur in two different ways. A theoretical explanation of +the main features of the helium spectrum has recently been attempted +in an interesting paper by Landé. He supposes the emission of the +orthohelium spectrum to be due to transitions between stationary +states where both electrons move in the same plane and revolve +in the same sense. The parhelium spectrum, on the other hand, is +ascribed by him to stationary states where the planes of the orbits +form an angle with each other. Dr~Kramers and I have made a +closer investigation of the interaction between the two orbits in +the different stationary states. The results of our investigation +which was begun several years before the appearance of Landé's +work have not yet been published. Without going into details +I may say, that even though our results in several respects differ +materially from those of Landé (compare Essay~II, \PageRef{56}), we agree +with his general conclusions concerning the origin of the orthohelium +and parhelium spectra. + +The final result of the binding of the second electron is intimately +related to the origin of the two helium spectra. Important +information on this point has been obtained recently by Franck +and his co-workers. As is well known he has thrown light upon +many features of the structure of the atom and of the origin +of spectra by observing the effect of bombarding atoms by +electrons of various velocities. A short time ago these experiments +showed that the impact of electrons could bring helium into a +``metastable'' state from which the atom cannot return to its +normal state by means of a simple transition accompanied by the +emission of radiation, but only by means of a process analogous to +a chemical reaction involving interaction with atoms of other +elements. This result is closely connected with the fact that the +binding of the second electron can occur in two different ways, as +is shown by the occurrence of two distinct spectra. Thus it is +evident from Franck's experiments that the normal state of the +atom is the last stage in the binding process involving the emission +of the parhelium spectrum by which the electron last captured as +well as the one first captured will be bound in a $1_{1}$~orbit. The +\PageSep{87} +metastable state, on the contrary, is the final stage of the process +giving the orthohelium spectrum. In this case the second electron, +as opposed to the first, will move in a $2_{1}$~orbit. This corresponds to +a firmness of binding which is about six times less than for the +electron in the normal state of the atom. + +If we now consider somewhat more closely this apparently +surprising result, it is found that a clear grasp of it may be obtained +from the point of view of correspondence. It can be shown that +the coherent class of motions to which the orthohelium orbits +belong does not contain a $1_{1}$~orbit. If on the whole we would claim +the existence of a state where the two electrons move in $1_{1}$~orbits +in the same plane, and if in addition it is claimed that the motion +should possess the periodic properties necessary for the definition +of stationary states, then there seems that no possibility is afforded +other than the assumption that the two electrons move around the +nucleus in one and the same orbit, in such a manner that at each +moment they are situated at the ends of a diameter. This extremely +simple ring-configuration might be expected to correspond to +the firmest possible binding of the electrons in the atom, and it +was on this account proposed as a model for the helium atom in +my first paper on atomic structure. If, however, we inquire about +the possibility of a transition from one of the orthohelium states +to a configuration of this type we meet conditions which are very +different from those which apply to transitions between two of +the orthohelium orbits. In fact, the occurrence of each of these +transitions is due to the existence of well-defined corresponding +constituent harmonic vibration in the central orbits which the outer +electron describes in the class of motions to which the stationary +states belong. The transition we have to discuss, on the other +hand, is one by which the last captured electron is transferred from +a state in which it is moving ``outside'' the other to a state in which +it moves round the nucleus on equal terms with the other electron. +Now it is impossible to find a series of simple intermediate forms +for the motion of those two electrons in which the orbit of the last +captured electron exhibits a sufficient similarity to a central motion +that for this transition there could be a correspondence of the +necessary kind. It is therefore evident, that where the two electrons +move in the same plane, the electron captured last \Chg{can not}{cannot} be +\PageSep{88} +bound firmer than in a $2_{1}$~orbit. If, on the other hand, we consider +the binding process which accompanies the emission of the parhelium +spectrum and where the electrons in the stationary states move in +orbits whose planes form angles with one another we meet essentially +different conditions. A corresponding intimate change in the +interaction between the electron last captured and the one previously +bound is not required here for the two electrons in the atom to +become equivalent. We may therefore imagine the last stage of +the binding process to take place in a manner similar to those +stages corresponding to transitions between orbits characterized by +greater values of $n$~and~$k$. + +In the \emph{normal state of the helium atom} the two electrons must +be assumed to move in equivalent $1_{1}$~orbits. As a first approximation +these may be described as two circular orbits, whose planes make +an angle of~$120°$ with one another, in agreement with the conditions +which the angular momentum of an atom according to the quantum +theory must satisfy. On account of the interaction between the +two electrons these planes at the same time turn slowly around +the fixed impulse axis of the atom. Starting from a distinctly +different point of view Kemble has recently suggested a similar +model for the helium atom. He has at the same time directed +attention to a possible type of motion of very marked symmetry +in which the electrons during their entire revolution assume +symmetrical positions with reference to a fixed axis. Kemble has +not, however, investigated this motion further. Previous to the +appearance of this paper Kramers had commenced a closer investigation +of precisely this type of motion in order to find out to what +extent it was possible from such a calculation to account for the +firmness with which the electrons are bound in the helium atom, +that is to account for the ionization potential. Early measurements +of this potential had given values corresponding approximately to +that which would result from the ring-configuration already mentioned. +This requires $17/8$~as much work to remove a single +electron as is necessary to remove an electron from the hydrogen +atom in its normal state. As the theoretical value for the latter +amount of work---which for the sake of simplicity will be represented +by~$W$---corresponds to an ionization potential of $13.53$~volts, +the ionization potential of helium would be expected to be $28.8$~volts. +\PageSep{89} +Recent and more accurate determinations, however, have +given a value for the ionization potential of helium which is considerably +lower and lies in the neighbourhood of $25$~volts. This +showed therefore the untenability of the ring-configuration quite +independently of any other considerations. A careful investigation of +the spatial atomic configuration requires elaborate calculation, and +Kramers has not yet obtained final results. With the approximation +to which they have been so far completed the calculations point to +the possibility of an agreement with the experimental results. The +final result may be awaited with great interest, since it offers in +the simplest case imaginable a test of the principles by which we +are attempting to determine stationary states of atoms containing +more than one electron. + +Hydrogen and helium, as seen in the survey of the periodic +system given in \Fig{1}, together form the first period in the system +of elements, since helium is the first of the inactive gases. The great +difference in the chemical properties of hydrogen and helium is +closely related to the great difference in the nature of the binding +of the electron. This is directly indicated by the spectra and +ionization potentials. While helium possesses the highest known +ionization potential of all the elements, the binding of the electron +in the hydrogen atom is sufficiently loose to account for the tendency +of hydrogen to form positive ions in aqueous solutions and chemical +combinations. Further consideration of this particular question +requires, however, a comparison between the nature and firmness +of the electronic configurations of other atoms, and it can therefore +not be discussed at the moment. + +\Section{Second Period. Lithium---Neon.} When considering the atomic +structure of elements which contain more than two electrons in the +neutral atom, we shall assume first of all that what has previously +been said about the formation of the helium atom will in the main +features also apply to the capture and binding of the first two +electrons. These electrons may, therefore, in the normal state of +the atom be regarded as moving in equivalent orbits characterized +by the quantum symbol~$1_{1}$. We obtain direct information about +the \emph{binding of the third electron} from the spectrum of lithium. +This spectrum shows the existence of a number of series of +\PageSep{90} +stationary states, where the firmness with which the last captured +electron is bound is very nearly the same as in the stationary states +of the hydrogen atom. These states correspond to orbits where $k$~is +greater than or equal to~$2$, and where the last captured electron +moves entirely outside the region where the first two electrons +move. But in addition this spectrum gives us information about a +series of states corresponding to $k = 1$ in which the energy differs +essentially from the corresponding stationary states of the hydrogen +atom. In these states the last captured electron, even if it remains +at a considerable distance from the nucleus during the greater part +of its revolution, will at certain moments during the revolution +approach to a distance from the nucleus which is of the same order +of magnitude as the dimensions of the orbits of the previously +bound electrons. On this account the electrons will be bound with +a firmness which is considerably greater than that with which the +electrons are bound in the stationary states of the hydrogen atom +corresponding to the same value of~$n$. + +Now as regards the lithium spectrum as well as the other alkali +spectra we are so fortunate (see \PageRef{32}) as to possess definite evidence +about the normal state of the atom from experiments on selective +absorption. In fact these experiments tell us that the first member +of the sequence of $S$-terms corresponds to this state. This term +corresponds to a strength of binding which is only a little more than +a third of that of the hydrogen atom. We must therefore conclude +that the outer electron in the normal state of the lithium atom +moves in a $2_{1}$~orbit, just as the outer electron in the metastable +state of the helium atom. The reason why the binding of the +outer electron \Chg{can not}{cannot} proceed to an orbit characterized by a smaller +value for the total quantum number may also be considered as +analogous in the two cases. In fact, a transition by which the third +electron in the lithium atom was ultimately bound in a $1_{1}$~orbit +would lead to a state in the atom in which the electron would play +an equivalent part with the two electrons previously bound. Such +a process would be of a type entirely different from the transitions +between the stationary states connected with the emission of the +lithium spectrum, and would, contrary to these, not exhibit a +correspondence with a harmonic component in the motion of the +atom. +\PageSep{91} + +We obtain, therefore, a picture of the formation and structure of +the lithium atom which offers a natural explanation of the great +difference of the chemical properties of lithium from those of helium +and hydrogen. This difference is at once explained by the fact that +the firmness by which the last captured electron is bound in its +$2_{1}$~orbit in the lithium atom is only about a third of that with which +the electron in the hydrogen atom is held, and almost five times +smaller than the firmness of the binding of the electrons in the +helium atom. + +What has been said here applies not alone to the formation of +the lithium atom, but may also be assumed to apply to the binding +of the third electron in every atom, so that in contrast to the first +two electrons which move in $1_{1}$~orbits this may be assumed to move +in a $2_{1}$~orbit. As regards the \emph{binding of the fourth, fifth and sixth +electrons} in the atom, we do not possess a similar guide as no simple +series spectra are known of beryllium, boron and carbon. Although +conclusions of the same degree of certainty \Chg{can not}{cannot} be reached it +seems possible, however, to arrive at results consistent with general +physical and chemical evidence by proceeding by means of considerations +of the same kind as those applied to the binding of the +first three electrons. In fact, we shall assume that the fourth, fifth +and sixth electrons will be bound in $2_{1}$~orbits. The reason why the +binding of a first electron in an orbit of this type will not prevent the +capture of the others in two quanta orbits may be ascribed to the fact +that $2_{1}$~orbits are not circular but very \Chg{excentric}{eccentric}; For example, the +$3$rd~electron cannot keep the remaining electrons away from the inner +system in the same way in which the first two electrons bound in +the lithium atom prevent the third from being bound in a +$1$-quantum orbit. Thus we shall expect that the $4$th, $5$th and $6$th +electrons in a similar way to the $3$rd will at certain moments of +their revolution enter into the region where the first two +bound electrons move. We must not imagine, however, that these +visits into the inner system take place at the same time, but +that the four electrons visit the nucleus separately at equal +intervals of time. In earlier work on atomic structure it was supposed +that the electrons in the various groups in the atom moved +in separate regions within the atom and that at each moment the +electrons within each separate group were arranged in configurations +\PageSep{92} +possessing symmetry like that of a regular polygon or polyhedron. +Among other things this involved that the electrons in each group +were supposed to be at the point of the orbit nearest the nucleus +at the same time. A structure of this kind may be described as one +where the motions of the electrons within the groups are coupled +together in a manner which is largely independent of the interaction +between the various groups. On the contrary, the characteristic +feature of a structure like that I have suggested is the \emph{intimate +coupling between the motions of the electrons in the various groups} +characterized by different quantum numbers, as well as the \emph{greater +independence in the mode of binding within one and the same group +of electrons} the orbits of which are characterized by the same +quantum number. In emphasizing this last feature I have two +points in mind. Firstly the smaller effect of the presence of previously +bound electrons on the firmness of binding of succeeding +electrons in the same group. Secondly the way in which the motions +of the electrons within the group reflect the independence both of +the processes by which the group can be formed and by which it +can be reorganized by change of position of the different electrons +in the atom after a disturbance by external forces. The last point +will be considered more closely when we deal with the origin and +nature of the X-ray spectra; for the present we shall continue the +consideration of the structure of the atom to which we are led by +the investigation of the processes connected with the successive +capture of the electrons. + +The preceding considerations enable us to understand the fact +that the two elements beryllium and boron immediately succeeding +lithium can appear electropositively with $2$~and $3$~valencies respectively +in combination with other substances. For like the third +electron in the lithium atom, the last captured electrons in these +elements will be much more lightly bound than the first two +electrons. At the same time we understand why the electropositive +character of these elements is less marked than in the case of +lithium, since the electrons in the $2$-quanta orbits will be much +more firmly bound on account of the stronger field in which they +are moving. New conditions arise, however, in the case of the +next element, carbon, as this element in its typical chemical combinations +\Chg{can not}{cannot} be supposed to occur as an ion, but rather as a +\PageSep{93} +neutral atom. This must be assumed to be due not only to the great +firmness in the binding of the electrons but also to be an essential +consequence of the symmetrical configuration of the electrons. + +With the binding of the $4$th, $5$th and $6$th electrons in $2_{1}$~orbits, +the spatial symmetry of the regular configuration of the orbits +must be regarded as steadily increasing, until with the binding of +the $6$th electron the orbits of the four last bound electrons may be +expected to form an exceptionally symmetrical configuration in +which the normals to the planes of the orbits occupy positions +relative to one another nearly the same as the lines from the centre +to the vertices of a regular tetrahedron. Such a configuration +of groups of $2$-quanta orbits in the carbon atom seems capable +of furnishing a suitable foundation for explaining the structure of +organic compounds. I shall not discuss this question any further, +for it would require a thorough study of the interaction between +the motions of the electrons in the atoms forming the molecule. +I might mention, however, that the types of molecular models to +which we are led are very different from the molecular models +which were suggested in my first papers. In these the chemical +``valence bonds'' were represented by ``electron rings'' of the same +type as those which were assumed to compose the groups of +electrons within the individual atoms. It is nevertheless possible +to give a general explanation of the chemical properties of the +elements without touching on those matters at all. This is largely +due to the fact that the structures of combinations of atoms of the +same element and of many organic compounds do not have the +same significance for our purpose as those molecular structures in +which the individual atoms occur as electrically charged ions. The +latter kind of compounds, to which the greater number of simple +inorganic compounds belong, is frequently called ``heteropolar'' and +possesses a far more typical character than the first compounds +which are called ``homoeopolar,'' and whose properties to quite a +different degree exhibit the individual peculiarities of the elements. +My main purpose will therefore be to consider the fitness which +the configurations of the electrons in the various atoms offer for +the formation of ions. + +Before leaving the carbon atom I should mention, that a model +of this atom in which the orbits of the four most lightly bound +\PageSep{94} +electrons possess a pronounced tetrahedric symmetry had already +been suggested by Landé. In order to agree with the measurements +of the size of the atoms he also assumed that these electrons moved +in $2_{1}$~orbits. There is, however, this difference between Landé's +view and that given here, that while Landé deduced the characteristic +properties of the carbon atom solely from an investigation of +the simplest form of motion which four electrons can execute +employing spatial symmetry, our view originates from a consideration +of the stability of the whole atom. For our assumptions about +the orbits of the electrons are based directly on an investigation of +the interaction between these electrons and the first two bound +electrons. The result is that our model of the carbon atom has +dynamic properties which are essentially different from the properties +of Landé's model. + +In order to account for the properties of \emph{the elements in the second +half of the second period} it will first of all be necessary to show +why the configuration of ten electrons occurring in the neutral atom +of neon possesses such a remarkable degree of stability. Previously +it has been assumed that the properties of this configuration were +due to the interaction between eight electrons which moved in +equivalent orbits outside the nucleus and an inner group of two +electrons like that in the helium atom. It will be seen, however, +that the solution must be sought in an entirely different direction. +It \Chg{can not}{cannot} be expected that \emph{the $7$th electron} will be bound in a $2_{1}$~orbit +equivalent to the orbits of the four preceding electrons. The occurrence +of five such orbits would so definitely destroy the symmetry +in the interaction of these electrons that it is inconceivable that a +process resulting in the accession of a fifth electron to this group +would be in agreement with the correspondence principle. On the +contrary it will be necessary to assume that the four electrons in +their exceptionally symmetrical orbital configuration will keep out +later captured electrons with the result that these electrons will be +bound in orbits of other types. + +The orbits which come into consideration for the $7$th electron in +the nitrogen atom and the $7$th, $8$th, $9$th and $10$th electrons in the +atoms of the immediately following elements will be circular orbits +of the type~$2_{2}$. The diameters of these orbits are considerably larger +than those of the $l_{1}$~orbits of the first two electrons; on the other +\PageSep{95} +hand the outermost part of the \Chg{excentric}{eccentric} $2_{1}$~orbits will extend some +distance beyond these circular $2_{2}$~orbits. I shall not here discuss the +capture and binding of these electrons. This requires a further investigation +of the interaction between the motions of the electrons +in the two types of $2$-quanta orbits. I shall simply mention, that +in the atom of neon in which we will assume that there are four +electrons in $2_{2}$~orbits the planes of these orbits must be regarded not +only as occupying a position relative to one another characterized +by a high degree of spatial symmetry, but also as possessing a +configuration harmonizing with the four elliptical $2_{1}$~orbits. An +interaction of this kind in which the orbital planes do not +coincide can be attained only if the configurations in both subgroups +exhibit a systematic deviation from tetrahedral symmetry. +This will have the result that the electron groups with $2$-quanta +orbits in the neon atom will have only a single axis of symmetry +which must be supposed to coincide with the axis of symmetry of +the innermost group of two electrons. + +Before leaving the description of the elements within the second +period it may be pointed out that the above considerations offer a +basis for interpreting that tendency of the neutral atoms of oxygen +and fluorine for capturing further electrons which is responsible for +the marked electronegative character of these elements. In fact, +this tendency may be ascribed to the fact that the orbits of +the last captured electrons will find their place within the region, +in which the previously captured electrons move in $2_{1}$~orbits. This +suggests an explanation of the great difference between the properties +of the elements in the latter half of the second period of the +periodic system and those of the elements in the first half, in whose +atoms there is only a single type of $2$-quanta orbits. + +\Section{Third Period. Sodium---Argon.} We shall now consider the +structure of atoms of elements in the third period of the periodic +system. This brings us immediately 'to the question of \emph{the binding +of the $11$th electron} in the atom. Here we meet conditions which +in some respects are analogous to those connected with the binding +of the $7$th electron. The same type of argument that applied to +the carbon atom shows that the symmetry of the configuration in +the neon atom would be essentially, if not entirely, destroyed by +\PageSep{96} +the addition of another electron in an orbit of the same type as +that in which the last captured electrons were bound. Just as in +the case of the $3$rd~and $7$th electrons we may therefore expect to +meet a new type of orbit for the 11th electron in the atom, and the +orbits which present themselves this time are the $3_{1}$~orbits. An +electron in such an orbit will for the greater part of the time remain +outside the orbits of the first ten electrons. But at certain moments +during the revolution it will penetrate not only into the region of +the $2$-quanta orbits, but like the $2_{1}$~orbits it will penetrate to +distances from the nucleus which are smaller than the radii of +the $1$-quantum orbits of the two electrons first bound. This fact, +which has a most important bearing on the stability of the atom, +leads to a peculiar result as regards the binding of the $11$th electron. +In the sodium atom this electron will move in a field which so far +as the outer part of the orbit is concerned deviates only very little +from that surrounding the nucleus in the hydrogen atom, but the +dimensions of this part of the orbit will, nevertheless, be essentially +different from the dimensions of the corresponding part of a $3_{1}$~orbit +in the hydrogen atom. This arises from the fact, that even +though the electron only enters the inner configuration of the first +ten electrons for short intervals during its revolution, this part of +the orbit will nevertheless exert an essential influence upon the +determination of the principal quantum number. This is directly +related to the fact that the motion of the electron in the first part +of the orbit deviates only a little from the motion which each of +the previously bound electrons in $2_{1}$~orbits executes during a complete +revolution. The uncertainty which has prevailed in the +determination of the quantum numbers for the stationary states +corresponding to a spectrum like that of sodium is connected with +this. This question has been discussed by several physicists. From +a comparison of the spectral terms of the various alkali metals, +Roschdestwensky has drawn the conclusion that the normal state +does not, as we might be inclined to expect a~priori, correspond to +a $1_{1}$~orbit as shown in \Fig{2} on \PageRef{79}, but that this state corresponds +to a $2_{1}$~orbit. Schrödinger has arrived at a similar result +in an attempt to account for the great difference between the +$S$~terms and the terms in the $P$~and $D$ series of the alkali spectra. +He assumes that the ``outer'' electron in the states corresponding +\PageSep{97} +to the $S$~terms---in contrast to those corresponding to the $P$~and +$D$ terms---penetrates partly into the region of the orbits of the +inner electrons during the course of its revolution. These investigations +contain without doubt important hints, but in reality the +conditions must be very different for the different alkali spectra. +Instead of a $2_{1}$~orbit as in lithium we must thus assume for +the spectrum of sodium not only that the first spectral term in +the $S$~series corresponds to a $3_{1}$~orbit, but also, as a more detailed +consideration shows, that the first term in the $P$~series corresponds +not to a $2_{2}$~orbit as indicated in \Fig{2}, but to a $3_{2}$~orbit. If the +numbers in this figure were correct, it would require among other +things that the $P$~terms should be smaller than the hydrogen terms +\Figure{3}{97} +corresponding to the same principal quantum number. This would +mean that the average effect of the inner electrons could be described +as a repulsion greater than would occur if their total electrical charge +were united in the nucleus. This, however, \Chg{can not}{cannot} be expected from +our view of atomic structure. The fact that the last captured electron, +at any rate for low values of~$k$, revolves partly inside the orbits of the +previously bound electrons will on the contrary involve that the +presence of these electrons will give rise to a virtual repulsion +which is considerably smaller than that which would be due to +their combined charges. Instead of the curves drawn between +points in \Fig{2} which represent stationary states corresponding +to the same value of the principal quantum number running from +right to left, we obtain curves which run from left to right, as +is indicated in \Fig{3}. The stationary states are labelled with +\PageSep{98} +quantum numbers corresponding to the structure I have described. +According to the view underlying \Fig{2} the sodium spectrum +might be described simply as a distorted hydrogen spectrum, +whereas according to \Fig{3} there is not only distortion but also +complete disappearance of certain terms of low quantum numbers. +It may be stated, that this view not only appears to offer an explanation +of the magnitude of the terms, but that the complexity +of the terms in the $P$~and $D$ series finds a natural explanation in +the deviation of the configuration of the ten electrons first bound +from a purely central symmetry. This lack of symmetry has its +origin in the configuration of the two innermost electrons and +``transmits'' itself to the outer parts of the atomic structure, since +the $2_{1}$~orbits penetrate partly into the region of these electrons. + +This view of the sodium spectrum provides at the same time an +immediate explanation of the pronounced electropositive properties +of sodium, since the last bound electron in the sodium atom is still +more loosely bound than the last captured electron in the lithium +atom. In this connection it might be mentioned that the increase +in atomic volume with increasing atomic number in the family of +the alkali metals finds a simple explanation in the successively +looser binding of the valency electrons. In his work on the X-ray +spectra Sommerfeld at an earlier period regarded this increase in +the atomic volumes as supporting the assumption that the principal +quantum number of the orbit of the valency electrons increases by +unity as we pass from one metal to the next in the family. His +later investigations on the series spectra have led him, however, +definitely to abandon this assumption. At first sight it might also +appear to entail a far greater increase in the atomic volume than +that actually observed. A simple explanation of this fact is however +afforded by realizing that the orbit of the electron will run +partly inside the region of the inner orbit and that therefore the +``effective'' quantum number which corresponds to the outer almost +elliptical loop will be much smaller than the principal quantum +number, by which the whole central orbit is described. It may +be mentioned that Vegard in his investigations on the X-ray spectra +has also proposed the assumption of successively increasing quantum +numbers for the electronic orbits in the various groups of the atom, +reckoned from the nucleus outward. He has introduced assumptions +\PageSep{99} +about the relations between the numbers of electrons in the various +groups of the atom and the lengths of the periods in the periodic +system which exhibit certain formal similarities with the results +presented here. But Vegard's considerations do not offer points of +departure for a further consideration of the evolution and stability +of the groups, and consequently no basis for a detailed interpretation +of the properties of the elements. + +When we consider the elements following sodium in the third +period of the periodic system we meet in \emph{the binding of the $12$th, +$13$th and $14$th electrons} conditions which are analogous to those +we met in the binding of the $4$th, $5$th and $6$th electrons. In the +elements of the third periods, however, we possess a far more +detailed knowledge of the series spectra. Too little is known +about the beryllium spectrum to draw conclusions about the +binding of the fourth electron, but we may infer directly from the +well-known arc spectrum of magnesium that the $12$th electron +in the atom of this element is bound in a $3_{1}$~orbit. As regards +the binding of the $13$th electron we meet in aluminium an +absorption spectrum different in structure to that of the alkali +metals. In fact here not the lines of the principal series but the +lines of the sharp and diffuse series are absorption lines. Consequently +it is the first member of the $P$~terms and not of the $S$~terms +which corresponds to the normal state of the aluminium +atom, and we must assume that the $13$th electron is bound in +a $3_{2}$~orbit. This, however, would hardly seem to be a general +property of the binding of the $13$th electron in atoms, but rather +to arise from the special conditions for the binding of the last +electron in an atom, where already there are two other electrons +bound as loosely as the valency electron of aluminium. At the +present state of the theory it seems best to assume that in the +silicon atom the four last captured electrons will move in $3_{1}$~orbits +forming a configuration possessing symmetrical properties +similar to the outer configuration of the four electrons in $2_{1}$~orbits +in carbon. Like what we assumed for the latter configuration we +shall expect that the configuration of the $3_{1}$~orbits occurring for the +first time in silicon possesses such a completion, that the addition +of a further electron in a $3_{1}$~orbit to the atom of the following elements +is impossible, and that \emph{the $15$th electron} in the elements of +\PageSep{100} +higher atomic number will be bound in a new type of orbit. In this +case, however, the orbits with which we meet will not be circular, +as in the capture of the $7$th electron, but will be rotating \Chg{excentric}{eccentric} +orbits of the type~$3_{2}$. This is very closely related to the fact, mentioned +above, that the non-circular orbits will correspond to a +firmer binding than the circular orbits having the same value for +the principal quantum number, since the electrons will at certain +moments penetrate much farther into the interior of the atom. +Even though a $3_{2}$~orbit will not penetrate into the innermost configuration +of $1_{1}$~orbits, it will penetrate to distances from the nucleus +which are considerably less than the radii of the circular $2_{}2$~orbits. +In the case of the $16$th, $17$th and $18$th electrons the conditions are +similar to those for the $15$th. So for argon we may expect a configuration +in which the ten innermost electrons move in orbits of +the same type as in the neon atom while the last eight electrons will +form a configuration of four $3_{1}$~orbits and four $3_{2}$~orbits, whose +symmetrical properties must be regarded as closely corresponding +to the configuration of $2$-quanta orbits in the neon atom. At the +same time, as this picture suggests a qualitative explanation of the +similarity of the chemical properties of the elements in the latter +part of the second and third periods, it also opens up the possibility +of a natural explanation of the conspicuous difference from a +quantitative aspect. + +\Section{Fourth Period. Potassium---Krypton.} In the fourth period +we meet at first elements which resemble chemically those at the +beginning of the two previous periods. This is also what we should +expect. We must thus assume that \emph{the $19$th electron} is bound in +a new type of orbit, and a closer consideration shows that this will +be a $4_{1}$~orbit. The points which were emphasized in connection +with the binding of the last electron in the sodium atom will be +even more marked here on account of the larger quantum number +by which the orbits of the inner electrons are characterized. In +fact, in the potassium atom the $4_{1}$~orbit of the $19$th electron will, +as far as inner loops are concerned, coincide closely with the shape +of a $3_{1}$~orbit. On this account, therefore, the dimensions of the +outer part of the orbit will not only deviate greatly from the +dimensions of a $4_{1}$~orbit in the hydrogen atom, but will coincide +\PageSep{101} +closely with a hydrogen orbit of the type~$2_{1}$, the dimensions of +which are about four times smaller than the $4_{1}$~hydrogen orbit. +This result allows an immediate explanation of the main features of +the chemical properties and the spectrum of potassium. Corresponding +results apply to calcium, in the neutral atom of which +there will be two valency electrons in equivalent $4_{1}$~orbits. + +After calcium the properties of the elements in the fourth period +of the periodic system deviate, however, more and more from the +corresponding elements in the previous periods, until in the family +of the iron metals we meet elements whose properties are essentially +different. Proceeding to still higher atomic numbers we again +meet different conditions. Thus we find in the latter part of the +fourth period a series of elements whose chemical properties approach +more and more to the properties of the elements at the end +of the preceding periods, until finally with atomic number~$36$ we +again meet one of the inactive gases, namely krypton. This is +exactly what we should expect. The formation and stability of the +atoms of the elements in the first three periods require that each +of the first $18$ electrons in the atom shall be bound in each succeeding +element in an orbit of the same principal quantum number +as that possessed by the particular electron, when it first appeared. +It is readily seen that this is no longer the case for the $19$th +electron. With increasing nuclear charge and the consequent +decrease in the difference between the fields of force inside and +outside the region of the orbits of the first $18$ bound electrons, the +dimensions of those parts of a $4_{1}$~orbit which fall outside will +approach more and more to the dimensions of a $4$-quantum orbit +calculated on the assumption that the interaction between the +electrons in the atom may be neglected. \emph{With increasing atomic +number a point will therefore be reached where a $3_{3}$~orbit will correspond +to a firmer binding of the $19$th electron than a $4_{1}$~orbit}, and +this occurs as early as at the beginning of the fourth period. This +cannot only be anticipated from a simple calculation but is confirmed +in a striking way from an examination of the series spectra. While +the spectrum of potassium indicates that the $4_{1}$~orbit corresponds +to a binding which is more than twice as firm as in a $3_{3}$~orbit +corresponding to the first spectral term in the $D$~series, the conditions +are entirely different as soon as calcium is reached. We +\PageSep{102} +shall not consider the arc spectrum which is emitted during the +capture of the $20$th electron but the spark spectrum which corresponds +to the capture and binding of the $19$th electron. While the +spark spectrum of magnesium exhibits great similarity with the +sodium spectrum as regards the values of the spectral terms in the +various series---apart from the fact that the constant appearing in +formula~\Eq{(12)} is four times as large as the Rydberg constant---we +meet in the spark spectrum of calcium the remarkable condition +\Figure{4}{102} +that the first term of the $D$~series is larger than the first term of +the $P$~series and is only a little smaller than the first term of the +$S$~series, which may be regarded as corresponding to the binding +of the $19$th electron in the normal state of the calcium atom. +These facts are shown in \Fig[figure]{4} which gives a survey of the +stationary states corresponding to the arc spectra of sodium and +potassium. As in figures \FigNum{2} and~\FigNum{3} of the sodium spectrum, we +have disregarded the complexity of the spectral terms, and the +numbers characterizing the stationary states are simply the quantum +\PageSep{103} +numbers $n$~and~$k$. For the sake of comparison the scale in which the +energy of the different states is indicated is chosen four times as +small for the spark spectra as for the arc spectra. Consequently +the vertical lines indicated with various values of~$n$ correspond for +the arc spectra to the spectral terms of hydrogen, for the spark +spectra to the terms of the helium spectrum given by formula~\Eq{(7)}. +Comparing the change in the relative firmness in the binding of +the $19$th electron in a $4_{1}$~and $3_{3}$~orbit for potassium and calcium we +see that we must be prepared already for the next element, +scandium, to find that the $3_{3}$~orbit will correspond to a stronger +binding of this electron than a $4_{1}$~orbit. On the other hand it +follows from previous remarks that the binding will be much lighter +than for the first $18$ electrons which agrees that in chemical combinations +scandium appears electropositively with three valencies. + +If we proceed to the following elements, a still larger number of +$3_{3}$~orbits will occur in the normal state of these atoms, since the +number of such electron orbits will depend upon the firmness of +their binding compared to the firmness with which an electron is +bound in a $4_{1}$~orbit, in which type of orbit at least the last captured +electron in the atom may be assumed to move. We therefore meet +conditions which are essentially different from those which we have +considered in connection with the previous periods, so that here +we have to do with \emph{the successive development of one of the inner +groups of electrons in the atom}, in this case with groups of electrons +in $3$-quanta orbits. Only when the development of this group has +been completed may we expect to find once more a corresponding +change in the properties of the elements with increasing atomic +number such as we find in the preceding periods. The properties +of the elements in the latter part of the fourth period show +immediately that the group, when completed, will possess $18$~electrons. +Thus in krypton, for example, we may expect besides +the groups of $1$,~$2$ and $3$-quanta orbits a markedly symmetrical +configuration of $8$~electrons in $4$-quanta orbits consisting of four $4_{1}$~orbits +and four $4_{2}$~orbits. + +The question now arises: In which way will the gradual formation +of the group of electrons having $3$-quanta orbits take place? +From analogy with the constitution of the groups of electrons with +$2$-quanta orbits we might at first sight be inclined to suppose that +\PageSep{104} +the complete group of $3$-quanta orbits would consist of three subgroups +of four electrons each in orbits of the types $3_{1}$,~$3_{2}$ and~$3_{3}$ +respectively, so that the total number of electrons would be $12$ +instead of~$18$. Further consideration shows, however, that such an +expectation would not be justified. The stability of the configuration +of eight electrons with $2$-quanta orbits occurring in neon must +be ascribed not only to the symmetrical configuration of the electronic +orbits in the two subgroups of $2_{1}$~and $2_{2}$ orbits respectively, +but fully as much to the possibility of bringing the orbits inside these +subgroups into harmonic relation with one another. The situation +is different, however, for the groups of electrons with $3$-quanta +orbits. Three subgroups of four orbits each \Chg{can not}{cannot} in this case be +expected to come into interaction with one another in a correspondingly +simple manner. On the contrary we must assume that +the presence of electrons in $3_{3}$~orbits will diminish the harmony of +the orbits within the first two $3$-quanta subgroups, at any rate +when a point is reached where the $19$th electron is no longer, as +was the case with scandium, bound considerably more lightly than +the previously bound electrons in $3$-quanta orbits, but has been +drawn so far into the atom that it revolves within essentially +the same region of the atom where these electrons move. We +shall now assume that this decrease in the harmony will so to +say ``open'' the previously ``closed'' configuration of electrons +in orbits of these types. As regards the final result, the number~$18$ +indicates that after the group is finally formed there will +be three subgroups containing six electrons each. Even if it has +not at present been possible to follow in detail the various +steps in the formation of the group this result is nevertheless +confirmed in an interesting manner by the fact that it is possible +to arrange three configurations having six electrons each in a simple +manner relative to one another. The configuration of the subgroups +does not exhibit a tetrahedral symmetry like the groups of $2$-quanta +orbits in carbon, but a symmetry which, so far as the relative +orientation of the normals to the planes of the orbits is concerned, +may be described as trigonal. + +In spite of the great difference in the properties of the elements +of this period, compared with those of the preceding period, the +completion of the group of $18$~electrons in $3$-quanta orbits in the +\PageSep{105} +fourth period may to a certain extent be said to have the same +characteristic results as the completion of the group of $2$-quanta +orbits in the second period. As we have seen, this determined not +only the properties of neon as an inactive gas, but in addition the +electronegative properties of the preceding elements and the +electropositive properties of the elements which follow. The fact +that there is no inactive gas possessing an outer group of $18$~electrons +is very easily accounted for by the much larger dimensions +which a $3_{3}$~orbit has in comparison with a $2_{2}$~orbit revolving in the +same field of force. On this account a complete $3$-quanta group +\Chg{can not}{cannot} occur as the outermost group in a neutral atom, but only +in positively charged ions. The characteristic decrease in valency +which we meet in copper, shown by the appearance of the singly +charged cuprous ions, indicates the same tendency towards the +completion of a symmetrical configuration of electrons that we +found in the marked electronegative character of an element like +fluorine. Direct evidence that a complete group of $3$-quanta orbits +is present in the cuprous ion is given by the spectrum of copper +which, in contrast to the extremely complicated spectra of the +preceding elements resulting from the unsymmetrical character of +the inner system, possesses a simple structure very much like that +of the sodium spectrum. This may no doubt be ascribed to a +simple symmetrical structure present in the cuprous ion similar to +that in the sodium ion, although the great difference in the constitution +of the outer group of electrons in these ions is shown +both by the considerable difference in the values of the spectral +terms and in the separation of the doublets in the $P$~terms of the +two spectra. The occurrence of the cupric compounds shows, however, +that the firmness of binding in the group of $3$-quanta orbits +in the copper atom is not as great as the firmness with which the +electrons are bound in the group of $2$-quanta orbits in the sodium +atom. Zinc, which is always divalent, is the first element in which +the groups of the electrons are so firmly bound that they \Chg{can not}{cannot} +be removed by ordinary chemical processes. + +The picture I have given of the formation and structure of the +atoms of the elements in the fourth period gives an explanation of +the chemical and spectral properties. In addition it is supported +by evidence of a different nature to that which we have hitherto +\PageSep{106} +used. It is a familiar fact, that the elements in the fourth period +differ markedly from the elements in the preceding periods +partly in their \emph{magnetic properties} and partly in the \emph{characteristic +colours} of their compounds. Paramagnetism and colours do occur +in elements belonging to the foregoing periods, but not in simple +compounds where the atoms considered enter as ions. Many +elements of the fourth period, on the contrary, exhibit paramagnetic +properties and characteristic colours even in dissociated +aqueous solutions. The importance of this has been emphasized +by Ladenburg in his attempt to explain the properties of the +elements in the long periods of the periodic system (see \PageRef{73}). +Langmuir in order to account for the difference between the fourth +period and the preceding periods simply assumed that the atom, +in addition to the layers of cells containing $8$~electrons each, possesses +an outer layer of cells with room for $18$~electrons which is completely +filled for the first time in the case of krypton. Ladenburg, +on the other hand, assumes that for some reason or other an +intermediate layer is developed between the inner electronic +configuration in the atom appearing already in argon, and the +external group of valency electrons. This layer commences with +scandium and is completed exactly at the end of the family of iron +metals. In support of this assumption Ladenburg not only mentions +the chemical properties of the elements in the fourth period, but +also refers to the paramagnetism and colours which occur exactly +in the elements, where this intermediate layer should be in +development. It is seen that Ladenburg's ideas exhibit certain +formal similarities with the interpretation I have given above of +the appearance of the fourth period, and it is interesting to note that +our view, based on a direct investigation of the conditions for the +formation of the atoms, enables us to understand the relation +emphasized by Ladenburg. + +Our ordinary electrodynamic conceptions are probably insufficient +to form a basis for an explanation of atomic magnetism. This is +hardly to be wondered at when we remember that they have not +proved adequate to account for the phenomena of radiation which +are connected with the intimate interaction between the electric +and magnetic forces arising from the motion of the electrons. In +whatever way these difficulties may be solved it seems simplest to +\PageSep{107} +assume that the occurrence of magnetism, such as we meet in the +elements of the fourth period, results from a lack of symmetry in +the internal structure of the atom, thus preventing the magnetic +forces arising from the motion of the electrons from forming a +system of closed lines of force running wholly within the atom. +While it has been assumed that the ions of the elements in the +previous periods, whether positively or negatively charged, contain +configurations of marked symmetrical character, we must, however, +be prepared to encounter a definite lack of symmetry in the +electronic configurations in ions of those elements within the fourth +period which contain a group of electrons in $3$-quanta orbits in the +transition stage between symmetrical configurations of $8$~and $18$ +electrons respectively. As pointed out by Kossel, the experimental +results exhibit an extreme simplicity, the magnetic moment of the +ions depending only on the number of electrons in the ion. Ferric +ions, for example, exhibit the same atomic magnetism as manganous +ions, while manganic ions exhibit the same atomic magnetism as +chromous ions. It is in beautiful agreement with what we have +assumed about the structure of the atoms of copper and zinc, that +the magnetism disappears with those ions containing $28$~electrons +which, as I stated, must be assumed to contain a complete group +of $3$-quanta orbits. On the whole a consideration of the magnetic +properties of the elements within the fourth period gives us a vivid +impression of how a wound in the otherwise symmetrical inner +structure is first developed and then healed as we pass from element +to element. It is to be hoped that a further investigation of the +magnetic properties will give us a clue to the way in which the +group of electrons in $3$-quanta orbits is developed step by step. + +Also the colours of the ions directly support our view of atomic +structure. According to the postulates of the quantum theory +absorption as well as emission of radiation is regarded as taking +place during transitions between stationary states. The occurrence +of colours, that is to say the absorption of light in the visible region +of the spectrum, is evidence of transitions involving energy changes +of the same order of magnitude as those giving the usual optical +spectra of the elements. In contrast to the ions of the elements of +the preceding periods where all the electrons are assumed to be very +firmly bound, the occurrence of such processes in the fourth period +\PageSep{108} +is exactly what we should expect. For the development and completion +of the electronic groups with $3$-quanta orbits will proceed, +so to say, in competition with the binding of electrons in orbits of +higher quanta, since the binding of electrons in $3$-quanta orbits +occurs when the electrons in these orbits are bound more firmly +than electrons in $4_{1}$~orbits. The development of the group will +therefore proceed to the point where we may say there is equilibrium +between the two kinds of orbits. This condition may be +assumed to be intimately connected not only with the colour of the +ions, but also with the tendency of the elements to form ions with +different valencies. This is in contrast to the elements of the first +periods where the charge of the ions in aqueous solutions is always +the same for one and the same element. + +\Section{Fifth Period. Rubidium---Xenon.} The structure of the atoms +in the remaining periods may be followed up in complete analogy +with what has already been said. Thus we shall assume that the +$37$th and $38$th electrons in the elements of the fifth period are +bound in $5_{1}$~orbits. This is supported by the measurements of the +arc spectrum of rubidium and the spark spectrum of strontium. +The latter spectrum indicates at the same time that $4_{3}$~orbits will +soon appear, and therefore in this period, which like the $4$th +contains $18$~elements, we must assume that we are witnessing a +\emph{further stage in the development of the electronic group of $4$-quanta +orbits}. The first stage in the formation of this group may be said +to have been attained in krypton with the appearance of a symmetrical +configuration of eight electrons consisting of two subgroups +each of four electrons in $4_{1}$~and $4_{2}$~orbits. A second preliminary +completion must be regarded as having been reached with the +appearance of a symmetrical configuration of $18$~electrons in the +case of silver, consisting of three subgroups with six electrons each +in orbits of the types $4_{1}$,~$4_{2}$ and~$4_{3}$. Everything that has been said +about the successive formation of the group of electrons with $3$-quanta +orbits applies unchanged to this stage in the transformation +of the group with $4$-quanta orbits. For in no case have we made +use of the absolute values of the quantum numbers nor of assumptions +concerning the form of the orbits but only of the number of +possible types of orbits which might come into consideration. At +\PageSep{109} +the same time it may be of interest to mention that the properties +of these elements compared with those of the foregoing period +nevertheless show a difference corresponding exactly to what would +be expected from the difference in the types of orbits. For instance, +the divergencies from the characteristic valency conditions of the +elements in the second and third periods appear later in the fifth +period than for elements in the fourth period. While an element +like titanium in the fourth period already shows a marked tendency +to occur with various valencies, on the other hand an element like +zirconium is still quadri-valent like carbon in the second period +and silicon in the third. A simple investigation of the kinematic +properties of the orbits of the electrons shows in fact that an +electron in an \Chg{excentric}{eccentric} $4_{3}$~orbit of an element in the fifth +period will be considerably more loosely bound than an electron in +a circular $3_{3}$~orbit of the corresponding element in the fourth +period, while electrons which are bound in \Chg{excentric}{eccentric} orbits of the +types $5_{1}$~and $4_{1}$ respectively will correspond to a binding of about +the same firmness. + +At the end of the fifth period we may assume that xenon, the +atomic number of which is~$54$, has a structure which in addition to +the two $1$-quantum, eight $2$-quanta, eighteen $3$-quanta and eighteen +$4$-quanta orbits already mentioned contains a symmetrical +configuration of eight electrons in $5$-quanta orbits consisting of two +subgroups with four electrons each in $5_{1}$~and $5_{2}$ orbits respectively. + +\Section{Sixth Period. Caesium---Niton.} If we now consider the atoms +of elements of still higher atomic number, we must first of all +assume that the $55$th and $56$th electrons in the atoms of caesium +and barium are bound in $6_{1}$~orbits. This is confirmed by the spectra +of these elements. It is clear, however, that we must be prepared +shortly to meet entirely new conditions. With increasing nuclear +charge we shall have to expect not only that an electron in a $5_{3}$~orbit +will be bound more firmly than in a $6_{1}$~orbit, but we must also +expect that a moment will arrive when during the formation of the +atom a $4_{4}$~orbit will represent a firmer binding of the electron than +an orbit of $5$~or $6$-quanta, in much the same way as in the elements +of the fourth period a new stage in the development of the $3$-quanta +group was started when a point was reached where for the first +\PageSep{110} +time the $19$th electron was bound in a $3_{3}$~orbit instead of in a $4_{1}$~orbit. +We shall thus expect in the sixth period to meet with a new +stage in the development of the group with $4$-quanta orbits. Once +this point has been reached we must be prepared to find with increasing +atomic number a number of elements following one another, +which as in the family of the iron metals have very nearly the same +properties. The similarity will, however, be still more pronounced, +since in this case we are concerned with the successive transformation +of a configuration of electrons which lies deeper in the interior +of the atom. You will have already guessed that what I have in view +is a simple explanation of the occurrence of the \emph{family of rare earths} +at the beginning of the sixth period. As in the case of the transformation +and completion of the group of $3$-quanta orbits in the fourth +period and the partial completion of groups of $4$-quanta orbits in +the fifth period, we may immediately deduce from the length of the +sixth period the number of electrons, namely~$32$, which are finally +contained in the $4$-quanta group of orbits. Analogous to what +applied to the group of $3$-quanta orbits it is probable that, when +the group is completed, it will contain eight electrons in each of the +four subgroups. Even though it has not yet been possible to follow +the development of the group step by step, we can even here give +some theoretical evidence in favour of the occurrence of a symmetrical +configuration of exactly this number of electrons. I shall +simply mention that it is not possible without coincidence of the +planes of the orbits to arrive at an interaction between four subgroups +of six electrons each in a configuration of simple trigonal +symmetry, which is equally simple as that shown by three subgroups. +The difficulties which we meet make it probable that a harmonic +interaction can be attained precisely by four groups each containing +eight electrons the orbital configurations of which exhibit axial +symmetry. + +Just as in the case of the family of the iron metals in the fourth +period, the proposed explanation of the occurrence of the family of +rare earths in the sixth period is supported in an interesting +manner by an investigation of the magnetic properties of these +elements. In spite of the great chemical similarity the members +of this family exhibit very different magnetic properties, so that +while some of them exhibit but very little magnetism others exhibit +\PageSep{111} +a greater magnetic moment per atom than any other element which +has been investigated. It is also possible to give a simple interpretation +of the peculiar colours exhibited by the compounds of these +elements in much the same way as in the case of the family of iron +metals in the fourth period. The idea that the appearance of the +group of the rare earths is connected with the development of inner +groups in the atom is not in itself new and has for instance been +considered by Vegard in connection with his work on X-ray spectra. +The new feature of the present considerations lies, however, in the +emphasis laid on the peculiar way in which the relative strength of +the binding for two orbits of the same principal quantum number +but of different shapes varies with the nuclear charge and with the +number of electrons previously bound. Due to this fact the presence +of a group like that of the rare earths in the sixth period may be +considered as a direct consequence of the theory and might actually +have been predicted on a quantum theory, adapted to the explanation +of the properties of the elements within the preceding periods +in the way I have shown. + +Besides \emph{the final development of the group of $4$-quanta orbits} we +observe in the sixth period in the family of the platinum metals \emph{the +second stage in the development of the group of $5$-quanta orbits}. +Also in the radioactive, chemically inactive gas niton, which completes +this period, we observe the first preliminary step in the +development of a group of electrons with $6$-quanta orbits. In the +atom of this element, in addition to the groups of electrons of two +$1$-quantum, eight $2$-quanta, eighteen $3$-quanta, thirty-two $4$-quanta +and eighteen $5$-quanta orbits respectively, there is also an outer +symmetrical configuration of eight electrons in $6$-quanta orbits, +which we shall assume to consist of two subgroups with four electrons +each in $6_{1}$~and $6_{2}$ orbits respectively. + +\Section{Seventh Period.} In the seventh and last period of the periodic +system we may expect the appearance of $7$-quanta orbits in the +normal state of the atom. Thus in the neutral atom of radium in +addition to the electronic structure of niton there will be two +electrons in $7_{1}$~orbits which will penetrate during their revolution +not only into the region of the orbits of electrons possessing lower +values for the principal quantum number, but even to distances +\PageSep{112} +from the nucleus which are less than the radii of the orbits of the +innermost $1$-quantum orbits. The properties of the elements in the +seventh period are very similar to the properties of the elements in the +fifth period. Thus, in contrast to the conditions in the sixth period, +there are no elements whose properties resemble one another like +those of the rare earths. In exact analogy with what has already +been said about the relations between the properties of the elements +in the fourth and fifth periods this may be very simply explained by +the fact that an \Chg{excentric}{eccentric} $5_{4}$~orbit will correspond to a considerably +looser binding of an electron in the atom of an element of the +seventh period than the binding of an electron in a circular $4_{4}$~orbit +in the corresponding element of the sixth period, while there will be +a much smaller difference in the firmness of the binding of these +electrons in orbits of the types $7_{1}$~and $6_{1}$ respectively. + +It is well known that the seventh period is not complete, for no atom +has been found having an atomic number greater than~$92$. This is +probably connected with the fact that the last elements in the +system are radioactive and that nuclei of atoms with a total charge +greater than~$92$ will not be sufficiently stable to exist under conditions +where the elements can be observed. It is tempting to +sketch a picture of the atoms formed by the capture and binding +of electrons around nuclei having higher charges, and thus to +obtain some idea of the properties which the corresponding hypothetical +elements might be expected to exhibit. I shall not develop +this matter further, however, since the general results we should +get will be evident to you from the views I have developed to +explain the properties of the elements actually observed. A survey +of these results is given in the following table, which gives a symbolical +representation of the atomic structure of the inactive gases +which complete the first six periods in the periodic system. In +order to emphasize the progressive change the table includes the +probable arrangement of electrons in the next atom which would +possess properties like the inactive gases. + +The view of atomic constitution underlying this table, which +involves configurations of electrons moving with large velocities +between each other, so that the electrons in the ``outer'' groups +penetrate into the region of the orbits of the electrons of the ``inner'' +groups, is of course completely different from such statical models +\PageSep{113} +of the atom as are proposed by Langmuir. But quite apart from this +it will be seen that the arrangement of the electronic groups in +the atom, to which we have been lead by tracing the way in which +each single electron has been bound, is essentially different from +the arrangement of the groups in Langmuir's theory. In order to +explain the properties of the elements of the sixth period Langmuir +assumes for instance that, in addition to the inner layers of cells +containing $2$,~$8$, $8$, $18$ and $18$ electrons respectively, which are +employed to account for the properties of the elements in the +earlier periods, the atom also possesses a layer of cells with room +for $32$~electrons which is just completed in the case of niton. + +\Figure{}{113} + +In this connection it may be of interest to mention a recent +paper by Bury, to which my attention was first drawn after the +deliverance of this address, and which contains an interesting +survey of the chemical properties of the elements based on similar +conceptions of atomic structure as those applied by Lewis and +Langmuir. From purely chemical considerations Bury arrives at +conclusions which as regards the arrangement and completion of +the groups in the main coincide with those of the present theory, +the outlines of which were given in my letters to Nature mentioned +in the introduction. + +\Section{Survey of the periodic table.} The results given in this address +are also illustrated by means of the representation of the periodic +system given in \Fig{1}. In this figure the frames are meant to +indicate such elements in which one of the ``inner'' groups is +in a stage of development. Thus there will be found in the +\PageSep{114} +fourth and fifth periods a single frame indicating the final completion +of the electronic group with $3$-quanta orbits, and the +last stage but one in the development of the group with $4$-quanta +orbits respectively. In the sixth period it has been necessary to +introduce two frames, of which the inner one indicates the last +stage of the evolution of the group with $4$-quanta orbits, giving rise +to the rare earths. This occurs at a place in the periodic system +where the third stage in the development of an electronic group +with $5$-quanta orbits, indicated by the outer frame, has already +begun. In this connection it will be seen that the inner frame +encloses a smaller number of elements than is usually attributed +to the family of the rare earths. At the end of this group an +uncertainty exists, due to the fact that no element of atomic +number~$72$ is known with certainty. However, as indicated in +\Fig{1}, we must conclude from the theory that the group with +$4$-quanta orbits is finally completed in lutetium~($71$). This element +therefore ought to be the last in the sequence of consecutive +elements with similar properties in the first half of the sixth period, +and at the place~$72$ an element must be expected which in its +chemical and physical properties is homologous with zirconium and +thorium. This, which is already indited on Julius Thomsen's old +table, has also been pointed out by Bury. [Quite recently Dauvillier +has in an investigation of the X-ray spectrum excited in preparations +containing rare earths, observed certain faint lines which he ascribes +to an element of atomic number~$72$. This element is identified by +him as the element celtium, belonging to the family of rare earths, +the existence of which had previously been suspected by Urbain. +Quite apart from the difficulties which this result, if correct, might +entail for atomic theories, it would, since the rare earths according +to chemical view possess three valencies, imply a rise in positive +valency of two units when passing from the element~$72$ to the +next element~$73$, tantalum. This would mean an exception from +the otherwise general rule, that the valency never increases by +more than one unit when passing from one element to the next in +the periodic table\Add{.}] In the case of the incomplete seventh period +the full drawn frame indicates the third stage in the development +of the electronic group with $6$-quanta orbits, which must begin in +actinium. The dotted frame indicates the last stage but one in +\PageSep{115} +the development of the group with $5$-quanta orbits, which hitherto +has not been observed, but which ought to begin shortly after +uranium, if it has not already begun in this element. + +With reference to the homology of the elements the exceptional +position of the elements enclosed by frames in \Fig{1} is further +emphasized by taking care that, in spite of the large similarity +many elements exhibit, no connecting lines are drawn between +two elements which occupy different positions in the system with +respect to framing. In fact, the large chemical similarity between, +for instance, aluminium and scandium, both of which are trivalent +and pronounced electropositive elements, is directly or indirectly +emphasized in the current representations of the periodic table. +While this procedure is justified by the analogous structure of the +trivalent ions of these elements, our more detailed ideas of atomic +structure suggest, however, marked differences in the physical +properties of aluminium and scandium, originating in the essentially +different character of the way in which the last three electrons +in the neutral atom are bound. This fact gives probably a direct +explanation of the marked difference existing between the spectra +of aluminium and scandium. Even if the spectrum of scandium is +not yet sufficiently cleared up, this difference seems to be of a much +more fundamental character than for instance the difference between +the arc spectra of sodium and copper, which apart from the large +difference in the absolute values of the spectral terms possess a +completely analogous structure, as previously mentioned in this +essay. On the whole we must expect that the spectra of elements +in the later periods lying inside a frame will show new features +compared with the spectra of the elements in the first three periods. +This expectation seems supported by recent work on the spectrum +of manganese by Catalan, which appeared just before the printing +of this essay. + +Before I leave the interpretation of the chemical properties by +means of this atomic model I should like to remind you once again +of the fundamental principles which we have used. The whole +theory has evolved from an investigation of the way in which +electrons can be captured by an atom. The formation of an atom +was held to consist in the successive binding of electrons, this +binding resulting in radiation according to the quantum theory. +\PageSep{116} +According to the fundamental postulates of the theory this binding +takes place in stages by transitions between stationary states +accompanied by emission of radiation. For the problem of the +stability of the atom the essential problem is at what stage such a +process comes to an end. As regards this point the postulates give +no direct information, but here the correspondence principle is +brought in. Even though it has been possible to penetrate considerably +further at many points than the time has permitted me +to indicate to you, still it has not yet been possible to follow in +detail all stages in the formation of the atoms. We cannot say, for +instance, that the above table of the atomic constitution of the +inert gases may in every detail be considered as the unambiguous +result of applying the correspondence principle. On the other hand +it appears that our considerations already place the empirical data +in a light which scarcely permits of an essentially different interpretation +of the properties of the elements based upon the postulates of +the quantum theory. This applies not only to the series spectra +and the close relationship of these to the chemical properties of the +elements, but also to the X-ray spectra, the consideration of which +leads us into an investigation of interatomic processes of an entirely +different character. As we have already mentioned, it is necessary +to assume that the emission of the latter spectra is connected with +processes which may be described as a reorganization of the completely +formed atom after a disturbance produced in the interior +of the atom by the action of external forces. + + +\Chapter{IV.}{Reorganization of Atoms and X-Ray Spectra} + +As in the case of the series spectra it has also been possible to represent +the frequency of each line in the X-ray spectrum of an element +as the difference of two of a set of spectral terms. We shall therefore +assume that each X-ray line is due to a transition between +two stationary states of the atom. The values of the atomic energy +corresponding to these states are frequently referred to as the +``energy levels'' of the X-ray spectra. The great difference between +the origin of the X-ray and the series spectra is clearly seen, however, +in the difference of the laws applying to the absorption of +radiation in the X-ray and the optical regions of the spectra. The +absorption by non-excited atoms in the latter case is connected +\PageSep{117} +with those lines in the series spectrum which correspond to combinations +of the various spectral terms with the largest of these +terms. As has been shown, especially by the investigations of +Wagner and de~Broglie, the absorption in the X-ray region, on +the other hand, is connected not with the X-ray lines but with +certain spectral regions commencing at the so-called ``absorption +edges.'' The frequencies of these edges agree very closely with the +spectral terms used to account for the X-ray lines. We shall now +see how the conception of atomic structure developed in the preceding +pages offers a simple interpretation of these facts. Let us +consider the following question: What changes in the state of the +atom can be produced by the absorption of radiation, and which +processes of emission can be initiated by such changes? + +\Section{Absorption and emission of X-rays and correspondence +principle.} The possibility of producing a change at all in the +motion of an electron in the interior of an atom by means of radiation +must in the first place be regarded as intimately connected +with the character of the interaction between the electrons within +the separate groups. In contrast to the forms of motion where at +every moment the position of the electrons exhibits polygonal or +polyhedral symmetry, the conception of this interaction evolved from +a consideration of the possible formation of atoms by successive +binding of electrons has such a character that the harmonic components +in the motion of an electron are in general represented in +the resulting electric moment of the atom. As a result of this it +will be possible to release a single electron from the interaction +with the other electrons in the same group by a process which +possesses the necessary analogy with an absorption process on +the ordinary electrodynamic view claimed by the correspondence +principle. The points of view on which we based the interpretation +of the development and completion of the groups during the +formation of an atom imply, on the other hand, that just as no +additional electron can be taken up into a previously completed +group in the atom by a change involving emission of radiation, +similarly it will not be possible for a new electron to be added +to such a group, when the state of the atom is changed by +absorption of radiation. This means that an electron which belongs +\PageSep{118} +to one of the inner groups of the atom, as a consequence of an +absorption process---besides the case where it leaves the atom +completely---can only go over either to an incompleted group, or +to an orbit where the electron during the greater part of its revolution +moves at a distance from the nucleus large compared to the +distance of the other electrons. On account of the peculiar conditions +of stability which control the occurrence of incomplete groups in +the interior of the atom, the energy which is necessary to bring +about a transition to such a group will in general differ very little +from that required to remove the particular electron completely +from the atom. We must therefore assume that the energy levels +corresponding to the absorption edges indicate to a first approximation +the amount of work that is required to remove an electron +in one of the inner groups completely from the atom. The +correspondence principle also provides a basis for understanding +the experimental evidence about the appearance of the emission +lines of the X-ray spectra due to transitions between the stationary +states corresponding to these energy levels. Thus the nature of the +interaction between the electrons in the groups of the atom implies +that each electron in the atom is so to say prepared, independently +of the other electrons in the same group, to seize any opportunity +which is offered to become more firmly bound by being taken up +into a group of electrons with orbits corresponding to smaller values +of the principal quantum number. It is evident, however, that on +the basis of our views of atomic structure, such an opportunity is +always at hand as soon as an electron has been removed from one +of these groups. + +At the same time that our view of the atom leads to a natural +conception of the phenomena of emission and absorption of X-rays, +agreeing closely with that by which Kossel has attempted to give +a formal explanation of the experimental observations, it also suggests +a simple explanation of those quantitative relations holding for the +frequencies of the lines which have been discovered by Moseley and +Sommerfeld. These researches brought to light a remarkable and +far-reaching similarity between the Röntgen spectrum of a given +element and the spectrum which would be expected to appear upon +the binding of a single electron by the nucleus. This similarity we +immediately understand if we recall that in the normal state of the +\PageSep{119} +atom there are electrons moving in orbits which, with certain +limitations, correspond to all stages of such a binding process and +that, when an electron is removed from its original place in the +atom, processes may be started within the atom which will correspond +to all transitions between these stages permitted by the +correspondence principle. This brings us at once out of those +difficulties which apparently arise, when one attempts to account +for the origin of the X-ray spectra by means of an atomic structure, +suited to explain the periodic system. This difficulty has been felt +to such an extent that it has led Sommerfeld for example in his +recent work to assume that the configurations of the electrons in +the various atoms of one and the same element may be different +even under usual conditions. Since, in contrast to our ideas, he +supposed all electrons in the principal groups of the atom to move +in equivalent orbits, he is compelled to assume that these groups +are different in the different atoms, corresponding to different +possible types of orbital shapes. Such an assumption, however, seems +inconsistent with an interpretation of the definite character of the +physical and chemical properties of the elements, and stands in marked +contradiction with the points of view about the stability of the atoms +which form the basis of the view of atomic structure here proposed. + +\Section{X-ray spectra and atomic structure.} In this connection it is +of interest to emphasize that the group distribution of the electrons +in the atom, on which we have based both the explanation of the +periodic system and the classification of the lines in the X-ray +spectra, shows itself in an entirely different manner in these two +phenomena. While the characteristic change of the chemical +properties with atomic number is due to the gradual development +and completion of the groups of the loosest bound electrons, the +characteristic absence of almost every trace of a periodic change in +the X-ray spectra is due to two causes. Firstly the electronic +configuration of the completed groups is repeated unchanged for +increasing atomic number, and secondly the gradual way in which +the incompleted groups are developed implies that a type of orbit, +from the moment when it for the first time appears in the normal +state of the neutral atom, always will occur in this state and will +correspond to a steadily increasing firmness of binding. The development +\PageSep{120} +of the groups in the atom with increasing atomic number, +which governs the chemical properties of the elements shows itself +in the X-ray spectra mainly in the appearance of new lines. Swinne +has already referred to a connection of this kind between the periodic +system and the X-ray spectra in connection with Kossel's theory. +We can only expect a closer connection between the X-ray phenomena +and the chemical properties of the elements, when the conditions +on the surface of the atom are concerned. In agreement +with what has been brought to light by investigations on absorption +of X-rays in elements of lower atomic number, such as have +been performed in recent years in the physical laboratory at Lund, +we understand immediately that the position and eventual structure +of the absorption edges will to a certain degree depend upon +the physical and chemical conditions under which the element +investigated exists, while such a dependence does not appear in +the characteristic emission lines. + +If we attempt to obtain a more detailed explanation of the +experimental observations, we meet the question of the influence +of the presence of the other electrons in the atom upon the firmness +of the binding of an electron in a given type of orbit. This influence +will, as we at once see, be least for the inner parts of the atom, +where for each electron the attraction of the nucleus is large in +proportion to the repulsion of the other electrons. It should also +be recalled, that while the relative influence of the presence of the +other electrons upon the firmness of the binding will decrease with +increasing charge of the nucleus, the effect of the variation in the +mass of the electron with the velocity upon the firmness of the +binding will increase strongly. This may be seen from Sommerfeld's +formula~\Eq{(11)}. While we obtain a fairly good agreement for the +levels corresponding to the removal of one of the innermost electrons +in the atom by using the simple formula~\Eq{(11)}, it is, however, already +necessary to take the influence of the other electrons into consideration +in making an approximate calculation of the levels corresponding +to a removal of an electron from one of the outer groups in the +atom. Just this circumstance offers us, however, a possibility of +obtaining information about the configurations of the electrons in +the interior of the atoms from the X-ray spectra. Numerous +investigations have been directed at this question both by +\PageSep{121} +Sommerfeld and his pupils and by Debye, Vegard and others. It +may also be remarked that de~Broglie and Dauvillier in a recent +paper have thought it possible to find support in the experimental +material for certain assumptions about the numbers of electrons in +the groups of the atom to which Dauvillier had been led by considerations +about the periodic system similar to those proposed by +Langmuir and Ladenburg. In calculations made in connection with +these investigations it is assumed that the electrons in the various +groups move in separate concentric regions of the atom, so that +the effect of the presence of the electrons in inner groups upon the +motion of the electrons in outer groups as a first approximation +may be expected to consist in a simple screening of the nucleus. +On our view, however, the conditions are essentially different, since +for the calculation of the firmness of the binding of the electrons +it is necessary to take into consideration that the electrons in the +more lightly bound groups in general during a certain fraction of +their revolution will penetrate into the region of the orbits of +electrons in the more firmly bound groups. On account of this +fact, many examples of which we saw in the series spectra, we \Chg{can not}{cannot} +expect to give an account of the firmness of the binding of the +separate electrons, simply by means of a ``screening correction'' +consisting in the subtraction of a constant quantity from the value +for~$N$ in such formulae as \Eq{(5)} and~\Eq{(11)}. Furthermore in the calculation +of the work corresponding to the energy levels we must take +account not only of the interaction between, the electrons in the +normal state of the atom, but also of the changes in the configuration +and interaction of the remaining electrons, which establish +themselves automatically without emission of radiation during the +removal of the electron. Even though such calculations have not +yet been made very accurately, a preliminary investigation has +already shown that it is possible approximately to account for the +experimental results. + +\Section{Classification of X-ray spectra.} Independently of a definite +view of atomic structure it has been possible by means of a formal +application of Kossel's and Sommerfeld's theories to disentangle +the large amount of experimental material on X-ray spectra. This +material is drawn mainly from the accurate measurements of +\PageSep{122} +Siegbahn and his collaborators. From this disentanglement of the +experimental observations, in which besides Sommerfeld and his +students especially Smekal and Coster have taken part, we have +obtained a nearly complete classification of the energy levels corresponding +to the X-ray spectra. These levels are formally referred +to types of orbits characterized by two quantum numbers $n$ and~$k$, +and certain definite rules for the possibilities of combination +between the various levels have also been found. In this way a +number of results of great interest for the further elucidation of +the origin of the X-ray spectra have been attained. First it has +not only been possible to find levels, which within certain limits +correspond to all possible pairs of numbers for $n$ and~$k$, but it has +been found that in general to each such pair more than one level +must be assigned. This result, which at first may appear very +surprising, upon further consideration can be given a simple +interpretation. We must remember that the levels depend not +only upon the constitution of the atom in the normal state, but +also upon the configurations which appear after the removal +of one of the inner electrons and which in contrast to the normal +state do not possess a uniquely completed character. If we thus +consider a process in which one of the electrons in a group +(subgroup) is removed we must be prepared to find that after the +process the orbits of the remaining electrons in this group may be +orientated in more than one way in relation to one another, and +still fulfil the conditions required of the stationary states by the +quantum theory. Such a view of the ``complexity'' of the levels, as +further consideration shows, just accounts for the manner in which +the energy difference of the two levels varies with the atomic +number. Without attempting to develop a more detailed picture +of atomic structure, Smekal has already discussed the possibility +of accounting for the multiplicity of levels. Besides referring to +the possibility that the separate electrons in the principal groups +do not move in equivalent orbits, Smekal suggests the introduction +of three quantum numbers for the description of the various groups, +but does not further indicate to what extent these quantum +numbers shall be regarded as characterizing a complexity in the +structure of the groups in the normal state itself or on the +contrary characterizing the incompleted groups which appear +when an electron is removed. +\PageSep{123} + +It will be seen that the complexity of the X-ray levels exhibits a +close analogy with the explanation of the complexity of the terms +of the series spectra. There exists, however, this difference between +the complex structure of the X-ray spectra and the complex +structure of the lines in the series spectra, that in the X-ray +spectra there occur not only combinations between spectral terms, +for which $k$~varies by unity, but also between terms corresponding +to the same value of~$k$. This may be assumed to be +due to the fact, that in the X-ray spectra in contrast to the series +\Figure{5}{123} +spectra we have to do with transitions between stationary states +where, both before and after the transition, the electron concerned +takes part in an intimate interaction with other electrons in orbits +with the same principal quantum number. Even though this +interaction may be assumed to be of such a nature that the +harmonic components which would appear in the motion of an +electron in the absence of the others will in general also appear +in the resulting moment of the atom, we must expect that the +interaction between the electrons will give rise to the appearance +in this moment of new types of harmonic components. +\PageSep{124} + +It may be of interest to insert here a few words about a new +paper of Coster which appeared after this address was given, +and in which he has succeeded in obtaining an extended and +detailed connection between the X-ray spectra and the ideas +of atomic structure given in this essay. The classification mentioned +above was based on measurements of the spectra of the +heaviest elements, and the results in their complete form, which +were principally due to independent work of Coster and Wentzel, +may be represented by the diagram in \Fig{5}, which refers to +elements in the neighbourhood of niton. The vertical arrows +\Figure{6}{124} +represent the observed lines arising from combinations between +the different energy levels which are represented by horizontal lines. +In each group the levels are arranged in the same succession as +their energy values, but their distances do not give a quantitative +picture of the actual energy-differences, since this would require a +much larger figure. The numbers~$n_{k}$ attached to the different levels +indicate the type of the corresponding orbit. The letters $a$ and~$b$ +refer to the rules of combination which I mentioned. According +to these rules the possibility of combination is limited (1)~by the +exclusion of combinations, for which $k$~changes by more than one +unit, (2)~by the condition that only combinations between an $a$- +and a $b$-level can take place. The latter rule was given in this +\PageSep{125} +form by Coster; Wentzel formulated it in a somewhat different +way by the formal introduction of a third quantum number. In +his new paper Coster has established a similar classification for the +lighter elements. For the elements in the neighbourhood of xenon +and krypton he has obtained results illustrated by the diagrams +given in \Fig{6}. Just as in \Fig{5} the levels correspond exactly to +those types of orbits which, as seen from the table on \PageRef[page]{113}, +according to the theory will be present in the atoms of these elements. +In xenon several of the levels present in niton have disappeared, +and in krypton still more levels have fallen away. Coster +has also investigated in which elements these particular levels +appear for the last time, when passing from higher to lower atomic +number. His results concerning this point confirm in detail the +predictions of the theory. Further he proves that the change in +the firmness of binding of the electrons in the outer groups in +the elements of the family of the rare earths shows a dependence +on the atomic number which strongly supports the assumption that +in these elements a completion of an inner group of $4$-quanta +orbits takes place. For details the reader is referred to Coster's +paper in the \Title{Philosophical Magazine}. Another important contribution +to our systematic knowledge of the X-ray spectra is +contained in a recent paper by Wentzel. He shows that various +lines, which find no place in the classification hitherto considered, +can be ascribed in a natural manner to processes of reorganization, +initiated by the removal of more than one electron from the +atom; these lines are therefore in a certain sense analogous to +the enhanced lines in the optical spectra. + +\Chapter{}{Conclusion} + +Before bringing this address to a close I wish once more to +emphasize the complete analogy in the application of the +quantum theory to the stability of the atom, used in explaining +two so different phenomena as the periodic system and X-ray +spectra. This point is of the greatest importance in judging the +reality of the theory, since the justification for employing considerations, +relating to the formation of atoms by successive capture +of electrons, as a guiding principle for the investigation of atomic +\PageSep{126} +structure might appear doubtful if such considerations could not +be brought into natural agreement with views on the reorganization +of the atom after a disturbance in the normal electronic +arrangement. Even though a certain inner consistency in this +view of atomic structure will be recognized, it is, however, hardly +necessary for me to emphasize the incomplete character of the +theory, not only as regards the elaboration of details, but also so +far as the foundation of the general points of view is concerned. +There seems, however, to be no other way of advance in atomic +problems than that which hitherto has been followed, namely to let +the work in these two directions go hand in hand. + +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% +\PGLicense +\begin{PGtext} +End of the Project Gutenberg EBook of The Theory of Spectra and Atomic +Constitution, by Niels (Niels Henrik David) Bohr + +*** END OF THIS PROJECT GUTENBERG EBOOK THEORY OF SPECTRA *** + +***** This file should be named 47464-pdf.pdf or 47464-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/4/7/4/6/47464/ + +Produced by Andrew D. 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