diff options
| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 19:05:54 -0700 |
|---|---|---|
| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 19:05:54 -0700 |
| commit | 9638e58f45d4e1a4277028e9bcf9651e334bec4f (patch) | |
| tree | 91c91aeacb0f42c7a34134d1925ba624c6363f2b /47464-t/47464-t.tex | |
Diffstat (limited to '47464-t/47464-t.tex')
| -rw-r--r-- | 47464-t/47464-t.tex | 6910 |
1 files changed, 6910 insertions, 0 deletions
diff --git a/47464-t/47464-t.tex b/47464-t/47464-t.tex new file mode 100644 index 0000000..f79a93e --- /dev/null +++ b/47464-t/47464-t.tex @@ -0,0 +1,6910 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% 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 *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{47464} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Latin-1 text encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. Required. %% +%% %% +%% alltt: Fixed-width font environment. Required. %% +%% %% +%% indentfirst: Indent first paragraph of each section. Optional. %% +%% array: Array enhancements. Required. %% +%% %% +%% footmisc: Start footnote numbering on each page. Required. %% +%% %% +%% graphicx: Standard interface for graphics inclusion. Required. %% +%% %% +%% calc: Length calculations. Required. %% +%% %% +%% fancyhdr: Enhanced running headers and footers. Required. %% +%% %% +%% geometry: Enhanced page layout package. Required. %% +%% hyperref: Hypertext embellishments for pdf output. 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). If true, print-optimized PDF file: Larger %% +%% text block, two-sided layout, black hyperlinks. %% +%% %% +%% %% +%% PDF pages: 209 (if ForPrinting set to false) %% +%% PDF page size: 4.5 x 6.5" (non-standard) %% +%% %% +%% Summary of log file: %% +%% * Two underfull hboxes. (No visual problems.) %% +%% %% +%% Compile History: %% +%% %% +%% November, 2014: (Andrew D. Hwang) %% +%% texlive2012, GNU/Linux %% +%% %% +%% Command block: %% +%% %% +%% pdflatex x4 %% +%% %% +%% %% +%% November 2014: pglatex. %% +%% Compile this project with: %% +%% pdflatex 47464-t.tex ..... FOUR times %% +%% %% +%% pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\listfiles +\documentclass[12pt]{book}[2005/09/16] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[utf8]{inputenc}[2006/05/05] + +\usepackage{ifthen}[2001/05/26] %% Logical conditionals + +\usepackage{amsmath}[2000/07/18] %% Displayed equations +\usepackage{amssymb}[2002/01/22] %% and additional symbols + +\usepackage{alltt}[1997/06/16] %% boilerplate, credits, license + +\IfFileExists{indentfirst.sty}{% + \usepackage{indentfirst}[1995/11/23] +}{} + +\usepackage{array}[2008/09/09] + +\usepackage[perpage,symbol]{footmisc}[2005/03/17] + +\usepackage{graphicx}[1999/02/16]%% For diagrams + +\usepackage{calc}[2005/08/06] + +\usepackage{fancyhdr} %% For running heads + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% Interlude: Set up PRINTING (default) or SCREEN VIEWING %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% ForPrinting=true false (default) +% Asymmetric margins Symmetric margins +% 1 : 1.6 text block aspect ratio 3 : 4 text block aspect ratio +% Black hyperlinks Blue hyperlinks +% Start major marker pages recto No blank verso pages +% +\newboolean{ForPrinting} + +%% UNCOMMENT the next line for a PRINT-OPTIMIZED VERSION of the text %% +%\setboolean{ForPrinting}{true} + +%% Initialize values to ForPrinting=false +\newcommand{\Margins}{hmarginratio=1:1} % Symmetric margins +\newcommand{\HLinkColor}{blue} % Hyperlink color +\newcommand{\PDFPageLayout}{SinglePage} +\newcommand{\TransNote}{Transcriber's Note} +\newcommand{\TransNoteCommon}{% + The camera-quality files for this public-domain ebook may be + downloaded \textit{gratis} at + \begin{center} + \texttt{www.gutenberg.org/ebooks/\ebook}. + \end{center} + + This ebook was produced using scanned images and OCR text generously + provided by the Brandeis University Library through the Internet Archive. + \bigskip + + Minor typographical corrections and presentational changes have been + made without comment. + \bigskip +} + +\newcommand{\TransNoteText}{% + \TransNoteCommon + + This PDF file is optimized for screen viewing, but may be recompiled + for printing. Please consult the preamble of the \LaTeX\ source file + for instructions and other particulars. +} +%% Re-set if ForPrinting=true +\ifthenelse{\boolean{ForPrinting}}{% + \renewcommand{\Margins}{hmarginratio=2:3} % Asymmetric margins + \renewcommand{\HLinkColor}{black} % Hyperlink color + \renewcommand{\PDFPageLayout}{TwoPageRight} + \renewcommand{\TransNote}{Transcriber's Note} + \renewcommand{\TransNoteText}{% + \TransNoteCommon + + This PDF file is optimized for printing, but may be recompiled for + screen viewing. Please consult the preamble of the \LaTeX\ source + file for instructions and other particulars. + } +}{% If ForPrinting=false, don't skip to recto + \renewcommand{\cleardoublepage}{\clearpage} +} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% End of PRINTING/SCREEN VIEWING code; back to packages %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\ifthenelse{\boolean{ForPrinting}}{% + \setlength{\paperwidth}{8.5in}% + \setlength{\paperheight}{11in}% +% 1:1.6 + \usepackage[body={5in,8in},\Margins]{geometry}[2002/07/08] +}{% + \setlength{\paperwidth}{4.5in}% + \setlength{\paperheight}{6.5in}% + \raggedbottom +% 3:4 + \usepackage[body={4.25in,5.6in},\Margins,includeheadfoot]{geometry}[2002/07/08] +} + +\providecommand{\ebook}{00000} % Overridden during white-washing +\usepackage[pdftex, + hyperfootnotes=false, + pdfauthor={Niels Henrik David Bohr}, + pdftitle={The Project Gutenberg eBook \#\ebook: The Theory of Spectra and Atomic Constitution.}, + pdfkeywords={Brandeis University, The Internet Archive, Andrew D. Hwang}, + pdfstartview=Fit, % default value + pdfstartpage=1, % default value + pdfpagemode=UseNone, % default value + bookmarks=true, % default value + linktocpage=false, % default value + pdfpagelayout=\PDFPageLayout, + pdfdisplaydoctitle, + pdfpagelabels=true, + bookmarksopen=true, + bookmarksopenlevel=0, + colorlinks=true, + linkcolor=\HLinkColor]{hyperref}[2007/02/07] + + +%% Fixed-width environment to format PG boilerplate %% +\newenvironment{PGtext}{% +\begin{alltt} +\fontsize{8.1}{10}\ttfamily\selectfont}% +{\end{alltt}} + +% Stylistic changes made for consistency +\newcommand{\Chg}[2]{#2} +%\newcommand{\Chg}[2]{#1} % Use this to revert inconsistencies in the original +\newcommand{\Add}[1]{\Chg{}{#1}} + +%% Miscellaneous global parameters %% +% No hrule in page header +\renewcommand{\headrulewidth}{0pt} + +% Loosen spacing +\setlength{\emergencystretch}{1em} + +\hyphenation{electro-dynamics electro-magnetic} + +% Scratch pad for length calculations +\newlength{\TmpLen} + +%% Running heads %% +\newcommand{\FlushRunningHeads}{% + \clearpage\fancyhf{}% + \ifthenelse{\boolean{ForPrinting}}{\cleardoublepage}{\clearpage}% +} +\newcommand{\InitRunningHeads}{% + \setlength{\headheight}{15pt} + \pagestyle{fancy} + \thispagestyle{empty} + \ifthenelse{\boolean{ForPrinting}} + {\fancyhead[RO,LE]{\thepage}} + {\fancyhead[R]{\thepage}} +} + +% Uniform style for running heads +\newcommand{\RHeads}[1]{\textsc{\MakeLowercase{#1}}} + +\newcommand{\SetRunningHeads}[2][C]{\fancyhead[#1]{\RHeads{#2}}} + +\newcommand{\BookMark}[2]{\phantomsection\pdfbookmark[#1]{#2}{#2}} + +%% Major document divisions %% +\newcommand{\PGBoilerPlate}{% + \pagenumbering{Alph} + \pagestyle{empty} + \BookMark{0}{PG Boilerplate.} +} +\newcommand{\FrontMatter}{% + \cleardoublepage + \FlushRunningHeads + \frontmatter + \BookMark{-1}{Front Matter.} +} +\newcommand{\MainMatter}{% + \FlushRunningHeads + \InitRunningHeads + \mainmatter + \BookMark{-1}{Main Matter.} +} +\newcommand{\PGLicense}{% + \FlushRunningHeads + \pagenumbering{Roman} + \InitRunningHeads + \BookMark{-1}{PG License.} + \SetRunningHeads{License} +} + +%% ToC formatting %% +\AtBeginDocument{\renewcommand{\contentsname}% + {\protect\thispagestyle{empty}% + \protect\SectTitle[\Large]{\textbf{CONTENTS}}\protect\vspace{-1.5\baselineskip}} +} + +\newcommand{\TableofContents}{% + \FlushRunningHeads + \InitRunningHeads + \SetRunningHeads{Contents} + \BookMark{0}{Contents.} + \tableofcontents +} + +% Set the section number in a fixed-width box +\newcommand{\ToCBox}[1]{\settowidth{\TmpLen}{III.}% + \makebox[\TmpLen][r]{#1}\hspace*{1em}% +} +% For internal use, to determine if we need the Sect./Page line +\newcommand{\ToCAnchor}{} + +% \ToCBox{Label}{Anchor}{Title} +\newcommand{\ToCLine}[3]{% + \label{toc:#2}% + \ifthenelse{\not\equal{\pageref{toc:#2}}{\ToCAnchor}}{% + \renewcommand{\ToCAnchor}{\pageref{toc:#2}}% + \noindent\makebox[\textwidth][r]{\null\hfill\scriptsize PAGE}\\[8pt]% + }{}% + \settowidth{\TmpLen}{9999}% + \noindent\strut\parbox[b]{\textwidth-\TmpLen}{\small% + \ToCBox{#1}\hangindent4em#3\dotfill}% + \makebox[\TmpLen][r]{\pageref{#2}}% + \smallskip +} + +%% Sectional units %% +% Typographical abstraction +\newcommand{\SectTitle}[2][\large]{% + \section*{\centering#1\normalfont #2} +} +\newcommand{\SubsectTitle}[2][\normalsize]{% + \subsection*{\centering#1\normalfont #2} +} +\newcommand{\SubsubsectTitle}[2][\small]{% + \subsubsection*{\centering#1\normalfont #2} +} + +% For internal cross-referencing +\newcommand{\EssayNo}{} +\newcounter{SecNo} + +% \Essay{Number}{Title}{Footnote} +\newcommand{\Essay}[3]{% + \FlushRunningHeads + \InitRunningHeads + \ifthenelse{\equal{#1}{III}}{% [** TN: Hack to handle long running head] + \ifthenelse{\boolean{ForPrinting}}{% + \fancyhead[CE]{\RHeads{The Structure of the Atom and the}}% + \fancyhead[CO]{\RHeads{Physical and Chemical Properties of the Elements}}% + }{% Not ForPrinting + \SetRunningHeads{The Structure of the Atom}% + }% + }{% Not Essay III + \SetRunningHeads{#2}% + } + \BookMark{0}{Essay #1.} + \renewcommand{\EssayNo}{#1}% For \Eq and \Tag + \addtocontents{toc}{\protect\SectTitle{ESSAY #1}} + \addtocontents{toc}{\protect\SubsectTitle[\protect\small]{\MakeUppercase{#2}}} + \SectTitle[\Large]{ESSAY #1\footnotemark} + \SubsectTitle{\MakeUppercase{#2}} + \footnotetext{#3}% +} + +% \Chapter{Number}{Title} +\newcommand{\Chapter}[2]{% + \BookMark{1}{#1 #2.} + \label{chapter:\EssayNo.#1} + \addtocontents{toc}{\protect\ToCLine{#1}{chapter:\EssayNo.#1}{\textsc{#2}}} + \SubsubsectTitle{#1 \MakeUppercase{#2}} +} + +\newcommand{\Section}[1]{ + \paragraph*{\indent #1} + \refstepcounter{SecNo} + \label{section:\theSecNo} + \addtocontents{toc}{\protect\ToCLine{}{section:\theSecNo}{#1}} +} + +\newcommand{\Preface}{% + \InitRunningHeads + \SetRunningHeads{Preface} + \BookMark{0}{Preface.} + \SectTitle[\Large]{\textbf{PREFACE}} +} + +\newcommand{\Signature}[4]{% + \par\null\hfill#1\hspace{\parindent} + + \settowidth{\TmpLen}{\small\textsc{#2}}% + \parbox{\TmpLen}{\centering\small\textsc{#2} \\ + \textit{#3} #4} + \normalsize +} + +%% Diagrams %% +\newcommand{\DefWidth}{0.95\textwidth}% Default figure width +\newcommand{\Graphic}[1]{% + \includegraphics[width=\DefWidth]{./images/fig_p#1.pdf}% +} +% \Figure[caption]{label}{page} +\newcommand{\Figure}[3][]{% + \begin{figure}[hbt!] + \centering + \phantomsection\Pagelabel{#3}% + \Graphic{#3}% + \ifthenelse{\not\equal{#1}{}}{% + \\ + #1% + }{% #1=[] + \ifthenelse{\not\equal{#2}{}}{% + \\ + \label{fig:#2}% + Fig.~#2.% + }{}% + }% + \end{figure}\ignorespaces% +} + +% \Fig[figure]{Number} +\newcommand{\Fig}[2][fig.]{\hyperref[fig:#2]{#1~#2}} +\newcommand{\FigNum}[1]{\hyperref[fig:#1]{#1}} + +\newcommand{\Pagelabel}[1]{\phantomsection\label{page:#1}} + +% Page separators +\newcommand{\PageSep}[1]{\ignorespaces} +\newcommand{\PageRef}[2][page]{\hyperref[page:#2]{#1~\pageref*{page:#2}}} +\newcommand{\PageNum}[1]{\hyperref[page:#1]{\pageref*{page:#1}}} + +% Miscellaneous textual conveniences (N.B. \emph, not \textit) +\newcommand{\ie}{i.e.} + +\newcommand{\First}[1]{\textsc{#1}} +\newcommand{\Title}[1]{\textit{#1}}% Journal title +\newcommand{\No}[1]{\textsc{#1}}% Journal number + +%% Miscellaneous mathematical formatting %% +\DeclareInputMath{176}{{}^{\circ}} +\DeclareInputMath{183}{\cdot} + +% Centered column heading in left-aligned array column +\newcommand{\ColHead}[1]{\multicolumn{1}{c}{#1}} +% Conditional (ForPrinting-dependent) breaking for wide math +\newcommand{\BreakMath}[2]{\ifthenelse{\boolean{ForPrinting}}{#1}{#2}} + +% Cross-ref-able equation tags +\newcommand{\Tag}[1]{% + \phantomsection\label{eqn:\EssayNo#1}\tag*{\ensuremath{#1}}% +} +\newcommand{\Eq}[1]{% + \hyperref[eqn:\EssayNo#1]{\ensuremath{#1}}% +} + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} +%% PG BOILERPLATE %% +\PGBoilerPlate +\begin{center} +\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. Hwang + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions 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. Special rules, +set forth in the General Terms of Use part of this license, apply to +copying and distributing Project Gutenberg-tm electronic works to +protect the PROJECT GUTENBERG-tm concept and trademark. Project +Gutenberg is a registered trademark, and may not be used if you +charge for the eBooks, unless you receive specific permission. If you +do not charge anything for copies of this eBook, complying with the +rules is very easy. You may use this eBook for nearly any purpose +such as creation of derivative works, reports, performances and +research. They may be modified and printed and given away--you may do +practically ANYTHING with public domain eBooks. Redistribution is +subject to the trademark license, especially commercial +redistribution. + + + +*** START: FULL LICENSE *** + +THE FULL PROJECT GUTENBERG LICENSE +PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK + +To protect the Project Gutenberg-tm mission of promoting the free +distribution of electronic works, by using or distributing this work +(or any other work associated in any way with the phrase "Project +Gutenberg"), you agree to comply with all the terms of the Full Project +Gutenberg-tm License available with this file or online at + www.gutenberg.org/license. + + +Section 1. General Terms of Use and Redistributing Project Gutenberg-tm +electronic works + +1.A. By reading or using any part of this Project Gutenberg-tm +electronic work, you indicate that you have read, understand, agree to +and accept all the terms of this license and intellectual property +(trademark/copyright) agreement. If you do not agree to abide by all +the terms of this agreement, you must cease using and return or destroy +all copies of Project Gutenberg-tm electronic works in your possession. +If you paid a fee for obtaining a copy of or access to a Project +Gutenberg-tm electronic work and you do not agree to be bound by the +terms of this agreement, you may obtain a refund from the person or +entity to whom you paid the fee as set forth in paragraph 1.E.8. + +1.B. "Project Gutenberg" is a registered trademark. It may only be +used on or associated in any way with an electronic work by people who +agree to be bound by the terms of this agreement. There are a few +things that you can do with most Project Gutenberg-tm electronic works +even without complying with the full terms of this agreement. See +paragraph 1.C below. There are a lot of things you can do with Project +Gutenberg-tm electronic works if you follow the terms of this agreement +and help preserve free future access to Project Gutenberg-tm electronic +works. See paragraph 1.E below. + +1.C. The Project Gutenberg Literary Archive Foundation ("the Foundation" +or PGLAF), owns a compilation copyright in the collection of Project +Gutenberg-tm electronic works. Nearly all the individual works in the +collection are in the public domain in the United States. If an +individual work is in the public domain in the United States and you are +located in the United States, we do not claim a right to prevent you from +copying, distributing, performing, displaying or creating derivative +works based on the work as long as all references to Project Gutenberg +are removed. Of course, we hope that you will support the Project +Gutenberg-tm mission of promoting free access to electronic works by +freely sharing Project Gutenberg-tm works in compliance with the terms of +this agreement for keeping the Project Gutenberg-tm name associated with +the work. You can easily comply with the terms of this agreement by +keeping this work in the same format with its attached full Project +Gutenberg-tm License when you share it without charge with others. + +1.D. The copyright laws of the place where you are located also govern +what you can do with this work. Copyright laws in most countries are in +a constant state of change. If you are outside the United States, check +the laws of your country in addition to the terms of this agreement +before downloading, copying, displaying, performing, distributing or +creating derivative works based on this work or any other Project +Gutenberg-tm work. The Foundation makes no representations concerning +the copyright status of any work in any country outside the United +States. + +1.E. Unless you have removed all references to Project Gutenberg: + +1.E.1. The following sentence, with active links to, or other immediate +access to, the full Project Gutenberg-tm License must appear prominently +whenever any copy of a Project Gutenberg-tm work (any work on which the +phrase "Project Gutenberg" appears, or with which the phrase "Project +Gutenberg" is associated) is accessed, displayed, performed, viewed, +copied or distributed: + +This eBook is for the use of anyone anywhere 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 + +1.E.2. If an individual Project Gutenberg-tm electronic work is derived +from the public domain (does not contain a notice indicating that it is +posted with permission of the copyright holder), the work can be copied +and distributed to anyone in the United States without paying any fees +or charges. If you are redistributing or providing access to a work +with the phrase "Project Gutenberg" associated with or appearing on the +work, you must comply either with the requirements of paragraphs 1.E.1 +through 1.E.7 or obtain permission for the use of the work and the +Project Gutenberg-tm trademark as set forth in paragraphs 1.E.8 or +1.E.9. + +1.E.3. If an individual Project Gutenberg-tm electronic work is posted +with the permission of the copyright holder, your use and distribution +must comply with both paragraphs 1.E.1 through 1.E.7 and any additional +terms imposed by the copyright holder. Additional terms will be linked +to the Project Gutenberg-tm License for all works posted with the +permission of the copyright holder found at the beginning of this work. + +1.E.4. Do not unlink or detach or remove the full Project Gutenberg-tm +License terms from this work, or any files containing a part of this +work or any other work associated with Project Gutenberg-tm. + +1.E.5. Do not copy, display, perform, distribute or redistribute this +electronic work, or any part of this electronic work, without +prominently displaying the sentence set forth in paragraph 1.E.1 with +active links or immediate access to the full terms of the Project +Gutenberg-tm License. + +1.E.6. You may convert to and distribute this work in any binary, +compressed, marked up, nonproprietary or proprietary form, including any +word processing or hypertext form. However, if you provide access to or +distribute copies of a Project Gutenberg-tm work in a format other than +"Plain Vanilla ASCII" or other format used in the official version +posted on the official Project Gutenberg-tm web site (www.gutenberg.org), +you must, at no additional cost, fee or expense to the user, provide a +copy, a means of exporting a copy, or a means of obtaining a copy upon +request, of the work in its original "Plain Vanilla ASCII" or other +form. Any alternate format must include the full Project Gutenberg-tm +License as specified in paragraph 1.E.1. + +1.E.7. Do not charge a fee for access to, viewing, displaying, +performing, copying or distributing any Project Gutenberg-tm works +unless you comply with paragraph 1.E.8 or 1.E.9. + +1.E.8. You may charge a reasonable fee for copies of or providing +access to or distributing Project Gutenberg-tm electronic works provided +that + +- You pay a royalty fee of 20% of the gross profits you derive from + the use of Project Gutenberg-tm works calculated using the method + you already use to calculate your applicable taxes. The fee is + owed to the owner of the Project Gutenberg-tm trademark, but he + has agreed to donate royalties under this paragraph to the + Project Gutenberg Literary Archive Foundation. Royalty payments + must be paid within 60 days following each date on which you + prepare (or are legally required to prepare) your periodic tax + returns. Royalty payments should be clearly marked as such and + sent to the Project Gutenberg Literary Archive Foundation at the + address specified in Section 4, "Information about donations to + the Project Gutenberg Literary Archive Foundation." + +- You provide a full refund of any money paid by a user who notifies + you in writing (or by e-mail) within 30 days of receipt that s/he + does not agree to the terms of the full Project Gutenberg-tm + License. You must require such a user to return or + destroy all copies of the works possessed in a physical medium + and discontinue all use of and all access to other copies of + Project Gutenberg-tm works. + +- You provide, in accordance with paragraph 1.F.3, a full refund of any + money paid for a work or a replacement copy, if a defect in the + electronic work is discovered and reported to you within 90 days + of receipt of the work. + +- You comply with all other terms of this agreement for free + distribution of Project Gutenberg-tm works. + +1.E.9. If you wish to charge a fee or distribute a Project Gutenberg-tm +electronic work or group of works on different terms than are set +forth in this agreement, you must obtain permission in writing from +both the Project Gutenberg Literary Archive Foundation and Michael +Hart, the owner of the Project Gutenberg-tm trademark. Contact the +Foundation as set forth in Section 3 below. + +1.F. + +1.F.1. Project Gutenberg volunteers and employees expend considerable +effort to identify, do copyright research on, transcribe and proofread +public domain works in creating the Project Gutenberg-tm +collection. Despite these efforts, Project Gutenberg-tm electronic +works, and the medium on which they may be stored, may contain +"Defects," such as, but not limited to, incomplete, inaccurate or +corrupt data, transcription errors, a copyright or other intellectual +property infringement, a defective or damaged disk or other medium, a +computer virus, or computer codes that damage or cannot be read by +your equipment. + +1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right +of Replacement or Refund" described in paragraph 1.F.3, the Project +Gutenberg Literary Archive Foundation, the owner of the Project +Gutenberg-tm trademark, and any other party distributing a Project +Gutenberg-tm electronic work under this agreement, disclaim all +liability to you for damages, costs and expenses, including legal +fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT +LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE +PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE +TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE +LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR +INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH +DAMAGE. + +1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a +defect in this electronic work within 90 days of receiving it, you can +receive a refund of the money (if any) you paid for it by sending a +written explanation to the person you received the work from. If you +received the work on a physical medium, you must return the medium with +your written explanation. The person or entity that provided you with +the defective work may elect to provide a replacement copy in lieu of a +refund. If you received the work electronically, the person or entity +providing it to you may choose to give you a second opportunity to +receive the work electronically in lieu of a refund. If the second copy +is also defective, you may demand a refund in writing without further +opportunities to fix the problem. + +1.F.4. Except for the limited right of replacement or refund set forth +in paragraph 1.F.3, this work is provided to you 'AS-IS', WITH NO OTHER +WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO +WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE. + +1.F.5. Some states do not allow disclaimers of certain implied +warranties or the exclusion or limitation of certain types of damages. +If any disclaimer or limitation set forth in this agreement violates the +law of the state applicable to this agreement, the agreement shall be +interpreted to make the maximum disclaimer or limitation permitted by +the applicable state law. The invalidity or unenforceability of any +provision of this agreement shall not void the remaining provisions. + +1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the +trademark owner, any agent or employee of the Foundation, anyone +providing copies of Project Gutenberg-tm electronic works in accordance +with this agreement, and any volunteers associated with the production, +promotion and distribution of Project Gutenberg-tm electronic works, +harmless from all liability, costs and expenses, including legal fees, +that arise directly or indirectly from any of the following which you do +or cause to occur: (a) distribution of this or any Project Gutenberg-tm +work, (b) alteration, modification, or additions or deletions to any +Project Gutenberg-tm work, and (c) any Defect you cause. + + +Section 2. Information about the Mission of Project Gutenberg-tm + +Project Gutenberg-tm is synonymous with the free distribution of +electronic works in formats readable by the widest variety of computers +including obsolete, old, middle-aged and new computers. It exists +because of the efforts of hundreds of volunteers and donations from +people in all walks of life. + +Volunteers and financial support to provide volunteers with the +assistance they need are critical to reaching Project Gutenberg-tm's +goals and ensuring that the Project Gutenberg-tm collection will +remain freely available for generations to come. In 2001, the Project +Gutenberg Literary Archive Foundation was created to provide a secure +and permanent future for Project Gutenberg-tm and future generations. +To learn more about the Project Gutenberg Literary Archive Foundation +and how your efforts and donations can help, see Sections 3 and 4 +and the Foundation information page at www.gutenberg.org + + +Section 3. Information about the Project Gutenberg Literary Archive +Foundation + +The Project Gutenberg Literary Archive Foundation is a non profit +501(c)(3) educational corporation organized under the laws of the +state of Mississippi and granted tax exempt status by the Internal +Revenue Service. The Foundation's EIN or federal tax identification +number is 64-6221541. Contributions to the Project Gutenberg +Literary Archive Foundation are tax deductible to the full extent +permitted by U.S. federal laws and your state's laws. + +The Foundation's principal office is located at 4557 Melan Dr. S. +Fairbanks, AK, 99712., but its volunteers and employees are scattered +throughout numerous locations. Its business office is located at 809 +North 1500 West, Salt Lake City, UT 84116, (801) 596-1887. Email +contact links and up to date contact information can be found at the +Foundation's web site and official page at www.gutenberg.org/contact + +For additional contact information: + Dr. Gregory B. Newby + Chief Executive and Director + gbnewby@pglaf.org + +Section 4. Information about Donations to the Project Gutenberg +Literary Archive Foundation + +Project Gutenberg-tm depends upon and cannot survive without wide +spread public support and donations to carry out its mission of +increasing the number of public domain and licensed works that can be +freely distributed in machine readable form accessible by the widest +array of equipment including outdated equipment. Many small donations +($1 to $5,000) are particularly important to maintaining tax exempt +status with the IRS. + +The Foundation is committed to complying with the laws regulating +charities and charitable donations in all 50 states of the United +States. Compliance requirements are not uniform and it takes a +considerable effort, much paperwork and many fees to meet and keep up +with these requirements. We do not solicit donations in locations +where we have not received written confirmation of compliance. To +SEND DONATIONS or determine the status of compliance for any +particular state visit www.gutenberg.org/donate + +While we cannot and do not solicit contributions from states where we +have not met the solicitation requirements, we know of no prohibition +against accepting unsolicited donations from donors in such states who +approach us with offers to donate. + +International donations are gratefully accepted, but we cannot make +any statements concerning tax treatment of donations received from +outside the United States. U.S. laws alone swamp our small staff. + +Please check the Project Gutenberg Web pages for current donation +methods and addresses. Donations are accepted in a number of other +ways including checks, online payments and credit card donations. +To donate, please visit: www.gutenberg.org/donate + + +Section 5. General Information About Project Gutenberg-tm electronic +works. + +Professor Michael S. Hart was the originator of the Project Gutenberg-tm +concept of a library of electronic works that could be freely shared +with anyone. For forty years, he produced and distributed Project +Gutenberg-tm eBooks with only a loose network of volunteer support. + +Project Gutenberg-tm eBooks are often created from several printed +editions, all of which are confirmed as Public Domain in the U.S. +unless a copyright notice is included. Thus, we do not necessarily +keep eBooks in compliance with any particular paper edition. + +Most people start at our Web site which has the main PG search facility: + + www.gutenberg.org + +This Web site includes information about Project Gutenberg-tm, +including how to make donations to the Project Gutenberg Literary +Archive Foundation, how to help produce our new eBooks, and how to +subscribe to our email newsletter to hear about new eBooks. +\end{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-t.tex or 47464-t.zip ***** % +% This and all associated files of various formats will be found in: % +% http://www.gutenberg.org/4/7/4/6/47464/ % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\end{document} +### +@ControlwordReplace = ( + ['\\ie', 'i.e.'] + ); + +@ControlwordArguments = ( + ['\\Signature', 1, 1, '', ' ', 1, 1, '', ' ', 1, 1, '', ' ', 1, 1, '', ''], + ['\\Figure', 0, 0, '', '', 1, 0, '', '', 1, 0, '<GRAPHIC>', ''], + ['\\BookMark', 1, 0, '', '', 1, 0, '', ''], + ['\\First', 1, 1, '', ''], + ['\\Essay', 1, 1, 'Essay ', ' ', 1, 1, '', '', 1, 0, '', ''], + ['\\Chapter', 1, 1, '', ' ', 1, 1, '', ''], + ['\\Section', 1, 1, '', ''], + ['\\Fig', 0, 0, '', '', 1, 1, 'figure ', ''], + ['\\FigNum', 1, 1, '', ''], + ['\\Pagelabel', 1, 0, '', ''], + ['\\PageRef', 1, 1, 'p. ', ''], + ['\\PageNum', 1, 1, '', ''], + ['\\Eq', 0, 0, '', '', 1, 1, '', ''], + ['\\Add', 1, 1, '', ''], + ['\\Chg', 1, 0, '', '', 1, 1, '', ''] + ); +$PageSeparator = qr/^\\PageSep/; +$CustomClean = 'print "\\nCustom cleaning in progress..."; +my $cline = 0; + while ($cline <= $#file) { + $file[$cline] =~ s/--------[^\n]*\n//; # strip page separators + $cline++ + } + print "done\\n";'; +### +This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2014.9.6) 26 NOV 2014 07:10 +entering extended mode + %&-line parsing enabled. +**47464-t.tex +(./47464-t.tex +LaTeX2e <2011/06/27> +Babel <3.9h> and hyphenation patterns for 78 languages loaded. +(/usr/share/texlive/texmf-dist/tex/latex/base/book.cls +Document Class: book 2007/10/19 v1.4h Standard LaTeX document class +(/usr/share/texlive/texmf-dist/tex/latex/base/bk12.clo +File: bk12.clo 2007/10/19 v1.4h Standard LaTeX file (size option) +) +\c@part=\count79 +\c@chapter=\count80 +\c@section=\count81 +\c@subsection=\count82 +\c@subsubsection=\count83 +\c@paragraph=\count84 +\c@subparagraph=\count85 +\c@figure=\count86 +\c@table=\count87 +\abovecaptionskip=\skip41 +\belowcaptionskip=\skip42 +\bibindent=\dimen102 +) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +Package: inputenc 2008/03/30 v1.1d Input encoding file +\inpenc@prehook=\toks14 +\inpenc@posthook=\toks15 +(/usr/share/texlive/texmf-dist/tex/latex/base/latin1.def +File: latin1.def 2008/03/30 v1.1d Input encoding file +)) (/usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty +Package: ifthen 2001/05/26 v1.1c Standard LaTeX ifthen package (DPC) +) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +Package: amsmath 2013/01/14 v2.14 AMS math features +\@mathmargin=\skip43 +For additional information on amsmath, use the `?' option. +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +Package: amstext 2000/06/29 v2.01 +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 +\@emptytoks=\toks16 +\ex@=\dimen103 +)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +Package: amsbsy 1999/11/29 v1.2d +\pmbraise@=\dimen104 +) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +Package: amsopn 1999/12/14 v2.01 operator names +) +\inf@bad=\count88 +LaTeX Info: Redefining \frac on input line 210. +\uproot@=\count89 +\leftroot@=\count90 +LaTeX Info: Redefining \overline on input line 306. +\classnum@=\count91 +\DOTSCASE@=\count92 +LaTeX Info: Redefining \ldots on input line 378. +LaTeX Info: Redefining \dots on input line 381. +LaTeX Info: Redefining \cdots on input line 466. +\Mathstrutbox@=\box26 +\strutbox@=\box27 +\big@size=\dimen105 +LaTeX Font Info: Redeclaring font encoding OML on input line 566. +LaTeX Font Info: Redeclaring font encoding OMS on input line 567. +\macc@depth=\count93 +\c@MaxMatrixCols=\count94 +\dotsspace@=\muskip10 +\c@parentequation=\count95 +\dspbrk@lvl=\count96 +\tag@help=\toks17 +\row@=\count97 +\column@=\count98 +\maxfields@=\count99 +\andhelp@=\toks18 +\eqnshift@=\dimen106 +\alignsep@=\dimen107 +\tagshift@=\dimen108 +\tagwidth@=\dimen109 +\totwidth@=\dimen110 +\lineht@=\dimen111 +\@envbody=\toks19 +\multlinegap=\skip44 +\multlinetaggap=\skip45 +\mathdisplay@stack=\toks20 +LaTeX Info: Redefining \[ on input line 2665. +LaTeX Info: Redefining \] on input line 2666. +) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +Package: amssymb 2013/01/14 v3.01 AMS font symbols +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support +\symAMSa=\mathgroup4 +\symAMSb=\mathgroup5 +LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' +(Font) U/euf/m/n --> U/euf/b/n on input line 106. +)) (/usr/share/texlive/texmf-dist/tex/latex/base/alltt.sty +Package: alltt 1997/06/16 v2.0g defines alltt environment +) (/usr/share/texlive/texmf-dist/tex/latex/tools/indentfirst.sty +Package: indentfirst 1995/11/23 v1.03 Indent first paragraph (DPC) +) (/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +Package: array 2008/09/09 v2.4c Tabular extension package (FMi) +\col@sep=\dimen112 +\extrarowheight=\dimen113 +\NC@list=\toks21 +\extratabsurround=\skip46 +\backup@length=\skip47 +) (/usr/share/texlive/texmf-dist/tex/latex/footmisc/footmisc.sty +Package: footmisc 2011/06/06 v5.5b a miscellany of footnote facilities +\FN@temptoken=\toks22 +\footnotemargin=\dimen114 +\c@pp@next@reset=\count100 +\c@@fnserial=\count101 +Package footmisc Info: Declaring symbol style bringhurst on input line 855. +Package footmisc Info: Declaring symbol style chicago on input line 863. +Package footmisc Info: Declaring symbol style wiley on input line 872. +Package footmisc Info: Declaring symbol style lamport-robust on input line 883. + +Package footmisc Info: Declaring symbol style lamport* on input line 903. +Package footmisc Info: Declaring symbol style lamport*-robust on input line 924 +. +) (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty +Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR) +(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +Package: keyval 1999/03/16 v1.13 key=value parser (DPC) +\KV@toks@=\toks23 +) (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty +Package: graphics 2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR) +(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty +Package: trig 1999/03/16 v1.09 sin cos tan (DPC) +) (/usr/share/texlive/texmf-dist/tex/latex/latexconfig/graphics.cfg +File: graphics.cfg 2010/04/23 v1.9 graphics configuration of TeX Live +) +Package graphics Info: Driver file: pdftex.def on input line 91. +(/usr/share/texlive/texmf-dist/tex/latex/pdftex-def/pdftex.def +File: pdftex.def 2011/05/27 v0.06d Graphics/color for pdfTeX +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty +Package: infwarerr 2010/04/08 v1.3 Providing info/warning/error messages (HO) +) (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty +Package: ltxcmds 2011/11/09 v1.22 LaTeX kernel commands for general use (HO) +) +\Gread@gobject=\count102 +)) +\Gin@req@height=\dimen115 +\Gin@req@width=\dimen116 +) (/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty +Package: calc 2007/08/22 v4.3 Infix arithmetic (KKT,FJ) +\calc@Acount=\count103 +\calc@Bcount=\count104 +\calc@Adimen=\dimen117 +\calc@Bdimen=\dimen118 +\calc@Askip=\skip48 +\calc@Bskip=\skip49 +LaTeX Info: Redefining \setlength on input line 76. +LaTeX Info: Redefining \addtolength on input line 77. +\calc@Ccount=\count105 +\calc@Cskip=\skip50 +) (/usr/share/texlive/texmf-dist/tex/latex/fancyhdr/fancyhdr.sty +\fancy@headwidth=\skip51 +\f@ncyO@elh=\skip52 +\f@ncyO@erh=\skip53 +\f@ncyO@olh=\skip54 +\f@ncyO@orh=\skip55 +\f@ncyO@elf=\skip56 +\f@ncyO@erf=\skip57 +\f@ncyO@olf=\skip58 +\f@ncyO@orf=\skip59 +) (/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +Package: geometry 2010/09/12 v5.6 Page Geometry +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifpdf.sty +Package: ifpdf 2011/01/30 v2.3 Provides the ifpdf switch (HO) +Package ifpdf Info: pdfTeX in PDF mode is detected. +) (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifvtex.sty +Package: ifvtex 2010/03/01 v1.5 Detect VTeX and its facilities (HO) +Package ifvtex Info: VTeX not detected. +) (/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty +Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional +) +\Gm@cnth=\count106 +\Gm@cntv=\count107 +\c@Gm@tempcnt=\count108 +\Gm@bindingoffset=\dimen119 +\Gm@wd@mp=\dimen120 +\Gm@odd@mp=\dimen121 +\Gm@even@mp=\dimen122 +\Gm@layoutwidth=\dimen123 +\Gm@layoutheight=\dimen124 +\Gm@layouthoffset=\dimen125 +\Gm@layoutvoffset=\dimen126 +\Gm@dimlist=\toks24 +) (/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty +Package: hyperref 2012/11/06 v6.83m Hypertext links for LaTeX +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty +Package: hobsub-hyperref 2012/05/28 v1.13 Bundle oberdiek, subset hyperref (HO) + +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty +Package: hobsub-generic 2012/05/28 v1.13 Bundle oberdiek, subset generic (HO) +Package: hobsub 2012/05/28 v1.13 Construct package bundles (HO) +Package hobsub Info: Skipping package `infwarerr' (already loaded). +Package hobsub Info: Skipping package `ltxcmds' (already loaded). +Package: ifluatex 2010/03/01 v1.3 Provides the ifluatex switch (HO) +Package ifluatex Info: LuaTeX not detected. +Package hobsub Info: Skipping package `ifvtex' (already loaded). +Package: intcalc 2007/09/27 v1.1 Expandable calculations with integers (HO) +Package hobsub Info: Skipping package `ifpdf' (already loaded). +Package: etexcmds 2011/02/16 v1.5 Avoid name clashes with e-TeX commands (HO) +Package etexcmds Info: Could not find \expanded. +(etexcmds) That can mean that you are not using pdfTeX 1.50 or +(etexcmds) that some package has redefined \expanded. +(etexcmds) In the latter case, load this package earlier. +Package: kvsetkeys 2012/04/25 v1.16 Key value parser (HO) +Package: kvdefinekeys 2011/04/07 v1.3 Define keys (HO) +Package: pdftexcmds 2011/11/29 v0.20 Utility functions of pdfTeX for LuaTeX (HO +) +Package pdftexcmds Info: LuaTeX not detected. +Package pdftexcmds Info: \pdf@primitive is available. +Package pdftexcmds Info: \pdf@ifprimitive is available. +Package pdftexcmds Info: \pdfdraftmode found. +Package: pdfescape 2011/11/25 v1.13 Implements pdfTeX's escape features (HO) +Package: bigintcalc 2012/04/08 v1.3 Expandable calculations on big integers (HO +) +Package: bitset 2011/01/30 v1.1 Handle bit-vector datatype (HO) +Package: uniquecounter 2011/01/30 v1.2 Provide unlimited unique counter (HO) +) +Package hobsub Info: Skipping package `hobsub' (already loaded). +Package: letltxmacro 2010/09/02 v1.4 Let assignment for LaTeX macros (HO) +Package: hopatch 2012/05/28 v1.2 Wrapper for package hooks (HO) +Package: xcolor-patch 2011/01/30 xcolor patch +Package: atveryend 2011/06/30 v1.8 Hooks at the very end of document (HO) +Package atveryend Info: \enddocument detected (standard20110627). +Package: atbegshi 2011/10/05 v1.16 At begin shipout hook (HO) +Package: refcount 2011/10/16 v3.4 Data extraction from label references (HO) +Package: hycolor 2011/01/30 v1.7 Color options for hyperref/bookmark (HO) +) (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/auxhook.sty +Package: auxhook 2011/03/04 v1.3 Hooks for auxiliary files (HO) +) (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty +Package: kvoptions 2011/06/30 v3.11 Key value format for package options (HO) +) +\@linkdim=\dimen127 +\Hy@linkcounter=\count109 +\Hy@pagecounter=\count110 +(/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def +File: pd1enc.def 2012/11/06 v6.83m Hyperref: PDFDocEncoding definition (HO) +) +\Hy@SavedSpaceFactor=\count111 +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/hyperref.cfg +File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive +) +Package hyperref Info: Option `hyperfootnotes' set `false' on input line 4319. +Package hyperref Info: Option `bookmarks' set `true' on input line 4319. +Package hyperref Info: Option `linktocpage' set `false' on input line 4319. +Package hyperref Info: Option `pdfdisplaydoctitle' set `true' on input line 431 +9. +Package hyperref Info: Option `pdfpagelabels' set `true' on input line 4319. +Package hyperref Info: Option `bookmarksopen' set `true' on input line 4319. +Package hyperref Info: Option `colorlinks' set `true' on input line 4319. +Package hyperref Info: Hyper figures OFF on input line 4443. +Package hyperref Info: Link nesting OFF on input line 4448. +Package hyperref Info: Hyper index ON on input line 4451. +Package hyperref Info: Plain pages OFF on input line 4458. +Package hyperref Info: Backreferencing OFF on input line 4463. +Package hyperref Info: Implicit mode ON; LaTeX internals redefined. +Package hyperref Info: Bookmarks ON on input line 4688. +\c@Hy@tempcnt=\count112 +(/usr/share/texlive/texmf-dist/tex/latex/url/url.sty +\Urlmuskip=\muskip11 +Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc. +) +LaTeX Info: Redefining \url on input line 5041. +\XeTeXLinkMargin=\dimen128 +\Fld@menulength=\count113 +\Field@Width=\dimen129 +\Fld@charsize=\dimen130 +Package hyperref Info: Hyper figures OFF on input line 6295. +Package hyperref Info: Link nesting OFF on input line 6300. +Package hyperref Info: Hyper index ON on input line 6303. +Package hyperref Info: backreferencing OFF on input line 6310. +Package hyperref Info: Link coloring ON on input line 6313. +Package hyperref Info: Link coloring with OCG OFF on input line 6320. +Package hyperref Info: PDF/A mode OFF on input line 6325. +LaTeX Info: Redefining \ref on input line 6365. +LaTeX Info: Redefining \pageref on input line 6369. +\Hy@abspage=\count114 +\c@Item=\count115 +) + +Package hyperref Message: Driver: hpdftex. + +(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def +File: hpdftex.def 2012/11/06 v6.83m Hyperref driver for pdfTeX +\Fld@listcount=\count116 +\c@bookmark@seq@number=\count117 +(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty +Package: rerunfilecheck 2011/04/15 v1.7 Rerun checks for auxiliary files (HO) +Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2 +82. +) +\Hy@SectionHShift=\skip60 +) +\TmpLen=\skip61 +\c@SecNo=\count118 +(./47464-t.aux) +\openout1 = `47464-t.aux'. + +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 465. +LaTeX Font Info: ... okay on input line 465. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 465. +LaTeX Font Info: ... okay on input line 465. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 465. +LaTeX Font Info: ... okay on input line 465. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 465. +LaTeX Font Info: ... okay on input line 465. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 465. +LaTeX Font Info: ... okay on input line 465. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 465. +LaTeX Font Info: ... okay on input line 465. +LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 465. +LaTeX Font Info: ... okay on input line 465. +(/usr/share/texlive/texmf-dist/tex/context/base/supp-pdf.mkii +[Loading MPS to PDF converter (version 2006.09.02).] +\scratchcounter=\count119 +\scratchdimen=\dimen131 +\scratchbox=\box28 +\nofMPsegments=\count120 +\nofMParguments=\count121 +\everyMPshowfont=\toks25 +\MPscratchCnt=\count122 +\MPscratchDim=\dimen132 +\MPnumerator=\count123 +\makeMPintoPDFobject=\count124 +\everyMPtoPDFconversion=\toks26 +) +*geometry* driver: auto-detecting +*geometry* detected driver: pdftex +*geometry* verbose mode - [ preamble ] result: +* driver: pdftex +* paper: <default> +* layout: <same size as paper> +* layoutoffset:(h,v)=(0.0pt,0.0pt) +* hratio: 1:1 +* modes: includehead includefoot twoside +* h-part:(L,W,R)=(9.03375pt, 307.14749pt, 9.03375pt) +* v-part:(T,H,B)=(1.26749pt, 466.58623pt, 1.90128pt) +* \paperwidth=325.215pt +* \paperheight=469.75499pt +* \textwidth=307.14749pt +* \textheight=404.71243pt +* \oddsidemargin=-63.23624pt +* \evensidemargin=-63.23624pt +* \topmargin=-71.0025pt +* \headheight=12.0pt +* \headsep=19.8738pt +* \topskip=12.0pt +* \footskip=30.0pt +* \marginparwidth=98.0pt +* \marginparsep=7.0pt +* \columnsep=10.0pt +* \skip\footins=10.8pt plus 4.0pt minus 2.0pt +* \hoffset=0.0pt +* \voffset=0.0pt +* \mag=1000 +* \@twocolumnfalse +* \@twosidetrue +* \@mparswitchtrue +* \@reversemarginfalse +* (1in=72.27pt=25.4mm, 1cm=28.453pt) + +\AtBeginShipoutBox=\box29 +(/usr/share/texlive/texmf-dist/tex/latex/graphics/color.sty +Package: color 2005/11/14 v1.0j Standard LaTeX Color (DPC) +(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/color.cfg +File: color.cfg 2007/01/18 v1.5 color configuration of teTeX/TeXLive +) +Package color Info: Driver file: pdftex.def on input line 130. +) +Package hyperref Info: Link coloring ON on input line 465. +(/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty +Package: nameref 2012/10/27 v2.43 Cross-referencing by name of section +(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty +Package: gettitlestring 2010/12/03 v1.4 Cleanup title references (HO) +) +\c@section@level=\count125 +) +LaTeX Info: Redefining \ref on input line 465. +LaTeX Info: Redefining \pageref on input line 465. +LaTeX Info: Redefining \nameref on input line 465. +(./47464-t.out) (./47464-t.out) +\@outlinefile=\write3 +\openout3 = `47464-t.out'. + + +Overfull \hbox (7.35703pt too wide) in paragraph at lines 475--475 +[]\OT1/cmtt/m/n/8 This eBook is for the use of anyone anywhere in the United St +ates and most[] + [] + + +Overfull \hbox (7.35703pt too wide) in paragraph at lines 477--477 +[]\OT1/cmtt/m/n/8 whatsoever. You may copy it, give it away or re-use it under + the terms of[] + [] + + +Overfull \hbox (15.85715pt too wide) in paragraph at lines 479--479 +[]\OT1/cmtt/m/n/8 www.gutenberg.org. If you are not located in the United Stat +es, you'll have[] + [] + + +Overfull \hbox (28.60733pt too wide) in paragraph at lines 480--480 +[]\OT1/cmtt/m/n/8 to check the laws of the country where you are located before + using this ebook.[] + [] + +LaTeX Font Info: Try loading font information for U+msa on input line 497. +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd +File: umsa.fd 2013/01/14 v3.01 AMS symbols A +) +LaTeX Font Info: Try loading font information for U+msb on input line 497. +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd +File: umsb.fd 2013/01/14 v3.01 AMS symbols B +) [1 + +{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] [1 + + + + +] [2] +LaTeX Font Info: Try loading font information for OMS+cmr on input line 664. + +(/usr/share/texlive/texmf-dist/tex/latex/base/omscmr.fd +File: omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions +) +LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <8> not available +(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 664. +LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <7> not available +(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 664. +[3] [4] (./47464-t.toc +Underfull \hbox (badness 1917) in paragraph at lines 18--18 +[][]| \OT1/cmr/m/sc/10.95 De-vel-op-ment of the Quan-tum The-ory of + [] + +[5 + + + +]) +\tf@toc=\write4 +\openout4 = `47464-t.toc'. + +[6] [7] [8] +LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <12> not available +(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 788. +[1 + + + + + +] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] +[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30 + + +] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] <./imag +es/fig_p30.pdf, id=538, 154.17601pt x 62.8749pt> +File: ./images/fig_p30.pdf Graphic file (type pdf) +<use ./images/fig_p30.pdf> +Package pdftex.def Info: ./images/fig_p30.pdf used on input line 2042. +(pdftex.def) Requested size: 291.78917pt x 119.00972pt. +[45] [46 <./images/fig_p30.pdf>] [47] [48] [49] [50] [51] [52] [53] [54] [55] [ +56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] +[72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] +[88] [89] [90] [91] [92 + + +] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] <./images/fi +g_p70.pdf, id=936, 154.17601pt x 157.5486pt> +File: ./images/fig_p70.pdf Graphic file (type pdf) +<use ./images/fig_p70.pdf> +Package pdftex.def Info: ./images/fig_p70.pdf used on input line 3686. +(pdftex.def) Requested size: 291.78917pt x 298.20833pt. +[105] [106 <./images/fig_p70.pdf>] [107] [108] [109] [110] [111] [112] [113] [1 +14] [115] [116] [117] <./images/fig_p79.pdf, id=1023, 154.17601pt x 62.8749pt> +File: ./images/fig_p79.pdf Graphic file (type pdf) +<use ./images/fig_p79.pdf> +Package pdftex.def Info: ./images/fig_p79.pdf used on input line 4026. +(pdftex.def) Requested size: 291.78917pt x 119.00972pt. +[118] [119 <./images/fig_p79.pdf>] [120] [121] [122] [123] [124] [125] [126] [1 +27] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [14 +0] [141] [142] [143] <./images/fig_p97.pdf, id=1170, 154.17601pt x 64.3203pt> +File: ./images/fig_p97.pdf Graphic file (type pdf) +<use ./images/fig_p97.pdf> +Package pdftex.def Info: ./images/fig_p97.pdf used on input line 4747. +(pdftex.def) Requested size: 291.78917pt x 121.74559pt. +[144] [145] [146 <./images/fig_p97.pdf>] [147] [148] [149] [150] +Underfull \hbox (badness 1292) in paragraph at lines 4867--4888 + [][]\OT1/cmr/bx/n/12 Fourth Pe-riod. Potassium---Krypton. \OT1/cmr/m/n/12 In t +he + [] + +<./images/fig_p102.pdf, id=1215, 154.17601pt x 133.2177pt> +File: ./images/fig_p102.pdf Graphic file (type pdf) +<use ./images/fig_p102.pdf> +Package pdftex.def Info: ./images/fig_p102.pdf used on input line 4930. +(pdftex.def) Requested size: 291.78917pt x 252.15475pt. +[151] [152] [153] [154 <./images/fig_p102.pdf>] [155] [156] [157] [158] [159] [ +160] [161] [162] [163] [164] [165] [166] [167] [168] [169] <./images/fig_p113.p +df, id=1314, 154.17601pt x 69.6201pt> +File: ./images/fig_p113.pdf Graphic file (type pdf) +<use ./images/fig_p113.pdf> +Package pdftex.def Info: ./images/fig_p113.pdf used on input line 5354. +(pdftex.def) Requested size: 291.78917pt x 131.77707pt. +[170 <./images/fig_p113.pdf>] [171] [172] [173] [174] [175] [176] [177] [178] [ +179] [180] [181] [182] [183] <./images/fig_p123.pdf, id=1395, 154.17601pt x 137 +.5539pt> +File: ./images/fig_p123.pdf Graphic file (type pdf) +<use ./images/fig_p123.pdf> +Package pdftex.def Info: ./images/fig_p123.pdf used on input line 5741. +(pdftex.def) Requested size: 291.78917pt x 260.3623pt. +[184] [185 <./images/fig_p123.pdf>] <./images/fig_p124.pdf, id=1420, 154.17601p +t x 93.951pt> +File: ./images/fig_p124.pdf Graphic file (type pdf) +<use ./images/fig_p124.pdf> +Package pdftex.def Info: ./images/fig_p124.pdf used on input line 5764. +(pdftex.def) Requested size: 291.78917pt x 177.83067pt. +[186 <./images/fig_p124.pdf>] [187] [188] [189] [1 + + +] +Overfull \hbox (3.10696pt too wide) in paragraph at lines 5902--5902 +[]\OT1/cmtt/m/n/8 1.C. The Project Gutenberg Literary Archive Foundation ("the + Foundation"[] + [] + + +Overfull \hbox (3.10696pt too wide) in paragraph at lines 5907--5907 +[]\OT1/cmtt/m/n/8 located in the United States, we do not claim a right to prev +ent you from[] + [] + + +Overfull \hbox (3.10696pt too wide) in paragraph at lines 5912--5912 +[]\OT1/cmtt/m/n/8 freely sharing Project Gutenberg-tm works in compliance with +the terms of[] + [] + +[2] [3] +Overfull \hbox (3.10696pt too wide) in paragraph at lines 5975--5975 +[]\OT1/cmtt/m/n/8 posted on the official Project Gutenberg-tm web site (www.gut +enberg.org),[] + [] + +[4] [5] [6] [7] [8] +Package atveryend Info: Empty hook `BeforeClearDocument' on input line 6204. +[9] +Package atveryend Info: Empty hook `AfterLastShipout' on input line 6204. +(./47464-t.aux) +Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 6204. + + *File List* + book.cls 2007/10/19 v1.4h Standard LaTeX document class + bk12.clo 2007/10/19 v1.4h Standard LaTeX file (size option) +inputenc.sty 2008/03/30 v1.1d Input encoding file + latin1.def 2008/03/30 v1.1d Input encoding file + ifthen.sty 2001/05/26 v1.1c Standard LaTeX ifthen package (DPC) + amsmath.sty 2013/01/14 v2.14 AMS math features + amstext.sty 2000/06/29 v2.01 + amsgen.sty 1999/11/30 v2.0 + amsbsy.sty 1999/11/29 v1.2d + amsopn.sty 1999/12/14 v2.01 operator names + amssymb.sty 2013/01/14 v3.01 AMS font symbols +amsfonts.sty 2013/01/14 v3.01 Basic AMSFonts support + alltt.sty 1997/06/16 v2.0g defines alltt environment +indentfirst.sty 1995/11/23 v1.03 Indent first paragraph (DPC) + array.sty 2008/09/09 v2.4c Tabular extension package (FMi) +footmisc.sty 2011/06/06 v5.5b a miscellany of footnote facilities +graphicx.sty 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR) + keyval.sty 1999/03/16 v1.13 key=value parser (DPC) +graphics.sty 2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR) + trig.sty 1999/03/16 v1.09 sin cos tan (DPC) +graphics.cfg 2010/04/23 v1.9 graphics configuration of TeX Live + pdftex.def 2011/05/27 v0.06d Graphics/color for pdfTeX +infwarerr.sty 2010/04/08 v1.3 Providing info/warning/error messages (HO) + ltxcmds.sty 2011/11/09 v1.22 LaTeX kernel commands for general use (HO) + calc.sty 2007/08/22 v4.3 Infix arithmetic (KKT,FJ) +fancyhdr.sty +geometry.sty 2010/09/12 v5.6 Page Geometry + ifpdf.sty 2011/01/30 v2.3 Provides the ifpdf switch (HO) + ifvtex.sty 2010/03/01 v1.5 Detect VTeX and its facilities (HO) + ifxetex.sty 2010/09/12 v0.6 Provides ifxetex conditional +hyperref.sty 2012/11/06 v6.83m Hypertext links for LaTeX +hobsub-hyperref.sty 2012/05/28 v1.13 Bundle oberdiek, subset hyperref (HO) +hobsub-generic.sty 2012/05/28 v1.13 Bundle oberdiek, subset generic (HO) + hobsub.sty 2012/05/28 v1.13 Construct package bundles (HO) +ifluatex.sty 2010/03/01 v1.3 Provides the ifluatex switch (HO) + intcalc.sty 2007/09/27 v1.1 Expandable calculations with integers (HO) +etexcmds.sty 2011/02/16 v1.5 Avoid name clashes with e-TeX commands (HO) +kvsetkeys.sty 2012/04/25 v1.16 Key value parser (HO) +kvdefinekeys.sty 2011/04/07 v1.3 Define keys (HO) +pdftexcmds.sty 2011/11/29 v0.20 Utility functions of pdfTeX for LuaTeX (HO) +pdfescape.sty 2011/11/25 v1.13 Implements pdfTeX's escape features (HO) +bigintcalc.sty 2012/04/08 v1.3 Expandable calculations on big integers (HO) + bitset.sty 2011/01/30 v1.1 Handle bit-vector datatype (HO) +uniquecounter.sty 2011/01/30 v1.2 Provide unlimited unique counter (HO) +letltxmacro.sty 2010/09/02 v1.4 Let assignment for LaTeX macros (HO) + hopatch.sty 2012/05/28 v1.2 Wrapper for package hooks (HO) +xcolor-patch.sty 2011/01/30 xcolor patch +atveryend.sty 2011/06/30 v1.8 Hooks at the very end of document (HO) +atbegshi.sty 2011/10/05 v1.16 At begin shipout hook (HO) +refcount.sty 2011/10/16 v3.4 Data extraction from label references (HO) + hycolor.sty 2011/01/30 v1.7 Color options for hyperref/bookmark (HO) + auxhook.sty 2011/03/04 v1.3 Hooks for auxiliary files (HO) +kvoptions.sty 2011/06/30 v3.11 Key value format for package options (HO) + pd1enc.def 2012/11/06 v6.83m Hyperref: PDFDocEncoding definition (HO) +hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive + url.sty 2013/09/16 ver 3.4 Verb mode for urls, etc. + hpdftex.def 2012/11/06 v6.83m Hyperref driver for pdfTeX +rerunfilecheck.sty 2011/04/15 v1.7 Rerun checks for auxiliary files (HO) +supp-pdf.mkii + color.sty 2005/11/14 v1.0j Standard LaTeX Color (DPC) + color.cfg 2007/01/18 v1.5 color configuration of teTeX/TeXLive + nameref.sty 2012/10/27 v2.43 Cross-referencing by name of section +gettitlestring.sty 2010/12/03 v1.4 Cleanup title references (HO) + 47464-t.out + 47464-t.out + umsa.fd 2013/01/14 v3.01 AMS symbols A + umsb.fd 2013/01/14 v3.01 AMS symbols B + omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions +./images/fig_p30.pdf +./images/fig_p70.pdf +./images/fig_p79.pdf +./images/fig_p97.pdf +./images/fig_p102.pdf +./images/fig_p113.pdf +./images/fig_p123.pdf +./images/fig_p124.pdf + *********** + +Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 6204. + +Package rerunfilecheck Info: File `47464-t.out' has not changed. +(rerunfilecheck) Checksum: 941BAAB177EAA6E2EBF87B376707898A;1582. +Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 6204. + ) +Here is how much of TeX's memory you used: + 7604 strings out of 493304 + 106244 string characters out of 6139872 + 214221 words of memory out of 5000000 + 10478 multiletter control sequences out of 15000+600000 + 17658 words of font info for 68 fonts, out of 8000000 for 9000 + 959 hyphenation exceptions out of 8191 + 28i,12n,44p,345b,485s stack positions out of 5000i,500n,10000p,200000b,80000s +</usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></us +r/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/sha +re/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/share/tex +live/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi12.pfb></usr/share/texlive/t +exmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr/share/texlive/texmf-di +st/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/share/texlive/texmf-dist/font +s/type1/public/amsfonts/cm/cmr12.pfb></usr/share/texlive/texmf-dist/fonts/type1 +/public/amsfonts/cm/cmr17.pfb></usr/share/texlive/texmf-dist/fonts/type1/public +/amsfonts/cm/cmr8.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfont +s/cm/cmsy10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/c +msy6.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pf +b></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb></usr +/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/share +/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti12.pfb></usr/share/texli +ve/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/share/texlive/tex +mf-dist/fonts/type1/public/amsfonts/cm/cmtt8.pfb> +Output written on 47464-t.pdf (209 pages, 1116135 bytes). +PDF statistics: + 1651 PDF objects out of 1728 (max. 8388607) + 1379 compressed objects within 14 object streams + 479 named destinations out of 1000 (max. 500000) + 193 words of extra memory for PDF output out of 10000 (max. 10000000) + |