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diff --git a/old/68530-0.txt b/old/68530-0.txt deleted file mode 100644 index 11b9f94..0000000 --- a/old/68530-0.txt +++ /dev/null @@ -1,9313 +0,0 @@ -The Project Gutenberg eBook of Crystals, by A. E. H. Tutton - -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 -will have to check the laws of the country where you are located before -using this eBook. - -Title: Crystals - -Author: A. E. H. Tutton - -Release Date: July 15, 2022 [eBook #68530] - -Language: English - -Produced by: Richard Tonsing and the Online Distributed Proofreading - Team at https://www.pgdp.net (This file was produced from - images generously made available by The Internet Archive) - -*** START OF THE PROJECT GUTENBERG EBOOK CRYSTALS *** - - - - - - INTERNATIONAL SCIENTIFIC SERIES - - VOLUME XCVIII. - - DIRECT REPRODUCTIONS OF AUTOCHROME PHOTOGRAPHS OF SCREEN PICTURES IN - POLARISED LIGHT. - -[Illustration: - - FIG. 90.—Screen Picture in Polarised Light, with Nicols crossed, of a - thick Plate perpendicular to the Axis of a naturally twinned Crystal - of Quartz, the left half being of right-handed Quartz and the right - half of alternately left and right-handed Quartz, the Planes of - Demarcation being oblique to the Plate. -] - -[Illustration: - - FIG. 97.—Crystals of Benzoic Acid in the Act of Growth, as seen on the - Screen in Polarised Light with crossed Nicols. -] - - THE INTERNATIONAL SCIENTIFIC SERIES - - - - - CRYSTALS - - - BY - A. E. H. TUTTON - - D.Sc., M.A. (NEW COLLEGE, OXON.), F.R.S. - - VICE-PRESIDENT OF THE MINERALOGICAL SOCIETY MEMBER OF THE COUNCILS OF - THE CHEMICAL SOCIETY AND THE BRITISH ASSOCIATION FOR THE ADVANCEMENT OF - SCIENCE - - - WITH 120 ILLUSTRATIONS - - - LONDON - KEGAN PAUL, TRENCH, TRÜBNER & CO. L^{TD} - DRYDEN HOUSE, GERRARD STREET, W. - 1911 - - - - - PREFACE - - -The idea underlying this book has been to present the phenomena of -crystallography to the general reading public in a manner which can be -comprehended by all. In the main the sequence is that of the author’s -evening discourse to the British Association at their meeting at -Winnipeg in the summer of 1909. It is hoped, however, that the book -combines the advantages of sufficient amplification of the story there -told to make it an adequately detailed account of the development of the -subject, and of the immense progress which has been made in it during -recent years, with a full description of the numerous experimental -illustrations given in the lecture, involving some of the most beautiful -phenomena displayed by crystals in polarised light. Such an account has -not been otherwise published, the brief abstract appearing in the Report -of the British Association for 1909 giving no account of the -experiments, which were a feature of the lecture, owing to the -employment of a fine projection polariscope of more or less novel -construction, and including two magnificent large Nicol prisms, a pair -of the original ones made by Ahrens. The author has been frequently -requested to publish a fuller account of this discourse, and as the -general plan of it so fully embodies the present aspect of this -fascinating science, it was determined, when invited by the publishers -to write a generally readable book on “Crystals,” to comply with these -requests. - -There is also included an account of the remarkable work of Lehmann and -his fellow workers on “Liquid Crystals,” and the bearing of these -discoveries on the nature of crystal structure is discussed in so far as -the experimental evidence has gone. Similarly, the theory of Pope and -Barlow, connecting crystalline structure with the chemical property of -valency, is referred to and explained, as this theory has called forth -deep and widespread interest. In both cases, however, the author has -been careful to avoid any expression of opinion on purely theoretical -questions for which there is as yet no definite experimental evidence, -and has confined himself strictly to indicating how far such interesting -theories are supported by actual experimental facts. - -No forbidding mathematical formulæ and no unessential technical terms -will be found in the book, the aim of the author being to make any -ordinarily cultured reader feel at the conclusion that the story has -been readily comprehensible, and that crystallography is not the -abstruse and excessively difficult subject which it has so generally -been supposed to be, but that, on the contrary, it is both simple and -straightforward, and full of the most enthralling interest, as well for -the exquisite phenomena with which it deals, as for the exceedingly -important bearing which it has on the nature, both chemical and -physical, of solid matter. - -If any of its readers should be so impressed with the value of work in -this domain of science as to be desirous of joining the very thin ranks -of the few who are engaged in it, they will find a guide to practical -goniometry and to the experimental investigation of crystals in all its -branches and details, as well as the necessary theoretical help, in the -author’s book on “Crystallography and Practical Crystal Measurement” -(Macmillan & Co., 1911), and also an account of the author’s own -contributions to the subject in a monograph entitled “Crystalline -Structure and Chemical Constitution” (Macmillan & Co., 1910). - - A. E. H. TUTTON. - - _January 1911._ - - - - - CONTENTS - - - PAGE - - Preface v - - CHAPTER - - I. Introduction 1 - - II. The Masking of Similarity of Symmetry and Constancy of - Angle by Difference of Habit, and its Influence on Early - Studies of Crystals 10 - - III. The prescient Work of the Abbé Haüy 22 - - IV. The Seven Styles of Crystal Architecture 33 - - V. How Crystals are Described. The Simple Law limiting the - Number of possible Forms 50 - - VI. The Distribution of Crystal Faces in Zones, and the Mode - of Constructing a Plan of the Faces 60 - - VII. The Work of Eilhardt Mitscherlich and his Discovery of - Isomorphism 70 - - VIII. Morphotropy as distinct from Isomorphism 98 - - IX. The Crystal Space-Lattice and its Molecular Unit Cell. The - 230 Point-Systems of Homogeneous Crystal Structure 111 - - X. Law of Variation of Angles in Isomorphous Series. Relative - Dimensions of Unit Cells. Fixity of Atoms in Crystal 121 - - XI. The Explanation of Polymorphism and the Relation between - Enantiomorphism and Optical Activity 133 - - XII. Effect of the Symmetry of Crystals on the Passage of Light - through them. Quartz, Calcite, and Gypsum as Examples 162 - - XIII. Experiments in Convergent Polarised Light with Quartz, as - an Example of Mirror-Image Symmetry and its accompanying - Optical Activity 183 - - XIV. Experiments with Quartz and Gypsum in Parallel Polarised - Light. General Conclusions from the Experiments with - Quartz 201 - - XV. How a Crystal Grows from a Solution 236 - - XVI. Liquid Crystals 255 - - XVII. The Chemical Significance of Crystallography. The Theory - of Pope and Barlow—Conclusion 283 - - Index 295 - - - - - CRYSTALS - - (INCLUDING LIQUID CRYSTALS) - - - - - CHAPTER I - INTRODUCTION. - - -It is a remarkable fact that no definition of life has yet been advanced -which will not apply to a crystal with as much veracity as to those -obviously animate objects of the animal and vegetable world which we are -accustomed to regard in the ordinary sense as “living.” A crystal -_grows_ when surrounded by a suitable environment, capable of supporting -it with its natural food, namely, its own chemical substance in the -liquid or vaporous state or dissolved in a solvent. Moreover, when a -crystal is broken, and then surrounded with this proper environment, it -grows much more rapidly at the broken part than elsewhere, repairing the -damage done in a very short space of time and soon presenting the -appearance of a perfect crystal once more. In this respect it is quite -comparable with animal tissue, the wonderful recuperative power of which -after injury, exhibited by special growth at the injured spot, is often -a source of such marvel to us. Indeed, a crystal may be broken in half, -and yet each half in a relatively very brief interval will grow into a -crystal as large as the original one again. The longevity and virility -of the spores and seeds of the vegetable kingdom have been the themes of -frequent amazement, although many of the stories told of them have been -unable to stand the test of strict investigation. The virility of a -crystal, however, is unchanged and permanent. - -A crystal of quartz, rock-crystal, for instance—detached, during the -course of the disintegration of the granitic rock of which it had -originally formed an individual crystal, by the denuding influences at -work in nature thousands of years ago, subsequently knocked about the -world as a rounded sand grain, blown over deserts by the wind, its -corners rounded off by rude contact with its fellows, and subjected to -every variety of rough treatment—may eventually in our own day find -itself in water containing in solution a small amount of the material of -which quartz is composed, silicon dioxide SiO_{2}. No sooner is this -favourable environment for continuing its crystallisation presented to -it, than, however old it may be, it begins to sprout and grow again. It -becomes surrounded in all probability by a beautiful coating of -transparent quartz, with exterior faces inclined at the exact angles of -quartz, although no sign of exterior faces had hitherto persisted -through all the stages of its varied adventures. Or it may grow chiefly -at two or three especially favourable places, and in the course of a few -weeks, under suitable conditions, at each place a perfect little quartz -crystal will radiate out from the sand grain, composed of a miniature -hexagonal prism terminated by the well-known pyramid, really consisting -of a pair of trigonal (rhombohedral) pyramids more or less equally -developed, and together producing an apparently hexagonal one. Four such -grains of sand, from which quartz crystals are growing, are shown in -Fig. 1, as they appear under a microscope magnifying about fifty -diameters. One of them shows a perfectly developed doubly terminated -crystal of quartz growing from the tip of a singly terminated one, -attached to and growing directly out of the grain. - -[Illustration: - - FIG. 1.—Sand Grains with Quartz Crystals growing from them. -] - -This marvellously everlasting power possessed by a crystal, of silent -imperceptible growth, that is, of adding to its own regular structure -further accretions of infinitesimal particles, the chemical molecules, -of its own substance, is one of the strangest functions of solid matter, -and one of the fundamental facts of science which is rarely realised, -compared with many of the more obvious phenomena of nature. - -A crystal in the ordinary sense of the word is solid matter in its most -perfectly developed and organised form. It is composed of the chemical -molecules of some definitely constituted substance, which have been laid -down in orderly sequence, in accordance with a specific architectural -plan peculiar to that particular chemical substance. The physical -properties of the latter are such that it assumes the solid form at the -ordinary temperature and pressure, leaving out of consideration for the -present the remarkable viscous and liquid substances which will be -specially dealt with in Chapter XVI. of this book, and which are -currently known as “liquid crystals.” This term is not perhaps a very -appropriate one. For the word “crystal” had much better be left to -convey the idea of rigidity of polyhedral form and internal structure, -which is the very basis of crystal measurement. - -The solid crystal may have been produced during the simple act of -congealment from the liquid state, on the cooling of the heated -liquefied substance to the ordinary temperature. Sulphur, for instance, -is well-known to crystallise in acicular crystals belonging to the -monoclinic system under such conditions, a characteristic crop being -shown in Fig. 2 (Plate I.); they were formed within an earthenware -crucible in which the fusion had occurred, and became revealed on -pouring out the remainder of the liquid sulphur when the crystallisation -had proceeded through about one-half of the original amount of the -“melt.” - -[Illustration: - - _PLATE I._ - - FIG. 2.—Monoclinic Acicular Crystals of Sulphur produced by - Solidification of Liquid. -] - -[Illustration: - - FIG. 3.—Octahedral Crystals of Arsenious Oxide produced by - Condensation of Vapour. - - CRYSTALS FORMED BY DIFFERENT PROCESSES. -] - -[Illustration: - - _PLATE II._ - - FIG. 4.—Cubic Octahedral Crystals of Potash Alum growing from - Solution. -] - -[Illustration: - - FIG. 10.—Micro-Chemical Crystals of Gypsum (Calcium Sulphate) produced - by Slow Precipitation (see p. 14). - - CRYSTALS FORMED BY DIFFERENT PROCESSES. -] - -Or the substance may be one which passes directly from the gaseous to -the solid condition, on the cooling of the vapour from a temperature -higher than the ordinary down to the latter, under atmospheric pressure. -Oxide of arsenic, As_{2}O_{3}, is a substance exhibiting this property -characteristically, and Fig. 3 (Plate I.) is a reproduction of a -photograph of crystals of this substance thus produced. The white solid -oxide was heated in a short test tube over a Bunsen flame, and the -vapour produced was allowed to condense on a microscope glass slip, and -the result examined under the microscope, using a 1½ inch objective. -Fig. 3 represents a characteristic field of the transparent octahedral -crystals. - -Or again, the crystal may have been deposited from the state of solution -in a solvent, in which case it is a question of the passage of the -substance from the liquid to the solid condition, complicated by the -presence of the molecules of the solvent, from which the molecules of -the crystallising solid have to effect their escape. Fig. 4 (Plate II.) -represents crystals of potash alum, for instance, growing from a drop of -saturated solution on a glass slip placed on the stage of the -microscope, the drop being spread within a hard ring of gold size and -under a cover-glass, in order to prevent rapid evaporation and avoid -apparent distortion by the curvature of an uncovered drop. The crystals -are of octahedral habit like those of oxide of arsenic, but many of them -also exhibit the faces of the cube. - -In any case, however it may be erected, the crystal edifice is produced -by the regular accretion of molecule on molecule, like the bricks or -stone blocks of the builder, and in accordance with an architectural -plan more elaborate and exact than that of any human architect. This -plan is that of one of the thirty-two classes into which crystals can be -naturally divided with respect to their symmetry. Which specific one is -developed, and its angular dimensions, are traits characteristic of the -substance. The thirty-two classes of crystals may be grouped in seven -distinctive systems, the seven styles of crystal architecture, each -distinguished by its own elements of symmetry. - -A crystal possesses two further fundamental properties besides its style -of architecture. The first is that it is bounded externally by plane -faces, arranged on the definite geometrical plan just alluded to and -mutually inclined at angles which are peculiar to the substance, and -which are, therefore, absolutely constant for the same temperature and -pressure. The second is that a crystal is essentially a homogeneous -solid, its internal structure being similar throughout, in such wise -that the arrangement about any one molecule is the same as about every -other. This structure is, in fact, that of one of the 230 homogeneous -structures ascertained by geometricians to be possible to crystals with -plane faces. The first property, that of the planeness of the crystal -faces, and their arrangement with geometrical symmetry, is actually -determined by the second, that of specific homogeneity. For, as with -human nature developed to its highest type, the external appearance is -but the expression of the internal character. - -When nature has been permitted to have fair play, and the crystal has -been deposited under ideal conditions, the planeness of its faces is -astonishingly absolute. It is fully equal to that attained by the most -skilled opticians after weeks of patient labour, in the production of -surfaces on glass or other materials suitable for such delicate optical -experiments as interference-band production, in which a distortion equal -to one wave-length of light would be fatal. In all such cases of ideal -deposition, those interfacial angles on the crystal which the particular -symmetry developed requires to be equal actually are so, to this same -high degree of refinement. This fact renders possible exceedingly -accurate crystal measurement, that is, the determination of the angles -of inclination of the faces to each other, provided refined measuring -instruments (goniometers), pure chemical substances, and the means of -avoiding disturbance, either material or thermal, during the deposition -of the crystal, are available. - -The study of crystals naturally divides itself into two more or less -distinct but mutually very helpful branches, and equally intimately -connected with the internal structure of crystals, namely, one which -concerns their exterior configuration and the structural morphology of -which it is the eloquent visible expression, and another which relates -to their optical characters. For the latter are so definitely different -for the different systems of crystal symmetry that they afford the -greatest possible help in determining the former, and give the casting -vote in all cases of doubt left after the morphological investigation -with the goniometer. It is, of course, their brilliant reflection and -refraction of light, with production of numerous scintillations of -reflected white light and of refracted coloured spectra, which endows -the hard and transparent mineral crystals, known from time immemorial as -gem-stones, with their attractive beauty. Indeed, their outer natural -faces are frequently, and unfortunately usually, cut away most -sacrilegiously by the lapidary, in order that by grinding and polishing -on them still more numerous and evenly distributed facets he may -increase to the maximum the magnificent play of coloured light with -which they sparkle. - -An interesting and very beautiful lecture experiment was performed by -the author in a lecture a few years ago at the Royal Institution, which -illustrated in a striking manner this fact that the light reaching the -eye from a crystal is of two kinds, namely, white light reflected from -the exterior faces and coloured light which has penetrated the crystal -substance and emerges refracted and dispersed as spectra. Two powerful -beams of light from a pair of widely separated electric lanterns were -concentrated on a cluster of magnificent large diamonds, kindly lent for -the purpose by Mr Edwin Streeter, and arranged in the shape of a crown, -it being about the time of the Coronation of His late Majesty King -Edward VII. The effect was not only to produce a blaze of colour about -the diamonds themselves, but also to project upon the ceiling of the -lecture theatre numerous images in white light of the poles of the -electric arc, derived by reflection from the facets, interspersed with -equally numerous coloured spectra derived from rays which had penetrated -the substance of the diamonds, and had suffered both refraction and -internal reflection. - - - - - CHAPTER II - THE MASKING OF SIMILARITY OF SYMMETRY AND CONSTANCY OF ANGLE BY - DIFFERENCE OF HABIT, AND ITS INFLUENCE ON EARLY STUDIES OF CRYSTALS. - - -[Illustration: - - FIG. 5.—Natural Rhombohedron of Iceland Spar with Subsidiary Faces. -] - -Nothing is more remarkable than the great variety of geometrical shapes -which the crystals of the same substance, derived from different -localities or produced under different conditions, are observed to -display. One of the commonest of minerals, calcite, carbonate of lime, -shows this feature admirably; the beautiful large rhombohedra from -Iceland, illustrated in Fig. 5, or the hexagonal prisms capped by low -rhombohedra from the Bigrigg mine at Egremont in Cumberland, shown in -Fig. 6, appear totally different from the “dog-tooth spar” so -plentifully found all over the world, a specimen of which from the same -mine is illustrated in Fig. 7. No mineral specimens could well appear -more dissimilar than these represented on Plate III. in Figs. 6 and 7, -when seen side by side in the mineral gallery of the British Museum -(Natural History) at South Kensington. But all are composed of similar -chemical molecules of calcium carbonate, CaCO_{3}; and when the three -kinds of crystals are investigated they are found to be identical in -their crystalline system, the trigonal, and indeed further as to the -subdivision or class of that system, which has come to be called the -calcite class from the importance of this mineral. - -[Illustration: - - _PLATE III._ - - FIG. 6.—Hexagonal Prisms of Calcite terminated by Rhombohedra. -] - -[Illustration: - - FIG. 7.—Scalenohedral Crystals of Calcite, “Dog-tooth Spar.” - - CRYSTALS OF CALCITE FROM THE SAME MINE, ILLUSTRATING DIVERSITY OF - HABIT. - - (Photographed from Specimens in the Natural History Department of the - British Museum, by kind permission.) -] - -Moreover, many of the same faces, that is, faces having the same -relation to the symmetry, are present on all three varieties, the -“forms” to which they equally belong being the common heritage of -calcite wherever found. A “form” is the technical term for a set of -faces having an equal value with respect to the symmetry. Thus the -prismatic form in Fig. 6 is the hexagonal prism, a form which is common -to the hexagonal and trigonal systems of symmetry, and the form -“indices” (numbers[1] inversely proportional to the intercepts cut off -from the crystal axes by the face typifying the form) of which are -{2̄1̄1}; the large development of this form confers the elongated -prismatic habit on the crystal. The terminations are faces of the flat -rhombohedron {110}. The pyramidal form of the dog-tooth spar shown in -Fig. 7 is the scalenohedron {20̄1}, and it is this form which confers -the tooth-like habit, so different from the hexagonal prism, upon this -variety of calcite. But many specimens of dog-tooth spar, notably those -from Derbyshire, consist of scalenohedra the middle portion of which is -replaced by faces of the hexagonal prism {2̄1̄1}, and the terminations -of which are replaced by the characteristic rhombohedron {100} of -Iceland spar; indeed, it is quite common to find crystals of calcite -exhibiting on the same individual all the forms which have been -mentioned, that is, those dominating the three very differently -appearing types. The author has quite recently measured such a crystal, -which, besides showing all these four forms well developed, also -exhibited the faces of two others of the well-known forms of calcite, -{3̄1̄1} and {310}, and a reproduction of a drawing of it to scale is -given in Fig. 8. Instead of indices the faces of each form bear a -distinctive letter; _m_ = {2̄1̄1}, _r_ = {100}, _e_ = {110}, _v_ = -{20̄1} (the faces of the scalenohedron are of somewhat small dimensions -on this crystal), _n_ = {3̄1̄1}, and _t_ = {310}. - -[Illustration: - - FIG. 8.—Measured Crystal of Calcite. -] - -It is obviously then the “habit” which is different in the three types -of calcite—Iceland spar, prismatic calc-spar, and dog-tooth -spar—doubtless owing to the different local circumstances of growth of -the mineral. Habit is simply the expression of the fact that a specific -“form,” or possibly two particular forms, is or are much more -prominently developed in one variety than in another. Thus the principal -rhombohedron _r_ = {100}, parallel to the faces of which calcite cleaves -so readily, is the predominating form in Iceland spar, while the -scalenohedron _v_ = {20̄1} is the habit-conferring form in dog-tooth -spar. Yet on the latter the rhombohedral faces are frequently developed, -blunting the sharp terminations of the scalenohedra, especially in -dog-tooth spar from Derbyshire or the Hartz mountains; and on the former -minute faces of the scalenohedron are often found, provided the -rhombohedron consists of the natural exterior faces of the crystal and -not of cleavage faces. In the same manner the prismatic crystals from -Egremont are characterised by two forms, the hexagonal prism _m_ = -{2̄1̄1} and the secondary rhombohedron _e_ = {110}, but both of these -forms, as we have seen on the actual crystal represented in Fig. 8, are -also found developed on other crystals of mixed habit. - -This illustration from the naturally occurring minerals might readily be -supplemented by almost any common artificial chemical preparation, -sulphate of potash for instance, K_{2}SO_{4}, the orthorhombic crystals -of which take the form of elongated prisms, even needles, on the one -hand, or of tabular plate-like crystals on the other hand, according as -the salt crystallises by the cooling of a supersaturated solution, or by -the slow evaporation of a solution which at first is not quite -saturated. In both cases, and in all such cases, whether of minerals or -chemical preparations, the same planes are present on the crystals of -the same substance, although all may not be developed on the same -individual except in a few cases of crystals particularly rich in faces; -and these same planes are inclined at the same angles. But their -relative development may be so very unlike on different crystals as to -confer habits so very dissimilar that the fact of the identity of the -substance is entirely concealed. - -[Illustration: - - FIG. 9.—Crystal of Gypsum. -] - -A further example may perhaps be given, that of a substance, hydrated -sulphate of lime, CaSO_{4}.2H_{2}O, which occurs in nature as the -beautiful transparent mineral gypsum or selenite—illustrated in Fig. 9, -and which is found in monoclinic crystals often of very large size—and -which may also be chemically prepared by adding a dilute solution of -sulphuric acid to a very dilute solution of calcium chloride. The -radiating groups of needles shown in Fig. 10 (Plate II.) slowly -crystallise out when a drop of the mixed solution is placed on a -microscope slip and examined under the microscope, using the one-inch -objective. These needles, so absolutely different in appearance from a -crystal of selenite, are yet similar monoclinic prisms, but in which the -prismatic form is enormously elongated compared with the other -(terminating) form. - -This difference of facial development, rendering the crystals of one and -the same substance from different sources so very unlike each other, was -apparently responsible for the very tardy discovery of the fundamental -law of crystallography, the constancy of the crystal angles of the same -substance. Gessner, sometime between the years 1560 and 1568, went so -far as to assert that not only are different crystals of the same -substance of different sizes, but that also the mutual inclinations of -their faces and their whole external form are dissimilar. - -What was much more obvious to the early students of crystals, and which -is, in fact, the most striking thing about a crystal after its regular -geometric exterior shape, was the obviously homogeneous character of its -internal structure. So many crystals are transparent, and so clear and -limpid, that it was evident to the earliest observers that they were at -least as homogeneous throughout as glass, and yet that at the same time -they must be endowed with an internal structure the nature of which is -the cause of both the exterior geometric regularity of form, so -different from the irregular shape of a lump of glass, and of the -peculiar effect on the rays of light which are transmitted through them. -From the earliest ages of former civilisations the behaviour of crystals -with regard to light has been known to be different for the different -varieties of gem-stones. - -About the year 1600 Cæsalpinus observed that sugar, saltpetre, and alum, -and also the sulphates of copper, zinc and iron, known then as blue, -white and green vitriol respectively, separate from their solutions in -characteristic forms. Had he not attributed this to the operation of an -organic force, in conformity with the curious opinion of the times -concerning crystals, he might have had the credit of being the pioneer -of crystallographers. The first two real steps in crystallography, -however, with which in our own historic times we are acquainted, were -taken in the seventeenth century within four years of each other, one -from the interior structural and the other from the exterior geometrical -point of view. For in 1665 Robert Hooke in this country made a study of -alum, which he appears to have obtained in good crystals, although he -was unacquainted with its true chemical composition. He describes in his -“Micrographia” how he was able to imitate the varying habits of the -octahedral forms of alum crystals by building piles of spherical musket -bullets, and states that all the various figures which he observed in -the many crystals which he examined could be produced from two or three -arrangements of globular particles. It is clear that the homogeneous -partitioning of space in a crystal structure by similar particles -building up the crystal substance was in Hooke’s mind, affording another -testimony to the remarkably prescient insight of our great countryman. - -Four years later, in 1669, Nicolaus Steno carried out in Florence some -remarkable measurements, considering the absence of proper instruments, -of the angles between the corresponding faces of different specimens of -rock-crystal (quartz, the naturally occurring dioxide of silicon, -concerning which there will be much to say later in this book), obtained -from different localities, and published a dissertation announcing that -he found these analogous angles all precisely the same. - -In the year 1688 the subject was taken up systematically by Guglielmini, -and in two memoirs of this date and 1705 he extended Steno’s conclusions -as to the constancy of crystal angles in the case of rock-crystal into a -general law of nature. Moreover, he began to speculate about the -interior structure of crystals, and, like Hooke, he took alum as his -text, and suggested that the ultimate particles possessed plane faces, -and were, in short, miniature crystals. He further announced the -constancy of the cleavage directions, so that to Guglielmini must be -awarded the credit for having, at a time when experimental methods of -crystallographic investigation were practically _nil_, discovered the -fundamental principles of crystallography. - -The fact that a perfect cleavage is exhibited by calcite had already -been observed by Erasmus Bartolinus in 1670, and in his “Experimenta -Crystalli Islandici” he gives a most interesting account of the great -discovery of immense clear crystals of calcite which had just been made -at Eskifjördhr in Iceland, minutely describing both their cleavage and -their strong double refraction. Huyghens in 1690 followed this up by -investigating some of these crystals of calcite still more closely, and -elaborated his laws of double refraction as the result of his studies. - -There now followed a century which was scarcely productive of any -further advance at all in our real knowledge of crystals. It is true -that Boyle in 1691 showed that the rapidity with which a solution cools -influences the habit of the crystals which are deposited from it. But -neither Boyle, with all his well-known ability, so strikingly displayed -in his work on the connection between the volume of a gas and the -pressure to which it is subjected, nor his lesser contemporaries Lemery -and Homberg, who produced and studied the crystals of several series of -salts of the same base with different acids, appreciated the truth of -the great fact discovered by Guglielmini, that the same substance always -possesses the same crystalline form the angles of which are constant. -Even with the growth of chemistry in the eighteenth century, the opinion -remained quite general that the crystals of the same substance differ in -the magnitude of their angles as well as in the size of their faces. - -We begin to perceive signs of progress again in the year 1767, when -Westfeld made the interesting suggestion that calcite is built up of -rhombohedral particles, the miniature faces of which correspond to the -cleavage directions. This was followed in 1780 by a treatise “De formis -crystallorum” by Bergmann and Gahn of Upsala, in which Guglielmini’s law -of the constancy of the cleavage directions was reasserted as a general -one, and intimately connected with the crystal structure. It was in this -year 1780 that the contact goniometer was invented by Carangeot, -assistant to Romé de l’Isle in Paris, and it at once placed at the -disposal of his master a weapon of research far superior to any -possessed by previous observers. - -[Illustration: - - FIG. 11.—Contact Goniometer as used by Romé de l’Isle. -] - -In his “Crystallographie,” published in Paris in 1783, Romé de l’Isle -described a very large number of naturally occurring mineral crystals, -and after measuring their angles with Carangeot’s goniometer he -constructed models of no less than 500 different forms. Here we have -work based upon sound measurement, and consequently of an altogether -different and higher value than that which had gone before. It was the -knowledge that his master desired to faithfully reproduce the small -natural crystals which he was investigating, on the larger scale of a -model, that led Carangeot to invent the contact goniometer, and thus to -make the first start in the great subject of goniometry. The principle -of the contact goniometer remains to-day practically as Carangeot left -it, and although replaced for refined work by the reflecting goniometer, -it is still useful when large mineral crystals have to be dealt with. An -illustration of a duplicate of the original instrument is shown in Fig. -11, by the kindness of Dr H. A. Miers. This duplicate was presented to -Prof. Buckland by the Duke of Buckingham in the year 1824, and is now in -the Oxford Museum. - -From the time that measurement of an accurate description was possible -by means of the contact goniometer, progress in crystallography became -rapid. Romé de l’Isle laid down the sound principle, as the result of -the angular measurements and the comparison of his accurate models with -one another, that the various crystal shapes developed by the same -substance, artificial or natural, were all intimately related, and -derivable from a primitive form, characteristic of the substance. He -considered that the great variety of form was due to the development of -secondary faces, other than those of the primitive form. He thus -connected together the work of previous observers, consolidated the -principles laid down by Guglielmini by measurements of real value, and -threw out the additional suggestion of a fundamental or primitive form. - -About the same time Werner was studying the principal forms of different -crystals of the same substance. The idea of a fundamental form appears -to have struck him also, and he showed how such a fundamental form may -be modified by truncating, bevelling, and replacing its faces by other -derived forms. His work, however, cannot possess the value of that of -Romé de l’Isle, as it was not based on exact measurement, and most of -all because Werner appears to have again admitted the fallacy that the -same substance could, in the ordinary way, and not in the sense now -termed polymorphism, exhibit several different fundamental forms. - -But a master mind was at hand destined definitely to remove these doubts -and to place the new science on a firm basis. An account of how this was -achieved is well worthy of a separate chapter. - - - - - CHAPTER III - THE PRESCIENT WORK OF THE ABBÉ HAÜY. - - -The important work of Romé de l’Isle had paved the way for a further and -still greater advance which we owe to the University of Paris, for its -Professor of the Humanities, the Abbé Réné Just Haüy, a name ever to be -regarded with veneration by crystallographers, took up the subject -shortly after Romé de l’Isle, and in 1782 laid most important results -before the French Academy, which were subsequently, in 1784, published -in a book, under the auspices of the Academy, entitled “Essai d’une -Théorie sur la Structure des Crystaux.” The author happens to possess, -as the gift of a kind friend, a copy of the original issue of this -highly interesting and now very rare work. It contains a brief preface, -dated the 26th November 1783, signed by the Marquis de Condorcet, -perpetual secretary to the Academy (who, in 1794, fell a victim to the -French revolution), to the effect that the Academy had expressed its -approval and authorised the publication “under its privilege.” - -The volume contains six excellent plates of a large number of most -careful drawings of crystals, illustrating the derivation from the -simple forms, such as the cube, octahedron, dodecahedron, rhombohedron, -and hexagonal prism, of the more complicated forms by the symmetrical -replacement of edges and corners, together with the drawings of many -structural lattices. In the text, Haüy shows clearly how all the -varieties of crystal forms are constructed according to a few simple -types of symmetry; for instance, that the cube, octahedron, and -dodecahedron all have the same high degree of symmetry, and that the -apparently very diverse forms shown by one and the same substance are -all referable to one of these simple fundamental or systematic forms. -Moreover, Haüy clearly states the laws which govern crystal symmetry, -and practically gives us the main lines of symmetry of five of the seven -systems as we now classify them, the finishing touch having been -supplied in our own time by Victor von Lang. - -Haüy further showed that difference of chemical composition was -accompanied by real difference of crystalline form, and he entered -deeply into chemistry, so far as it was then understood, in order to -extend the scope of his observations. It must be remembered that it was -only nine years before, in 1774, that Priestley had discovered oxygen, -and that Lavoisier had only just (in the same year as Haüy’s paper was -read to the Academy, 1782) published his celebrated “Elements de -Chimie”; and further, that Lavoisier’s memoir “Reflexions sur le -Phlogistique” was actually published by the Academy in the same year, -1783, as that in which this book was written by Haüy. Moreover, it was -also in this same year, 1783, that Cavendish discovered the compound -nature of water. - -Considering, therefore, all these facts, it is truly surprising that -Haüy should have been able to have laid so accurately the foundations of -the science of crystallography. That he undoubtedly did so, thus -securing to himself for all time the term which is currently applied to -him of “father of crystallography,” is clearly apparent from a perusal -of his book and of his subsequent memoirs. - -The above only represents a small portion of Haüy’s achievements. For he -discovered, besides, the law of rational indices, the generalisation -which is at the root of crystallographic science, limiting, as it does, -the otherwise infinite number of possible crystal forms to comparatively -few, which alone are found to be capable of existence as actual -crystals. The essence of this law, which will be fully explained in -Chapter V., is that the relative lengths intercepted along the three -principal axes of the crystal, by the various faces other than those of -the fundamental form, the faces of which are parallel to the axes, are -expressed by the simplest unit integers, 1, 2, 3, or 4, the latter being -rarely exceeded and then only corresponding to very small and altogether -secondary faces. - -This discovery impressed Haüy with the immense influence which the -structure of the crystal substance exerts on the external form, and how, -in fact, it determines that form. For the observations were only to be -explained on the supposition that the crystal was built up of structural -units, which he imagined to be miniature crystals shaped like the -fundamental form, and that the faces were dependent on the step-like -arrangement possible to the exterior of such an assemblage. This brought -him inevitably to the intimate relation which cleavage must bear to such -a structure, that it really determined the shape of, and was the -expression of the nature of, the structural units. Thus, before the -conception of the atomic theory by Dalton, whose first paper (read 23rd -October 1803), was published in the year 1803 in the Proceedings of the -Manchester Literary and Philosophical Society, two years after the -publication of Haüy’s last work (his “Traité de Minéralogie,” Paris, -1801), Haüy came to the conclusion that crystals were composed of units -which he termed “_Molécules Intégrantes_,” each of which comprised the -whole chemical compound, a sort of gross chemical molecule. Moreover, he -went still further in his truly original insight, for he actually -suggested that the _molécules intégrantes_ were in turn composed of -“_Molécules Elémentaires_,” representing the simple matter of the -elementary substances composing the compound, and hinted further that -these elementary portions had properly orientated positions within the -_molécules intégrantes_. - -He thus not only nearly forestalled Dalton’s atomic theory, but also our -recent work on the stereometric orientation of the atoms in the molecule -in a crystal structure. Dalton’s full theory was not published until the -year 1811, in his epoch-making book entitled “A New System of Chemical -Philosophy,” although his first table of atomic weights was given as an -appendix to the memoir of 1803. Thus in the days when chemistry was in -the making at the hands of Priestley, Lavoisier, Cavendish, and Dalton -do we find that crystallography was so intimately connected with it that -a crystallographer well-nigh forestalled a chemist in the first real -epoch-making advance, a lesson that the two subjects should never be -separated in their study, for if either the chemist or the -crystallographer knows but little of what the other is doing, his work -cannot possibly have the full value with which it would otherwise be -endowed. - -The basis of Haüy’s conceptions was undoubtedly cleavage. He describes -most graphically on page 10 of his “Essai” of 1784 how he was led to -make the striking observation that a hexagonal prism of calcite, -terminated by a pair of hexagons normal to the prism axis, similar to -the prisms shown in Fig. 6 (Plate III.) except that the ends were flat, -showed oblique internal cleavage cracks, by enhancing which with the aid -of a few judicious blows he was able to separate from the middle of the -prism a kernel in the shape of a rhombohedron, the now well-known -cleavage rhombohedron of calcite. He then tried what kinds of kernels he -could get from dog-tooth spar (illustrated in Fig. 7) and other -different forms of calcite, and he was surprised to find that they all -yielded the same rhombohedral kernel. He subsequently investigated the -cleavage kernels of other minerals, particularly of gypsum, fluorspar, -topaz, and garnet, and found that each mineral yielded its own -particular kernel. He next imagined the kernels to become smaller and -smaller, until the particles thus obtained by cleaving the mineral along -its cleavage directions _ad infinitum_ were the smallest possible. These -miniature kernels having the full composition of the mineral he terms -“_Molécules Constituantes_” in the 1784 “Essai,” but in the 1801 -“Traité” he calls them “_Molécules Intégrantes_” as above mentioned. He -soon found that there were three distinct types of _molécules -intégrantes_, tetrahedra, triangular prisms, and parallelepipeda, and -these he considered to be the crystallographic structural units. - -[Illustration: - - FIG. 12. -] - -Having thus settled what were the units of the crystal structure, Haüy -adopted Romé de l’Isle’s idea of a primitive form, not necessarily -identical with the _molécule intégrante_, but in general a -parallelepipedon formed by an association of a few _molécules -intégrantes_, the parallelepipedal group being termed a “_Molécule -Soustractive_.” The primary faces of the crystal he then supposed to be -produced by the simple regular growth or piling on of _molécules -intégrantes_ or _soustractives_ on the primitive form. The secondary -faces not parallel to the cleavage planes next attracted his attention, -and these, after prolonged study, he explained by supposing that the -growth upon the primitive form eventually ceased to be complete at the -edges of the primary faces, and that such cessation occurred in a -regular step by step manner, by the suppression of either one, two, or -sometimes three _molécules intégrantes_ or _soustractives_ along the -edge of each layer, like a stepped pyramid, the inclination of which -depends on how many bricks or stone blocks are intermitted in each layer -of brickwork or masonry. Fig. 12 will render this quite clear, the face -AB being formed by single block-steps, and the face CD by two blocks -being intermitted to form each step. The plane AB or CD containing the -outcropping edges of the steps would thus be the secondary plane face of -the crystal, and the _molécules_ _intégrantes_ or _soustractives_ (the -steps can only be formed by parallelepipedal units) being -infinitesimally small, the re-entrant angles of the steps would be -invisible and the really furrowed surface appear as a plane one. Haüy is -careful to point out, however, that the crystallising force which causes -this stepped development (or lack of development) is operative from the -first, for the minutest crystals show secondary faces, and often better -than the larger crystals. - -[Illustration: - - FIG. 13. -] - -An instance of a mineral with tetrahedral _molécules intégrantes_ Haüy -gives in tourmaline, and the primitive form of tourmaline he considered -to be a rhombohedron, conformably to the well-known rhombohedral -cleavage of the mineral, made up of six tetrahedra. Again, hexagonal -structures formed by three prismatic cleavage planes inclined at 60° are -considered by him as being composed of _molécules intégrantes_ of the -form of 60° triangular prisms, or _molécules soustractives_ of the shape -of 120° rhombic prisms, each of the latter being formed by two -_molécules intégrantes_ situated base to base. This will be clear from -Figs. 13 and 14, the former representing the structure as made up of -equilateral prismatic structural units, and the latter portraying the -same structure but composed of 120°-parallelepipeda by elimination of -one cleavage direction; each unit in the latter case possesses double -the volume of the triangular one, and being of parallelepipedal section -is capable of producing secondary faces when arranged step-wise, whereas -the triangular structure is not. The points at the intersections in -these diagrams should for the present be disregarded; they will shortly -be referred to for another purpose. - -[Illustration: - - FIG. 14. -] - -Probably, the most permanent and important of Haüy’s achievements was -the discovery of the law of rational indices. At first this only took -the form of the observation of the very limited number of rows of -_molécules intégrantes_ or _soustractives_ suppressed. In introducing it -on page 74 of his 1784 “Essai” he says: “_Quoique je n’aie observé -jusqu’ici que des décroissemens qui se sont par des soutractions d’une -ou de deux rangées de molécules, et quelquefois de trois rangées, mais -très rarement, il est possible qu’il se trouve des crystaux dans -lesquels il y ait quatre ou cinq rangées de molécules supprimées à -chaque décroissement, et même un plus grand nombre encore. Mais ces cas -me semblent devoir être plus rares, à proportion que le nombre des -rangées soutraites sera plus considérable. On conçoit donc comment le -nombre des formes secondaires est néçessairement limité._” - -The essential difference between Haüy’s views and our present ones, -which will be explained in Chapter IX., is that Haüy takes cleavage -absolutely as his guide, and considers the particles, into which the -ultimate operation of cleavage divides a crystal, as the solid -structural units of the crystal, the unit thus having the shape of at -least the _molécule intégrante_. Now every crystalline substance does -not develop cleavage, and others only develop it along a single plane, -or along a couple of planes parallel to the same direction, that of -their intersection and of the axis of the prism which two such cleavages -would produce, and which prism would be of unlimited length, being -unclosed. - -Again, in other cases cleavage, such as the octahedral cleavage of -fluorspar, yields octahedral or tetrahedral _molécules intégrantes_ -which are not congruent, that is to say, do not fit closely together to -fill space, as is the essence of Haüy’s theory. Hence, speaking -generally, partitioning by means of cleavage directions does not -essentially and invariably yield identical plane-faced molecules which -fit together in contact to completely fill space, although in the -particular instances chosen from familiar substances by Haüy it often -happens to do so. Haüy’s theory is thus not adequately general, and the -advance of our knowledge of crystal forms has rendered it more and more -apparent that Haüy’s theory was quite insufficient, and his _molécules -intégrantes_ and _soustractives_ mere geometrical abstractions, having -no actual basis in material fact; but that at the same time it gave us a -most valuable indication of where to look for the true conception. - -This will be developed further into our present theory of the -homogeneous partitioning of space, in Chapter IX. But it may be stated -here, in concluding our review of the pioneer work of Haüy, that in the -modern theory all consideration of the shape of the ultimate structural -units is abandoned as unnecessary and misleading, and that each chemical -molecule is considered to be represented by a point, which may be either -its centre of gravity, a particular atom in the molecule (for we are now -able in certain cases to locate the orientation of the spheres of -influence of the elementary atoms in the chemical molecules), or a -purely representative point standing for the molecule. The only -condition is that the points chosen within the molecules shall be -strictly analogous, and similarly orientated. The dots at the -intersections of the lines in Figs. 13 and 14 are the representative -points in question. We then deal with the distances between the points, -the latter being regarded as molecular centres, rather than with the -dimensions of the cells themselves regarded as solid entities. We thus -avoid the as yet unsolved question of how much is matter and how much is -interspace in the room between the molecular centres. In this form the -theory is in conformity with all the advances of modern physics, as well -as of chemistry. And with this reservation, and after modifying his -theory to this extent, one cannot but be struck with the wonderful -perspicacity of Haüy, for he appears to have observed and considered -almost every problem with which the crystallographer is confronted, and -his laws of symmetry and of rational indices are perfectly applicable to -the theory as thus modernised. - - - - - CHAPTER IV - THE SEVEN STYLES OF CRYSTAL ARCHITECTURE. - - -It is truly curious how frequently the perfect number, seven, is endowed -with exceptional importance with regard to natural phenomena. The seven -orders of spectra, the seven notes of the musical octave, and the seven -chemical elements, together with the seven vertical groups to which by -their periodic repetition they give rise, of the “period” of -Mendeléeff’s classification of the elements, will at once come to mind -as cases in point. This proverbial importance of the number seven is -once again illustrated in regard to the systems of symmetry or styles of -architecture displayed by crystals. For there are seven such systems of -crystal symmetry, each distinguished by its own specific elements of -symmetry. - -It is only within recent years that we have come to appreciate what are -the real elements of symmetry. For although there are but seven systems, -there are no less than thirty-two classes of crystals, and these were -formerly grouped under six systems, on lines which have since proved to -be purely arbitrary and not founded on any truly scientific basis. It -was supposed that those classes in any system which did not exhibit all -the faces possible to the system owed this lack of development to the -suppression of one-half or three-quarters of the possible number, and -such classes were consequently called “hemihedral” and “tetartohedral” -respectively. As in the higher systems of symmetry there were usually -two or more ways in which a particular proportionate suppression of -faces could occur, it happened that several classes, and not merely -three—holohedral (possessing the full number of faces), hemihedral, and -tetartohedral—constituted each of these systems. - -Thanks largely to the genius of Victor von Lang, who was formerly with -us in England at the Mineral Department of the British Museum, and to -his successor there, Nevil Story Maskelyne, we have at last a much more -scientific basis for our classification of crystals, and one which is in -complete harmony with the now perfected theory of possible homogeneous -structures. Victor von Lang showed that the true elements of symmetry -are planes of symmetry and axes of symmetry. A crystal possessing a -plane of symmetry is symmetrical on both sides of that plane, both as -regards the number of the faces and their precise angular disposition -with respect to one another. - -It is quite possible, and even the usual case, that the relative -development of the faces, that is their actual sizes, may prevent the -symmetry from being at first apparent; but when we come to measure the -angles between the faces, by use of the reflecting goniometer, and to -plot their positions out on the surface of a sphere, or on a plane -representation of the latter on paper, the exceedingly useful -“stereographic projection,” we at once perceive the symmetry perfectly -plainly. - -[Illustration: - - FIG. 15.—Crystal of Potassium Nickel Sulphate. -] - -[Illustration: - - FIG. 16.—Projection of Potassium Nickel Sulphate and its Isomorphous - Analogues. -] - -Thus in Fig. 15 is represented a crystal of the salt potassium nickel -sulphate, K_{2}Ni(SO_{4})_{2}.6H_{2}O, belonging to the monoclinic -system of symmetry, and which, therefore, possesses only one plane of -symmetry. In Fig. 16 its stereographic projection is shown, in which -each face in one of the symmetrical halves is represented by a dot, the -plane of symmetry, parallel to the face _b_, being the plane of the -paper, so that each dot not on the circumference really represents two -symmetrical faces, one above and one below the paper, while the -circumferential dots represent faces perpendicular to the symmetry plane -and paper. The mode of arriving at such a useful projection, or plan of -the faces, will be discussed more fully later in Chapter VI. But for the -present purpose it will be sufficient to note that the right and left -halves of the crystal shown in Fig. 15 are obviously symmetrical to each -other, and that the plan of either half, projected on the dividing plane -of symmetry itself, may be taken as given in Fig. 16; that is, we may -imagine the crystal shown in Fig. 15 to be equally divided by a section -plane which is vertical and perpendicular to the paper when the latter -is held up behind the crystal and in front of the eye, this section -plane being the plane of symmetry and parallel to the face _b_ = (010). -It may thus be imagined as the plane of projection of Fig. 16. - -An axis of symmetry is a direction in the crystal such that when the -latter is rotated for an angle of 60°, 90°, 120°, or 180° around it, the -crystal is brought to look exactly as it did before such rotation. When -a rotation for 180° is necessary in order to reproduce the original -appearance, the axis is called a “digonal” axis of symmetry, for two -such rotations then complete the circle and bring the crystal back to -identity, not merely to similarity. When the rotation into a position of -similarity is for 120°, three such rotations are required to restore -identity, and the axis is then termed a “trigonal” one. Similarly, four -rotations to positions of similarity 90° apart are essential to complete -the restoration to identity, and the axis is then a “tetragonal” one, -each rotation of a right angle causing the crystal to appear as at -first, assuming, as in all cases, the ideal equality of development of -faces. Lastly, if 60° of rotation bring about similarity, six such -rotations are required in order to effect identity of position, and the -axis is known as a “hexagonal” one. - -Now, there is one system of symmetry which is characterised by the -presence of a single hexagonal axis of symmetry, and this is the -_hexagonal system_. A crystal of this system, one of the naturally -occurring mineral apatite, which has been actually measured by the -author, is shown in Fig. 17. There is another system, the chief property -of which is to possess a tetragonal axis of symmetry, and which is -therefore termed the _tetragonal system_. A tetragonal crystal of -anatase, titanium dioxide, TiO_{2}, which has likewise been measured on -the goniometer by the author, is shown in Fig. 18. And there is yet -another system, the trigonal, the chief attribute of which is the -possession of a single trigonal axis of symmetry, and which is -consequently named the trigonal system. In Fig. 19 is shown a crystal of -calcite, within which the directions of the three rhombohedral -crystallographic axes of the trigonal system, and that of the vertical -trigonal axis of symmetry, are indicated in broken-and-dotted lines. - -[Illustration: - - FIG. 17.—Measured Crystal of Apatite. -] - -[Illustration: - - FIG. 18.—Measured Crystal of Anatase. -] - -But there is one system of symmetry, the highest possible, and which has -already been referred to as the _cubic system_, which combines in itself -all but one (the hexagonal axis) of the elements of symmetry. Indeed, -not only does it possess a tetragonal, a trigonal, and a digonal axis of -symmetry, but also ten other symmetry axes; for these three -automatically involve altogether the presence of no less than three -tetragonal, four trigonal, and six digonal axes of symmetry, together -with nine planes of symmetry, twenty-two elements of symmetry being thus -present in all. - -The perfections of the cube, the simple lines of which are illustrated -in Fig. 20, as the expression of the highest kind of symmetry, with -angles all right angles and sides and edges all equal, were so fully -appreciated by the geometrical minds of the ancient Greek philosophers, -imbued with the innate love of symmetry characteristic of their nation, -that to them the cube became the emblem of perfection. We are reminded -of this interesting fact in the Book of Revelation, which, in describing -in its inimitable language the wonders of the Holy City, speaks of it as -“lying foursquare,” and attributes to it the properties of the cube, -that “The length and the breadth and the height of it are equal.” - -[Illustration: - - FIG. 19.—Crystal of Calcite. -] - -[Illustration: - - FIG. 20.—The Cube. -] - -[Illustration: - - FIG. 21.—The Hexakis Octahedron. -] - -The full symmetry of the cubic system is not realised, however, by a -study of the cube alone; we only appreciate it when we come to examine -the general form of the cubic system, that which is produced by starting -with a face oblique to all three axes, and with different amounts of -obliquity to each, and seeing how many repetitions of the face the -symmetry demands. The presence of such a face involves as a matter of -fact, when all the elements of symmetry are satisfied, the presence also -of no less than forty-seven others, symmetrically situated, the -forty-eight-sided figure produced being the hexakis octahedron shown in -Fig. 21, and which is occasionally actually found developed in nature as -the diamond. All diamonds do not by any means exhibit this form so -wonderfully rich in faces, but diamonds are from time to time found -which do show all the forty-eight faces well developed. - -[Illustration: - - FIG. 22.—Measured Crystal of Topaz. -] - -Besides these four more highly symmetrical systems or styles of crystal -architecture, a fifth, the _monoclinic system_, characterised by a -single plane of symmetry and one axis of digonal symmetry perpendicular -thereto, has already been alluded to, and a typical crystal illustrated -in Fig. 15. A sixth, the _rhombic system_, perhaps in some ways the most -interesting of all, and certainly so optically, possesses three -rectangular axes of symmetry, identical in direction with the -crystallographic axes, and three mutually rectangular planes of -symmetry, coincident with the axial planes and intersecting each other -in the axes. The lengths of the three crystal axes are unequal, however, -and herein lies the essential difference from the cube. A very typical -rhombic substance is topaz, a crystal of which, about three millimetres -in diameter, is shown very much enlarged in Fig. 22. Every face on this -crystal has been actually investigated on the goniometer, and the -interfacial angles measured. - -[Illustration: - - FIG. 23.—Measured Crystal of Copper Sulphate. -] - -Lastly, there is the seventh, the _triclinic system_, in which there are -neither planes nor axes of symmetry, but, even in its holohedral class, -only symmetry about the centre, each face having a parallel fellow. -Sulphate of copper, blue vitriol, CuSO_{4}.5H_{2}O, shows this type of -symmetry, or rather lack of it, very characteristically, and a crystal -of this beautiful deep blue salt, measured by the author, is represented -in Fig. 23. - -Hence, we have arrived logically at seven systems of symmetry or styles -of crystal architecture, distinguished by the nature of their essential -axes of symmetry, and the planes of symmetry which may accompany them. -Now the full degree of symmetry of each system may be reduced to a -certain minimum without lowering the system, and in all the systems but -the triclinic there are several definite stages of reduction before the -minimum is reached, each stage corresponding to one of the thirty-two -classes of crystals. Thus in the cubic system there are four classes -besides the holohedral, in the tetragonal six, in the hexagonal four, in -the trigonal six, in the rhombic and monoclinic two each, and in the -triclinic one. - -[Illustration: - - _PLATE IV._ - - FIG. 24.—Octahedra of Potassium Cadmium Cyanide. -] - -[Illustration: - - FIG. 25.—Octahedra of Cæsium Alum. - - CUBIC CRYSTALS GROWING FROM SOLUTION. -] - -We have thus attained at length to a truly scientific classification of -crystal forms, by using axes and planes of symmetry as _criteria_. There -is no occasion whatever to imagine suppression of faces in the classes -of lower than the holohedral or highest symmetry of any system. In these -classes it is simply the fact that less than the full number of elements -of symmetry possible to the system are present and characterise the -class, which still conforms, however, to the minimum symmetry absolutely -essential to the system. - -The drawings of crystals of the seven systems in the foregoing -illustrations will have given a correct idea of the nature of the -symmetry in each case. But now it may be much more interesting to -present a series of reproductions of photographs of some actual crystals -of the different systems. Such a series is given in Figs. 24 to 33, -Plates IV. to VIII. They were taken with the aid of the microscope, the -substances being crystallised from a slightly supersaturated solution in -each case, on a microscope slip. A ring of gold size was first laid on -the slip, and allowed to dry for several days. The drop of solution, in -the metastable supersaturated condition (corresponding to the region of -solubility which lies between the solubility and supersolubility curves, -Fig. 98, page 240), was placed in the middle of the ring, and -crystallisation just allowed to start, either owing to evaporation and -consequent production of the labile condition for spontaneous -crystallisation, or by access of a germ crystal from the air. It was -then covered with a cover-glass, which had the desired effect of -enclosing the solution in a parallelsided cell, a film of the thickness -of thick paper, suitable for undistorted microscopic observation and -photomicrography, and also the effect of arresting evaporation and -therefore the rapidity of the growth of the crystals, so that a -photomicrograph taken with the minimum necessary exposure was quite -sharp. - -The crystals shown in the accompanying photographic reproductions, Figs. -24 to 33 (Plates IV. to VIII.), as well as Fig. 4 (Plate II.), already -described, were thus photographed in the very act of slow growth, -employing a one-inch objective very much stopped down. Such photographs -are infinitely sharper and more beautifully and delicately shaded than -those taken of dry crystals. - -Fig. 24, Plate IV., represents cubic octahedra of the double cyanide of -potassium and cadmium, 2KCN.Cd(CN)_{2}, a salt which crystallises out in -relatively large and wonderfully transparent and well-formed single -octahedra on a micro-slip, and is particularly suitable for -demonstrating the character of this highest system, the cubic, of -crystal symmetry. Special development of the pair of faces of the -octahedron parallel to the glass surfaces has occurred, owing to greater -freedom of growth at the boundaries of these faces, as is usual in such -circumstances of deposition, but the other pairs of faces are quite -large enough to show their nature clearly. - -Fig. 25, on the same Plate IV., shows a slide of cæsium alum, -Cs_{2}SO_{4}.Al_{2}(SO_{4})_{3}.24H_{2}O, in which the octahedra are -smaller, and some of them, notably one in the centre of the field, are -perfectly proportioned. - -[Illustration: - - _PLATE V._ - - FIG. 26.—Octahedra of Ammonium Iron Alum crystallising on a Hair. -] - -[Illustration: - - FIG. 27.—Tetragonal Crystals of Potassium Ferrocyanide. - - CRYSTALS GROWING FROM SOLUTION. -] - -[Illustration: - - _PLATE VI._ - - FIG. 28.--Rhombic Crystals of Potassium Hydrogen Tartrate. -] - -[Illustration: - - FIG. 29.--Rhombic Crystals of Ammonium Magnesium Phosphate, showing - Special Growth along Line of Scratch. - - RHOMBIC CRYSTALS GROWING BY SLOW PRECIPITATION. -] - -Fig. 26, Plate V., represents octahedra of ammonium iron alum (formula -like that of cæsium alum, but with NH_{4} replacing Cs and Fe replacing -Al) crystallising on a hair. It illustrates the interesting manner in -which crystallisation will sometimes occur, under conditions of -quietude, when some object or other on which the crystals can readily -deposit themselves is present or introduced, such as a silk or cotton -thread, or a hair as in this case. - -Fig. 27, on the same Plate V., represents tetragonal crystals of -potassium ferrocyanide, K_{4}Fe(CN)_{6}, composed of tabular crystals -parallel to the basal pinakoid, bounded by faces of one order, first or -second, of tetragonal prism, the corners being modified at 45° by -smaller faces of the other order of tetragonal prism. - -Fig. 28, Plate VI., is a photograph of large rhombic crystals of -hydrogen potassium tartrate, HKC_{4}H_{4}O_{6}, obtained by addition of -tartaric acid to a dilute solution of potassium chloride. They are -rectangular rhombic prisms capped by pyramidal forms, and also modified -by other prismatic and domal forms. - -Fig. 29, also on Plate VI., represents another rhombic substance, -ammonium magnesium phosphate, NH_{4}MgPO_{4}.6H_{2}O, obtained by very -slow precipitation of a dilute solution of magnesium sulphate containing -ammonium chloride and ammonia with hydrogen disodium phosphate. It -illustrates in an interesting manner how, when a saturated solution is -kept quiet, and then the surface of the vessel containing it is -scratched by a needle point, a line of small crystals at once starts -forming along the line of scratch, even although the latter has made no -actual impression on the glass itself. Such a line of crystals will be -observed running across the middle of the slide. - -Fig. 30, Plate VII., shows a monoclinic substance, ammonium magnesium -sulphate (NH_{4})_{2}Mg(SO_{4})_{2}.6H_{2}O, which crystallises out -splendidly on a micro-slip. The field includes several very well-formed -typical crystals of the salt, which is one of the same exceedingly -important isomorphous series to which potassium nickel sulphate, Fig. -15, belongs; it is obtained by mixing solutions containing molecularly -equivalent quantities of ammonium and magnesium sulphates. The primary -monoclinic prism is the chief form, terminated by clinodome faces and -smaller strip-faces of the basal plane, the latter, however, being -occasionally the chief end form. Small pyramid faces are also seen here -and there modifying the solid angles. - -Another beautifully crystallising monoclinic substance is shown in the -next slide, Fig. 31, on the same Plate VII., namely, potassium sodium -carbonate, KNaCO_{3}.6H_{2}O, obtained from a solution of molecular -proportions of potassium and sodium carbonates. Numerous forms of the -monoclinic system are developed, on relatively large and perfectly -transparent and delicately shaded individuals. - -A triclinic substance is represented in Fig. 32, Plate VIII., potassium -ferricyanide, K_{6}F_{2}(CN)_{12}. The triply oblique nature of the -symmetry is clearly exhibited by this salt, the absence of any right -angles being very marked. - -[Illustration: - - _PLATE VII._ - - FIG. 30.—Monoclinic Crystals of Ammonium Magnesium Sulphate. -] - -[Illustration: - - FIG. 31.—Monoclinic Crystals of Sodium Potassium Carbonate. - - MONOCLINIC CRYSTALS GROWING FROM SOLUTION. -] - -[Illustration: - - _PLATE VIII._ - - FIG. 32.—Triclinic Crystals of Potassium Ferricyanide. -] - -[Illustration: - - FIG. 33.—Tetrahedral Crystals of Sodium Sulphantimoniate, Cubic Class - 28. - - CRYSTALS GROWING FROM SOLUTION. -] - -Fig. 33, also on Plate VIII., illustrates more particularly a class of -one of the systems, the cubic, which is of lower than holohedral (full) -systematic symmetry. This is the case also with hydrogen potassium -tartrate and ammonium magnesium phosphate, but the forms shown of those -salts on the slides represented in Figs. 28 and 29 are chiefly those -which are also common to the holohedral classes of their respective -systems, and the lower class symmetry is not emphasised. But here in -Fig. 33, representing Schlippe’s salt, sodium sulphantimoniate, -Na_{3}SbS_{4}.9H_{2}O, we have very clear development of the -tetrahedron, belonging to the lowest of the five classes (class 28) of -the cubic system. The crystals are almost all combinations of two -complementary tetrahedra, one of which is developed so very much more -than the other that the faces of the latter only appear as minute -replacements at the corners of the predominating tetrahedron. - -This is the last for the present of these fascinating growths of -crystals under the microscope, but three more will be given -subsequently, in Figs. 99 and 100, on Plate XXI., and Fig. 101, Plate -XI., to illustrate crystallisation from metastable and labile solutions. - -Fig. 34, Plate IX., represents another kind of phenomenon, equally -instructive. It shows a field in a crystal of quartz, as seen under the -same power of the microscope, a one-inch objective with small stop and -an ordinary low power eyepiece. Just above and to the left of the centre -of the field is a cavity, the shape of which is remarkable, for it is -that of a quartz crystal, a hexagonal prism terminated by rhombohedral -faces. The cavity is filled with a saturated solution of salt, except -for a bubble of water vapour, and a beautiful little cube of sodium -chloride which has crystallised out from the solution. This slide, -therefore, gives us an example of a natural cubic crystal, and also an -indication of the shape of quartz crystals, the cavity itself being a -kind of negative quartz crystal. The crystal in which it occurs must -have been formed very deep down in a reservoir of molten material -beneath a volcano, under the great pressure of superincumbent rock -masses. It was probably one of the quartz crystals of a granite rock -which had crystallised under these conditions. Almost every crystal of -quartz found in such granite rocks displays thousands of small cavities -filled with liquid and a bubble, although it is very rare to find one -with so good a cube of salt and having the configuration of a quartz -crystal for the shape of the cavity. Many such cavities, however, -contain as the liquid compressed carbonic acid, the very fact of the -carbonic acid being in the liquefied state affording ample evidence of -the pressure under which the crystal was formed. The proof that the -liquid is carbonic acid in these cases is afforded by the fact that when -the crystal is warmed to 32°C., the critical temperature of carbon -dioxide, under which it can no longer remain liquid, but must become a -gas, the bubble disappears and the cavity becomes filled with gas. -Carbonic acid cavities are readily recognised, inasmuch as the bubble is -extremely mobile, and is normally in a state of movement on the very -slightest provocation. - -[Illustration: - - _PLATE IX._ - - FIG. 34.—Liquid Cavities in quartz Crystal (Trigonal) containing - Saturated Solution and Cubic Crystals of Sodium Chloride. -] - -[Illustration: - - FIG. 36.—Two characteristic Forms of Snow Crystals (Trigonal). -] - -[Illustration: - - FIG. 35.—Negative Ice Crystals, or “Water Flowers,” in Ice. -] - -The liquid cavity in the remarkable quartz crystal illustrated in Fig. -34, and the bubble of vapour formed on cooling, and consequent -contraction of the liquid more than the solid quartz (the thermal -dilatation of liquids being usually greater than that of solids) when it -was no longer able to fill the cavity, remind one of the beautiful water -flowers formed for the contrary reason in ice on passing a beam of light -through a slab, owing to the warming effect of the accompanying heat -rays. Water crystallises like quartz, in the trigonal system, its normal -forms being the hexagonal prism and the rhombohedron. A slab of lake ice -is generally a huge crystal plate perpendicular to the trigonal axis, or -in the case of disturbed growth an interlacing mass of such crystals, -all perpendicular to the optic axis, the axis of the hexagonal prism and -of trigonal symmetry. When the heat rays from the lantern pass through -such a slab of ice, the surface of which is focussed on the screen by a -projecting lens, they cause the ice to begin to melt in numerous spots -in the interior of the slab simultaneously; and the structure of the -crystal is revealed by the operation occurring with production of -cavities taking the shape of hexagonal stars, which when focussed appear -on the screen as shown in Fig. 35. They are filled with water except for -a bubble (vacuole), which contains only water vapour. For the liquid -water occupies less room than did the ice from which it was produced, -owing to the well-known fact that water expands on freezing. This -abnormal expansion with cooling begins at the temperature of the maximum -density of water, 4° C., and proceeds steadily until the freezing point -0° is reached, when, at the moment of crystallisation, the mass suddenly -increases in volume by as much as 10 per cent. This expansive leap when -the molecules of water marshal themselves into the organised order of -the homogeneous structure, that of the space-lattice of the trigonal -(rhombohedral) system, is one of the most remarkable phenomena in -nature, and its exceptional character, so contrary to the usual -contraction on solidification of a liquid, is of vital moment to aquatic -life. For the layer of ice formed, being lighter than water, floats on -the surface of the latter, and thus forms a protective layer and -prevents to a large extent further freezing, except as a slow thickening -of the layer, the total freezing of the water of a lake or river being -rendered practically impossible, an obvious provision for the security -of life of the piscatorial and other inhabitants of the waters. - -Hence, as the molecules of the substance H_{2}O are one by one detached -from their solid assemblage as ice, and become more loosely associated -as the less voluminous liquid water, they cannot occupy the whole of the -cavity formed in the solid ice, and a small vacuous space, occupied only -by water vapour at its ordinary low tension corresponding to the low -temperature, is formed and appears as the bubble. Moreover, the cavity -itself takes the shape of a hexagonal star-shaped flower, the bubble -showing at its centre, the cavity being thus a kind of negative ice -crystal, like the negative quartz crystal shown in Fig. 34. Apparently -in the production of these cavities, just as in the production of the -well-known etched figures on crystal faces by the application of a -minute quantity of a solvent for the crystal substance, the crystal -edifice is taken down, molecule by molecule, in a regular manner, -resulting in the formation of a cavity showing the symmetry of the -space-lattice which is present in the crystal structure. - -[Illustration: - - _PLATE X._ - - FIG. 37.—Piz Palü and Snow-field of the Pers Glacier, from the - Diavolezza Pass, Upper Engadine. - - (From a Photograph by the author.) -] - -The water flowers of Fig. 35 remind one very much of snow crystals, two -of which, re-engraved from the wonderfully careful drawings of the late -Mr Glaisher, are represented in Fig. 36, Plate IX. They all exhibit the -symmetry of the hexagonal prism, which is equally a form of the trigonal -system as it is of the hexagonal system. The snow crystals, being formed -from water vapour condensed in the cold upper layers of the atmosphere, -appear more or less as skeleton crystals, owing to the rarity of the -semi-gaseous material condensed, compared with the extent of the space -in which the crystallisation occurs. Indeed the exquisite tracery of -these snow crystals appears to afford a visual proof of the existence of -the trigonal-hexagonal space-lattice as the framework of the crystal -structure of ice. When one considers the countless numbers of such -beautiful gems of nature’s handiwork massed together on an extensive -snow-field of the higher Alps—such as that of the Piz Palü in the Upper -Engadine, shown in Fig. 37, Plate X., as seen from the Diavolezza -Pass—produced in the pure air of the higher regions of the atmosphere, -and frequently seen by the early morning climber lying uninjured in all -their beauty on the surface of the snow-field, one is lost in amazement -at the prodigality displayed in the broadcast distribution of such -peerless gems. - - - - - CHAPTER V - HOW CRYSTALS ARE DESCRIBED. THE SIMPLE LAW LIMITING THE NUMBER OF - POSSIBLE FORMS. - - -The most wonderful of all the laws relating to crystals is the one -already briefly referred to which limits and regulates the possible -positions of faces, within the lines of symmetry which have been -indicated in the last chapter. Having laid down the rules of symmetry, -it might be thought that any planes which obey these laws, as regards -their mode of repetition about the planes and axes of symmetry, would be -possible. But as a matter of fact this is not so, only a very few planes -inclined at certain definite angles, repeated in accordance with the -symmetry, being ever found actually developed. The reason for this is of -far-reaching importance, for it reveals to us the certainty that a -crystal is a homogeneous structure composed of definite structural units -of tangible size, probably the chemical molecules, built up on the plan -of one of the fourteen space-lattices made known to us by Bravais, and -to be referred to more fully in Chapter VIII. In order to render this -fundamental law comprehensible, it will be essential to explain in a few -simple words how the crystallographer identifies and labels the numerous -faces on a crystal, in short, how he describes a crystal, in a manner -which shall be understood immediately by everybody who has studied the -very simple rules of the convention. - -It is a matter of common knowledge that the mathematical geometrician -defines the position of any point in space with reference to three -planes, which in the simplest case are all mutually at right angles to -each other like the faces of a cube, and which intersect in three -rectangular axes _a_, _b_, _c_, the third _c_ being the vertical axis, -_b_ the lateral one, and _a_ the front-and-back axis. The distances of -the point from the three reference planes, as measured by the lengths of -the three lines drawn from the point to the planes parallel to the three -axes of intersection, at once gives him what he calls the “co-ordinates” -of the point, which absolutely define its position. In the same way we -can imagine three axes drawn within the crystal, by which not only the -position of any point on any face of the crystal may be located, but -which may be used more simply still to fix the position of the face -itself. The directions chosen as those of the three axes are the edges -of intersection of three of the best developed faces. - -If there are three such faces inclined at right angles they would be -chosen in preference to all others, as they would certainly prove to be -faces of prime significance as regards the symmetry of the crystal. If -there are no such rectangularly inclined faces developed on the crystal, -then the three best developed faces nearest to 90° to each other are -chosen, the two factors of nearness to rectangularity and excellence of -development being simultaneously borne in mind in making the choice of -axial planes, and discretion used. - -[Illustration: - - FIG. 38.—The Cube and its Three Equal Rectangular Axes. -] - -[Illustration: - - FIG. 39.—Tetragonal Prism and its three Rectangular Axes. -] - -If the crystal belong to the cubic, tetragonal, or rhombic systems, for -instance, three faces meeting each other rectangularly are possible -planes on the crystal, and will very frequently be found actually -developed; such would obviously be chosen as the axial planes. The edges -of the cube, or of the tetragonal or rectangular rhombic prism, will be -the directions of the crystallographic axes in this case, and we can -imagine them moved parallel to themselves until the common centre of -intersection, the “origin” of the analytical geometrician, will occupy -the centre of the crystal, and the faces of the latter be built up -symmetrically about it. When the crystal is cubic, the three axes will -be of equal length as shown in Fig. 38; if tetragonal, the two -horizontal axes will be equal, but will differ in length from the -vertical axis, as represented in Fig. 39. If the crystal be rhombic, all -three axes will be of different lengths, as indicated in Fig. 40, which -represents the axes and axial planes of an actual rhombic substance, -topaz, for which the lateral axis _b_ and vertical axis _c_ are nearly -but not quite equal, while the front-and-back axis _a_ is very -different. - -When the crystal is of monoclinic symmetry, as in Fig. 41, three axes -will similarly be found as the intersection of three principal parallel -pairs of faces, but two of them will be inclined at an angle other than -90° to each other, while the third, the lateral one in Fig. 41, will be -at right angles to those first two and to the plane containing them; -moreover, all three are unequal in length. In the case of a triclinic -crystal, shown in Fig. 42, however, there can be no right angles, and -the intersections of three important faces meeting each other at angles -as near 90° as possible are chosen as the axes, regard being had to both -factors of approximation to rectangularity and importance of -development. These triclinic axes are the most general type of crystal -axes, for not only are the angles not right angles, but the lengths of -the axes are also unequal. - -[Illustration: - - FIG. 40.—Axial Planes of a Rhombic Crystal. -] - -[Illustration: - - FIG. 41.—Axial Planes of a Monoclinic Crystal. -] - -[Illustration: - - FIG. 42.—Axial Planes of a Triclinic Crystal. -] - -[Illustration: - - FIG. 43.—Hexagonal Prism of the First Order and its Four Axes. -] - -[Illustration: - - FIG. 44.—Hexagonal Prism of the Second Order. -] - -[Illustration: - - FIG. 45.—The Rhombohedron and its Three Equal Axes. -] - -The cases of the hexagonal and trigonal systems are somewhat special. -The hexagonal has four such axes, as represented in Fig. 43, the lines -of intersection of the faces of the hexagonal prism closed by a pair of -perpendicular terminal planes, namely, one vertical axis parallel to the -vertical edges, and three horizontal axes inclined at 120° to each -other, and parallel to the pair of basal plane faces, equal to each -other in length, but different from the length of the vertical axis. The -hexagonal axial-plane prism shown in Fig. 43 is known as one of the -first order. The hexagonal prism corresponding to the tetragonal one of -Fig. 39, in which the axes emerge in the centres of the faces, is said -to be of the second order, and is shown in Fig. 44. The trigonal system -of crystals is best described with reference to three equal but not -rectangular axes, parallel to the faces of the rhombohedron, one of the -principal forms of the system, so well seen in Iceland spar, and -illustrated in Fig. 45. The rhombohedron may be regarded as a cube -resting on one of its corners (solid angles), with the diagonal line -joining this to the opposite corner vertical, and the cube then deformed -by flattening or elongating it along the direction of this diagonal. The -edges meeting at the ends of this vertical diagonal are then the -directions of the three trigonal crystallographic axes. - -In this last illustration the vertical direction of the altered diagonal -is that of the trigonal axis of symmetry, for the rhombohedron is -brought into apparent coincidence with itself again if rotated for 120° -round this direction. But although a symmetry axis, this is not a -crystallographic axis of reference. It is not shown in Fig. 45, -therefore, but is given in Fig. 19. On the other hand, the singular -vertical axis of reference of the tetragonal and hexagonal systems is -identical with the tetragonal or hexagonal axis of symmetry of these -systems, and the three crystallographic axes of reference of the cube -are identical with the three tetragonal axes of symmetry of the cubic -system. In the rhombic system also, the three rectangular axes of -reference are identical with the three digonal axes of symmetry, and in -the monoclinic system the one axis of reference which is normal to the -plane of the two inclined axes is the unique digonal axis of symmetry of -that system. - -Having thus evolved a scientific scheme of reference axes for the faces -of a crystal, it is necessary in all the systems other than the cubic -and trigonal, in which the axes are of equal lengths, to devise a mode -of arriving at the relative lengths of the axes; for on this depends the -mode of determining the positions of the various faces, other than the -three parallel pairs (or four in the case of the hexagonal system) -chosen as the axial planes. This is very simply done by choosing a -fourth important face inclined to all three axes, when one of this -character is developed, as very frequently happens, as the determinative -face or plane fixing the unit lengths of the axes. When no such face is -present on the crystal, two others can usually be found, each of which -is inclined to two different axes, so that between them all three axial -lengths are determined. The faces of the octahedron, of the primary -tetragonal pyramid and the primary rhombic pyramid, and of the -corresponding forms of the other systems, are such determinative planes, -fixing the lengths of the axes. This fact will be clear from the typical -illustration of the most general of these primary or “parametral” forms, -the triclinic equivalent of the octahedron, given in Fig. 46, the faces -being obviously obtained by joining the points marking unit lengths of -the three axes. - -[Illustration: - - FIG. 46.—Triclinic Equivalent of the Octahedron. -] - -Having thus settled the directions of the crystallographic axes and -their lengths, it is the next step which reveals the remarkable law to -which reference was made at the opening of this chapter. For we find -that all other faces on the crystal, however complicated and rich in -faces it may be, cut off lengths from the axes which are represented by -low whole numbers, that is, either 2, 3, 4, or possibly 5, and very -rarely more than 6 unit lengths. By far the greater number of faces do -not cut off more than three unit lengths from any axis. Prof. Miller of -Cambridge, in the year 1839, gave us a most valuable mode of labelling -and distinguishing the various faces by a symbol involving these three -values, employed, however, not directly but in an indirect yet very -simple manner. If _m_, _n_, _r_ be the three numbers expressing the -intercepts cut off by a face on the three axes, _a_, _b_, _c_ -respectively, and if the Millerian index numbers be represented by _h_, -_k_, _l_, then— - - _m_ = _a_/_h_, _n_ = _b_/_k_, _r_ = _c_/_l_, - or, _h_ = _a_/_m_, _k_ = _b_/_n_, _l_ = _c_/_r_. - -Each figure or “index” of the Millerian symbol is thus inversely -proportional to the length of the intercept on the axis concerned. The -intercepts themselves are used as symbols in another mode of labelling -crystal faces, suggested by Weiss, but this method proves too cumbersome -in practice. - -The Millerian symbol of a face is always placed within ordinary curved -brackets ( ), but if the symbol is to stand for the whole set of faces -composing the form, the brackets are of the type { }. Thus the -Millerian symbol of the fourth face (that in the top-right front -octant), determinative of the unit axial lengths, is (111), as shown in -Fig. 46, the face in question being marked with this symbol; while the -symbol {111} indicates the set of faces of the whole or such part of the -double pyramid as composes the unit form. In the triclinic system this -form only consists of the face (111) and the parallel one (̄1̄1̄1), but -in the case of the regular octahedron of the cubic system it embraces -all the eight faces. The triclinic octahedron, Fig. 46, is thus made up -of four forms of two faces each. A negative sign over an index indicates -interception on the axis _a_ behind the centre, on the axis _b_ to the -left of the centre, or on the vertical axis _c_ below the centre. - -To take an actual example, suppose a face other than the primary one to -make the intercepts on the axes 4, 2, 1; in this case _h_ = _a_/4, _k_ = -_b_/2, and _l_ = _c_/1, that is, when referred to the fundamental -primary form for which _a_, _b_, _c_ are each unity, _h_ = ¼, _k_ = ½, -_l_ = 1, or, bringing them to whole numbers by multiplying by 4, _h_ = -1, _k_ = 2, _c_ = 4, and the symbol in Millerian notation is (124). -Again, suppose we wish to find the intercepts on the three cubic axes -made by the face (321) of the hexakis octahedron shown in Fig. 21. To -get each intercept we multiply together the two other Millerian indices, -and if necessary afterwards reduce the three figures obtained to their -simplest relative values. For the face (321) we obtain 2, 3, 6. This -means that the face (321) in the top-right-front octant of the hexakis -octahedron cuts off two unit lengths of axis _a_, three unit lengths of -axis _b_, and six unit lengths of axis _c_. No fractional parts thus -ever enter into the relations of the axial lengths intercepted by any -face on a crystal, and the whole numbers representing these relations -are always small, the number 6 being the usual limit. - -This important law is known as the “Law of Rational Indices,” and is the -corner-stone of crystallography. A forecast of it was given in Chapter -III., in describing how it was first discovered by Haüy, and it was -shown how impressed Haüy was with its obvious significance as an -indication of the brick-like nature of the crystal structure. What the -“bricks” were, Haüy was not in a position to ascertain with certainty, -as chemistry was in its infancy, and Dalton’s atomic theory had not then -been proposed. - -That Haüy had a shrewd idea, however, that the structural units were the -chemical molecules, and that while the main lines of symmetry were -determined by the arrangement of the molecules its details were settled -by the arrangement of the atoms in the molecules, is clear to any one -who reads his 1784 “Essai” and 1801 “Traité,” and interprets his -_molécules intégrantes_ and _élémentaires_ in the light of our knowledge -of to-day. - -Before we pass on, however, to consider the modern development of the -real meaning of the law of rational indices, as revealed by recent work -on the internal structure of crystals, it will be well to consider -first, in the next chapter, a few more essential facts as to crystal -symmetry, and the current mode of constructing a comprehensive, yet -simple, plan of the faces present on a crystal. - - - - - CHAPTER VI -THE DISTRIBUTION OF CRYSTAL FACES IN ZONES, AND THE MODE OF CONSTRUCTING - A PLAN OF THE FACES. - - -It will have been clear from the facts related in the previous chapters -that the salient property possessed by all crystals, when ideal -development is permitted by the circumstances of their growth, and the -substance is not one of unusual softness or liable to ready distortion, -is that the exterior form consists of and is defined by truly plane -faces inclined to each other at angles which are specific and -characteristic for each definite chemical substance; and that these -angles are in accordance with the symmetry of some particular one of the -thirty-two classes of crystals, and are such as cause the indices of the -faces concerned to be rational small numbers. - -It will also be clear that, given the presence of any face other than -the three axial planes, the symmetry of the class—supposing the crystal -to exhibit some development of symmetry and not to belong to class 32, -the general case possessing no symmetry—will require the repetition of -this face a definite number of times on other parts of the crystal. Such -a set of faces possessing the same symmetry value we have already learnt -to call a “Form,” and the faces composing it will have the same -Millerian index numbers in their symbols, but differently arranged and -with negative signs over those which relate to the interception of the -back part of the _a_ axis, the left part of the _b_ axis, or the lower -part of the vertical _c_ axis; that is, parts to the front and right, -and above, the centre of intersection of the three crystal axes are -considered as the positive parts of those axes. - -A form, if of general character, that is, if composed of faces each of -which is inclined to all three axes, will comprise more faces the higher -the symmetry. Thus, in the cubic system, the form shown in Fig. 21, the -hexakis octahedron, comprises as many as forty-eight faces, all covered -by the form symbol {321}; while in the rhombic system the highest number -of faces in a form is eight, in the monoclinic only four, and in the -triclinic system two. It will also have become clear that the law of -rational indices limits the number of forms possible of any one type. -For instance, very few hexakis octahedra are known, the most frequently -occurring ones besides {321} being {421}, {531}, and {543}. Forms, of -any class, possessing higher indices than these are very rare, -especially in the systems of lower symmetry. - -[Illustration: - - FIG. 47.—The Spherical Projection. -] - -We next come to a further very interesting fact about crystals. Let us -imagine a crystal, on which the faces are fairly evenly developed, to be -placed in the middle of a sphere of jelly, as indicated in Fig. 47 -(reproduced from a Memoir by the late Prof. Penfield), so that the -centre or origin of the axial system of the crystal and the centre of -the sphere coincide. Let us now further imagine that long needles are -stuck through the jelly and the crystal, one perpendicular to each -crystal face, and so as to reach the centre. The crystal represented in -Fig. 47 is a combination of the cube _a_, octahedron _o_, and rhombic -dodecahedron _d_. If such a thing as we have imagined were possible, we -should find that the needles would emerge at the surface of the sphere -in points which would lie on great circles, that is, on circles which -represent the intersection of the sphere by planes passing through the -centre. Moreover, the points would be distributed along these circles at -regularly recurring angular positions, corresponding to the symmetry of -the crystal. If the crystal belonged to one of the higher systems of -symmetry, it would happen that four of the points on at least one of -these great circles, and possibly on three of them, would be 90° apart, -that is, would be at the ends of rectangular diameters, which would most -likely be the axes of reference. The other points would be distributed -symmetrically on each side of these four points. - -The great circles on which the points are thus symmetrically -distributed—and they may legitimately be taken to represent the faces, -for tangent planes to the sphere at these points would be parallel to -the faces—are known as “zone circles,” and the faces represented by the -points on any one of them form a “zone.” Now a zone of faces has this -practical property, that when the crystal is supported so as to be -rotatable about the zone axis—which is parallel to the edges of -intersection of all the faces composing the zone, and is the normal to -the plane of the great circle representing the zone—and a telescope is -directed towards the crystal perpendicularly to the zone axis, while a -bright object such as an illuminated slit is arranged conveniently so as -to be reflected from any face of the crystal into the telescope, an -image of it being thus visible in the latter, then it will be found that -on rotating the crystal a similar image will be seen reflected in the -telescope from every face of the zone in turn. Moreover, when the -crystal is mounted on a graduated circle, the angle of rotation between -the positions of adjustment to the cross-wires of the telescope of any -two successive images, reflected from adjacent faces of the crystal, is -actually the angle between the two points representing the faces -concerned on the zone circle, and is the supplement of the internal -dihedral angle between the two crystal faces themselves. It is, in fact, -the angle between the normals (perpendiculars) to the two faces, the -angle which is measured on the goniometer. - -This is, indeed, the very simple principle of the reflecting goniometer, -invented by Wollaston in the year 1809, and which in its modern improved -form is the all-important principal instrument of the crystallographer’s -laboratory. The work with it consists largely in the measurement of the -angles between the faces in all the principal zones developed on the -crystal. The very fact, however, that crystal faces occur so absolutely -accurately in zones immeasurably lightens the labours of the -crystallographer, and is one of prime importance. - -[Illustration: - - FIG. 48.—The Reflecting Goniometer. -] - -The most accurate and convenient modern form of reflecting goniometer, -reading to half-minutes of arc, and provided with a delicate adjusting -apparatus for the crystal, is shown in Fig. 48. It is constructed by -Fuess of Berlin. - -The graduated circle _a_ is horizontal and is divided directly to 15′, -the verniers enabling the readings to be carried further either to -single minutes, which is all that is usually necessary, or to -half-minutes in the cases of very perfect crystals. The divided circle -is rotated by means of the ring _b_ situated below, and the reading of -the verniers is accomplished with the aid of the microscopes _c_. The -circle which carries the verniers is not fixed, except when this is done -deliberately by means of the clamping screw _d_, but rotates with the -telescope _e_ to which it is rigidly attached by means of an arm and a -column _f_. A fine adjustment is provided with the clamping arrangement, -so that the telescope can be adjusted delicately with respect to the -divided circle. Both telescope and collimator are rigidly fixed at about -120° from each other during the actual measurements. The collimator _g_ -is carried on a column _h_ definitely fixed to one of the legs (the back -one in Fig. 48) of the main basal tripod of the instrument. The signal -slit of the collimator is carried at the focus of the objective about -the middle of the tube _g_, the outer half of the latter being an -illumination tube carrying a condensing lens to concentrate the rays of -light from the goniometer lamp on the slit. The latter is not of the -usual rectilinear character, but composed of two circular-arc jaws, so -that, while narrow in the middle part like an ordinary spectroscope -slit, it is much broader at the two ends in order to be much more -readily visible; the central part is narrow in order to enable fine -adjustment to the vertical cross-wire of the telescope to be readily and -accurately carried out. The shape of this signal-slit will be gathered -from the images of the slit shown in Fig. 61 (page 126) in Chapter X. -The telescope carries an additional lens _k_ at its inner, objective, -end, in order that when this lens is rotated into position the telescope -may be converted into a low power microscope for viewing the crystal and -thus enabling its adjustment to be readily carried out. - -The crystal _l_ is mounted on a little cone of goniometer wax (a mixture -of pitch and beeswax) carried by the crystal holder. The latter fits in -the top of the adjusting movements, which consist of a pair of -rectangularly arranged centring motions, and a pair of cylindrical -adjusting movements; the milled-headed manipulating centring screws of -the former are indicated by the letters _m_ and _n_ in Fig. 48, and -those which move the adjusting segments are marked _o_ and _p_. The top -screw fixes the crystal holder. - -The crystal on its adjusting apparatus can be raised or lowered to the -proper height, level with the axes of the telescope and collimator, by -means of a milled head at the base of the instrument, there being an -inner crystal axis moving (vertically only) independently of the circle. -Moreover, a second axis outside this enables the crystal to be rotated -independently of the circle, the conical axis of which is outside this -again. The two can be locked together when desired, however, by a -clamping screw provided with a fine adjustment _q_. Freedom of movement -of the crystal axis, unimpeded by the weight of the circle, is thus -permitted for all adjusting purposes, the circle being only brought into -play when measurement is actually to occur. With this instrument the -most accurate work can be readily carried out, and for ease of -manipulation and general convenience it is the best goniometer yet -constructed. - -The idea of regarding the centre of the crystal as the centre of a -sphere, within which the crystal is placed (Fig. 47, page 62), gives -crystallographers a very convenient method of graphically representing a -crystal on paper, by projecting the sphere on to the flat surface of the -paper, the eye being supposed to be placed at either the north or south -pole of the sphere, and the plane of projection to be that of the -equatorial great circle. The faces in the upper hemisphere are -represented by dots which are technically known as the “poles” of the -faces, corresponding to the points where the needles normal to the faces -emerge from the imaginary globe, and all these points or poles lie on a -few arcs of great circles, which appear in the projection either also as -circular arcs terminating at diametrically opposite points on the -circumference of the equatorial circle, which forms the outer boundary -of the figure and is termed the “primitive circle,” or else, when the -planes of the great circles are at right angles to the equatorial -primitive circle, they appear as diametral straight lines passing -through the centre of the primitive circle. - -Such a stereographic projection offers a comprehensive plan of the whole -of the crystal faces, which at once informs us of the symmetry in all -cases other than very complicated ones. A typical one, that of the -rhombic crystal of topaz shown in Fig. 22 (page 40), is given in Fig. -49. - -It will happen in all cases of higher symmetry, as in that of topaz, for -instance, that the poles in the lower hemisphere will project into the -same points as those representing the faces in the upper hemisphere; but -in cases of lower symmetry, where they are differently situated, they -are usually represented by miniature rings instead of dots. From the -interfacial angles measured on the goniometer the relative lengths and -angular inclinations (if other than 90°) of the crystal axes can readily -be calculated, by means of the simple formulæ of spherical trigonometry; -and the stereographic projection constructed from the measurements as -just described proves an inestimable aid to these calculations, by -affording a comprehensive diagram of all the spherical triangles -required in making the calculations. - -[Illustration: - - FIG. 49.—Stereographic Projection of Topaz. -] - -The relative axial lengths _a_ : _b_ : _c_ (in which _b_ is always -arranged to be = 1), and the axial angles α (between _b_ and _c_), β -(between _a_ and _c_), and γ (between _a_ and _b_), form the “elements” -of a crystal. These, together with a list of the “forms” observed, and a -table of the interfacial angles, define the morphology of the crystal, -and are included in every satisfactory description of a crystallographic -investigation. They are preceded by a statement of the name and chemical -composition and formula of the substance, the system and the class of -symmetry, and the habit or various habits developed by crystals from a -considerable number of crops. An example of the mode of setting out such -a description will be found on pages 157 to 160. - -Having thus made ourselves acquainted with the real nature of the -distribution of faces on a crystal, and learnt how the crystallographer -measures the angles between the faces by means of the reflecting -goniometer, plots them out graphically on a stereographic projection, -and calculates therefrom the “elements” of the crystal, it will be -convenient again to take up the historical development of the subject so -far as it relates to crystal forms and angles, and their bearing on the -chemical composition of the substance composing the crystal, by -introducing the reader to the great work of Mitscherlich, whose -influence in the domain of chemical crystallography was as profound as -that of Haüy proved to be as regards structural crystallography. - - - - - CHAPTER VII - THE WORK OF EILHARDT MITSCHERLICH AND HIS DISCOVERY OF ISOMORPHISM. - - -During the height of the French Revolution, which caused the work of the -Abbe Haüy to be suspended for a time, although he was fortunately not -one of the many scientific victims of that terrible period, there was -born, on the 7th of January 1794, in the village of Neuende, near Jever, -in Oldenburg, the man who was destined to continue that work on its -chemical side. Eilhardt Mitscherlich was the son of the village pastor, -and nephew of the celebrated philologer, Prof. Mitscherlich of -Göttingen. His uncle’s influence appears to have given young -Mitscherlich a leaning towards philological studies, for during his -later terms at the Gymnasium at Jever, where he received his early -education, he devoted himself with great energy to the study of history -and languages, for which he had a marked talent, under the able -direction and kind solicitude of the head of the Gymnasium at that time, -the historian Schlosser. He eventually specialised on the Persian -language, and when Schlosser was promoted to Frankfort young -Mitscherlich accompanied him, and there prosecuted these favourite -studies until the year 1811, when he went to the university of -Heidelberg. - -For some time now he had cherished the hope of proceeding to Persia and -conducting philological investigations on the spot, and in 1813, an -opportunity presenting itself in the prospect of an embassy being -despatched to Persia by Napoleon, he transferred himself to the -university of Paris, with the object of obtaining permission from -Napoleon to accompany the embassy. This visit to Paris must have been -one of Mitscherlich’s most exciting and interesting experiences. For -Napoleon had just returned from the disastrous Russian campaign of 1812, -and was feverishly engaged in raising a new army wherewith to stem the -great rise of the people which was now re-awakening patriotic spirit -throughout the whole of Germany, and which threatened to sweep away, as -it eventually did, the huge fabric of his central European Empire. - -Indeed Mitscherlich appears to have been detained in Paris during the -exciting years 1813 and 1814, and with the abdication of Napoleon on -April 4th of that year, he was obliged to give up all idea of proceeding -to Persia. He decided that the only way of accomplishing his purpose was -to attempt to travel thither as a doctor of medicine. He therefore -returned to his native Germany during the summer of 1814, and proceeded -to Göttingen, which was then famous for its medical school. Here he -worked hard at the preliminary science subjects necessary for the -medical degree, while still continuing his philology to such serious -purpose as to enable him to publish, in 1815, the first volume of a -history of the Ghurides and Kara-Chitayens, entitled “Mirchondi historia -Thaheridarum.” It is obvious from the sequel, however, that he very soon -began to take much more than a merely passing interest in his scientific -studies, and he eventually became so fascinated by them, and -particularly chemistry, as to abandon altogether his cherished idea of a -visit to Persia. Europe was now settling down after the stormy period of -the hundred days which succeeded Napoleon’s escape from Elba, -terminating in his final overthrow on June 18th, 1815, at Waterloo, and -Mitscherlich was able to devote himself to the uninterrupted prosecution -of the scientific work now opening before him. He had the inestimable -advantage of bringing to it a culture and a literary mind of quite an -unusually broad and original character; and if the fall of Napoleon -brought with it the loss to the world of an accomplished philologist, it -brought also an ample compensation in conferring upon it one of the most -erudite and broad-minded of scientists. - -In 1818 Mitscherlich went to Berlin, and worked hard at chemistry in the -university laboratory under Link. It was about the close of this year or -the beginning of 1819 that he commenced his first research, and it -proved to be one which will ever be memorable in the annals both of -chemistry and of crystallography. He had undertaken the investigation of -the phosphates and arsenates, and his results confirmed the conclusions -which had just been published by Berzelius, then the greatest chemist of -the day, namely, that the anhydrides of phosphoric and arsenic acids -each contain five equivalents of oxygen, while those of the lower -phosphorous and arsenious acids contain only three. But while making -preparations of the salts of these acids, which they form when combined -with potash and ammonia, he observed a fact which had escaped Berzelius, -namely, that the phosphates and arsenates of potassium and ammonium -_crystallise in similar forms_, the crystals being so like each other, -in fact, as to be indistinguishable on a merely cursory inspection. - -Being unacquainted with crystallography, and perceiving the importance -of the subject to the chemist, he acted in a very practical and sensible -manner, which it is more than singular has not been universally imitated -by all chemists since his time. He at once commenced the study of -crystallography, seeing the impossibility of further real progress -without a working knowledge of that subject. He was fortunate in finding -in Gustav Rose, the Professor of Geology and Mineralogy at Berlin, not -merely a teacher close at hand, but also eventually a life-long intimate -friend. Mitscherlich worked so hard under Rose that he was very soon -able to carry out the necessary crystal measurements with his newly -prepared phosphates and arsenates. He first established the complete -morphological similarity of the acid phosphates and arsenates of -ammonium, those which have the composition NH_{4}H_{2}PO_{4} and -NH_{4}H_{2}AsO_{4} and crystallise in primary tetragonal prisms -terminated by the primary pyramid faces; and then he endeavoured to -produce other salts of ammonia with other acids which should likewise -give crystals of similar form. But he found this to be impossible, and -that only the phosphates and arsenates of ammonia exhibited the same -crystalline forms, composed of faces inclined at similar angles, which -to Mitscherlich at this time appeared to be identical. He next tried the -effect of combining phosphoric and arsenic acids with other bases, and -he found that potassium gave salts which crystallised apparently exactly -like the ammonium salts. - -He then discovered that not only do the acid phosphates and arsenates of -potassium and ammonium, H_{2}KPO_{4}, H_{2}(NH_{4})PO_{4}, -H_{2}KAsO_{4}, and H_{2}(NH_{4})AsO_{4} crystallise in similar -tetragonal forms, but also that the four neutral di-metallic salts of -the type HK_{2}PO_{4} crystallise similarly to each other. - -He came, therefore, to the conclusion that there do exist bodies of -dissimilar chemical composition having the same crystalline form, but -that these substances are of similar constitution, in which one element, -or group of elements, may be exchanged for another which appears to act -analogously, such as arsenic for phosphorus and the ammonium group -(although its true nature was not then determined) for potassium. He -observed that certain minerals also appeared to conform to this rule, -such as the rhombohedral carbonates of the alkaline earths, calcite -CaCO_{3}, dolomite CaMg(CO_{3})_{2}, chalybite FeCO_{3}, and dialogite -MnCO_{3}; and the orthorhombic sulphates of barium (barytes, BaSO_{4}), -strontium (celestite, SrSO_{4}), and lead (anglesite, PbSO_{4}). -Wollaston, who, in the year 1809, had invented the reflecting -goniometer, and thereby placed a much more powerful weapon of research -in the hands of crystallographers, had already, in 1812, shown this to -be a fact as regards the orthorhombic carbonates (witherite, -strontianite, and cerussite) and sulphates (barytes, celestite, and -anglesite) of barium, strontium, and lead, as the result of the first -exact angular measurements made with his new instrument; but his -observations had been almost ignored until Mitscherlich reinstated them -by his confirmatory results. - -While working under the direction of Rose, Mitscherlich had become -acquainted with the work of Haüy, whose ideas were being very much -discussed about this time, Haüy himself taking a very strong part in the -discussion, being particularly firm on the principle that every -substance of definite chemical composition is characterised by its own -specific crystalline form. Such a principle appeared to be flatly -contradicted by these first surprising results of Mitscherlich, and it -naturally appeared desirable to the latter largely to extend his -observations to other salts of different groups. It was for this reason -that he had examined the orthorhombic sulphates of barium, strontium, -and lead, and the rhombohedral carbonates of calcium, magnesium, iron, -and manganese, with the result already stated that the members of each -of these groups of salts were found to exhibit the same crystalline -form, a fact as regards the former group of sulphates which had already -been pointed out not only by Wollaston but by von Fuchs (who appears to -have ignored the work of Wollaston) in 1815, but had been explained by -him in a totally unsatisfactory manner. Moreover, about the same time -the vitriols, the sulphates of zinc, iron, and copper, had been -investigated by Beudant, who had shown that under certain conditions -mixed crystals of these salts could be obtained; but Beudant omitted to -analyse his salts, and thus missed discovering the all-important fact -that the vitriols contain water of crystallisation, and in different -amounts under normal conditions. Green vitriol, the sulphate of ferrous -iron, crystallises usually with seven molecules of water of -crystallisation, as does also white vitriol, zinc sulphate; but blue -vitriol, copper sulphate, crystallises with only five molecules of water -under ordinary atmospheric conditions of temperature and pressure. -Moreover, copper sulphate forms crystals which belong to the triclinic -system, while the sulphates of zinc and iron are dimorphous, the common -form of zinc sulphate, ZnSO_{4}.7H_{2}O, being rhombic, like Epsom -salts, the sulphate of magnesia which also crystallises with seven -molecules of water, MgSO_{4}.7H_{2}O, while that of ferrous sulphate, -FeSO_{4}.7H_{2}O, is monoclinic, facts which still further complicate -the crystallography of this group and which were quite unknown to -Beudant and were unobserved by him. But Beudant showed that the addition -of fifteen per cent. of ferrous sulphate to zinc sulphate, or nine per -cent. to copper sulphate, caused either zinc or copper sulphate to -crystallise in the same monoclinic form as ferrous sulphate. He also -showed that all three vitriols will crystallise in mixed crystals with -magnesium or nickel sulphates, the ordinary form of the latter salt, -NiSO_{4}.7H_{2}O, being rhombic like that of Epsom salts. - -The idea that two chemically distinct substances not crystallising in -the cubic system, where the high symmetry determines identity of form, -can occur in crystals of the same form, was most determinedly combated -by Haüy, and the lack of chemical analyses in Beudant’s work, and the -altogether incorrect “vicarious” explanation given by von Fuchs, gave -Haüy very grave cause for suspicion of the new ideas. The previous -observations of Rome de l’Isle in 1772, Le Blanc in 1784, Vauquelin in -1797, and of Gay-Lussac in 1816, that the various alums, potash alum, -ammonia alum, and iron alum, will grow together in mixed crystals or in -overgrowths of one crystal on another, when a crystal of any one of them -is hung up in the solution of any other, does not affect the question, -as the alums crystallise in the cubic system, the angles of the highly -symmetric forms of which are absolutely identical by virtue of the -symmetry itself. - -It was while this interesting discussion was proceeding that -Mitscherlich was at work in Berlin, extending his first researches on -the phosphates and arsenates to the mineral sulphates and carbonates. -But he recognised, even thus early, what has since become very clear, -namely, that owing to the possibility of the enclosure of impurities and -of admixture with isomorphous analogues, minerals are not so suitable -for investigation in this regard as the crystals of artificially -prepared chemical salts. For the latter can be prepared in the -laboratory in a state of definitely ascertained purity, and there is no -chance of that happening which Haüy was inclined to think was the -explanation of Mitscherlich’s results, namely, that certain salts have -such an immense power of crystallisation that a small proportion of them -in a solution of another salt may coerce the latter into crystallisation -in the form of that more powerfully crystallising salt. Mitscherlich -made a special study, therefore, of the work of Beudant, and repeated -the latter observer’s experiments, bringing to the research both his -crystallographic experience and that of a skilful analyst. He prepared -the pure sulphates of ferrous iron, copper, zinc, magnesium, nickel and -cobalt, all of which form excellent crystals. He soon cleared up the -mystery in which Beudant’s work had left the subject, by showing that -the crystals contained water of crystallisation, and in different -amounts. He found what has since been abundantly verified, that the -sulphates of copper and manganese crystallise in the triclinic system -with five molecules of water, CuSO_{4}.5H_{2}O and MnSO_{4}.5H_{2}O; in -the case of manganese sulphate, however, this is only true when the -temperature is between 7° and 20°, for if lower than 7° rhombic crystals -of MnSO_{4}.7H_{2}O similar to those of the magnesium sulphate group are -deposited, and if higher than 20° the crystals are tetragonal and -possess the composition MnSO_{4}.4H_{2}O. The Epsom salts group -crystallising in the rhombic system with seven molecules of water -consists of magnesium sulphate itself, MgSO_{4}.7H_{2}O, zinc sulphate -ZnSO_{4}.7H_{2}O, and nickel sulphate NiSO_{4}.7H_{2}O. The third group -of Mitscherlich consists of sulphate of ferrous iron FeSO_{4}.7H_{2}O -and cobalt sulphate CoSO_{4}.7H_{2}O, and both crystallise at ordinary -temperatures with seven molecules of water as indicated by the formulæ, -but in the monoclinic system. Thus two of the groups contain the same -number of molecules of water, yet crystallise differently. But -Mitscherlich next noticed a very singular fact, namely, that if a -crystal of a member of either of these two groups be dropped into a -saturated solution of a salt of the other group, this latter salt will -crystallise out in the form of the group to which the stranger crystal -belongs. Hence he concluded that both groups are capable of -crystallising in two different systems, rhombic and monoclinic, and that -under the ordinary circumstances of temperature and pressure three of -the salts form most readily the rhombic crystals, while the other two -take up most easily the monoclinic form. Mitscherlich then mixed the -solutions of the different salts, and found that the mixed crystals -obtained presented the form of some one of the salts employed. Thus even -so early in his work Mitscherlich indicated the possibility of -dimorphism. Moreover, before the close of the year 1819 he had satisfied -himself that aragonite is a second distinct form of carbonate of lime, -crystallising in the rhombic system and quite different from the -ordinary rhombohedral form calcite. Hence this was another undoubted -case of dimorphism. - -During this same investigation in 1819, Mitscherlich studied the effect -produced by mixing the solution of each one of the above-mentioned seven -sulphates of dyad-acting metals with the solution of sulphate of potash, -and made the very important discovery that a double salt of definite -composition was produced, containing one equivalent of potassium -sulphate, one equivalent of the dyad sulphate (that of magnesium, zinc, -iron, manganese, nickel, cobalt, or copper), and six equivalents of -water of crystallisation, and that they all crystallised well in similar -forms belonging to the monoclinic system. Some typical crystals of one -of these salts, ammonium magnesium sulphate, are illustrated in Fig. 30 -(Plate VII., facing page 44). This is probably the most important series -of double salts known to us, and is the series which has formed the -subject of prolonged investigation on the part of the author, no less -than thirty-four different members of the series having been studied -crystallographically and physically since the year 1893, and many other -members still remain to be studied. An account of this work is given in -a Monograph published in the year 1910 by Messrs Macmillan & Co., and -entitled, “Crystalline Structure and Chemical Constitution.” - -This remarkable record for a first research was presented by -Mitscherlich to the Berlin Academy on the 9th December 1819. During the -summer of the same year Berzelius visited Berlin, and was so struck with -the abilities of Mitscherlich, then twenty-five years old, that he -persuaded him to accompany him on his return to Stockholm, and -Mitscherlich continued his investigations there under the eye of the -great chemist. His first work at Stockholm consisted of a more complete -study of the acid and neutral phosphates and arsenates of potash, soda, -ammonia, and lead. He showed that in every case an arsenate crystallises -in the same form as the corresponding phosphate. Moreover, in 1821 he -demonstrated that sodium dihydrogen phosphate, NaH_{2}PO_{4}, -crystallises with a molecule of water of crystallisation in two -different forms, both belonging to the rhombic system but with quite -different axial ratios; this was consequently a similar occurrence to -that which he had observed with the sulphates of the iron and zinc -groups. - -It was while Mitscherlich was in Stockholm that Berzelius suggested to -him that a name should be given to the new discovery that analogous -elements can replace each other in their crystallised compounds without -any apparent change of crystalline form. Mitscherlich, therefore, termed -the phenomenon “isomorphism,” from ἰσός, equal to, and μορφή, shape. The -term “isomorphous” thus strictly means “equal shaped,” implying not only -similarity in the faces displayed, but also absolute equality of the -crystal angles. The fact that the crystals of isomorphous substances are -not absolutely identical in form, but only very similar, was not likely -to be appreciated by Mitscherlich at this time. For the reflecting -goniometer had only been invented by Wollaston in 1809, and accurate -instruments reading to minutes of arc were mechanical rarities. It will -be shown in the sequel, as the result of the author’s investigations, -that there _are_ angular differences, none the less real because -relatively very small, between the members of such series. But -Mitscherlich was not in the position to observe them. It must be -remembered, moreover, that he was primarily a chemist, and that he had -only acquired sufficient crystallographic knowledge to enable him to -detect the system of symmetry, and the principal forms (groups of faces -having equal value as regards the symmetry) developed on the crystals -which he prepared. His doctrine of isomorphism, accepted in this broad -sense, proved of immediate and important use in chemistry. For there -were uncertainties as to the equivalents of some of the chemical -elements, as tabulated by Berzelius, then the greatest authority on the -subject, and these were at once cleared up by the application of the -principle of isomorphism. - -The essence of Mitscherlich’s discovery was, that the chemical nature of -the elements present in a compound influences the crystalline form by -determining the number and the arrangement of the atoms in the molecule -of the compound; so that elements having similar properties, such for -instance as barium, strontium, and calcium, or phosphorus and arsenic, -combine with other elements to form similarly constituted compounds, -both as regards number of atoms and their arrangement in the molecule. -Number of atoms alone, however, is no criterion, for the five atoms of -the ammonium group NH_{4} replace the one atom of potassium without -change of form. - -This case of the base ammonia had been one of Mitscherlich’s greatest -difficulties during the earlier part of his work, and remained a -complete puzzle until about this time, when its true chemical character -was revealed. For until the year 1820 Berzelius believed that it -contained oxygen. Seebeck and Berzelius had independently discovered -ammonium amalgam in 1808, and Davy found, on repeating the experiment, -that a piece of sal-ammoniac moistened with water produced the amalgam -with mercury just as well as strong aqueous ammonia. Both Berzelius and -Davy came to the conclusion that ammonia contains oxygen, like potash -and soda, and that a metallic kind of substance resembling the alkali -metals, potassium and sodium, was isolated from this oxide or hydrate by -the action of the electric current, which Seebeck had shown facilitated -the formation of the so-called ammonium amalgam. Davy, however, accepted -in part the views of Gay-Lussac and Thénard, who, in 1809, concluded -from their experiments that ammonium consisted of ammonia gas NH_{3} -with an additional atom of hydrogen, the group NH_{4} then acting like -an alkali metal, views which time has substantiated. But their further -erroneous conclusion that sodium and potassium also contained hydrogen -was rejected by him. Berzelius, however, set his face both against this -latter fallacy and the really correct NH_{4} theory, and it was not -until four years after Ampère, in 1816, had shown that sal-ammoniac was, -in fact, the compound of the group NH_{4} with chlorine, that Berzelius, -about the year 1820, after thoroughly sifting the work of Ampère, -accepted the view of the latter that in the ammonium salts it is the -group NH_{4}, acting as a radicle capable of replacing the alkali -metals, which is present. - -The fact that this occurred at this precise moment, four years after the -publication of Ampère’s results, leads to the conclusion that the -observation of Mitscherlich, that the ammonium compounds are isomorphous -with the potassium compounds, was the compelling argument which caused -Berzelius finally to admit what has since proved to be the truth. - -While still at Stockholm Mitscherlich showed that the chromates and -manganates are isomorphous with the sulphates, and also that the -perchlorates and permanganates are isomorphous with each other. Although -these facts could not be properly explained at the time, owing to the -inadequate progress of the chemistry of manganese, it was seen that -potassium chromate, K_{2}CrO_{4}, contained the same number of atoms as -potassium sulphate, K_{2}SO_{4}, and that potassium permanganate -KMnO_{4} and perchlorate KClO_{4} likewise resembled each other in -regard to the number of atoms contained in the molecule. - -As a good instance of the use of the principle of isomorphism, we may -recall that when Marignac, in 1864, found himself in great difficulty -about the atomic weights of the little known metals tantalum and niobium -which he was investigating, he discovered that their compounds are -isomorphous; the pentoxides of the two metals occur together in -isomorphous mixture in several minerals, and the double fluorides with -potassium fluoride, K_{2}TaF_{7} and K_{2}NbF_{7} are readily obtained -in crystals of the same form. The specific heat of tantalum was then -unknown, so that the law of Dulong and Petit connecting specific heat -with atomic weight could not be applied, and the vapour density of -tantalum chloride, as first determined by Deville and Troost with impure -material, did not indicate an atomic weight for tantalum which would -give it the position among the elements that the chemical reactions of -the metal indicated. Yet Marignac was able definitely to decide, some -time before the final vapour density determinations of Deville and -Troost with pure salts, from the fact of the isomorphism of their -compounds, that the only possible positions for tantalum and niobium -were such as corresponded with the atomic weights 180 and 93 -respectively. Time has only confirmed this decision, and we now know -that niobium and tantalum belong to the same family group of elements as -that to which vanadium belongs, and the only difference which modern -research has introduced has been to correct the decimal places of the -atomic weights, that of niobium (now also called columbium, the name -given to it by its discoverer, Hatchett, in 1801) being now accepted as -92.8 and that of tantalum 179.6, when that of hydrogen = 1. - -Applying the law of isomorphism in a similar manner, Berzelius was -enabled to fix the atomic weights of copper, cadmium, zinc, nickel, -cobalt, iron, manganese, chromium, sulphur, selenium, and chlorine, the -numbers accepted to-day differing only in the decimal places, in -accordance with the more accurate results acquired by the advance of -experimental and quantitative analytical methods. But with regard to -several other elements, owing to inadequate data, Berzelius made serious -mistakes, showing how very great is the necessity for care and for ample -experimental data and accurate measurements, before the principle of -isomorphism can be applied with safety. Given these, and we have one of -the most valuable of all the aids known to us in choosing the correct -atomic weight of an element from among two or three possible -alternatives. We are only on absolutely sure ground when we are dealing -not only with a series of compounds consisting of the same number of -atoms, but when also the interchangeable elements are the intimately -related members of a family group, such as we have since become familiar -with in the vertical groups of elements in the periodic table of -Mendeléeff. - -Before leaving Stockholm Mitscherlich showed, from experiments on the -crystallisation of mixtures of the different sulphates with which he had -been working, that isomorphous substances intermix in crystals in all -proportions, and that they also replace one another in minerals in -indefinite proportions, a fact which has of recent years been -wonderfully exemplified in the cases of the hornblende (amphibole) and -augite (pyroxene) groups. - -In November 1821 Mitscherlich closed these memorable labours at -Stockholm and returned to Berlin, where he acted as extraordinary -professor of the university until 1825, when he was elected professor in -ordinary. His investigations for a time were largely connected with -minerals, but on July 6th, 1826, he presented a further most important -crystallographic paper to the Berlin Academy, in which he announced his -discovery of the fact that one of the best known chemical elements, -sulphur, is capable of crystallising in two distinct forms. The ordinary -crystals found about Etna and Vesuvius and in other volcanic regions -agree with those deposited from solution in carbon bisulphide in -exhibiting rhombic symmetry. But Mitscherlich found that when sulphur is -fused and allowed to cool until partially solidified, and the still -liquid portion is then poured out of the crucible, the walls of the -latter are found to be lined with long monoclinic prisms. These have -already been illustrated in Fig. 2, Plate I., in Chapter I. - -Here was a perfectly clear case of an element—not liable to any charge -of difference of chemical composition such as might have applied to the -cases of sodium dihydrogen phosphate, carbonate of lime, and iron -vitriol and its analogues, which he had previously described as cases of -the same substance crystallising in two different forms—which could be -made to crystallise in two different systems of symmetry at will, by -merely changing the circumstances under which the crystallisation -occurred. His explanation being thus proved absolutely, he no longer -hesitated, but at once applied the term “dimorphous” to these substances -exhibiting two different forms, and referred to the phenomenon itself as -“dimorphism.” The case of carbonate of lime had given rise to prolonged -discussion, for the second variety, the rhombic aragonite, had been -erroneously explained by Stromeyer, after Mitscherlich’s first -announcement in 1819, as being due to its containing strontia as well as -lime, and the controversy raged until Buchholz discovered a specimen of -aragonite which was absolutely pure calcium carbonate, so that -Mitscherlich’s dimorphous explanation was fully substantiated. - -Dimorphism is very beautifully illustrated by the case of the trioxide -of antimony, Sb_{2}O_{3}, a slide of which, obtained by sublimation of -the oxide from a heated tube on to the cool surface of a glass -microscope slip, is seen reproduced in Fig. 50, Plate XI. The two forms -are respectively rhombic and cubic. The rhombic variety usually takes -the form of long needle-shaped crystals, which are shown in Fig. 50 -radiating across the field and interlacing with one another; the cubic -variety crystallises in octahedra, of which several are shown in the -illustration, perched on the needles, one interesting individual being -poised on the end of one of the needles. The two forms occur also in -nature as the rhombic mineral valentinite and the cubic mineral -senarmontite, which latter crystallises in excellent regular octahedra. -Antimonious oxide, moreover, is not only isomorphous, but isodimorphous -with arsenious oxide, a slide of octahedra of which has already been -reproduced in Fig. 3, Plate I., in Chapter I. For besides this common -octahedral form of As_{2}O_{3} artificial crystals of arsenious oxide -have been prepared of rhombic symmetry, resembling valentinite. Hence -the two lower oxides of arsenic and antimony afford us a striking case -of the simultaneous display of Mitscherlich’s two principles of -isomorphism and dimorphism. - -Thus the position in 1826 was that Mitscherlich had discovered the -principle of isomorphism, and had also shown the occurrence of -dimorphism in several well-proved specific cases, and that he regarded -at this time isomorphism as being a literal reality, absolute identity -of form. - -[Illustration: - - _PLATE XI._ - - FIG. 50.—Rhombic Needles and Cubic Octahedra of Antimony Trioxide - obtained by Sublimation. An interesting Example of Dimorphism. -] - -[Illustration: - - FIG. 101.—Ammonium Chloride crystallising· from a Labile - Supersaturated Solution (see p. 248). - - REPRODUCTIONS OF PHOTOMICROGRAPHS. -] - -These striking results appeared at once to demolish the theory that any -one substance of definite chemical composition is characterised by a -specific crystalline form, which was Haüy’s most important -generalisation. Mitscherlich, however, soon expressed doubts as to the -absolute identity of form of his isomorphous crystals, and saw that it -was quite possible that in the systems other than the cubic (in which -latter system the highly perfect symmetry itself determines the form, -and that the angles shall be identically constant), there might be -slight distinctive differences in the crystal angles. For he caused to -be constructed, by the celebrated optician and mechanician, Pistor, the -most accurate goniometer which had up till then been seen, provided with -four verniers, each reading to ten seconds of arc, and with a telescope -magnifying twenty times, for viewing the reflections of a signal, -carried by a collimator, from the crystal faces. Unfortunately in one -respect, he was almost at once diverted, by the very excess of -refinement of this instrument, to the question of the alteration of the -crystal angles by change of temperature, and lost the opportunity, never -to recur, of doing that which would at once have reconciled his views -with those of Haüy in regard to this important matter, namely, the -determination of these small but real differences in the crystal angles -of the different members of isomorphous series, and the discovery of the -interesting law which governs them, a task which in these later days has -fallen to the lot of the author. - -Another remarkable piece of crystallographic work, this time in the -optical domain, which has rendered the name of Mitscherlich familiar, -was his discovery of the phenomenon of crossed-axial-plane dispersion of -the optic axes in gypsum. (The nature and meaning of “optic axes” will -be explained in Chapter XIII., page 185.) During the course of a lecture -to the Berlin Academy in the year 1826 Mitscherlich, always a brilliant -lecturer and experimenter at the lecture table, exhibited an experiment -with a crystal of gypsum (selenite) which has ever since been referred -to as the “Mitscherlich experiment.” He had been investigating the -double refraction of a number of crystalline substances at different -temperatures, and had observed that gypsum, hydrated calcium sulphate, -CaSO_{4}.2H_{2}O, was highly sensitive in this respect, especially as -regards the position of its optic axes. At the ordinary temperature it -is biaxial, with an optic axial angle of about 60°, but on heating the -crystal the angle diminishes, until just above the temperature of -boiling water the axes become identical, as if the crystal were -uniaxial, and then they again separate as the temperature rises further, -but in the plane at right angles to that which formerly contained them; -hence the phenomenon is spoken of as “crossed-axial-plane dispersion.” -Mitscherlich employed a plate of the crystal cut perpendicularly to the -bisectrix of the optic axial angle, and showed to the Academy the -interference figures (see Plate XII.) which it afforded in convergent -polarised light with rising temperature. At first, for the ordinary -temperature, the usual rings and lemniscates surrounding the two optic -axes were apparent at the right and left margins of the field; as the -crystal was gently heated (its supporting metallic frame being heated -with a spirit lamp) the axes approached each other, with ever changing -play of colour and alteration of shape of the rings and lemniscates, -until eventually the dark hyperbolic brushes, marking by their well -defined vertices the positions of the two optic axes within the -innermost rings, united in the centre of the field to produce the -uniaxial dark rectangular cross; the rings around the centre had now -become circles, the lemniscates having first become ellipses which more -and more approximated, as the temperature rose, to circles. Then the -dark cross opened out again, and the axial brushes separated once more, -but in the vertical direction, and the circles became again first -ellipses and then lemniscates, eventually developing inner rings around -the optic axes; if the source of heat were not removed at this stage the -crystal would suddenly decompose, becoming dehydrated, and the field on -the screen would become dark. If, however, the spirit lamp were removed -before this occurred, the phenomena were repeated in the reverse order -as the crystal cooled. - -This beautiful experiment is now frequently performed, as gypsum is -perhaps the best example yet known which exhibits the phenomenon of -crossed-axial-plane dispersion by change of temperature alone. A -considerable number of other cases are known, such as brookite, the -rhombic form of titanium dioxide TiO_{2}, and the triple tartrate of -potassium, sodium, and ammonium, but these are more sensitive to change -of wave-length in the illuminating light than to change of temperature. - -[Illustration: - - FIG. 51.—The Mitscherlich Experiment with Gypsum. -] - -[Illustration: - - _PLATE XII._ - - FIG. 52.—Appearance of the Interference Figure half a Minute after - commencing the Experiment. Temperature of Crystal about 40° C. -] - -[Illustration: - - FIG. 53.—Appearance a Minute or so later, the Axes approaching the - Centre. Temperature of Crystal about 85° C. -] - -[Illustration: - - FIG. 54.—The Two Optic Axes coincident in the Centre of the Figure, - two or three Minutes from the commencement. Temperature of Crystal - 106° C. -] - -[Illustration: - - FIG. 55.—The Axes re-separated in the Vertical Plane a Minute or two - later. Temperature of Crystal about 125° C. - - THE MITSCHERLICH EXPERIMENT WITH GYPSUM. - - FOUR STAGES IN THE TRANSFORMATION OF THE INTERFERENCE FIGURE IN - CONVERGENT POLARISED LIGHT, FROM HORIZONTALLY BIAXIAL THROUGH - UNIAXIAL TO VERTICALLY BIAXIAL, ON RAISING THE TEMPERATURE TO 125° - C. - - (From Photographs by the author.) -] - -The author has recently exhibited the “Mitscherlich experiment” to the -Royal Society,[2] and also in his Evening Discourse to the British -Association at their 1909 meeting in Winnipeg, in a new and more elegant -manner, employing the large Nicolprism projection polariscope shown in -Fig. 51, and a special arrangement of lenses for the convergence of the -light, which is so effective that no extraneous heating of the crystal -is required. The convergence of the rays is so true on a single spot in -the centre of the crystal plate about two millimetres diameter, that a -crystal plate not exceeding 6 mm. is of adequate size, mounted in a -miniature holder-frame of platinum or brass with an aperture not more -than 3 mm; the thickness of the crystal should remain about 2 mm., in -order that the rings round the axes may not be too large and diffuse, -the crystal being endowed with very feeble double refraction, which is -one of the causes of the phenomenon. Such a small crystal heats up so -rapidly in the heat rays accompanying the converging light rays—even -with the essential cold water cell two inches thick between the lantern -condenser and the polarising Nicol, for the protection of the balsam of -the latter—that any extraneous heating by a spirit or other lamp is -entirely unnecessary. The moment the electric arc of the lantern is -switched on, the optic axial rings appear at the right and left margins -of the screen, when the crystal is properly adjusted and the arc -correctly centred, and they march rapidly to the crossing point in the -centre, where the dark hyperbolæ unite to produce the rectangular St -Andrew’s cross, the rings, figure-eight curves, and other lemniscates -passing through the most exquisite evolutions and colour changes all the -time until they form the circular Newton’s rings, around the centre of -the cross; after this the cross and circles again open out, but along -the vertical diameter of the screen, into hyperbolæ and rings and -loop-like lemniscates surrounding two axes once more. It is wise as soon -as the separation in this plane is complete and the first or second -separate rings have appeared round the axes, to arrest the heating by -merely interposing intermittently a hand screen between the lantern and -polariser, or by blowing a current of cool air past the crystal, which -will cause the axes to recede again, and the phenomena to be reversed, -the crossing point being repassed, and the axes brought into the -original horizontal plane again. By manipulation of the screen, or -air-current, the axes can thus be caused to approach or to recede from -the centre at will, along either the horizontal or vertical diameter. -Four characteristic stages of the experiment are shown in Figs. 52 to -55, Plate XII. Fig. 52 exhibits the appearance just after commencing the -experiment, the optic axes being well in the field of view. Fig. 53 -shows the axes horizontally approaching the centre. Fig. 54 shows the -actual crossing, which occurs for different crystals at temperatures -varying from 105°.5 to 111°.5 C.; and Fig. 55 represents the axes again -separated, but vertically. - -The experiment as thus performed is one of the most beautiful -imaginable, and it can readily be understood how delighted were -Mitscherlich’s audience on the occasion of its first performance by him. -The author has since discovered no less than six other cases of -substances which exhibit crossed-axial-plane dispersion of the optic -axes, in the course of his investigations, one of which is illustrated -in Plate XIII., facing page 108; and, moreover, has arrived at a general -explanation of the whole phenomenon, the main points of which are that -such substances, besides showing very feeble double refraction (the two -extreme of the three refractive indices being very close together), also -exhibit very close approximation of the intermediate refractive index β -to either the minimum index α or the maximum index γ. Also, change of -temperature, or of wave-length, or most usually both, must so operate as -to bring the two indices closest together into actual identity and then -to pass beyond each other, these two indices thus exchanging positions, -the extreme one becoming the intermediate index. In other words, the -uniaxial cross and circular rings are produced owing to two of the three -refractive indices (corresponding to the directions of the three -rectangular axes of the ellipsoid which, in general, expresses the -optical properties of a crystal) becoming equal at the particular -temperature at which the phenomenon is observed to occur, and for light -of the specific wave-length in question. The ellipsoid of general form -which represents the optical properties of a biaxial crystal thus -becomes converted into a rotation ellipsoid corresponding to a uniaxial -crystal. Brookite and the triple tartrate are excellent examples of the -predominance of the effect of change of wave-length, for the optic axes -are separated in both cases widely in one plane for red light and almost -equally widely in the perpendicular plane for blue light. The new cases -observed by the author are sensitive both to change of wave-length and -to change of temperature, and so fall midway between the cases just -quoted and the case of gypsum. The cause of it, in four of these new -instances, is a very interesting one, connected with the regular change -of the refractive indices in accordance with the law of progression in -an isomorphous series according to the atomic weight of the alkali metal -present, which will be discussed in Chapter X. - -A further most important discovery was made by Mitscherlich in the year -1827, which also profoundly concerns the work of the author, namely, -that of selenic acid, H_{2}SeO_{4}, analogous to sulphuric acid, and of -the large group of salts derived from it, the selenates, analogous to -the sulphates. He showed first that potassium selenate, K_{2}SeO_{4}, is -isomorphous with potassium sulphate, K_{2}SO_{4}, and subsequently that -the selenates in general are isomorphous with the corresponding -sulphates; consequently it followed that selenium is a member of the -sulphur family of elements. This element selenium had only been -discovered ten years previously by his friend Berzelius, and doubtless -Mitscherlich had seen a great deal of the work in connection with it -during the two years which he spent in the laboratory of Berzelius at -Stockholm, and was deeply interested in it. - -The discovery has proved a most fruitful one, for the selenates are -beautifully crystalline salts, particularly suitable for -crystallographic researches, and their detailed investigation has -afforded a most valuable independent confirmation of the important -results obtained for the sulphates. - -Again in 1830 Mitscherlich, following up the preliminary work already -referred to, definitely established another fact bearing on the same -series, namely, the isomorphism of potassium manganate K_{2}MnO_{4} with -the sulphate and selenate of potash; moreover, on continuing his study -of the manganese salts he further substantiated the isomorphism of the -permanganates with the perchlorates, and isolated permanganic acid. This -also proved a most important step forward, as these salts likewise -afford admirable material for crystallographic investigation, and such -an examination, carried out by Muthmann and Barker, has yielded most -valuable results. - -Much later in his career Mitscherlich also described the dimorphous -iodide of mercury, HgI_{2}, one of the most remarkable and interesting -salts known to us, the unstable yellow rhombic modification being -converted into the more stable red tetragonal form by merely touching -with a hard substance. Also we are indebted to him at the same later -period for our knowledge of the crystalline forms of the elements -phosphorus, iodine, and selenium, when crystallised from solution in -bisulphide of carbon. - -From the record of achievements which has now been given in this chapter -it will be obvious how much chemical crystallography owes to -Mitscherlich. The description of his work has taken us into almost every -branch of the subject, morphological, optical, and thermal, and although -it has consequently been necessary to refer to phenomena which have not -yet been explained in this book, it has doubtless proved on the whole -most advantageous thus to present the life work of this great master as -a complete connected story. - - - - - CHAPTER VIII - MORPHOTROPY AS DISTINCT FROM ISOMORPHISM. - - -It has been shown in the last chapter how Mitscherlich discovered the -principle of isomorphism, as applying to the cases of substances so -closely related that their interchangeable chemical elements are members -of the same family group; and also how the principle enabled him to -determine the chemical constitution of two hitherto unknown acids which -he isolated, selenic H_{2}SeO_{4} and permanganic HMnO_{4}. For he -observed that the selenates were isomorphous with the sulphates, and the -permanganates with the perchlorates. It was further made clear that the -principle as bequeathed to us by Mitscherlich was only defined in very -general terms, and its details have only recently been precisely -decided. - -Before proceeding further (in Chapter X.) with the elucidation of the -true nature of isomorphism, however, some important crystallographic -relationships between substances less closely related than family -analogues must be referred to, as the outcome of a series of -investigations by von Groth, chiefly between the derivatives of the -hydrocarbon benzene. Also, some suggestive results obtained by the -author from an investigation of an organic homologous series, that is, -one the members of which differ by the regular addition of a CH_{3} -group, may be briefly referred to. - -The interval between the work of Mitscherlich and that of von Groth was -one of doubt, discouragement, and somewhat of discredit for chemical -crystallography. The chemists Laurent[3] and Nicklès[4] carried out -during the years from 1842 to 1849 measurements of numerous organic -substances and of some inorganic compounds, the former chiefly halogen -or other derivatives of particular hydrocarbons or salts of homologous -fatty acids. Laurent, for instance, found that naphthalene -tetrachloride, C_{10}H_{8}.Cl_{4}, and chloronaphthalene tetrachloride, -C_{10}H_{7}Cl.Cl_{4}, crystallise in different systems, the former in -the monoclinic and the latter in the rhombic system. Yet the primary -prism angles of the two are less than a degree different, namely, 109° -0′ and 109° 45′. Laurent named this kind of similarity “hemimorphism,” a -most unfortunate term as it was already employed in crystallography in -its other well-known geometrical significance, that is, to denote a -crystal differently terminated at the two ends of an axis. Many other -like similarities were discovered by Laurent, and he again coined an -objectionable term, now discarded, to represent the cases of similarity -extending over more than the same system, namely, “isomeromorphism.” - -Nicklès observed similar facts in connection with the barium salts of -the fatty acids, which crystallise in different systems with different -amounts of water of crystallisation. But their prism angles are all -within a couple of degrees of each other, varying from 98° to 100°. Thus -the phenomenon of “isogonism,” a term much less objectionable than those -invented by Laurent, appears to be a common observance not only for -different kinds of derivatives of the same original hydrocarbon or other -organic nucleus, but also for the case of homologous series. But Nicklès -missed the real point by including salts with different amounts of -water, which, it will be shown later, entirely upset the crystalline -structure. When this is eliminated the resemblance between true -similarly constituted homologues, differing by regular increments of -CH_{3}, is very much closer than would appear from Nicklès’ results. - -Unfortunately, some of the work of Laurent and Nicklès was not carried -out with the care and accuracy which is indispensable for researches -which are to retain permanent value, and critics were not slow to arise. -Kopp,[5] in 1849, unmercifully exposed these failings, so that the real -kernel of the work, which was of considerable value, came into -discredit. - -Pasteur,[6] however, in 1848, besides the important observations -regarding enantiomorphism, to be described in Chapter XI., had noticed -similar zonal likenesses between related tartrates, amounting only -therefore to isogonism and not to isomorphism; for here again the system -often differed, particularly when the members of a series compared -differed in their water of crystallisation. Thus there was ample -evidence of a really significant series of facts in the work of these -authors, but they were not properly arranged and explained. - -So high was the feeling against the whole subject carried, however, -after Kopp’s memoir, that had it not been for the steadying influence of -Rammelsberg and Marignac, who themselves carried out many -crystallographic measurements as new substances continued to be -discovered with great rapidity, the science would have suffered a -serious set-back. Moreover, even Rammelsberg was led astray in the -direction of the views of the chemists of the time, that isomorphism -could be extended over the crystal system. Frankenheim, whose discovery -of the space-lattice, to be referred to in the next chapter, will ever -render his name famous, strongly opposed this view. Delafosse, on the -other hand, recognised some truth in both views, and assumed that there -were two kinds of isomorphism, that of Mitscherlich on the one hand, and -the broader one of Laurent on the other hand, and that in the case of -the latter kind the overstepping of the system is no bar. - -Hjortdahl,[7] in the year 1865, supported the views of Delafosse more or -less, at any rate so far as to assume the possibility of the existence -of partial isomorphism, that is, of isogonism. He was very definite, -however, against accepting the proposition that any general law could be -applied. He himself discovered a partial similarity of angles in several -homologous series of organic compounds. - -About this time Sella[8] uttered a warning which is one worthy of being -prominently posted in every research laboratory, namely, that _It is -unwise to make hasty generalisations from the results of a small number -of observations_. Were this principle more generally followed, much -greater progress would in the end be achieved, and without the -discouragement and discredit which inevitably follows the detection of -errors due to lack of broad experimental foundation. It is certainly an -incontrovertible fact that only such generalisations as find themselves -in accordance with all new but well-verified experimental facts as they -are revealed can stand the test of time and become accepted universally -as true laws of nature. And it is unreasonable to expect any -generalisation to be of such a character unless it is already based on -so large a number of facts that there is little fear of other new ones -upsetting them. - -Some order was, however, introduced into this chaotic state of chemical -crystallography in the year 1870 by P. von Groth.[9] He investigated -systematically the derivatives of the hydrocarbon benzene, C_{6}H_{6}, -many of which are excellently crystallising solids suitable for -goniometrical measurement. He showed that although the crystal system -may be and often is altered, yet there is a striking similarity in the -angles between the faces of certain zones, which for the purposes of -comparison he arranged to be parallel to each other in his descriptions -of the crystals, so that the relationship would then consist in an -elongation or a shortening of this particular zone axis, which was -usually a crystallographic axis. He recognised that this was a totally -different phenomenon from isomorphism, and called it “morphotropy.” -Although it may possibly be permissible from one point of view to regard -isomorphism as a particular case of complete morphotropy along all -zones, such a course is not advisable, as morphotropic similarities are -frequently of a comparatively loose and often indeed of a somewhat vague -character, while isomorphous relationships are governed by very precise -laws. - -Thus von Groth showed first that benzene, C_{6}H_{6}, crystallises in -the rhombic system with axial ratios _a_ : _b_ : _c_ = 0.891 : 1 : -0.977. Next, that when one or two of the hydrogen atoms are replaced by -hydroxyl OH groups the substances produced, phenol C_{6}H_{5}.OH and -resorcinol C_{6}H_{4}(OH)_{2}, are found also to crystallise in the -rhombic system, and in the second case, for which alone the axial ratios -could be determined, the ratio _a_ : _b_ proved to be very similar, but -the ratio _c_ : _b_ was different, the actual values being _a_ : _b_ : -_c_ = 0.910 : 1 : 0.540. Pyrocatechol, the isomer of resorcinol, also -crystallises in the rhombic system, but the crystals have not been -obtained sufficiently well formed to enable any deductions to be made -from any measurements carried out with them. - -Similarly, the nitro-derivatives of phenol, orthonitrophenol -C_{6}H_{4}.OH.NO_{2}, dinitrophenol C_{6}H_{3}.OH.(NO_{2})_{2}, and -trinitrophenol C_{6}H_{2}.OH.(NO_{2})_{3}, also crystallise in the -rhombic system, and with the following respective axial ratios: 0.873 : -1 : 0.60; 10.933 : 1 : 0.753; 0.937 : 1 : 0.974. Again, the value for -the ratio _a_ : _b_ is not very different from that of benzene itself, -while the ratio _c_ : _b_ differs considerably in the first two cases. -Similar relations were also found to hold good in the cases of -meta-dinitrobenzene, C_{6}H_{4}(NO_{2})_{2}, axial ratios 0.943 : 1 : -0.538, and trinitrobenzene, C_{6}H_{3}(NO_{2})_{3}, which possesses the -axial ratios 0.954 : 1 : 0.733. - -The introduction of a chlorine or bromine atom or a CH_{3} group in -place of hydrogen was found by von Groth to produce more than the above -effect, the symmetry being often lowered to monoclinic, a fact which had -also been observed to occur in the cases of certain isomers of the -substances quoted above, ortho-dinitrobenzene for instance. But it was -nevertheless observed that the angles between the faces in the prism -zone remained very similar, the angles between the faces of the primary -prism (110) and (1̄10), for instance, only varying in eight such -derivatives of all three types, whether rhombic or monoclinic, from 93° -45′ to 98° 51′. - -The crystallographic relationships of organic substances, however, are -very much complicated by the possibilities of isomerism, the ortho, -meta, and para compounds—corresponding to the replacement of the two -hydrogen atoms attached to two adjacent, alternate, or opposite carbon -atoms respectively, of the six forming the benzene ring—generally -differing extensively and sometimes completely in crystalline form. -Consequently, the phenomenon of morphotropy is best considered quite -independently of isomorphism. - -An interesting intermediate case between morphotropy and true -isomorphism was investigated by the author in the year 1890, namely, a -series of homologous organic compounds differing by regular increments -of the organic radicle CH_{3}. They were prepared by Prof. Japp and Dr -Klingemann, and consisted of the methyl, CH_{3}, ethyl, C_{2}H_{5}, and -propyl, C_{3}H_{7}, derivatives of the substance triphenyl pyrrholone, -all of them being solids crystallising well. The problem was somewhat -complicated by the development of polymorphism, the methyl, ethyl, and -propyl compounds having each been found to be dimorphous, and not -improbably trimorphous, but only two varieties of each salt were -obtained in crystals adequately perfect for measurement. That the -production of these different forms was due to polymorphism and not to -chemical isomerism (different arrangement of the chemical atoms in the -molecule) was shown by the fact that one variety could be obtained from -the other by simply altering the conditions of crystallisation from the -same solvent. Their identical chemical composition was established by -direct analysis. - -The methyl (CH_{3}) compound crystallised in rhombohedra and in -triclinic prisms. The ethyl (C_{2}H_{5}) derivative was deposited in -triclinic prisms exactly resembling those of the methyl compound in -habit and disposition of faces. A crystal of the triclinic methyl -derivative which would represent equally well the ethyl compound is -shown in Fig. 56. The angles also of the crystals of the two substances -are so similar that one might infer the existence of true and complete -isomorphism. The actual angular differences rarely exceeded three -degrees. - -[Illustration: - - FIG. 56.—Crystal of Methyl Triphenyl Pyrrholone. -] - -Besides the triclinic form the ethyl derivative was also obtained in -monoclinic crystals, one of which is represented in Fig. 57. This -illustration might serve equally well, however, for a corresponding -monoclinic form of the propyl (C_{3}H_{7}) derivative, and the angles of -these two monoclinic ethyl and propyl compounds are even closer than -those of the triclinic methyl and ethyl derivatives, the closeness -increasing with the advent of symmetry. - -[Illustration: - - FIG. 57.—Crystal of Ethyl Triphenyl Pyrrholone. -] - -This similarity of angles in the cases of the two pairs of triclinic and -monoclinic compounds is not only true about particular zones, but about -all the zones, so that it is a case isomorphism rather than of isogonism -(morphotropy). The similarity of optical properties is also very close, -and so much so in the cases of the monoclinic crystals of ethyl and -propyl triphenyl pyrrholone that both exhibit very high dispersion of -the optic axes. In the case of the propyl derivative the difference -between the apparent angle of the optic axes for red lithium light and -for green thallium light amounts to 11°. In the case of the ethyl -compound this difference is enhanced so considerably that the crystals -afford a remarkable instance of dispersion of the optic axes in crossed -axial planes, resembling the case of gypsum discovered by Mitscherlich -and described in the last chapter, except that the sensitiveness is to -change of wave-length in the illuminating light rather than to change of -temperature. The optic axial plane is perpendicular to the symmetry -plane for lithium and sodium light, as it is also in the case of the -propyl compound; but in the ethyl derivative it crosses over for -thallium light and rays beyond that towards the violet, into a plane at -right angles to the former plane, namely, the symmetry plane itself. The -total dispersion between the two axes as separated in the one plane for -red light, and as separated in the other perpendicular plane for blue -light, is more than 70°. Fig. 58, Plate XIII., shows the nature of the -interference figures afforded in convergent polarised light of different -wave-lengths by a section-plate perpendicular to the first median line. -The figure at _f_ represents what is observed in white light, as far as -is possible by a drawing in black and white. It consists of a series of -concave coloured curves, falling in between the arms of the cross, and -looping round the axes, a figure very much like that afforded by -brookite and triple tartrate of ammonium, potassium, and sodium, the -substances already mentioned in Chapter VII. as being similarly very -sensitive to change of wave-length. The figure in red monochromatic -lithium light is shown at _a_ in Fig. 58, and that for yellow sodium -light at _b_, the axes being now much closer together. On changing to -green thallium light the line joining the optic axes becomes vertical -instead of horizontal, as shown at _d_. - -When, instead of employing monochromatic flames, the spectroscopic -monochromatic illuminator (Fig. 75, page 193), described by the author -some years ago to the Royal Society, is employed to illuminate the -polariscope, the source of light being the electric arc, the change of -the figure from that given by the extreme red of the spectrum to that -afforded by the violet may be beautifully followed, and the exact -wave-length in the greenish yellow determined for which the crossing -occurs and an apparently uniaxial figure of circular rings and -rectangular cross is produced. For it is possible with the aid of this -illuminator directly to observe the production of the uniaxial figure. -The wave-length is either directly afforded by the graduation of the -fine-adjustment micrometric drum or is obtained from a curve of -wave-lengths, constructed to correspond to the circle readings of the -illuminator. The appearance of the interference figure for this critical -wave-length is shown at _c_ in Fig. 58. The remaining figure at _e_ -represents the appearance when a mixture of sodium and thallium light is -employed, which clearly indicates the four extreme axial positions, and -assists in elucidating the nature of the figure _f_ exhibited in white -light. - -The second form of the propyl derivative belongs to the rhombic system, -and a similar rhombic form of the ethyl compound was once obtained, but -lost again on attempting to recrystallise. - -These interesting relationships of the homologous methyl, ethyl, and -propyl derivatives of triphenyl pyrrholone thus appear to form a -connecting link between cases of isogonism or morphotropy and of true -isomorphism. - -[Illustration: - - _PLATE XIII._ - - FIG. 58.—Interference Figures in Convergent Polarised Light of - different Wave-lengths afforded by the Monoclinic Variety of Ethyl - Triphenyl Pyrrholone; _a_, in Red Lithium Light; _b_, in Yellow - Sodium Light; _c_, in Greenish-Yellow Light of the Critical - Wave-length for Production of the Uniaxial Figure; _d_, in Green - Thallium Light; _e_, in mixed Sodium and Thallium Light; and _f_, in - White Light. - - (Reproductions of Drawings by the author.) -] - -We are now, therefore, in a position to approach the question of true -isomorphism, and as leading up to the fuller treatment of the subject in -Chapter X. we may conclude this chapter by referring first to one -important investigation in which the necessity for extreme accuracy of -measurement and perfection of material was fully appreciated. This was -an admirable research carried out in the years 1887 and 1888 by H. A. -Miers[10] on the red silver minerals, proustite, sulpharsenite of -silver, Ag_{3}AsS_{3}, and pyrargyrite, the analogous sulphantimonite of -silver, Ag_{3}SbS_{3}, which afforded a further indication of the -existence of real small differences of angle between the members of -truly isomorphous series. These two minerals form exceptionally -beautiful crystals belonging to the trigonal system, the hexagonal prism -being always a prominent form, terminated by the primary and other -rhombohedra, scalenohedra and various pyramidal forms, many of the -crystals being exceedingly rich in faces. When the crystals are freshly -obtained from the dark recesses of the silver mine they are very -lustrous and transparent, but they are gradually affected by light, like -many silver compounds, and require to be stored in the dark in order to -preserve their transparency. A magnificent crystal of proustite from -Chili is one of the finest objects in the British Museum at South -Kensington, but is rarely seen on account of the necessity for -preservation from light. Pyrargyrite is generally dark grey in -appearance, and affords a reddish-purple “streak” (colour of the powder -on scratching or pulverising). Proustite, however, possesses a beautiful -scarlet-vermilion colour, and affords a very bright red streak. - -Now these two beautiful minerals are obviously analogous compounds of -the same metal, silver, with the sulpho-acid of two elements, arsenic -and antimony, belonging strictly to the same family group, the -nitrogen-phosphorus group, of the periodic classification of the -elements according to Mendeleéff. Consequently, they should be perfectly -isomorphous. Miers has shown in a most complete manner that they are so, -that they occur in very perfect crystals of similar habit belonging to -the same class of the trigonal system, the ditrigonal polar class, both -minerals being hemimorphic, that is, showing different forms at the two -terminations, in accordance with the symmetry of the polar class of the -trigonal system. But the angles of the two substances were not found to -be identical, although constant for each compound within one minute of -arc, there being slight but very real differences, which are very well -typified by the principal angle in each case, that of the primary -rhombohedron. In the case of proustite it is 72° 12′, while the -rhombohedron angle of pyrargyrite is 71° 22′. - -This interesting and beautiful investigation of Miers thus gave us an -inkling of the truth, that small angular differences do exist between -the members of isomorphous compounds. It paved the way for, and indeed -partly suggested, the author’s systematic investigation of the -sulphates, selenates, and double salts of the alkali series of metals, a -brief account of the main results of which will be given in Chapter X. - - - - - CHAPTER IX - THE CRYSTAL SPACE-LATTICE AND ITS MOLECULAR UNIT CELL. THE 230 - POINT-SYSTEMS OF HOMOGENEOUS CRYSTAL STRUCTURE. - - -The interval between the morphotropic work described in the last chapter -and the present time has been remarkable for the completion of the -geometrical and mathematical investigation, and the successful -identification, of all the possible types of homogeneous structures -possessing the essential attributes of crystals. It has now been -definitely established that there are 230 such types of homogeneous -structures possible, and the whole of them conform to the conditions of -symmetry of one or other of the thirty-two classes of crystals. This -fact is now thoroughly agreed upon by all the authorities who have made -the subject their special study, and may truly be considered as -fundamental. - -There has long been a consensus of opinion that the crystal edifice is -built up of structural units which can be likened to the bricks or stone -blocks of the builder, but which in the case of the crystal are so small -as to be invisible even under the highest power of the microscope. The -conceptions of their nature, however, have been almost as numerous as -the investigators themselves, everyone who has thought over the subject -forming his own particular ideas concerning them. We have had the -“Molécules intégrantes” of Haüy, the “Polyhédres” of Bravais, the -“Fundamentalbereich” of Schönflies, the “Parallelohedra” of von Fedorow, -and the fourteen-walled cell, the “Tetrakaidecahedron” of Lord Kelvin, -and again the “Polyhedra” of Pope and Barlow. Ideas have thus been -extremely fertile, and indeed almost every variety of speculation has -been indulged in as to the shape and nature of the unit of the structure -which can exhibit such remarkable evidences of organisation and such -extraordinary optical and other physical properties as those of a -crystal. - -There is one inherent difficulty, however, which renders all such -speculations more or less chimerical, until we know very much more as to -the structure of the chemical atom, and the organisation of the -corpuscles composing it. Such speculations, however, are deeply -interesting, and the difficulty alluded to accounts largely for the -great variety of conception possible. It is this, that the matter of the -molecules, and again that of the atoms composing them, is not -necessarily, nor even probably, continuous and in contact throughout, -but that on the contrary the space which may legitimately be assigned to -the unit of the structure is partly void. How much of this unit space is -matter and how much is unoccupied, and how the one is related to the -other as regards its position or distribution in space, we have yet no -means of knowing, although there are signs that the day is not far -distant when we shall know at least something concerning it. The recent -brilliant work of Sir J. J. Thomson and his school of physicists has -rendered it clear that the chemical atom is composed of cycles of -electronic corpuscles, the orbital motions of which determine its -boundaries. - -In this condition of our knowledge obviously the only safe course is -to consider each atom of the chemical molecule as occupying a “sphere -of influence,” within the limits of which the material parts of the -atom, the corpuscles in organised motion, are confined. The -“Fundamentalbereich” of Schönflies and the “Sphere of Influence” of -Barlow, are the conceptions which in all probability have the greatest -value in the present state of our knowledge, and if we adopt the -latter we shall not be committing ourselves to anything more than the -experimental facts fully warrant. - -It may be quite definitely stated, however, that there is a considerable -amount of experimental evidence that the unit of the space-lattice of -the crystal structure is certainly not more complex than the chemical -molecule, the idea of an aggregation of chemical molecules to form a -“physical molecule” acting as a structural unit having proved to be a -misleading myth. - -Fortunately, however, there is no necessity whatever to introduce the -subject of the actual shape of the unit, and the greatest progress has -been effected by disregarding it altogether, and agreeing to the -representation of the unit by a point. This leads us at once to perceive -the importance of the brilliant work of the geometricians, who have now -completed their theory of the homogeneous partitioning of space into -point-systems possible to crystals, the structural units of the latter -being regarded as points. The investigations extend from those of -Frankenheim in the year 1830 to the finishing touches given by Barlow in -1894, and prominently standing forth as those of the greatest -contributors to the subject, besides the two investigators just -mentioned, are the names of Bravais, Sohncke, Schönflies and von -Fedorow. - -Bravais, perfecting the work of his predecessor Frankenheim, made us -acquainted with the fourteen fundamentally important space-lattices, or -same-ways orientated arrangements of points. If we regard each chemical -molecule as represented by a point, disregarding the separate atoms of -which it is composed, then these fourteen space-lattices represent the -possible arrangements of the molecules in the crystal in all the simpler -cases; three of these lattices have cubic symmetry; the tetragonal, -hexagonal, trigonal, rhombic and monoclinic systems claim two -space-lattices each; while one space-lattice conforms to the lack of -symmetry of the triclinic system. - -The fourteen space-lattices of Bravais thus represent the arrangement of -the chemical molecules in the crystal, and determine the systematic -symmetry. The points being taken absolutely analogously in all the -molecules, and the whole assemblage being homogeneous, that is, such -that the environment about any one spot is the same as about every -other, the arrangement is obviously a same-ways orientated one, the -molecules being all arranged parallel-wise to each other. - -[Illustration: - - FIG. 59.—Triclinic Space-Lattice. -] - -But the fact that the structure is that of a space-lattice also causes -the crystal to obey the law of rational indices. To enable us to see how -this comes about it is only necessary to regard a space-lattice. In Fig. -59 is represented the general form of space-lattice, that which -corresponds to triclinic symmetry. It is obviously built up of -parallelepipeda, the edges of which are proportional to the lengths of -the three triclinic axes, and their mutual inclinations are those of the -latter. As we may take our representative point anywhere in the -molecule, so long as the position chosen is the same for all the -molecules of the assemblage, we may imagine the points occupying the -centres of the parallelepipeda instead of the corners if we choose, for -that would only be equivalent to moving the whole space-lattice slightly -parallel to itself. Hence, each cell may be regarded as the habitat of -the chemical molecule. - -Now the faces of the crystal parallel to each two of the three sets of -parallel lines forming the space-lattice will be the three pairs of -axial-plane faces, and any fourth face inclined to them must be got by -removing parallelepipedal blocks in step-wise fashion, precisely like -bricks, as already shown in Fig. 12 (page 28) in Chapter III., in order -to illustrate the step by step removal of Haüy’s unit blocks. It will -readily be seen that if one more cell be removed from each row than from -the row below it, the line of contact touching the projecting corner of -the last block of each row will be inclined more steeply than if two -more cells were removed from each row. Moreover, the angle varies -considerably between the two cases, and if three blocks are removed at a -time the angle gets very small indeed. Hence, there cannot be many such -planes possible, and we see at once why the indices of the faces -developed on a crystal are composed of low whole numbers and why the -forms are so relatively few in number. Owing to the minuteness of a -chemical molecule, all the irregularities of such a surface are -submicroscopic, and the general effect to the eye is that of a smooth -plane surface. - -The space-lattice arrangement of the molecules in the crystal structure -thus causes the crystal to follow the law of rational indices, by -limiting and restricting the number of possible facial forms which can -be developed. It also determines which one of the seven systems of -symmetry or styles of crystal architecture the crystal shall adopt. It -does not determine the details of the architecture, however, that is, to -which of the thirty-two classes it shall conform, this not being the -function of the molecular arrangement but of the atomic arrangement that -is, of the arrangement of the cluster of atoms which form the molecule, -and this leads us to the next step in the unravelling of the internal -structure of crystals. - -The credit of this next stage of further progress is due to Sohncke, -whose long labours resulted in the discrimination and description of -sixty-five “Regular Point-Systems,” homogeneous assemblages of points -symmetrically and identically arranged about axes of symmetry, which are -sometimes screw axes, that is, axes about which the points are spirally -distributed. Sohncke’s point-systems express the number of ways in which -symmetrical repetition can occur. Moreover, the points may always be -grouped in sets or clusters, the centres of gravity of which form a -Bravais space-lattice. - -This latter fact is of great interest, for it means that Sohncke’s -points may represent the chemical atoms, and that the stereometric -arrangement of the atoms in the molecule is that which produces the -point-system and determines the crystal class, while the whole cluster -of atoms forming the molecule furnishes, as above stated, the -representative point of the space-lattice. - -This, however, is not the whole story, for the sixty-five Sohnckian -regular point-systems only account for twenty-one of the thirty-two -crystal classes, the remaining eleven being those of lower than full -holohedral systematic symmetry, and which are characterised by showing -complementary right and left-handed forms. In other words, they exhibit -two varieties, on one of which faces of low symmetry are developed on -the right, while on the other symmetrically complementary faces are -developed on the left; that is, these little faces modify on the right -and left respectively the solid angles formed by those faces of the -crystal which are common to both the holohedral class of the system and -to the lower symmetry class in question. In some cases, moreover, these -two complementary forms are known to exist alone, without the presence -of faces common to both the holohedral class and the class of lower -symmetry. The two varieties of the crystals are the mirror images of -each other, being related as a right-hand glove is to a left-hand one. - -Further, the crystals of these eleven classes very frequently exhibit -the power of rotating the plane of polarised light to the right or to -the left, and complementarily in the cases of the two varieties of any -one substance, corresponding to the complementariness of the two crystal -forms. The converse is even more absolute, for no optically active -crystal has yet been discovered which does not belong to one or other of -these eleven classes of lower than holohedral symmetry. - -The final step of accounting for the structure of these highly -interesting eleven classes of crystals was taken simultaneously by a -German, Schönflies, a Russian, von Fedorow, and an Englishman, Barlow, -who quite independently and by totally different lines of reasoning and -of geometrical illustration showed that they were entirely accounted for -by the introduction of a new element of symmetry, that of mirror-image -repetition, or “enantiomorphous similarity” as distinguished from -“identical similarity.” These three investigators all united in finally -concluding that when the definition of symmetrical repetition is thus -broadened to include enantiomorphous similarity, 165 further -point-systems are admitted, and the whole 230 point-systems then account -for the whole of the thirty-two classes of crystals. - -Schönflies’ simple definition of the nature of the structure is that -every molecule is surrounded by the rest collectively in like manner, -when likeness may be either identity or mirror-image resemblance. Von -Fedorow finds the extra 165 types to be comprised in “double systems” -consisting of two “analogous systems” which are the mirror images of -each other. Barlow proceeds to find in how many ways the two -mirror-image forms can be combined together, there being in general -three distinct modes of duplication, including the insertion of one -inside the other. He also shows that all homologous points in a -structure of the type of one of these additional 165 point-systems -together form one of the 65 Sohnckian point-systems, the structure being -capable of the same rotations or translations, technically known as -“coincidence movements” (movements which bring the structure to exhibit -the same appearance as at first), as those which are characteristic of -that point-system. - -This fascinating subject of mirror-image symmetry, and the optical -activity connected with it, will be reverted to and the latter explained -in Chapter XI. - -We have thus seen how satisfactorily the geometrical theory of the -homogeneous partitioning of space has been worked out, and how admirably -it agrees with our preliminary supposition that a crystal is a -homogeneous structure. The fact that the 230 homogeneous point-systems -all fall into and distribute themselves among the thirty-two classes of -crystals, the symmetry of which has also now been fully established, -affords undeniable proof that as regards this branch of the subject -something like finality and clearness of vision has now been arrived at. - - - - - CHAPTER X -LAW OF VARIATION OF ANGLES IN ISOMORPHOUS SERIES. RELATIVE DIMENSIONS OF - UNIT CELLS. FIXITY OF ATOMS IN CRYSTAL. - - -We are now in a position to approach the conclusion of the long -controversy as to the constancy or otherwise of crystal angles in -the cases of greatest similarity, those of isomorphous substances, -and to appreciate how the conflicting views of Haüy and Mitscherlich -and their schools of thought have at length been reconciled. As the -result of a comprehensive study, on the part of the author, of the -sulphates and selenates of the rhombic series R_{2}S/SeO_{4}, and of -the double sulphates and selenates of the monoclinic series -R_{2}M(S/SeO_{4})_{2}.6H_{2}O, in which R represents the alkali -metals, potassium, rubidium and cæsium, and in which M may be -magnesium, zinc, iron, nickel, cobalt, manganese, copper or cadmium, -four facts of prime significance have been definitely established. - -(1) The crystals of the different members of an isomorphous series -exhibit slight but real differences in their interfacial angles, the -magnitude of the angle changing regularly with the alteration of the -atomic weight of the interchangeable metals or negative elements of the -same family group which give rise to the series, as one metal or -acid-forming element is replaced by another. The amount of the -difference increases as the symmetry of the system diminishes. Thus the -maximum difference for the more symmetrical rhombic series of sulphates -and selenates is 56′, which occurs in the case of one angle between -potassium and cæsium selenates, and it is usually much less than this; -in the case of the less symmetrical monoclinic series of double salts -the maximum angular difference observed was 2° 21′, between potassium -and cæsium magnesium sulphates. - -(2) The physical properties of the crystals, such as their optical and -thermal constants, are also functions of the atomic weights of the -elements of the same family group which by their interchange produce the -series. - -(3) The dimensions of the elementary parallelepipedon of the -space-lattice, or in other words, the separation of the molecular -centres of gravity, the points or nodes of the space-lattice, along the -three directions of the crystal axes, also vary with the atomic weight -of the interchangeable elements. - -(4) Specific chemical replacements are accompanied by clearly defined -changes in the crystal structure along equally specific directions. -Thus, when the metal, say potassium, in an alkali sulphate or selenate -is replaced by another of the same alkali-family group, rubidium or -cæsium, there is a marked alteration in the crystal angles and in the -dimensions of the space-lattice, corresponding to elongation of the -vertical axis; and when the acid-forming element sulphur is replaced by -selenium, its family analogue, a similar very definite change occurs, -but the expansion in this case takes place in the horizontal plane of -the crystals. - -Confirmatory results have also been obtained as regards the -morphological constants, the investigations not extending to the optical -or thermal properties, by Muthmann for the permanganates, and by Barker -for the perchlorates, of the alkali metals. Hence, there can be no doubt -whatever that, as regards the various series investigated, which are -such as would be expected to afford the most definite results owing to -the electro-positive nature of metals being at its maximum strength in -the alkali group, the above rules are definite laws of nature. - -Thus it is clear that in the cases of isomorphous substances, which were -the only possible exceptions to the generalisation that _to every -chemically distinct solid substance of other than perfect cubic symmetry -there appertains a specific crystalline form, endowed with its own -particular angles and morphological crystal elements, which are -absolutely constant for the same temperature_, the law does really hold, -and isomorphous substances are no exceptions. The law of progression of -the crystal properties according to the atomic weight of the -interchangeable elements affords indeed at the same time both an -amplification of the generalisation and a precise explanation of its -mode of operation in these cases. - -The discovery of the local effect produced by the two kinds, positive -and negative, of chemical replacement, has a profound bearing on crystal -structure. For it is thereby rendered certain that the atoms are fixed -in the crystal edifice, and therefore in the molecule in the solid -state. It becomes obvious that the atoms—in their stereometric positions -in the molecule, being thus fixed in the solid crystal when the -molecules set themselves rigidly in the regular organisation of the -space-lattice—form the points of the regular point-system of the crystal -structure, which determines to which of the thirty-two classes of -symmetry the crystal shall belong. Any movement of the atoms in the -crystal, other than that which accompanies change of temperature, and -possibly change of pressure, is thus improbable; and this experimental -proof of their fixity, afforded by the fact that definitely orientated -changes accompany the replacement of particular atoms, also doubtless -indicates that the latter are located in the particular directions along -which the changes of exterior angle and of internal structural -dimensions are observed to occur. Stereo-chemistry, which has made such -enormous advances during the last few years, thus becomes of even -greater importance than Wislicenus and its other originators ever dreamt -of. - -Within the atoms in the crystal the constituent electronic corpuscles -may be and probably are in rapid movement, and such physical effects as -have hitherto been ascribed to movement of the atoms within the crystal -are doubtless due to movement of the electronic corpuscles within them, -the sphere of influence of the atom itself being fixed in space in the -solid crystal, and being doubtless defined by the area within which the -corpuscular movements occur. - -[Illustration: - - FIG. 60.—Diagram illustrating Progressive Change of Crystal Angles in - Isomorphous Series. -] - -Three illustrations of the law of change of the crystal properties with -variation of the atomic weight of the determinative elements of an -isomorphous series may be given, and will serve to render the practical -meaning of the generalisation clearer. The first is a diagrammatic -representation, in Fig. 60 (in a very exaggerated manner as the real -change would be inappreciable on the scale drawn), of the change of -angle on replacing the potassium in potassium sulphate, K_{2}SO_{4}, or -selenate, K_{2}SeO_{4}, by rubidium or cæsium. The inner crystal -outline, a vertical section, is that of the potassium salt. The vertical -lines represent the intersections of the two faces of the brachypinakoid -_b_ = {010} with a vertical plane parallel to the macropinakoid _a_ = -{100}; the horizontal lines represent the intersection of the two faces -of the basal plane _c_ = {001} with the same vertical plane; and the -oblique lines represent the intersection of the vertical plane with the -four faces of the dome form _q_ = {011}, which are inclined to both _b_ -and _c_ planes. The diagram is thus designed to show the variation of -the inclination of these latter dome faces to the two rectangular axial -plane faces _b_ and _c_. The outer crystal outline represents a similar -section of a crystal of the corresponding cæsium salt, and the middle -outline that of a crystal of the rubidium salt. - -The progressive alteration of the angle of the q-face will be obvious, -the direction of the change being correct, but the amount of change, as -already stated, being much exaggerated; in reality it never reaches a -degree between the two extreme (potassium and cæsium) salts. It will be -remembered that the respective atomic weights of potassium, rubidium, -and cæsium are 38·85, 84·9 and 131·9, when hydrogen equals 1, that of -rubidium being almost exactly the mean. - -[Illustration: - - FIG. 61.—Diagram illustrating Progressive Change of Double Retraction - in Isomorphous Series. -] - -The second illustration is taken from the optical properties. Fig. 61 -represents graphically the regular diminution of double refraction (the -difference between the two extreme indices of refraction α and γ) which -accompanies increase of the atomic weight of the metal present. The -diagram exhibits the closing up of the two spectra afforded by three -analogously orientated 60°-prisms, one of each of the three salts, such -as was used in determining two of the refractive indices of the salt. -Each prism produces two refracted rays from the single ray furnished by -the collimator of the spectrometer, and consequently two images of the -signal-slit of the collimator when monochromatic light is used, or two -spectra if white light be employed. The Websky signal-slit is narrow at -the centre to enable an accurate allocation to the vertical cross-wire -of the telescope to be made, but wide at its top and bottom ends, in -order to transmit ample light, and Fig. 61 shows four images of this -signal produced by each prism, namely, one R in red C-hydrogen light and -another B in greenish-blue F-hydrogen light belonging to each of the two -spectra, in order to locate the two ends of each of the latter, coloured -monochromatic light of each of the two colours in turn and of the exact -C and F wave-lengths having been fed to the spectrometer from the -spectroscopic illuminator. It will be observed in the case of the top -row that the two spectra, each indicated by the adjacent red and -greenish-blue images, are well apart, the relative distance being about -that actually observed in the case of potassium sulphate. They are -nearer together, however, in the second row, which indicates what is -observed in the case of the analogous rubidium salt, and in the lowest -row representing the relative distances of the two spectra apart in the -case of the cæsium salt, they are so close together as to overlap; for -in this latter case the greenish-blue image of the left-hand spectrum, -corresponding to the a index of refraction, occupies the same position -as the image for yellow sodium light of the right-hand spectrum -corresponding to γ would occupy in the case of cæsium sulphate, the a -refractive index for F-light being 1·5660 and the γ index for Na-light -being 1·5662. The progression of the alteration of the amount of the -double refraction is thus very striking, as the atomic weight of the -metal is varied. - -The third illustration of the law of progression with atomic weight is -also an optical one, and is taken from the monoclinic series of double -sulphates and selenates. It indicates the rotation, with increase of the -atomic weight of the metal, of the ellipsoid which graphically -represents the optical properties, about the unique axis of symmetry, -which is likewise an axis of optical symmetry, of the crystal. In the -potassium salt the ellipsoid occupies the position indicated by the -ellipse drawn in continuous line in Fig. 62, the section of the -ellipsoid by the symmetry plane; the outline of a tabular crystal -parallel to the symmetry plane is also given, as well as the axes of the -crystal and of the ellipsoid lying in that plane. - -[Illustration: - - FIG. 62.—Diagram illustrating Progressive Rotation of Optical - Ellipsoid in Monoclinic Isomorphous Series. -] - -In the rubidium salt the ellipsoid has rotated over to the left, as -indicated by the dotted ellipse, for a few degrees, the number of which -varies slightly for the different groups of double salts; while in the -cæsium salt it has swung over much more still, to the place marked by -the ellipse drawn in broken line. In both this and the last illustration -it will be remarked that the optical change is greater between the -rubidium and cæsium salts than it is between the potassium and rubidium -salts, the reason being that the optical properties are usually -functions (of the atomic weight of the interchangeable elements) which -are of an order higher than the first corresponding to simple -proportionality. - -These three ocular illustrations may serve to render this interesting -law of progression, according to the atomic weight of the -interchangeable elements which give rise to the isomorphous series, -clearer to the mind, by placing before it concrete instances of the -operation of the law. - -The generalisation itself may be very concisely expressed in the -statement that: - -_The whole of the properties, morphological and physical, of the -crystals of an isomorphous series of salts are functions of the atomic -weights of the interchangeable chemical elements of the same family -group which give rise to the series._ - -The fact that this law extends to the structural dimensions, equally -with all other morphological properties, as stated under (3) at the -beginning of this chapter, is of especial interest. For it has actually -been found possible to determine the relations of the dimensions of the -unit parallelepipeda of the space-lattices of the various salts, that -is, the separation of the molecular points of the space-lattice in the -directions of the three crystal axes, for the various salts of the -isomorphous series. This is achieved by combining in suitable formulæ -the volume of the unit cell of the space-lattice with the relative -lengths of the three crystal axes, _a_, _b_, _c_. - -The axial ratios _a_ : _b_ : _c_ are calculated from the measurements of -the crystal angles, as explained in Chapter VI., page 68, and the volume -is the physical constant long known as “molecular volume,” but now for -the first time understood as regards its meaning in the case of solid -substances. It is the quotient of the chemical constant molecular weight -(the sum of the atomic weights, taking into account the number of atoms -of each element present) by the specific gravity of the substance, here -the solid crystal. Very great care has been taken to obtain absolutely -accurate determinations of the specific gravities of the salts, as much -depends on this now very valuable physical constant, and all the values -obtained were reduced to the constant reference temperature of 20°, as -the density notoriously alters rapidly with change of temperature. - -We have thus arrived at morphological constants of very considerable -importance, which are best termed “_Molecular Distance Ratios_,” as they -express the relative distances apart in the three directions of space of -the centres of gravity or other representative points of contiguous -chemical molecules. They are dependent on three experimental -determinations, atomic weight, specific gravity, and crystal angles, all -of which have now been brought to the highest pitch of refinement and -accuracy; hence the molecular distance ratios are particularly -trustworthy constants. If it were only known how much is matter and how -much is space in the molecular parallelepipedal cell, we should actually -have in these constants a relative measure of the sizes of the -molecules. They do give us, however, the relative directional dimensions -of the molecular unit parallelepipedal cells of the space-lattices of -the various members of the isomorphous series, just as the molecular -volumes give us the relative volumes of these cells. For in an -isomorphous series we are absolutely sure that the plan on which the -space-lattice is constructed, its style of architecture, is identical -for all the members of the isomorphous series. Hence, the molecular -distance ratios are in these cases absolutely valid and strictly -comparable. The ratios are generally expressed by the Greek letters χ : -ψ : ω. - -On comparing the molecular distance ratios for a potassium, a rubidium, -and a cæsium salt of any of the series of sulphates, selenates, -permanganates, perchlorates, double sulphates or double selenates -investigated, we invariably find that the values of χ, ψ, and ω for the -rubidium salt (rubidium having the intermediate atomic weight) lie -between the analogous sets of three values for the potassium and cæsium -salts respectively, in complete accordance with the law. - -For the generalisation to apply absolutely it is essential that the -interchangeable elements shall belong strictly to the same family group -of the periodic classification of Mendeleéff. Potassium, rubidium, and -cæsium fulfil this condition absolutely, and so the law of progression -of the crystal properties with the atomic weight of the interchangeable -elements applies rigidly to their salts. Now there are two bases, the -metal thallium and the complex radicle group ammonium NH_{4}, which are -not thus related to the group of three alkali metals just mentioned, but -which are yet capable of replacing those metals isomorphously in their -crystals without more change of angle or of structural constants than is -provoked by the replacement of potassium by cæsium; and often indeed the -amount of change has been singularly like the lesser amount observed -when rubidium has been interchanged for potassium. But although this is -so, the directions of the changes are irregular, being sometimes the -same as when rubidium or cæsium is introduced, and sometimes -contrariwise, and in the case of thallium there are also striking -optical differences, the thallium salts being exceptionally highly -refractive. Still, morphologically the ammonium and thallium salts may -legitimately be included in the same isomorphous series with the salts -of potassium, rubidium, and cæsium, and a somewhat wider interpretation -has to be given to the term “isomorphism” in order to admit these cases. -To distinguish the inner group formed by family analogues, that is, the -more exclusive group obeying the law of progression according to the -atomic weight, the term “eutropic” is employed. - -Thus the “isomorphous series” of rhombic sulphates, selenates, -permanganates, and perchlorates, and the monoclinic series of double -sulphates and double selenates, comprise the potassium, rubidium, -cæsium, thallium and ammonium salts and double salts of sulphuric, -selenic, permanganic, and perchloric acids, while the inner more -exclusive “eutropic series,” following the law absolutely, comprises in -each case only the salts containing the family analogues, potassium, -rubidium, and cæsium. - -In this beautiful manner has the controversy between the schools of Haüy -and Mitscherlich now been settled, the interesting law described in this -chapter having definitely laid down the true nature and limitations of -isomorphism, while at the same time absolutely proving as a law of -nature the constancy and specific character of the crystal angles of -every definitely chemically constituted substance. - - - - - CHAPTER XI - THE EXPLANATION OF POLYMORPHISM, AND THE RELATION BETWEEN - ENANTIOMORPHISM AND OPTICAL ACTIVITY. - - -_Polymorphism._ It has been shown in Chapter VII. that Mitscherlich had -in several instances proved the possibility of the occurrence of the -same substance in two different forms, notably sodium dihydrogen -phosphate NaH_{2}PO_{4}.H_{2}O, calcium carbonate CaCO_{3} (as calcite -and aragonite), the metallic sulphates known as vitriols, and the -chemical element sulphur, and that he gave to the phenomenon the name -“dimorphism.” Since that time large numbers of dimorphous substances -have been discovered, and several which occur in three forms and even a -few in no less than four totally distinct forms. Until the establishment -of the geometrical theory of crystal structure, as expounded in Chapter -IX., this phenomenon of polymorphism gave rise to endless fruitless -discussion. It was most generally attributed to the different nature of -the so-called “physical molecule,” which was supposed to be an aggregate -of chemical molecules and the unit of the space-lattice determining the -crystal system; the different polymorphous varieties were supposed to be -built up of structural units or physical molecules consisting of an -aggregation of a different number of chemical molecules. Several -attempts were made by various investigators, notably by Muthmann and by -Fock, to determine the number of chemical molecules constituting the -physical molecule. - -All these efforts, however, ended unsatisfactorily, and in the year 1896 -the author showed, in a memoir[11] on “The Nature of the Structural -Unit,” that in general the physical molecule is a myth, and that the -chemical molecule is the only structural unit possessing the full -chemical composition of the substance in question; and that its centre -of gravity, or better, any representative point within it, such as a -particular atom, is the unit point of the Bravais space-lattice of the -crystal structure, while the atoms of which the chemical molecule are -composed, arranged stereometrically identically similarly in all the -molecules, are the points of the individual point-systems which make up -the combined point-system. This does not imply a necessarily parallel -and identically orientated arrangement of all the molecules, as at first -postulated by Sohncke and which is a fact for his sixty-five -point-systems; for in accordance with the conclusions of Schönflies, von -Fedorow, and Barlow discussed in Chapter IX., cases are possible in -which alternate molecules may be arranged as each other’s mirror images. -Such are the cases of external molecular compensation or molecular -combination, two oppositely enantiomorphous sets of molecules balancing -each other within the structure, but by exterior compensation as regards -the molecule itself. Moreover, the principle of mirror-image symmetry -enters, as stated in Chapter IX, altogether into the constitution of no -less than 165 of the 230 types of homogeneous structure possible to -crystals. - -Hence the conception of a physical molecule is totally unnecessary and, -moreover, erroneous. The alkali sulphates and selenates exhibit -dimorphism, one member of the series, ammonium selenate, having only -hitherto been observed in the pure state in the second, monoclinic, -form, and never in the ordinary rhombic form; and the author has -conclusively proved for these salts, and also for the double salts which -they form with the sulphates and selenates of magnesium, zinc, iron, -nickel, cobalt, manganese, copper, and cadmium, that the chemical -molecule is the only kind of molecule present, and that its -representative points are, as just stated, the nodes of the Bravais -space-lattice of the crystal structure, determining both the system of -the crystal and its obedience to the law of rational indices. - -The explanation of polymorphism thus proves, in the light of the results -which have now been laid before the reader, to be a remarkably simple -one. Special pains were taken in explaining those results to show that -the temperature had a great deal to do with the conditions of -equilibrium of the crystal structure, for it determines the -intermolecular distances, that is, the amount of separation of the -molecules, and thus controls their possibility of movement with respect -to one another. Now the behaviour of the chemical molecules on the -advent of crystallisation is undoubtedly largely influenced by the -stereometric arrangement of the atoms composing them, and it is possible -for the latter to be such that the molecules may take up several -different parallel or enantiomorphously related positions; or as we have -just seen, a regular alternation within the crystal structure of such -mirror-image positions may be taken up. These different arrangements, -whether parallel or enantiomorphously opposite, may be, and probably -will be, of different degrees of stability, each of these different -forms finding its maximum stability of equilibrium at some particular -temperature, which is different for the different varieties. Hence, at a -series of ascending or descending temperatures, assuming the pressure to -remain the ordinary atmospheric, these different types of homogeneous -crystal structures will be most liable to be produced, each at its own -particular temperature, for which stable equilibrium of that crystal -structure occurs. - -These different assemblages are as a rule quite dissimilar, certainly in -the crystal elements, often in class and not infrequently in system. -Generally two such different crystalline forms are all that are possible -within the life-range of temperature of the substance. But occasionally -three or even as many as four such different forms are found to be -capable of existence within the temperature life-limits of the -substance. - -Polymorphism is thus completely and simply explained as a direct result -of the establishment of the geometrical theory of crystal structure as -laid down in Chapter IX. The equilibrium of the homogeneous structure is -a function of the temperature, and the stereometric arrangement of the -atoms in the chemical molecule of a substance may be such as permits of -two or more homogeneous arrangements of the molecules in assemblages of -varying degrees of stability, but each of which has a maximum stability -at a particular temperature. Hence, within any given range of -temperature such a substance will assume that type of homogeneous -arrangement of its molecules in a crystal which corresponds to the -stablest equilibrium within these temperature limits, assuming the -pressure constant within the bounds of the usual atmospheric variations. -Employing the language of physical chemistry, such a substance will thus -present two or more different solid “phases,” each characterised by its -specific crystalline form, the elementary parallelepipedon of which is -quite a distinct one. Each phase possesses also its own specific optical -and other physical properties, such as melting point, solubility, -thermal expansion, and elasticity. - -It would appear as if the element sulphur is also polymorphous in this -sense, for the monoclinic prismatic form (Fig. 2, Plate I.)—the best -known and most easily prepared, from the state of fusion, of all the -forms other than the common rhombic form, in which sulphur is found in -the neighbourhood of volcanoes and in which it is also deposited from -solution in carbon bisulphide—is of distinctly lower stability, the -crystals passing in a few days into powder composed of minute crystals -of the stable rhombic variety. But in the case of carbon, with its -totally different and apparently at ordinary temperatures equally stable -varieties of octahedral-cubic diamond (Fig. 82, Plate XVI.) and -hexagonal graphite, there is some doubt; for although the diamond is -converted into graphite at a red heat in the electric arc, it is -doubtful whether we are not in the presence of a case of chemical -polymerism or allotropy, like the case of ozone, where three atoms of -oxygen compose the molecule, instead of the two atoms in the molecule of -ordinary oxygen. The fact that the negatively electrified electronic -corpuscles of the Crookes tube cause the same conversion of diamond into -graphite, producing according to Parsons and Swinton a temperature of -4,890° C. in the act, is evidence in favour of allotropy, as the charged -corpuscles are a very likely agent for breaking down such atomic -combinations. Moreover, diamond is volatilised out of contact with air -at 3,600° C. without liquefaction, and the vapour when cold condenses as -graphite. But there is reason to believe, from experiments by Sir Andrew -Noble and Sir William Crookes, that under great pressure carbon does -liquefy at 3,600° C., and that the liquid drops on cooling crystallise -as diamond. - -The yellow and red varieties of phosphorus may also be due to a similar -cause, the yellow variety, which forms excellent crystals, corresponding -to P_{4}, while the red variety may correspond to a molecule composed of -a different number of atoms than four. - -Another view of the nature of polymorphism has lately been brought -forward by Lehmann, as the result of his remarkable experimental -discovery of “liquid crystals,” to which fuller reference will be made -in Chapter XVI. This new view is, however, but an amplification of the -foregoing explanation of polymorphism, indicating the possible mode in -which the stereometric position of the atoms in the molecule does -actually influence and even determine the particular homogeneous -structure which shall be erected, and explains why the temperature plays -such an important rôle. Lehmann’s theory is that any one definitely -stereometrically constituted chemical molecule can only display one -particular homogeneous structure and form of crystal, and that when at a -particular temperature the system or class of symmetry is altered, this -occurs because the stereometric arrangement of the atoms within the -molecule is altered, that is, a new form of molecule is produced, which -naturally gives rise to a new form of crystal. As far as the author -understands it, this does not mean an isomeric change from the chemical -point of view, the chemical compound remaining the same, but that the -stereometric positions of the atoms have been changed, without altering -their chemical attachments, but sufficiently to change the nature of the -point-system which they produce. A significant fact in support of this -view is that the molecules of the substances forming liquid crystals are -usually very complicated and extended ones, comprising a large number of -atoms, the molecules, in fact, corresponding in length with the long -names of the organic substances of which they are generally composed. - -Lehmann’s work has certainly proved that the molecule is endowed -with more individuality than has hitherto been ascribed to it, and -he even shows that there is some ground for believing that his -liquid crystals are such because this directive orientative force -resident in the molecules themselves maintains them in their -mutually crystallographically orientated positions even in the -liquid state, which may be and sometimes is as mobile as water. It -thus appears that any general acceptance of Lehmann’s ideas will -only tend to amplify and further explain the nature of polymorphism -on the lines here laid down, the temperature of conversion of one -form into another being merely that at which either a different -homogeneous packing is possible, or that at which the stereometric -relations of the atoms in the molecule are so altered as to produce -a new form of point-system without forming a new chemical compound. - -_Enantiomorphism of Crystalline Form and Optical Activity._ It has -already been stated that two supplementary forms which are similar but -not identical, the one being the inverse or mirror-image reflection of -the other, as a right-hand glove is to a left-hand one, are termed -“enantiomorphous.” Also it has been shown that all those crystal forms -which have no plane of symmetry, either of simple symmetry or -alternating symmetry (which is equivalent to saying that no centre of -symmetry is present in addition to no plane of symmetry), are -enantiomorphous, and that such forms belong to eleven specific classes. -It has further been shown that the introduction of this principle of -mirror-image symmetry or enantiomorphism into the conditions already -laid down by Bravais and Sohncke for a homogeneous structure, by von -Fedorow, Schönflies, and Barlow, enabled those investigators to derive -the remaining 165 of the 230 possible types of homogeneous structures -compatible with crystal structure, over and above the 65 already -established by Bravais and Sohncke, and thus to complete the geometry of -crystal structure, when the units of such structure are represented by -points. Sohncke subsequently accepted the new principle, and modified -his own theory so as to bring it into line with it. He exhibited some -disinclination, however, at first, to accept the idea—which is a part of -the assumption of the other three authors just referred to, and which -appears to be absolutely necessary to explain one or two of the most -complicated of the crystal classes—of the possibility of two -enantiomorphous kinds of molecule being present in the crystal of the -same single substance, the balancing of the two sets having the effect -of producing mirror-image symmetry of the whole crystal, that is, the -development of a plane of symmetry. - -Now the whole subject is of deep interest, both physical and chemical as -well as crystallographical, inasmuch as it is precisely such substances -as show enantiomorphism,—and can thus exist in two forms, one of which -is the mirror-image of the other and not its identical counterpart, the -two being like a pair of gloves,—which are found to possess the property -of rotating the plane of polarised light and which are therefore said to -be “optically active.” Moreover, the property may be displayed by both -the crystals and their respective solutions, or by the crystals only. -If, therefore, two optical antipodes of the same substance are known, -one rotating the plane of polarisation to the right and the other -rotating it to the same extent to the left, their crystals invariably -exhibit mirror-image symmetry with respect to each other. The converse -does not necessarily hold good, however, that a crystal possessing the -symmetry of one of these eleven classes will always exhibit optical -activity. - -Pasteur[12] was the first to recognise this important relation between -enantiomorphous crystalline form and optical activity, in the case of -tartaric acid, which has the empirical formula C_{4}H_{6}O_{6} and the -constitution: - - COOH - | - CHOH - | - CHOH - | - COOH - -Tartaric acid was isolated by Scheele in 1769, and its discovery was -described in the very first memoir of that distinguished chemist. -Another very similar acid, as regards some of its more apparent -properties, was afterwards, in 1819, described by John of Berlin, and -investigated by Gay-Lussac in 1826; the latter obtained it from the -grape juice deposits of the wine manufactory of Kestner at Thann in the -Vosges. It was still more fully investigated by Gmelin in 1829, who -called it racemic acid (Traubensäure). But it needed the genius of -Berzelius to prove that it really had the same composition as tartaric -acid, although so different to that acid in some of its properties. - -We have here as a matter of fact, the first instance brought to light -involving the principle of isomerism, the existence of two or more -distinct compounds having the same chemical composition as regards the -numbers of atoms of the same elements present, but differing in chemical -or physical properties, or both, owing to the different arrangement of -those atoms within the molecule. The “isomers” may be chemical or purely -physical; the latter involves no alteration of the linking of the atoms, -but merely of their disposition in space, and is the kind met with in -the case of the tartaric acids. - -Biot, so noted for his optical researches, showed afterwards that -tartaric and racemic acids behave optically differently in solution, an -aqueous solution of the former rotating the plane of polarisation to the -right whilst that of racemic acid is optically inactive, not rotating -the plane of polarisation at all. That is, if the dark field be produced -in the polariscope, by crossing the polarising and analysing Nicol -prisms at right angles, tartaric acid solution will restore the light -again, and the analyser will have to be rotated to the right in order to -reproduce darkness. In the case of tartaric acid, the crystals -themselves also rotate the plane of polarisation, the amount being as -much as 11°.4 in sodium fight for a plate of the crystal one millimetre -thick. On the other hand, neither the solution nor the crystals of -racemic acid rotate the plane of polarisation at all. - -Pasteur’s discovery, made in the year 1848, consisted in finding that -racemic acid is really a molecular compound of two physical “isomers,” -namely, of ordinary tartaric acid, which, as we have seen, rotates the -plane of polarisation to the right, and of another variety of tartaric -acid which rotates the beam of polarised fight to the same extent to the -left. The latter and ordinary tartaric acid he therefore distinguished -as lævo tartaric acid and dextro-tartaric acid respectively. Pasteur -went even further than this, in discovering yet a fourth variety of -tartaric acid, which is optically inactive like racemic acid, but which -cannot be split up into two optically active antipodes. - -Indeed, it has since been shown that there are three varieties of this -truly inactive tartaric acid; they are cases of isomerism of the -chemical molecule itself, that is, the stereometric arrangement of the -atoms in the molecule is different in the three cases. For the molecule -of tartaric acid—in common with the molecules of all carbon compounds -the solutions of which, or which themselves in the liquid state, rotate -the plane of polarisation—possesses an asymmetric carbon atom, an atom -of carbon which is linked by its four valency attachments to four -different kinds of atoms or radicle groups; indeed, the molecule of -tartaric acid contains two such asymmetric carbon atoms, namely, the two -in the pair of CHOH groups. For each of these carbon atoms is linked by -one attachment to the carbon atom of the outer COOH group, by another to -an atom of hydrogen, by a third to the oxygen of the group OH, and by -its fourth attachment to the carbon atom of the other group CHOH, which -carries the rest of the molecule, that is, this attachment is to the -other half-molecule CHOH.COOH. Hence, it is quite obvious that there can -be two different dispositions of the atoms in space, one of which would -be the mirror-image of the other, while leaving the arrangement of the -atoms about the two asymmetric carbon atoms dissimilar and not -symmetrical in mirror-image fashion. That is, the two dispositions would -render the molecules in the two cases enantiomorphous with respect to -each other, and these two would be the arrangements respectively in the -two optically active varieties. That this is the correct explanation of -the ordinary dextro variety and the lævo variety of tartaric acid can -now admit of no doubt. - -But if the groups round the two asymmetric carbon atoms are symmetrical -in mirror-image fashion, there will be compensation within the molecule -itself, and the substance will be optically inactive from internal -reasons. This is the explanation of the optically inactive variety which -is unresolvable into any components. The different varieties of this -inactive form are doubtless due to the different possibilities of -arrangement of the atoms in each half, while leaving the two halves -round each asymmetric carbon atom symmetrical to each other. - -We now know that the decomposable inactive variety, racemic acid, may -be readily obtained by· dissolving equal weights of the ordinary -dextro and lævo varieties in water and crystallising the solution by -slow evaporation at the ordinary temperature. For further -investigation has fully borne out the conclusion of Pasteur, that -racemic acid simply consists of a molecular compound of the two active -varieties. It is thus itself inactive because it is externally -compensated, the two kinds of enantiomorphous molecules being -alternately regularly distributed throughout the whole crystal -structure, the very case which von Fedorow, Schönflies, and Barlow -assumed to be possible, and which Sohncke only tardily admitted. The -crystalline form of racemic acid is, as was to be expected, quite -different from the monoclinic form of the active tartaric acids, being -triclinic; and indeed it is not crystallographically comparable with -the active form, inasmuch as the crystals of racemic acid contain a -molecule of water of crystallisation, whereas the active varieties -crystallise anhydrous. - -Ordinary dextro and lævo tartaric acids crystallise in identical forms -of the sphenoidal or monoclinic-hemimorphic class of the monoclinic -system, the class which is only symmetrical about a digonal axis, the -unique symmetry plane of the monoclinic system, which also operates when -full monoclinic symmetry is developed, being absent in this class. Hence -the interfacial crystal angles, the monoclinic axial angle, and the -axial ratios are identical for the two varieties. But the crystals are -hemimorphic, owing to the absence of the symmetry plane, and -complementarily so, the dextro variety being distinguished by the -presence of only the right clino-prism {011}, while the lævo variety is -characterised by the presence only of the left-clino-prism {0̄11}, these -two complementary forms, each composed of only two faces and which on a -holohedral crystal exhibiting the full symmetry of the monoclinic system -would both be present as a single form of four faces, being never both -developed on the same optically active crystal. - -This hemimorphism of the two kinds of crystals will be rendered clear by -Figs. 63 and 64, representing typical crystals of dextro and lævo -tartaric acids which are obviously the mirror images of each other. - -[Illustration: - - FIG. 63. -] - -[Illustration: - - FIG. 64. - - Crystals of Dextro and Lævo Tartaric Acids. -] - -A remarkable discovery was made by Pasteur in connection with one -of the salts of racemic acid, sodium ammonium racemate, -Na(NH_{4})C_{4}H_{4}O_{6}, or - - COONH_{4} - | - CHOH - | - CHOH - | - COONa - -which is obtained by adding ammonia to the readily procurable salt -hydrogen sodium racemate. Sodium ammonium racemate was found by Pasteur -to be decomposable into the salts of dextro and lævo tartaric acids, on -crystallisation of a solution saturated at 28° C. by inoculation with a -crystal of either of those active salts. The solution on cooling being -in the state of slight supersaturation, which we now know from the work -of Ostwald and of Miers as the metastable condition, corresponding to -the interval between the solubility and supersolubility curves (see Fig. -98), if a crystal say of sodium ammonium lævo-tartrate be introduced, -this variety crystallises out first and can be separated from the -residual dextro-salt, which can then be subsequently crystallised. -Moreover, in certain direct crystallisations of sodium ammonium racemate -without such specialised inoculation, Pasteur found all the crystals -hemimorphic, some right-handed and some left-handed, and he was actually -able to isolate from each other crystals of the two varieties. On -separate recrystallisation of these two sets of crystals, he found them -to retain permanently their right or left-handed character, indicating -that the molecules themselves composing these crystals were -enantiomorphous. Their solutions correspondingly rotated the plane of -polarisation of light in opposite directions. Pasteur afterwards -obtained from the dextro-salt pure ordinary (dextro) tartaric acid, and -from the lævo-salt the lævo-acid, by converting them first into the lead -salts and then precipitating the lead as sulphide by sulphuretted -hydrogen. - -In the case of lævo tartaric acid, this was its first isolation, as it -had hitherto been unknown. Gernez afterwards independently found that a -saturated solution of sodium ammonium racemate affords crystals of the -lævo-salt just as readily as of the dextro-salt; if a crystal of either -salt be introduced, crystals corresponding to that variety are produced. - -Another most fruitful observation of Pasteur, the principle of which has -since been the means of isolating one of the two constituents of many -racemic compounds, was that when the spores of _Penicillium glaucum_ are -added to a solution of racemic acid containing traces of phosphates the -ordinary dextro component is destroyed by the organism, while the lævo -component is unattacked so long as any dextro remains; hence, if the -fermentation operation be stopped in time the lævo-acid may be isolated -and crystallised. Why a living organism thus eats up by preference one -variety only, possessing a particular right or left-handed screw -structure, of a compound containing the same elementary constituents -chemically united in the same manner, remains a most interesting -biological mystery. - -The crystals of both dextro and lævo tartaric acids prove to be -pyro-electric, that is, develop electric excitation when slightly -heated. The end which exhibits the development of the clinodome develops -positive electricity in each case, when the crystal is allowed to cool -after warming, so that the two varieties are oppositely pyro-electric, -just as they are oppositely optically active. The most convenient method -of demonstrating the fact is to dust a little of Kundt’s powder, a -mixture of finely powdered red lead and sulphur, through a fine muslin -sieve on to the crystal as it cools. The sulphur becomes negatively -electrified and the red lead positively by mutual friction of the -particles in the sifting, and the sulphur thus attaches itself to the -positively electrified part of the crystal and the red lead to the -negatively electrified end. This phenomenon of the development by the -two varieties of an optically active substance of opposite electrical -polarity has since been shown to be a general one. - -Finally, on mixing concentrated solutions containing equivalent weights -of dextro and lævo tartaric acid Pasteur observed that heat was evolved, -a sign of chemical combination, and the solution afterwards deposited on -cooling crystals of racemic acid. Hence, the only conclusion possible is -that racemic acid must be a molecular compound of the two oppositely -optically active tartaric acids. It thus partakes of the character of a -double salt, analogous to potassium magnesium sulphate for instance. -Consequently the crystal structure is one in which alternating molecules -of the two acids are uniformly distributed, and the case is actually -presented of two oppositely enantiomorphous sets of molecules producing -a homogeneous structure. - -This interesting pioneer case of tartaric acid has been the cause of -the term “racemic” being applied to the inactive form of a substance -when it is decomposable into two oppositely optically active -enantiomorphous varieties of the substance. No well authenticated -exception has been found, in all the many instances which have been -observed of the phenomenon since Pasteur’s time, to the fact that -optically active substances exhibit what was formerly termed -hemihedrism; that is, expressing the case in accordance with our later -more accurate ideas of crystal structure as elucidated in previous -chapters, such substances invariably belong to classes of symmetry -possessing less than the full number of elements of symmetry possible -to the system to which the class belongs. These classes are eleven in -number, those possessing no plane of symmetry; they are, namely, the -asymmetric class of the triclinic system, the sphenoidal class of the -monoclinic system (to which the two tartaric acids, dextro and lævo, -belong), the bisphenoidal class of the rhombic system, the pyramidal -and trapezohedral classes of the trigonal, tetragonal, and hexagonal -systems, and the tetrahedral-pentagonal-dodecahedral and -pentagonal-icositetrahedral classes of the cubic system. - -The optical activity has been proved by Le Bel and Van t’Hoff to be due -in most cases to enantiomorphism of the chemical molecules, that is, to -the enantiomorphous stereometric arrangement of the atoms in the -molecules, and therefore also,—as we have just seen, in accordance with -the geometrical theory of crystal structure,—of the combined -point-system in the case of each of the two varieties. - -The point-systems are probably of a spiral screw-like character, either -right-handed or left-handed, as has been shown by Sohncke to be the case -for the two varieties of quartz, which crystallises in the trapezohedral -class of the trigonal system, one of the eleven classes just enumerated. -The example afforded by quartz will be developed fully in the next two -chapters, as this beautifully crystallised mineral enables us to study -and to demonstrate the phenomena of optical activity in a unique manner -and on the large scale. - -The solutions as well as the crystals are usually optically active in -the cases where, as in the instance of the tartaric acids, the -substances are soluble in water or other solvent. Occasionally, however, -the optical activity is lost by dissolving in a solvent, and in such -cases it is the point-system only, and not the molecules themselves, -which is enantiomorphous. Sodium chlorate, NaClO_{3}, is an instance of -this kind. Moreover, a crystal can belong, as already mentioned, to one -of the eleven above enumerated classes of symmetry without displaying -optical activity, as all the point-systems possessing the symmetry of -these eleven classes do not exhibit screw-coincidence movements. Barium -nitrate, Ba(NO_{3})_{2}, is such a case. - -The two “optical antipodes,” as the dextro and lævo varieties are -conveniently termed, of an optically active substance thus possess an -enantiomorphous crystal structure; but they are alike in their physical -properties such as density, melting point, optical refraction and optic -axial angle, cleavage, and elasticity. The crystal angles are identical -for the forms which are developed in common by them, and which are -usually those which the particular low class of symmetry possesses in -common with the holohedral class of the system. The crystallographic -difference between the two varieties comes in with respect to the -specific forms characteristic of the particular class of lower than full -systematic symmetry, and these forms are never displayed in common by -the two varieties, this being the essence of the enantiomorphism. When -the crystals are not rich in faces, however, it frequently happens that -only the common forms of higher symmetry just referred to are developed -on the crystals, and the two varieties are then indistinguishable in -exterior configuration; it is only on testing their rotatory power, -either by means of a section-plate of the crystal or by means of a -solution, or their pyro-electric properties, or, lastly, their -etch-figures afforded by a trace of a solvent (which etchings on the -crystal faces are enantiomorphous and an excellent indication of the -true symmetry), that their real character can be ascertained. Many -mistakes have been made in the past, and crystals assigned to a higher -than their true class of symmetry, owing to the investigation of only a -single crop of crystals fortuitously poor in the number of forms -displayed. - -In the racemic form, if one should be deposited from the mixed solutions -of the two optical antipodes as a molecular compound of the latter, we -have an occurrence akin to polymerism, that is, the combination into a -single whole entity of a number of molecules, essentially two in the -case of racemism. Just as polymeric varieties of organic substances are -always found to have quite different crystalline forms, so an optically -inactive racemic form of a substance is generally quite different -crystallographically to the dextro and lævo varieties. But there is -usually some similarity along specific zones of the crystals, a kind of -isogonism or morphotropy being developed, such as has been shown to -occur, for instance, by Armstrong and Pope in the case of the substance -sobrerol.[13] - -Besides the true racemic form it is often observed that under certain -conditions crystals are obtained which appear to combine the characters -of both the dextro and lævo varieties, exhibiting both series of -distinguishing hemimorphic or hemihedral forms on the same crystal; that -is, they show the full, holohedral, symmetry of the system. This has -been shown by Kipping and Pope[14] to be due to repeated twinning, thin -layers of the right and left-handed varieties being alternated, just, in -fact, as in the interesting form of quartz known as amethyst, to which -reference with experimental demonstration will be made in Chapter XIV.; -the whole structure assumes in consequence the simulated higher symmetry -which usually accompanies laminated twinning. Such forms have been -termed “pseudo-racemic.” In their memoir (_loc. cit._, p. 993) Kipping -and Pope summarise a large amount of highly interesting work on this -chemico-crystallographic subject which has been carried out by them, and -it may be useful to quote their precise definition of the relationship -between racemic and pseudo-racemic substances. They say: - -“We define a pseudo-racemic substance as an intercalation of an equal, -or approximately equal, proportion of two enantiomorphously related -components, each of which preserves its characteristic type of -crystalline structure, but is so intercalated with the other as to form -a crystalline individual of non-homogeneous structure. A solid racemic -compound, on the other hand, may be defined as a crystalline substance -of homogeneous structure which contains an equal proportion of two -enantiomorphously related isomerides. - -“The relations holding between a mere mixture of optical antipodes, a -pseudo-racemic substance, and a racemic compound, are closely parallel -to those existing between a crystalline mixture, an isomorphous mixture, -and a double salt. The crystallographic methods, by which a double salt -can be distinguished from an isomorphous mixture, may be directly -applied to distinguish between racemic and pseudo-racemic substances. -Thus, according as the crystalline substance obtained from a mixture of -two salts resembles or differs from either of its components -crystallographically, it is regarded either as an isomorphous mixture or -a double salt; similarly, an inactive externally compensated substance, -which closely resembles its active isomerides crystallographically, is -to be considered as pseudo-racemic, whereas when the contrary is true, -it is to be regarded as racemic.” - -The work of Kipping and Pope may be regarded as having finally -vindicated and substantiated the law of Pasteur, that substances of -enantiomorphous molecular configuration develop enantiomorphous -crystalline structures, and that the crystal structures assumed by -enantiomorphously related molecular configurations are themselves -enantiomorphously related. - -This subject, the main results and principles of which have now been -elucidated, may well be closed with a reference to an interesting case -of enantiomorphism and optical activity which the author has himself -investigated,[15] and which is very similar to the case of the tartaric -acids. It had been previously shown[16] by P. F. Frankland and W. Frew, -that when calcium glycerate was submitted to the fermenting action of -the _Bacillus ethaceticus_ one-half only of the glyceric acid was -destroyed, and that the remaining half was optically active, rotating -the plane of polarisation to the right. - -Now glyceric acid, - - CH_{2}.OH - | - CH.OH, - | - COOH - -has manifestly one so-called asymmetric carbon atom (that is, a carbon -atom the four valencies of which are satisfied by attachment to four -different monad elements or groups), that belonging to the CHOH group. -There are consequently two possible arrangements of the molecule in -space, probably corresponding to the two optically active varieties, -namely, those represented, as far as is possible in one plane, as below, -the asymmetric carbon atom (not shown in the graphic representation) -being supposed to be at the centre of the tetrahedron, which is usually -taken to represent a carbon atom with its four valencies. - -[Illustration] - -Dextro-glyceric acid itself proved to be an uncrystallisable syrup, but -the calcium salt, Ca(C_{3}H_{5}O_{4})_{2}.2H_{2}O, was obtained in -crystals sufficiently well-formed to permit of a complete -crystallographic investigation, which the author undertook by friendly -arrangement with Prof. Frankland. Although the acid itself is -dextro-rotatory, aqueous solutions of the calcium salt are lævo-rotatory -to the extent of –12.09 units of “specific rotation” for sodium light. - -The crystals were colourless well-formed prisms which proved to be of -monoclinic symmetry, the best individuals being formed by very slow -evaporation of the aqueous solution. They were terminated at both ends -by pyramid and dome faces, and sometimes grew to the length of a -centimetre. The actual crystal elements found after a full series of -measurements were as under:— - - - CALCIUM DEXTRO-GLYCERATE. - -_Crystal system_: monoclinic. - -_Class of Monoclinic System_: sphenoidal or monoclinic-hemimorphic. - -_Habit_: prismatic. - -_Monoclinic axial angle_: β=69° 6′. - -_Ratio of axes_: _a_ : _b_ : _c_ = 1.4469 : 1 : 0.6694. - -_Forms observed_: - - _a_ = {100}, _c_ = {001}, _r′_ = {̄201}, _p_ = {110}, - _m_ = {011}, _o_ = {111} _s_ = {̄1̄11}, _n_ = {̄2̄11}. - -It will thus be seen that the system and the class are precisely those -of the two active tartaric acids, which renders the case the more -interesting. The usual appearance of the crystals is shown in Fig. 65, -and the stereographic projection is given in Fig. 66, which will -elucidate the symmetry more clearly, the plane of projection being the -plane of symmetry The latter, however, in this class is inoperative, the -two ends of the digonal symmetry axis, which runs perpendicularly to the -plane of the paper, being differently terminated, as in the tartaric -acids. The faces of the forms _o_ = {111} and _m_ = {011} were never -found developed on the left side of the symmetry plane, that is, on the -left side of the crystal as drawn in Fig. 65, the symmetry plane running -perpendicularly to the paper vertically from front to back; they were -only present on the right. Conversely, the faces of _s_ = {̄1̄11} and -_n_ = {̄2̄11} were never found developed on the right, but only on the -left of the plane of possible symmetry. - -Moreover, it was frequently observed that the right-hand faces (110) and -(̄110) of the primary prismform _p_ were much more brilliant and truly -plane than those on the left hand, (1̄10) and (̄1̄10), which were -usually dull and often curved, as were also frequently the faces of the -left-hand forms _s_ and _n_. The right-hand distinguishing forms _m_ and -_o_, on the contrary, were generally most brilliant and gave admirable -reflections of the goniometer signal-slit. - -[Illustration: - - FIG. 65.—Crystal of Calcium Dextro-Glycerate. -] - -[Illustration: - - FIG. 66.—Stereographic Projection of Calcium Dextro-Glycerate. -] - -The following table represents the results of the angular measurements, -twelve different well-formed individual crystals having been employed. -The angles marked with an asterisk were the important angles the mean -observed values of which were accepted as correct, being the best -measured angles, and which were therefore used as the basis of the -calculations. - - _Table of Interfacial Angles of Calcium Glycerate._ - - Angle measured. No. of Limits. Mean Calculated. - measurements. observed. - {_ap_ = 100 : 110 42 52° 32′ − 54° 16′ 53° 29′ * - {_pp_ = 110 : ̄110 20 72 7 − 73 33 73 4 73° 2′ - - {_ac_ = 100 : 001 13 68 22 − 69 42 69 3 69 6 - {_cr′_ = 001 : ̄201 13 52 4 − 52 31 52 13 * - {_r′a_ = ̄201 : ̄100 13 58 35 − 58 46 58 41 * - - _cm_ = 001 : 011 10 31 47 − 32 19 32 3 32 2 - - _r′n_ = ̄201 : ̄2̄11 2 29 43 − 29 48 29 45 29 47 - - {_ao_ = 100 : 111 7 53 59 − 54 10 54 3 53 54 - {_om_ = 111 : 011 7 18 20 − 18 35 18 26 18 29 - {_ma_ = 011 : ̄100 13 107 22 − 108 24 107 41 107 37 - {_an_ = ̄100 : ̄2̄11 11 62 32 − 63 44 63 6 63 10 - {_ns_ = ̄2̄11 : ̄1̄11 1 − 21 35 21 27 - {_sa_ = ̄1̄11 : 100 3 94 49 − 95 34 95 18 95 23 - - {_po_ = 110 : 111 9 43 51 − 44 44 44 35 44 38 - {_oc_ = 111 : 001 9 32 56 − 33 15 33 7 33 7 - {_cs_ = 001 : ̄1̄11 7 41 32 − 43 2 42 10 42 17 - {_sp_ = ̄1̄11 : ̄1̄10 7 59 15 − 60 54 59 59 59 58 - {_pc_ = ̄1̄10 : 00̄1 14 77 2 − 78 21 77 42 77 45 - {_cp_ = 00̄1 : 110 16 101 39 − 103 36 102 16 102 15 - - {_pm_ = 110 : 011 9 52 15 − 53 25 52 42 52 41 - {_mn_ = 011 : ̄2̄11 5 79 5 − 79 26 79 15 79 12 - {_np_ = ̄2̄11 : ̄1̄10 5 47 41 − 48 23 48 4 48 7 - - {_pr′_ = 110 : ̄201 14 106 35 − 108 42 108 5 108 1 - {_r′p_ = ̄201 : ̄1̄10 26 70 54 − 73 32 71 55 71 59 - {_ps_ = ̄1̄10 : 1̄1̄1 5 66 42 − 67 17 67 4 67 5 - {_sr′_ = 1̄1̄1 : 20̄1 6 40 36 − 41 29 41 5 40 56 - - _pm_ = ̄110 : 011 3 75 5 − 76 21 75 37 75 45 - -There is a moderately good cleavage parallel to the basal plane _c_ = -{001}. - -The optical properties afford conclusive proof of the monoclinic nature -of the symmetry. The plane of the optic axes is perpendicular to the -possible symmetry plane, _b_ = {010}, and the first median line makes an -angle of 23° with the vertical axis c, emerging consequently nearly -normal to the basal plane _c_ = {001}, so that a section-plate parallel -to the _c_-faces, or a tabular crystal or cleavage plate parallel to -_c_, shows the optic axial rings and brushes well. The values of the -apparent optic axial angle in air, 2E, and of the true optic axial angle -within the crystal, 2V_{a}, the latter measured with the aid of a pair -of accurately ground section-plates perpendicular to the first and -second median lines and immersed in oil, are given in the next table. - - 2E 2V_{a} - For lithium light 51° 35′ 34° 56′ - „ sodium „ 52° 30′ 35° 28′ - „ thallium „ 53° 50′ 36° 16′ - -The intermediate refractive index β was found to be as under— - - For red lithium light 1.4496 - „ yellow sodium „ 1.4521 - „ green thallium „ 1.4545 - -The double refraction was also determined and found to be of positive -sign. - -The optical properties of calcium dextro-glycerate thus confirm -absolutely the monoclinic nature of the symmetry, as regards the crystal -system. And it was conclusively demonstrated by the goniometrical part -of the investigation that the exterior symmetry was not such as agreed -with holohedral monoclinic symmetry, but with that of the sphenoidal -class, in which the only one of the two elements of monoclinic symmetry -(the plane of symmetry and the digonal axis of symmetry) in operation is -the digonal axis, thus leaving the two terminations of that axis, at -opposite sides, right and left, of the possible symmetry plane, -unsymmetrical. And this is precisely the symmetry which is -characteristic of an enantiomorphous optically active substance. - -Unfortunately, the corresponding lævo-salt has not yet been obtained in -measurable crystals, but there can be no doubt that whenever such are -forthcoming they will display enantiomorphism in the precisely opposite -and complementary sense, the facial forms characteristic in this -dextro-salt of the right termination of the digonal axis being absent on -that side of the systematic symmetry plane but developed on the left -side instead, and _vice versa_, and that the two enantiomorphous forms -will together make up the whole of the faces required by the full -symmetry of the monoclinic system. - -A concrete instance like this, worked out practically in the laboratory, -brings home the precise nature of this interesting relationship, between -crystallographic and molecular enantiomorphism on the one hand and -optical activity on the other hand, in a particularly clear and forcible -manner. It is hoped that this brief account of it will also consequently -have been of assistance to the reader, in more clearly appreciating the -main points of this chapter. - - - - - CHAPTER XII -EFFECT OF THE SYMMETRY OF CRYSTALS ON THE PASSAGE OF LIGHT THROUGH THEM. - QUARTZ, CALCITE, AND GYPSUM AS EXAMPLES. - - -The action of transparent crystals on the rays of light which they -transmit is a subject not only of the deepest interest, but also of the -utmost importance. For it is immediately possible to detect a cubic -crystal, and to discriminate between two groups, optically uniaxial and -biaxial respectively, of the other six systems of symmetry, three -systems going to each group, by this means alone. For a cubic crystal is -singly refractive in all directions. A 60°-prism, for instance, cut from -a cube of rock-salt, for the purpose of obtaining the refractive index -of the mineral by the ordinary method of producing a spectrum and -arranging it for minimum deviation of the refracted rays, affords but a -single spectrum, or a single sharp image of the spectrometer slit when -the latter is fed by pure monochromatic light instead of ordinary white -light. This is true however the prism may have been cut, as regards its -orientation with respect to the natural crystal faces. - -But a 60°-prism cut from a crystal belonging to the optically biaxial -group, composed of the rhombic, monoclinic, and triclinic systems of -symmetry, will always afford two images of the slit or two spectra, -corresponding to two indices of refraction; and, when the orientation of -the prism is arranged so that the refracting angle is bisected by a -principal plane of the ellipsoid which represents the optical -properties, and the refracting edge is parallel to one of the principal -axes of the optical ellipsoid, the prism, when arranged for minimum -deviation of the light rays, will at once afford two of the three -refractive indices, α, β, γ, corresponding to light vibrations along two -of the three principal axial directions of the ellipsoid. The two -indices which the prism affords will be (1) the one which corresponds to -vibrations parallel to the refracting edge, and (2) that which -corresponds to undulations perpendicular to the edge and to the -direction of transmission of the light through the prism (the third axis -of the ellipsoid). For the vibrations of the light in the two rays into -which the beam is divided on entering the crystal are both perpendicular -to the direction of transmission and to each other; the two images or -spectra produced owing to the double refraction, that is, owing to the -different velocities of the two mutually rectangularly vibrating rays, -thus correctly afford the means of determining two of the three -principal (axial) refractive indices. - -Gypsum, the monoclinic hydrated sulphate of lime, CaSO_{4}.2H_{2}O, -already referred to in connection with the Mitscherlich experiment in -Chapter VII., is an excellent substance to employ for the demonstration -of this fact, by cutting and polishing a 60°-prism out of a clear -transparent crystal of the mineral as above described; and if a Nicol -prism be introduced in the path of the rays, one spectrum or -monochromatic image will be extinguished when the Nicol is arranged at -its 0° position, and the other when the Nicol is rotated 90° from this -position. This proves that the two rays affording the two refractive -indices are polarised in planes at right angles to each other, and, -moreover, enables us to verify that the planes in which the vibrations -of the two rays occur are actually parallel and perpendicular -respectively to the refracting edge of the prism. For the two -extinctions occur when the vibration plane of the Nicol is either -vertical, parallel to the prism edge, or horizontal, perpendicular -thereto. - -If a second prism be cut complementarily to the first, that is, so that -the refracting edge is parallel to the third axis of the ellipsoid (the -direction of transmission through the first prism) and the bisecting -plane again parallel to one of the three axial planes of the ellipsoid, -such a prism will also yield two refracted images corresponding to two -indices; one of them, that particular image the vibrations of which are -parallel to the refracting edge, will correspond to that one of the -three principal indices which was not given by the first prism, while -the other one will afford a duplicate determination of one of the two -indices afforded by the first prism. Hence, a couple of such axially -orientated prisms of a rhombic, monoclinic, or triclinic crystal will -enable us to determine all three refractive indices, and one of them in -duplicate, which latter fact will enable us to check the accuracy of our -work. - -If the 60°-prism be cut from a crystal of the uniaxial group, that is, -from a hexagonal, tetragonal, or trigonal crystal—quartz or calcite -being admirable examples of the latter and particularly suitable for -demonstration purposes—it will generally afford two spectra in the same -manner as a crystal of the three birefringent systems of lower symmetry. -But there is one special mode of cutting which results in the prism -exhibiting only a single spectrum, namely, when the hexagonal, -tetragonal, or trigonal axis of symmetry, which is also the unique -“optic axis” of the crystal along which there is no double refraction, -is arranged to be perpendicular to the bisecting plane of the 60°-prism. -For then the light is transmitted along this unique axial direction when -the prism is arranged for the minimum deviation of the refracted rays -out of their original path, and as it may vibrate in any direction -perpendicular thereto with equal velocity there is no separation into -two rays, that is, no double refraction, and thus only a single spectrum -is afforded by such a prism in white light, or a single image of the -slit in monochromatic light, and this latter will at once yield the -refractive index which is generally indicated conventionally by the -letter ω, corresponding to light vibrations perpendicular to the axis. - -Spectroscopists take advantage of this interesting fact, when they -employ a train of quartz prisms so cut in order to explore the violet -and ultra-violet region of the spectrum; for quartz transmits many of -the ultra-violet rays which glass absorbs. Each prism gives only a -single image like glass, whereas if it were otherwise cut it would give -two spectra, which would so complicate matters as to render quartz -useless for the purpose. - -When the prism of quartz or calcite, or of any hexagonal, tetragonal, or -trigonal substance, is cut so that the rays of light are transmitted -through it perpendicularly to the axis, and so that the refracting edge -is parallel to the axis, the light is broken up into two rays, one of -which is composed of light vibrating parallel to the edge and therefore -to the axis, and the other of light vibrating perpendicularly to the -axis. Such a prism consequently affords the two principal extreme -refractive indices of the crystal, ω and ε, the latter letter being -always assigned to the refractive index of a uniaxial crystal -corresponding to vibrations parallel to the axis. - -A uniaxial crystal, one belonging to the hexagonal, tetragonal, or -trigonal systems, has thus two principal refractive indices, ω and ε, -while a biaxial crystal, one belonging to the rhombic, monoclinic, or -triclinic systems of symmetry, has three, α, β, γ, corresponding to -vibrations respectively parallel to the three rectangular axial -directions of the optical ellipsoid, which are also the crystallographic -axial directions in the case of a rhombic crystal. The index α is the -minimum, and γ the maximum refractive index,the β index being -intermediate; when the latter lies nearer to α in value, the crystal is -said to be a positive one, but when nearer to γ the crystal is -conventionally supposed to be negative. Similarly, when in a uniaxial -crystal ε is the greater, as it is in the case of quartz, the crystal is -termed positive, but if ω be the greater index, as happens in the case -of calcite, then the crystal is by convention considered negative. - -Just as in the case of gypsum, which is a positive biaxial crystal (the -reason for the term biaxial will presently be more fully explained), -when the two spectra afforded by a prism of calcite or quartz cut to -afford both ε and ω are examined in plane polarised light, by -introducing a Nicol prism somewhere in the path of the light, the two -images corresponding respectively to ε and ω will be found to be -produced by light polarised in two planes at right angles to each other. -For when the Nicol is at its 0° position one will be extinguished, and -when it is at 90° the other will be quenched. At the 45° position of the -Nicol both images will be visible with their partial intensities, as -happens also in the cases of biaxial prisms. - -This behaviour of 60°-prisms of crystals belonging to the seven -different styles of crystal architecture, as compared with a prism of -glass or other transparent non-crystalline substance, is extremely -instructive. For not only is the optical constant refractive index—the -measure of the power exhibited by the crystal of bending light, -corresponding to its effect in retarding by the nature of its internal -structure the velocity of the light vibrations—the most important of all -the optical constants, but also in the course of its determination we -learn more of the behaviour of crystals towards light than from any -other type of optical experiment. - -[Illustration: - - FIG. 67.—Experiment to show Rectangular Polarisation of the two - Spectra afforded by a 60°-Prism of a Doubly Refracting Crystal cut - to afford two Indices of Refraction. -] - -In Fig. 67 is shown a convenient mode of demonstrating the experiment -with the aid of the electric lantern and one of the large Nicol prisms -of the projection polariscope, already briefly described in Chapter VII. -in connection with the Mitscherlich experiment. The 60°-prism is -arranged on a small adjustable stand nearest the screen; then comes the -Nicol polarising prism of 2½ to 3 inches clear aperture, behind which is -the projecting lens, at the focus of which is placed the adjustable slit -on a separate stand. The slit is filled with light from the condenser of -the electric lantern, and in the lantern front a thick water cell is -arranged, in order to remove sufficient of the heat rays which accompany -the light beam to avoid damage to the balsam joint of the calcite Nicol. -When all the parts are properly arranged a sharp image of the slit -should first be thrown on the screen directly, in the temporary absence -of the 60°-prism, and then on replacing the latter at the proper angle -for minimum deviation, when the light traverses the prism parallel to -its third unused side, a spectrum or pair of spectra—according to the -position of the Nicol and to the nature of the 60°-prism as explained in -the foregoing discussion of the possibilities—will be projected on a -second screen (or the same one if movable) arranged at the proper angle -to receive the refracted rays. - -If a single spectrum be afforded, which remains single on rotation of -the Nicol, the prism is of glass or of a uniaxial crystal cut so that -the light passes along the optic axis. If two spectra be shown when the -Nicol is arranged in the neighbourhood of its 45° position, the crystal -is a doubly refracting one, and if orientated so that the single optic -axis, if the crystal be uniaxial, is parallel to the refracting edge, -or, if the crystal be biaxial, so that the refracting edge is parallel -to one of the three principal axes of the optical ellipsoid and its -bisecting plane is parallel not only to this but also to a second -principal axis, then one spectrum, corresponding to one principal -refractive index, will extinguish when the Nicol is rotated to its 0° -position, and the other spectrum, corresponding to a second principal -refractive index, will be quenched on rotation of the Nicol to its 90° -position. - -The separation of the two spectra on the screen depends on the amount of -the double refraction, and in the case of calcite this is exceptionally -large, so that the two spectra are widely separated on the screen. They -differ also considerably in dispersion. In the case of quartz the double -refraction is very small, and the spectral images of the slit are -consequently so close together as almost to touch one another. The pair -of spectra afforded by gypsum are similarly very close together, owing -also to weak double refraction. The amount of the double refraction is -measured by the difference between the uniaxial indices ε and ω, or that -between the minimum and maximum biaxial indices α and γ. The two spectra -given by quartz and calcite will correspond to ε and ω, and the greatest -separation of spectra occurs in the case of gypsum when the spectra are -those corresponding to α and γ, and not to α and β or β and γ. - -It will now be useful and very helpful to examine more closely into the -nature of the beautiful mineral quartz, in order that a series of -interesting experiments may be described with it, which will assist -largely in rendering the optical characters of crystals clear to us. - -_Quartz_, rock-crystal, although perhaps the commonest and best known of -all crystallised substances, the naturally occurring dioxide of silicon -SiO_{2}, is yet one of the most remarkable and fascinatingly -interesting. To begin with, as explained in the last chapter, quartz -belongs to one of the eleven enantiomorphous classes of lower than full -systematic symmetry, those which exhibit two mirror-image forms related -to one another like a pair of gloves. The particular class of the eleven -to which quartz belongs is the trapezohedral class of the trigonal -system, and two typical left-handed and right-handed crystals are shown -in Fig. 68 and Fig. 69 respectively. - -There is one principal form which is common to both the hexagonal and -trigonal systems, namely, the hexagonal prism, and this is the chief -form exhibited by quartz crystals. They are terminated by an apparently -hexagonal pyramid, but which really consists of a pair of complementary -rhombohedra, which are purely trigonal forms; three upper faces of each -rhombohedron are developed at one end of the prism which may be regarded -as the upper, and the three lower faces of each of the two individual -rhombohedra likewise at the lower end of a fully developed doubly -terminated crystal. The rhombohedron is the characteristic form of the -trigonal system of crystal symmetry, the systematic crystallographic -axes being parallel to its edges. It is like a cube deformed by -extension or compression along a diagonal, which latter is arranged -vertically, and becomes the trigonal axis of symmetry (not a -crystallographic axis), as shown in Fig. 70. - -[Illustration: - - FIG. 68. -] - -[Illustration: - - FIG. 69. - - Left-handed and Right-handed Crystals of Quartz. -] - -When two rhombohedra are equally developed, one being rotated with -respect to the other 60° round the vertical trigonal axis of symmetry, -they together resemble a hexagonal pyramid, and crystals of quartz thus -terminated at both ends are not uncommon, so that at first sight a -quartz crystal might be mistaken for a hexagonal prism doubly terminated -by the hexagonal pyramid, and the mineral considered, in error, to -belong to the hexagonal system. - -[Illustration: - - FIG. 70.—The Rhombohedron and its Axes. -] - -But one alternate set of three faces of the hexagonal pyramid at one -end, and the oppositely alternate set of three similar faces at the -other end, will usually be found to be much less brilliant (indeed often -quite dull) than the other alternate three, and very frequently also the -amount of development is markedly different, both facts indicating that -the terminal faces belong to two different but complementary -rhombohedral forms, and that the system of symmetry is the trigonal and -not hexagonal. - -But there is much stronger evidence than this for trigonal symmetry. For -the little faces marked _s_ and _x_ on Figs. 68 and 69 are -characteristic of the trapezohedral class of the trigonal system, and it -will be observed that on one crystal, Fig. 68, these faces occupy and -modify a left-hand corner or solid angle on the crystal, while on the -other crystal, Fig. 69, they occupy and replace a right-hand solid -angle. Now, if a plate be cut out of the former crystal perpendicularly -to the axis of the hexagonal prism, that is, to the optic axis of the -trigonal uniaxial crystal, it will be found to rotate the plane of -polarisation to the left, the direction in which the small faces are -situated; while if a similar plate be cut out of the right-handed -crystal shown in Fig. 69, that is, one which has the small faces on the -right, it will be observed to rotate the plane of polarisation to the -right. - -As quartz possesses the symmetry of the trigonal system and is thus -optically uniaxial, its optical properties are expressed, in common with -those of all trigonal, tetragonal, and hexagonal crystals, by an -ellipsoid of revolution, an ellipsoid the section of which perpendicular -to the principal axis—that of revolution, the maximum or minimum -diameter of the ellipsoid—is a circle. The optical properties are -consequently the same in all directions round this axis, which has -already been referred to by its common appellation of the “optic axis.” - -The optic axis is identical in direction with the trigonal axis of -symmetry in the case of quartz or other trigonal crystal, and in the -cases of hexagonal and tetragonal crystals with the axes of hexagonal -and tetragonal symmetry, these three axes of specific symmetry being the -distinctive property of these three respective systems, which are thus -known in common as optically “uniaxial.” - -Consequently, no double refraction is suffered by a ray transmitted -parallel to the optic axis, and the refractive index is equal in all -directions perpendicular to the optic axis, that is, for all rays -vibrating perpendicularly to the axis; hence the value of the refractive -index obtained along any such direction is one extreme value for the -whole crystal, and as already mentioned is distinguished by the letter -ω. The refractive index along the direction of the axis itself is the -other extreme value, and is labelled ε. It must be clearly appreciated, -however, that it is not the direction of transmission but that of -vibration perpendicular thereto, that is meant when it is said that, for -instance, the direction of the axis corresponds to the index ε. That is -to say, a ray the _vibrations_ of which occur parallel to the optic axis -of a uniaxial crystal is refracted to an amount which corresponds to the -refractive index ε, while a ray the vibrations of which occur -perpendicularly to the axis affords ω. The difference between ε and ω is -the measure of the double refraction of the crystal. - -In the case of quartz ε is the greater, being 1.5534 for sodium light, -quartz being thus positive according to the convention already alluded -to; while ω is the smaller, namely, 1.5443. In the case of the other -widely distributed trigonally uniaxial mineral calcite, carbonate of -lime CaCO_{3}, the opposite is the case, ω being the greater, having the -value 1.6583 for sodium light, and ω the less, namely, 1.4864, calcite -being thus a negatively uniaxial substance. The amount of the double -refraction in the cases of the two minerals is very different, ε-ω for -quartz being 0.0091, and ω-ε for calcite being as much as 0.1719. -Calcite is indeed a mineral endowed with an especially large amount of -double refraction, a property which renders it so eminently suitable for -use in demonstrating the phenomenon, and for the construction of the -Nicol polarising prism, in which one of the two mutually perpendicularly -polarised rays, that which affords the index ω, is got rid of by total -reflection at a balsam joint, a large rhomb of calcite being cut in half -along a particular diagonal plane and the two halves cemented together -again with Canada balsam; the other ray, which affords ε (but not at its -minimum value), is transmitted as a beam of perfectly polarised light. - -The result of this difference in the amount of the double refraction of -the two minerals quartz and calcite is very interesting as regards their -behaviour with polarised light. A thin plate of quartz, such as is often -found in the slices of rock sections employed for microscopic -investigation, of muscovite granite or quartz porphyry for instance, and -which is usually about one-fiftieth of a millimetre in thickness, shows -brilliant colours in a parallel beam of polarised light, the Nicol -prisms of the polarising microscope being crossed for the production of -the dark field before the introduction of the section-plate on the -stage. This is only true, however, when the plate has not been cut -perpendicularly to the axis, for such a thin plate thus cut does not -perceptibly affect the dark field, there being no double refraction of -rays transmitted along the axis, and the interference colours afforded -by crystal plates in polarised light being due to the interference of -the two rays produced by double refraction, one of which is retarded -behind the other so as to be in a different phase of vibration. Also, -the plate, even when cut obliquely, and best of all parallel, to the -axis, has to be rotated in its own plane (perpendicular to the optical -axis of the microscope), to the favourable position for the production -of the most brilliant colour. This especially favourable position is -halfway between (at 45° to) the positions at which darkness is afforded -by the plate. For on rotating the plate between the crossed Nicols it -becomes four times dark during a complete revolution, and at places -exactly 90° apart, known as the “extinction positions,” whenever, in -fact, that plane perpendicular to the plate which contains the optic -axis is parallel to the plane of polarisation of either the polarising -or analysing Nicol. At the intermediate 45° positions the maximum colour -is produced. - -The colour owes its origin, as already mentioned, to the interference of -the two rays, corresponding to the two refractive indices, into which -the light is divided on entering the crystal in any direction except -along the axis. For one of the rays is retarded behind the other owing -to the difference in velocity which is expressed reciprocally -(inversely) by the refractive indices, and thus a difference of phase is -produced between the two light-wave motions, with the inevitable result -of interference when the vibrations have been reduced to the same plane -by the analyser; light of one particular wave-length is then -extinguished, and the plate therefore exhibits a tint in which the -complementary colour to that extinguished predominates. The light which -leaves the polarising Nicol is vibrating in one plane, but on reaching -the crystal this is resolved into two rays vibrating at right angles to -each other, and at 45° on each side of its previous direction of -vibration, supposing the crystal to be arranged for the production of -most brilliant colour. On reaching the analysing Nicol, the function of -which is to bring the two vibrations again into the same plane, these -two rays are each separately resolved back to the planes of vibration of -the two Nicols, and that pair (one from each ray) vibrating parallel to -the analysing Nicol are transmitted, while the other pair are -extinguished. The two former rays thus surviving, one individual ray of -the two having one refractive index and the other individual the other -index, are thus in a position to interfere; for they are composed of -vibrations in the same plane and of practically the same intensity, and -differ only in phase. Extinction occurs when this amounts to half a -wave-length, or an odd multiple of this, to which, however, requires to -be added half a wave difference of phase which is introduced by the -operation of the analyser. This explanation is a general one, applicable -to thin plates of crystals belonging to all the six systems of symmetry -other than the cubic. For plates of the latter, unless they are in an -abnormal condition of strain, do not polarise. - -When we take a plate of calcite of the same small thickness as that of -the quartz in a rock section, thinner than a sheet of thin paper, we -find that the calcite does not polarise. So great is the retardation of -one of the two rays behind the other in calcite, that a plate -excessively thin is required in order that colour shall be observed. For -the colours of crystal plates under the polariscope, due to double -refraction, are subject to the same laws as the colours of thin films, -namely, that as the thickness increases—introducing more and more -retardation in the case of a crystal, just as in a thin film greater -length of path is introduced with increase of thickness—the various -tints of all the seven orders of Newton’s spectra are exhibited in turn, -each spectrum differing by one further wave-length of retardation, and -after the seventh the white of the higher orders (white light mixed with -colour, the latter thus appearing only as a faint tint) gives place to -true white light, colour being no longer perceptible. Hence with -calcite, owing to the extremely powerful double refraction, and -therefore very considerable retardation of the slower ray behind the -quicker, a plate a fiftieth of a millimetre only in thickness already -shows the white of the higher orders, that is, appears only very feebly -tinted with colour, and a plate of calcite very much thinner still is -required to show brilliant colours. A plate of calcite, therefore, cut -obliquely or parallel to the optic axis, of the thickness of a rock -section or thicker, simply appears four times dark and four times light -alternately, at positions 45° apart, as the section-plate is rotated in -its own plane perpendicular to the axis of the polariscope. - -When a plate of either quartz or calcite one-fiftieth of an inch thick, -cut perpendicularly to the optic axis, is examined under the polariscope -or polarising microscope, the dark field is unaffected by its -introduction on the stage, remaining dark on a complete rotation of the -crystal plate in its own plane. Moreover, the calcite plate continues to -behave similarly however much the thickness is increased, the field -remaining dark. But when quartz is examined as regards the effect of -thickness an extraordinary thing happens. As the plate is thickened, -that is, as a series of plates of gradually increasing thickness are -successively placed on the stage, the dark field begins to brighten, and -eventually colour makes its appearance. Moreover, rotation of the plate -in its own plane—supposing the latter to be strictly perpendicular to -the axis of the polariscope and the plate itself to have been truly cut -perpendicularly to the optic axis of the quartz crystal—produces no -change whatever, the colour remaining the same and evenly distributed -over the plate, thus differing from the previous phenomena of -interference due to double refraction. When monochromatic light is -employed, yellow sodium light for instance, it is found that if the -plate be not too thick, say a millimetre in thickness, the dark field is -restored when the analyser is rotated in a particular direction, either -to the right or to the left, for a specific angle, which is 21° 42′ for -a plate of quartz one millimetre thick. Moreover, if the plate has been -cut from a crystal showing the distinctive trapezohedral-class faces s -and x on the right (Fig. 69) the analysing Nicol requires to be rotated -to the right; whereas if the plate has been cut from a crystal showing -these little determinative faces on the left (Fig. 68) the analyser has -to be rotated to the left in order to quench the light. - -It is obvious, therefore, that the colours of these thicker plates of -quartz are due to the phenomenon of “optical activity.” The original -plane of polarisation of the light received from the polarising Nicol is -rotated by the quartz plate, and to an extent which is directly -proportional to the thickness. When white light is used a particular -colour is extinguished for each position of the analyser, and the -complementary colour therefore predominates in the tint actually -exhibited. Now the most intensely luminous part of the spectrum is about -wave-length 0.000550 millimetre in the yellow, and in the case of a -plate of quartz 7.5 millimetres thick this colour is extinguished when -the Nicols are crossed, while a plate of half this thickness, 3.75 mm., -actually exhibits the colour under crossed Nicols and extinguishes it -under parallel Nicols. For the angle of rotation of the plane of -polarisation for light of this wave-length is 90° for a plate 3.75 mm. -thick, so that the analyser has to be turned through a right angle from -the crossed position, that is, placed parallel to the polariser, in -order to extinguish this colour. A plate of double the thickness, 7.5 -mm., will require the analyser to be rotated through 180°, the angle of -rotation for this thickness of plate, in order to extinguish this yellow -ray. But 180° rotation simply brings the Nicol again to the crossed -position, so that no rotation is really necessary at all. - -Now the complementary colour to the yellow of wave-length 0.000550 mm. -is the transition violet tint, the well-known “tint of passage” between -the brilliant red end of the first order spectrum of Newton and the deep -blue of the beginning of the second order. Hence, this violet tint is -afforded by a plate of 7.5 mm. thickness when the Nicols are crossed, -and by a plate of 3.75 mm. thickness when they are parallel. When, -therefore, these plates are examined respectively under crossed and -parallel Nicols, and the analysing Nicol is turned ever so little, the -tint changes remarkably rapidly into brilliant red or blue, according to -the direction of the rotation of the Nicol and the nature, whether right -or left-handed, of the quartz. Moreover, when two complementary plates -of each thickness are thus examined, one of each pair being cut from a -right-handed crystal and the other from a left-handed one, the colour -will be red in one case and blue in the other for the same direction of -rotation of the analyser. - -A composite plate is frequently found very useful in work in connection -with optical rotation, and is known as a “biquartz,” two plates of -opposite rotations being cemented together by Canada balsam, the plane -of junction being made perpendicular to the plate so as to be almost -invisible when the plate is examined normally. When polarised light is -employed, the least rotation of the analyser from exact crossing with -the polariser, for which the violet transition tint is evenly produced -over the whole composite plate, causes the half on one side of the plane -of junction (appearing as a fine line) to turn red and the other half to -turn blue or green. - -This, in essence, is the nature of the optical activity of quartz, and -the secondary effects derived from it influence all the optical -phenomena afforded by this interesting mineral. Owing to the fact that -quartz crystals are practically unendowed with any facility for -cleavage, the natural rhombohedral cleavage being very imperfectly -developed and rarely seen, it is possible to cut, grind, and polish -large plates of this beautiful, colourless, and limpidly transparent -mineral without a trace of flaw. Such quartz plates of large size, -adequate to fill the field of a large projection polariscope, the stage -aperture of which is nearly 2 inches in diameter, form magnificent -polarising objects for the projection on the screen of the effects -observed in polarised light. As many of the optical properties of -crystals may be illustrated with their aid, it is proposed in the next -two chapters to describe a few of the more interesting screen -experiments which can be performed with quartz, first (Chapter XIII.) in -convergent polarised light, and then (Chapter XIV.) in parallel -polarised light, and thus to illustrate the facts relating to the -connection between optical activity and the internal structure of -crystals in a manner which will at the same time be interesting and will -lead to their much clearer comprehension. - -The experiments described are largely those with which the author -illustrated his lecture to the British Association for the Advancement -of Science during their 1909 meeting at Winnipeg. - - - - - CHAPTER XIII -EXPERIMENTS IN CONVERGENT POLARISED LIGHT WITH QUARTZ, AS AN EXAMPLE OF - MIRROR-IMAGE SYMMETRY AND ITS ACCOMPANYING OPTICAL ACTIVITY. - - -It has already been shown that crystals are optically divisible into two -classes characterised respectively by single and by double refraction. -Singly refractive crystals belong exclusively to the system of highest -symmetry, the cubic. They afford obviously only one index of refraction, -which is generally symbolised by the Greek letter μ, the value of this -constant being the same for all directions throughout the crystal. -Crystals of the other six systems of symmetry are all doubly refractive. -Those of the trigonal, tetragonal, and hexagonal systems have been shown -in the last chapter to possess two refractive indices, a maximum and a -minimum, one represented by ε corresponding to light vibrating parallel -to the singular axis of the system, the trigonal, tetragonal, or -hexagonal axis of symmetry, and another signified by ω corresponding to -light vibrations perpendicular to that axis. For the properties are -identical in all directions around this axis, which is thus the optic -axis as well as the predominating crystallographic one. Such crystals -are consequently known as “uniaxial.” When ε is the larger refractive -index the crystal is positive, while if ω be the maximum the crystal is -said to be negative. It has been shown in the last chapter that quartz -belongs to the positive category, while calcite is negative. Along the -one direction of the optic axis these uniaxial crystals behave like -singly refractive crystals do in all directions. - -Crystals of the rhombic, monoclinic, and triclinic systems of symmetry -have also a minimum refractive index, symbolised by α, and a maximum -index indicated by γ, corresponding to light vibrating parallel to two -directions at right angles to each other; the third direction -perpendicular to both these and normal to their plane does not afford an -index of refraction equal to either of these, however, as in the case of -a uniaxial crystal, but one of an intermediate value, for which the -second letter β of the Greek alphabet is reserved. Whether this value β -is nearer to the minimum α or to the maximum γ determines the -conventional optical sign of the crystal, whether positive or negative. -In the case of the rhombic system the three rectangular directions in -question are identical with the three rectangular crystallographic axes. -In the monoclinic system the single symmetry axis normal to the unique -plane of symmetry is identical in direction with either the α, β, or γ -optical direction, but in the triclinic system there are no coincidences -between the crystal axes and those of the optical ellipsoid. Along none -of these axial directions of the optical ellipsoid which can be imagined -to express graphically the refractive index—an ellipsoid known as the -optical “indicatrix,” and which has been shown by Fletcher to be a more -convenient mode of expressing the optical characters of a crystal than -the vibration-velocity ellipsoid of Fresnel—do the optical properties -resemble those of a uniaxial crystal along the optic axis, or of a cubic -singly refractive crystal, the crystal being doubly refractive along all -three axes. - -But it is a remarkable fact, nevertheless, that there are two directions -in such a crystal along which the latter is apparently singly -refractive, and these two directions are known as the “optic axes,” and -the crystals of the three systems of lower symmetry are consequently -said to be “biaxial.” These two singular directions are symmetrical to -two of the three rectangular axes of the ellipsoid, those corresponding -to the extreme indices α and γ, in the plane containing which two axes -they lie, and they are perpendicular to the third β. For if we draw the -ellipse of which the minimum and maximum axes are represented in length -by α and γ, there will obviously be four symmetrical positions on the -curve where a line drawn to the centre of the ellipse would be equal to -the intermediate value β. If we join opposite pairs of these four points -by diameters (lines passing through the centre of the ellipse) we have -two directions each of which, together with the perpendicular direction -of the β axis, lies on a circular section of the ellipsoid, for all -radii from the centre lying in each of these sections are alike equal to -β. Consequently, light transmitted along the two directions in the -crystal normal (perpendicular) to these two circular sections will -suffer no apparent double refraction, the refractive index being the -same, namely β, and the velocity of vibration equal in all directions in -the crystal parallel to the two circular sections. Hence, we have two -directions in biaxial crystals in which the optical properties are -similar to those of uniaxial crystals along their singular optic axis. -But the optical properties along the two optic axes of a biaxial crystal -are advisedly stated to be “similar” to, and not “identical” with those -along the optic axis of a uniaxial crystal; for although they are -identical to all ordinary experimental tests, they are not quite so when -we come to ultimate details, which, however, are beyond the purview of -this book, but an account of which will be found in the author’s -“Crystallography and Practical Crystal Measurement” (Macmillan & Co., -1911). - -[Illustration: - - FIG. 71.—Projection Polariscope arranged for Convergent Light. -] - -With these prefatory theoretical remarks, which are necessary in order -that the experiments now to be described should be understood, we may -proceed to consider a graduated series of experimental demonstrations -which it is hoped will render clear some of the more important features -of crystal structure which have been dealt with in previous chapters. -Our principal agent will be polarised light, that is, light which has -been reduced to vibration in a single plane by means of the well-known -Nicol’s prism. This latter is a rhomb of calcite which has been cut in -two parts along a specific diagonal direction, and the two parts of -which have been re-cemented together with Canada balsam, in such a -manner that one of the two rays, known as the “ordinary” and which -corresponds to the ω refractive index, into which the doubly refracting -calcite crystal divides the ordinary light which it receives from the -lantern or other source of light, is totally reflected at the layer of -balsam, while the other ray, known as the “extraordinary” and -corresponding to a refractive index of intermediate value between ω and -ε, and composed of vibrations at right angles to those of the totally -reflected ray, is alone transmitted, as a ray of plane polarised light. - -We employ a pair of such Nicol prisms (a very large pair being shown in -Fig. 71), together with a convenient system of lenses for focussing -either the object-crystal or the phenomena displayed by it, as a -“polariscope,” which is the most powerful weapon of optical research on -crystals which has ever been invented. When the two prisms are arranged -so that the vibration planes of the polarised light which they would -singly transmit are parallel, we speak of them as “parallel Nicols,” and -light is transmitted unimpeded through the pair thus placed in -succession; but when one of them is rotated the light diminishes, until -when the vibration planes are at right angles no light escapes at all if -the Nicols are properly constructed, there being produced what is known -as the “dark field” of the “crossed Nicols.” For the plane polarised -light reaching the analyser from the polariser cannot get through the -former, its plane of possible light vibration being perpendicular to -that of the already polarised beam. - -The phenomena exhibited by crystals in polarised light are of two kinds, -namely, those observed when a parallel (cylindrical) beam of fight is -passed through the crystal, and those exhibited when a converging -(conical) beam of fight is employed and concentrated on the crystal, the -centre of which should occupy the apex of the cone. The disposition of -apparatus in the former case of parallel light will be described in the -next chapter and illustrated in Fig. 79. The arrangement for convergent -light, as employed for projections on the screen, has already been -referred to in connection with the Mitscherlich experiment with gypsum, -and illustrated in Fig. 51 (page 92). The arrangement is shown again -here for convenience, in Fig. 71. The parts of the apparatus are briefly -as follows: (1) the electric lantern with self-adjusting Brockie-Pell or -Oliver arc lamp and a 4½ or 5–inch set of condensers; (2) the water -cell; (3) the polarising Nicol with a parallelising concave lens at its -divided-circle end; (4) a condensing lens; (5) the convergent system of -three lenses closely mounted in succession; (6) the crystal; (7) the -collecting system of three lenses equal and similar to the convergent -system; (8) the field lens; (9) the projection lens; and (10) the -analysing Nicol. The ten parts are separately mounted in the author’s -apparatus, which confers greater freedom in experimenting and more power -of varying the conditions; the converging and collecting lens systems, -however, are mounted in a separately adjustable manner on a common -standard, which carries in the centre complete goniometrical adjustments -for the crystal. - -When we place on the stage of the polariscope, the Nicols being crossed, -a plate of a uniaxial crystal cut perpendicularly to the optic axis, and -subsequently a similar plate of a biaxial crystal cut perpendicularly to -that axis of the optical ellipsoid, either α or γ, which is the -bisectrix of the acute angle between the two optic axes, and use the -system of lenses which converges the light rays received from the -polarising Nicol prism on the crystal, as shown in Fig. 71, we observe -in the two cases quite different and very beautiful interference -phenomena, which at once distinguish a uniaxial from a biaxial crystal. -The two appearances are illustrated in Plate XIV., by Figs. 72, 73, and -74, which are reproductions of the author’s direct photographs. Fig. 72 -shows the interference figure afforded by uniaxial calcite, which is the -same for all positions of the crystal plate when rotated in its own -plane by the rotation of the stage. Figs. 73 and 74 represent the -interference figures given by biaxial aragonite, the orthorhombic form -of carbonate of lime, calcite and aragonite being the two forms of this -substance, which has been shown in Chapter VII. to be dimorphous. The -effect shown in Fig. 73 is afforded when the line joining the two optic -axes is parallel to the plane of vibration of either of the crossed -Nicols, and the interference figure represented in Fig. 74 is given when -the stage and crystal (or the two Nicols simultaneously) are rotated -45°. - -The uniaxial calcite figure (Fig. 72) consists of circular -spectrum-coloured rings resembling the well-known Newton’s rings, but -with a dark cross, fairly sharp near the centre but shading off towards -the margin of the field, marking the directions of the vibration planes -of the Nicols. - -The biaxial aragonite figures (Figs. 73 and 74) show two series of rings -surrounding the two optic axes and thus locating the positions of their -emergence, equidistant from the centre of the field, where the bisectrix -emerges. They are not circular, but are curves known as lemniscates, -which are complete rings nearest to the two optic axes, but soon pass -into figure-of-eight loops, and eventually into ellipse-like lemniscates -enveloping both optic axes, and more and more approaching circles in -their curvature as the margin of the field is approached. Moreover, when -the direction of the fine joining the two optic axes is parallel to the -vibration plane of either of the Nicols, as was the case when Fig. 73 -was produced and photographed, a black rectangular cross is seen, one -bar, which is much the sharper one, passing through the optic axes and -the other lying between them at right angles to the first bar, the -centre of the cross being in the middle of the field. - -[Illustration: - - _PLATE XIV._ - - FIG. 72.—Crystal Plate cut perpendicularly to the Axis. - - Uniaxial Interference Figure afforded by Calcite (Trigonal) in - Convergent Polarized Light, with Crossed Nicols. -] - -[Illustration: - - FIG. 73.—Crystal Plate cut perpendicularly to the Bisectrix of the - Acute Optic Axial Angle. - - Biaxial Interference Figure afforded by Aragonite (Rhombic) in - Convergent Polarised Light, with Nicols crossed and parallel to the - Vibration Directions of the Crystal. -] - -[Illustration: - - FIG. 74.—The same Plate as for the previous Figure. - - The same when the two Nicols have been rotated in the same direction - for 45°, still remaining crossed. - - CHARACTERISTIC UNIAXIAL AND BIAXIAL INTERFERENCE FIGURES IN CONVERGENT - POLARISED LIGHT. - - (Reproductions of direct Photographs by the author.) -] - -On rotating the crystal plate in its own plane, while no change occurs -with the calcite, the aragonite figure changes as regards the black -cross, which breaks up into hyperbolic curves currently spoken of as -“brushes,” until when the plate has been rotated 45° the appearance is -that shown in Fig. 74, the eye being supposed to have followed the -rotation. Or, keeping the eye still, the effect shown in Fig. 74 is -equally produced by the simultaneous rotation of both Nicols for 45°. -The vertices of the hyperbolæ now mark the positions of the optic axes, -and the angle between them is the apparent angle of the optic axes as -seen in air, which is considerably different from the true angle between -the optic axes within the crystal, owing to the very different -refraction of light in air and in the crystal substance. - -Now some crystals exhibit a very different optic axial angle at -different temperatures, and one of the most beautiful experiments which -have ever been performed is the Mitscherlich experiment with gypsum, -which has already been described in Chapter VII. in connection with the -work of Mitscherlich, and illustrated in Plate XII., Figs. 52 to 55. -Other substances, on the other hand, show a marked change of optic axial -angle as the wave-length of the light is changed, and such a case has -already been described in Chapter VIII. and illustrated in Plate XIII., -Fig. 58. The figure afforded by such a substance in ordinary white light -is, however, a complicated one, quite different from the normal one of -Fig. 73 afforded by aragonite, as will be clear on reference to the -interference figure shown at _f_ in Fig. 58, which represents the figure -given by ethyl triphenyl pyrrholone in white light. - -In order to understand such biaxial interference figures thoroughly, -they should be studied in monochromatic light, when one obtains a clear -and sharp figure consisting of black curves as well as the cross or -brushes, and very sharp vertices to the brushes when the crystal is -arranged as in Fig. 74. The optic axial angle can then be measured for -each important wave-length of light in turn, and the variation for -wave-length followed throughout the whole spectrum. For this purpose it -is very convenient to have a source of monochromatic light of any or -every wave-length always at hand, and the author some years ago devised -a spectroscopic monochromatic illuminator,[17] for use with any -observing instrument, and which is particularly convenient for use with -the polariscopical goniometer which is employed in practice for the -measurement of optic axial angles. It is shown, along with the latter -instrument, in Fig. 75. The spectroscope has a single but very large -prism of heavy but colourless flint glass, and the spectrum produced—the -electric lantern being the source of light, its rays being concentrated -on the slit—is filtered through a second slit at the other end of the -spectroscope, where the detachable eyepiece is situated when the -instrument is used as an ordinary spectroscope, and for the calibration -(with the Fraunhofer solar lines) of the circle on which the prism is -mounted. The escaping narrow slit of monochromatic light includes only -the 250th part of the spectrum, so is monochromatic in a high sense of -the word. It impinges on a little ground glass diffuser carried in a -very short tube in front of this exit slit, and the optic axial angle -polariscope is brought up almost into contact with the ground glass, and -is thus supplied with an even field of pure monochromatic light. With -this apparatus it is easy to observe the exact crossing wave-length in -all cases of crossed-axial-plane dispersion such as that illustrated in -Fig. 58; for the reading of the graduated circle on which the prism is -mounted, and which is rotated in order to cause monochromatic light of -the different wave-lengths in turn to stream through the exit slit, -affords the exact wave-length with the aid of the calibration curve once -for all prepared. This calibration of the graduations is readily carried -out by using sunlight, and determining the readings corresponding to the -adjustment of the principal Fraunhofer lines in the middle of the exit -slit. - -[Illustration: - - FIG. 75.—Optic Axial Angle Goniometer and Spectroscopic Monochromatic - Illuminator. -] - -Having thus rendered clear the nature of ordinary interference figures -afforded by crystals of the two types, uniaxial and biaxial, in -convergent polarised light, we may pass on to see what happens when we -take a number of plates of quartz of different thicknesses, cut -perpendicularly to the optic axis in all cases, instead of a plate of -calcite. We will examine first a fine pair of hexagonal quartz plates so -cut, each 1 millimetre thick exactly, and about 2 inches in diameter. -One was cut from a right-handed hexagonal prism, and the other from a -similar left-handed one. - -Employing the lantern projection polariscope shown in Fig. 71, arranged -for convergent light just as for the Mitscherlich experiment, and with -the Nicols crossed, we will now see what happens when each of these -plates in turn is placed at the focus of the light rays, between the two -convergent systems of lenses. On the screen we observe in each case a -somewhat similar interference figure to that given by calcite, a black -cross and rainbow coloured circular rings, the smallest ring, however, -being very large relatively to the innermost ring given by calcite, and -the other rings being also further separated from each other. Moreover, -the black cross appears broadened out, this spreading of both rings and -cross being due to the thinness of the plate combined with the low -double refraction of quartz. Further, the right-handed and left-handed -plates both afford apparently identical figures. In order to obtain a -sharp figure like that of calcite we require to add a fourth lens, kept -in reserve for such cases, to each of the two similar convergent lens -systems, one on each side of the crystal plate, in order to increase the -convergence of the light rays. The figure then obtained with one of the -two plates is reproduced in Fig. 76, Plate XV. - -[Illustration: - - _PLATE XV._ - - FIG. 76. - - Interference Figure afforded by a Quartz Plate, 1 Millimetre thick, in - strongly Convergent Polarised Light. -] - -[Illustration: - - FIG. 77. - - Interference Figure afforded by a Quartz Plate, 3·75 Millimetres - thick, in moderately Convergent Polarised Light. -] - -[Illustration: - - FIG. 78. - - Interference Figure (Airy’s Spirals) afforded by two superposed Quartz - Plates, 3·75 Millimetres thick, one of Right-handed Quartz and the - other of Left-handed Quartz, in moderately Convergent Polarised - Light. - - INTERFERENCE FIGURES IN CONVERGENT POLARISED LIGHT AFFORDED BY QUARTZ - PLATES CUT PERPENDICULARLY TO THE AXIS. -] - -Let us now observe, however, what occurs when a thicker plate of quartz -is used. Taking one of 7.5 mm. thickness, and placing it in the focus of -the converging rays, after removing the two extra lenses, we see on the -screen quite a different effect, an attempt to reproduce which -photographically in black and white is made in Fig. 77 on the same Plate -XV. The rings are closer together (using the same degree of -convergence), and the innermost is smaller; moreover, within it all -signs of the central part of the black cross have disappeared, and -instead a brilliant violet colour is shown, which alters to bright red -of the first order spectrum with the least rotation of the analysing -Nicol in one direction from its crossed position with respect to the -polarising Nicol, while if the rotation be in the opposite direction the -deep blue of Newton’s second order is produced. The arms of the cross, -however, appear towards the margin of the field. The violet colour shown -for the exact position of crossing of the Nicols is the tint of passage -between the first and second orders of Newton’s spectra, and this -illumination of the central part of the interference figure is obviously -the effect of the optical activity of quartz, for the tint is the same -as is produced with the plate in ordinary parallel plane polarised -fight, and is, in fact, due to the central axial rays of the convergent -cone being practically parallel. - -On rotating the analysing Nicol for a few more degrees to the right we -observe that the innermost ring widens out and that the red passes into -orange and yellow, the quartz plate being a right-handed one. But when a -similar plate cut from a left-handed quartz crystal is used instead, the -inner ring closes up somewhat for the same rotation of the analyser, -moving inwards instead of outwards, and the blue colour given with the -first slight rotation passes into green and yellow as the rotation is -continued. Moreover, the circular character of the rings is altered, and -so much so that when the rotation has proceeded as far as 45° the shape -of the rings has changed almost to a square. These alterations in the -interference figure are characteristic of the two varieties of quartz -crystals. A useful rule to remember is, that for a right-handed crystal -rotation of the analyser to the right causes the colours to appear in -the order of their refrangibility, namely, the least refrangible red -first, then orange, yellow, green, blue and violet in their order; while -for a left-handed crystal the converse is true when the direction of -rotation of the analyser is the same, that is, to the right, clockwise; -obviously also the colours appear in the opposite order when the -rotation of the analyser is to the left. - -It will now prove of interest to examine the effects produced by two -plates of opposite varieties of quartz of half this thickness, namely, -3.75 mm. The phenomena are very similar to those just described, but the -rings are a little wider, and the larger area within the innermost ring -is now filled with yellow light instead of violet, when the analyser is -exactly crossed to the polariser. It passes into a bright green when the -analyser is rotated slightly on one side, and into orange when the Nicol -is rotated in the reverse direction. But the most interesting thing of -all is to observe what occurs when these two plates of 3.75 mm. -thickness, one of right-handed quartz and the other of left-handed, are -superposed and placed in contact together as one plate, of double the -thickness, 7.5 mm., at the convergent focus. A beautiful spiral figure -is produced on the screen, composed of the celebrated “Airy’s spirals” -as if the black cross were being reproduced in the central part, but -with each of its bars distorted into the shape of the letter S, as shown -in Fig. 78 at the foot of Plate XV. The contrary effects of the two -opposing rotations are thus extraordinarily indicated visually in the -interference figure afforded by the composite plate. - -Now, it is of great practical interest that certain quartz crystals are -found in nature which show Airy’s spirals directly, on cutting a plate -7.5 mm. thick or thereabouts, perpendicular to the optic axis. For -instance, one in the author’s collection of quartzes, a single plate of -an apparently homogeneous and perfectly limpid crystal, shows the -spirals exceedingly well and clearly defined. As a matter of fact, it is -a twin, a right and a left-handed crystal being twinned together with an -invisible plane of composition, which is only revealed on examining the -crystal in polarised light, as will be demonstrated in the next chapter -by the use of parallel polarised light. The fact of such a plate of -quartz affording Airy’s spirals in convergent polarised light is, -however, of itself an excellent proof of the twinning of two crystal -individuals of the opposite varieties. - -Now the very shape of these spiral figures suggests screw action of the -molecular structure of the crystals on the waves of light passing -through them, and moreover, of the action of two screws of opposite -directions of winding, one clockwise and the other anti-clockwise, thus -remarkably confirming the supposition that the point-systems of the -structure of the right and left-handed varieties of quartz are of a -helical nature and respectively of opposite modes of winding. - -Another experiment, devised by Reusch, which still further enhances the -probability that this supposition as to the structure of quartz crystals -is correct, may next be introduced. A thin film of biaxial mica has been -cut into twenty-four narrow strips, which have been laid over each other -at angles of 60°, so that a screw-shaped pile has been formed of the -central overlapping parts, consisting of four complete rotations; that -is, there are four repetitions of the “pitch” of the screw, each -composed of six films. On placing this composite plate of mica at the -convergent focus of the lantern polariscope, so that the overhanging -ends of any four identically superposed strips occupy the focus, the -ordinary biaxial interference figure of mica—two sets of rings and -hyperbolic brushes, very much like Fig. 52, Plate XII.—is observed on -the screen. But when the plate is moved so that the central part comes -into the focus, where all the twenty-four films overlap in their six -different orientations 60° apart, and so that all the light rays have to -traverse the whole helical pile of the twenty-four films, a uniaxial -figure exactly like that of quartz is produced, namely, one composed of -circular rings, with a black cross only visible, however, at the -marginal part, and with the inner ring filled with brightly coloured -light. Moreover, on slightly rotating the analysing Nicol the innermost -ring moves outwards or inwards and the colour changes to blue or red, -according to the direction in which the helix had been wound, in exact -accordance with the rule stated above for quartz. - -If now a second such helical pile of mica films, but one for which the -opposite manner of winding has been adopted, anti-clockwise if the first -had been clockwise, be examined at the convergent focus, precisely the -same appearance will be observed with crossed Nicols, but the opposite -changes will occur on rotating the analyser. Finally, to complete the -interesting proof of the helical nature of quartz crystals, when these -two oppositely wound composite mica plates are superposed—each being -marked carefully to indicate the direction of the helix and the proper -mode of superposition in order to effect precise oppositeness of -arrangement, mirror-image symmetry, in fact, about the plane of -contact—and placed in the convergent beam near its focus, there is at -once seen on the screen a magnificent display of Airy’s spirals, as -perfect as those afforded by the fine natural twin last experimented -with. Hence, there can be no doubt whatever that the remarkable optical -behaviour of quartz is due to its point-system being of a helical -nature, a right or a left-handed screw structure being apparently -produced in nature with equal facility. The circumstances of environment -during the formation of the crystal probably determine which variety -shall be produced, and when the nature of the environment becomes -changed during the operation of formation either twins are produced of -the two varieties, or separate individual crystals. - -This may well conclude our experiments in convergent polarised light, -which—including the beautiful Mitscherlich experiment described in -Chapter VII., of exhibiting the crossing of the optic axial plane in the -case of gypsum, and the production of all the types of interference -figures in succession, as the crystal becomes warmed by the heat rays -accompanying the beam of convergent light—will have introduced the -reader to a typical series of such experiments, and such as were -actually exhibited by the author to the British Association at Winnipeg. -We may pass, therefore, in the next chapter to the consideration of an -equally interesting series in which a parallel beam of polarised light -will be used, which will still further elucidate the internal structure -in the especially instructive case of quartz crystals, and that of -crystals in general. - - - - - CHAPTER XIV -EXPERIMENTS WITH QUARTZ AND GYPSUM IN PARALLEL POLARISED LIGHT. GENERAL - CONCLUSIONS FROM THE EXPERIMENTS WITH QUARTZ. - - -In order to rearrange the projection polariscope for experiments in -parallel light, we simply remove the three lenses on separate stands -(Fig. 71), and the convergent systems of lenses on their special -adjustable stand with goniometrical crystal holder, from between the two -Nicol prisms, and replace them by two other separately mounted lenses, -acting together as an achromatic projecting objective, and a rotatable -object stage. The whole arrangement as thus altered for experiments in -parallel polarised light is shown in position in Fig. 79. The change is -readily made, a gap in the plinth-bed guides near the analysing Nicol -enabling it to be effected without removing either of the prisms, the -analyser being simply drawn along a few inches nearer the end in order -to expose the changing gap. The pair of lenses consists of a -plano-convex lens of 5 inches focus and 2¼ inches diameter, and another -plano-convex lens of 8½ inches focus and 2 inches aperture, with their -convex faces turned towards each other. Together they produce on the -screen an excellent image of the object on the stage, and the size of -the image can be varied at will by regulating the relative positions of -the two lenses with respect to each other and to the object stage. If -found more suitable for the particular screen distance available, the -5–inch lens may be replaced by a 6–inch lens also provided as an -alternative. - -[Illustration: - - FIG. 79.—Projection Polariscope arranged for Parallel Light. -] - -When the analysing Nicol is arranged with its vibration direction -parallel to that of the polariser, we obtain bright light on the screen -on actuating the electric lantern, and the image of an object on the -stage can thus be projected on the screen on a bright ground. But when -the analyser is crossed to the polariser, that is, rotated to the -position 90° from this parallel position, the two planes of vibration of -the Nicols being then at right angles, the screen is quite dark. Before -continuing in this dark field our experimental study of quartz, which is -obviously a type of the more exceptionally behaving substances owing to -its special structure, it will be wise to examine a more ordinary kind -of crystalline substance. For this purpose gypsum—better known in -optical work as selenite, hydrated sulphate of lime, CaSO_{4}.2H_{2}O, -crystallising in beautifully transparent and often large crystals -belonging to the monoclinic system, a typical one of which has been -illustrated in Fig. 9 (page 14), and which we have already referred to -in connection with the Mitscherlich experiment described in Chapter -VII.—is especially suitable, on account of its clear and colourless -transparency, the large size of crystals available, and the brilliancy -of the polarisation colours which they afford when adequately thin. A -very perfect cleavage being developed parallel to the symmetry plane, -the clinopinakoid {010}, such thin films, of even thickness throughout, -can be readily prepared. - -Such a very thin cleavage plate, about 1½ inches in its longest -dimension, is mounted with Canada balsam between a pair of circular -glass plates 1⅞ inches in diameter, the standard size of object plates -for the projection polariscope; the double plate is then supported in a -mahogany frame also of the standard size—4 by 2¼ inches, with clear -aperture of 1⅝ inches diameter and supporting rabbet for the plate 1⅞ to -2 inches diameter—on the rotating stage by a pair of spring clips. The -Nicols being arranged with their vibration directions parallel, in order -to permit light to travel to the screen, and the lenses being arranged -properly for a sharply focussed picture of suitable size, the outline of -the crystal plate will be seen on the screen, and the whole area of the -crystal will either at once appear coloured, or will do so on more or -less rotation of the stage carrying the crystal, which rotates the -latter in its own plane. The crystal outline is of the character shown -in Fig. 80, which also gives the positions of the crystal axes a and c, -and a simple stereographic projection of the faces of the crystal, from -which the nature of the faces bounding the section-plate will be clear. - -[Illustration: - - FIG. 80.—Section of Gypsum Crystal showing the Extinction Directions. -] - -On rotating the Nicol analyser the colours change, and appear at their -maximum brilliancy when the field is dark and the Nicols crossed. -Leaving the analyser crossed to the polariser, and rotating the stage -and therefore the crystal, the colours again change, and at certain -positions 90° apart during the rotation, marked by the two strong lines -in Fig. 80, they disappear altogether, and the crystal becomes dark like -the rest of the field, while the positions of maximum brilliancy of -colour are found to be situated at the 45°-positions intermediate -between these positions of “extinction.” When the quenching occurs the -vibration planes of the two rays, travelling by virtue of double -refraction through the crystal, are parallel to the planes of vibration -of the rays transmitted through the two Nicols, and the fact is a very -important one, enabling us to determine the directions of light -vibration in the crystal. In the case of our gypsum plate, the cleavage -of gypsum being parallel to the unique plane of symmetry of the -monoclinic crystal, these two positions are the directions of the two -axes of the optical ellipsoid which lie in the symmetry plane, and they -correspond to the vibration directions of rays affording the refractive -indices α and γ. The direction corresponding to γ is that of the “first -median line,” the bisectrix of the acute angle between the optic axes; -while α corresponds to the obtuse bisectrix or “second median line.” -These directions are clearly marked by the strong lines in Fig. 80. The -third axis of the optical ellipsoid is obviously perpendicular to the -plate and to the symmetry plane, and corresponds to the intermediate -refractive index β. Thus this simple observation of the extinction -directions in such a case as gypsum enables us at once to fix completely -the orientation of the optical ellipsoid, a fundamental optical -determination. - -A second thin plate of gypsum may next be examined, similarly prepared -and mounted. It is clearly a composite one, being composed of a pair of -twins. For when placed on the stage in the dark field of the crossed -Nicols, and rotated to the position for maximum brilliancy of colour, it -shows different colours in the two halves, as indicated by different -shading in Fig. 81. If, however, the analysing Nicol prism be withdrawn -from the plinth-bed and removed altogether the crystal appears in its -natural colourless condition as a single one, with no indication -whatever of any line of division. - -[Illustration: - - FIG. 81.—Twin of Gypsum as seen in Parallel Polarised Light. -] - -Some exceedingly brilliant polarisation effects are afforded by a number -of objects exhibited by the author in his lecture at Winnipeg, composed -of selenite (gypsum) twins and triplets, some arranged to cross one -another like the mica films of Reusch described in the last chapter, but -only for a single rotation, three twin strips going to a rotation, at -angular distances of 120°; others are arranged in geometrical patterns, -and in circles overlapping one another, and the whole series afford the -most gorgeous and variegated display of colour imaginable, the colours, -moreover, altering either on rotation of the stage or of the analysing -Nicol, and thus passing through every tint conceivable. - -Having thus demonstrated the usual effect afforded by a doubly -refracting crystal plate in parallel polarised light, we may next -illustrate two special cases, which will lead us up to the case of -quartz once more. The first relates to a crystal belonging to the cubic -system, which is theoretically singly refractive or “isotropic”; the -second concerns a plate of a uniaxial crystal cut perpendicularly to the -optic axis, the unique direction of single refraction of such a crystal. -A plate of fluorspar affords a good example of the first case. When -placed on the stage of the polariscope it shows no colour at all in -polarised light, whatever be the position of the two Nicols with respect -to each other, and the field remains dark when they are crossed, the -crystal, in fact, behaving just like so much glass. - -A word of caution, however, is here necessary, for natural mineral -crystals are not infrequently formed under conditions of considerable -strain, at high temperatures or under great pressure, as in the case of -the diamond for instance. So that we must be careful to choose a normal -and well-formed crystal of fluorspar for our experiment. This point may -be well illustrated by placing on the stage a thick circular plate of -glass, an inch or more in diameter, which has been purposely heated and -then suddenly cooled in order to evoke such a condition of strain. -Crossing the Nicols so as to obtain the dark field, there is at once -produced on the screen a black cross and circular concentric -spectrum-coloured rings, resembling with wonderful simulation the -interference figure, shown in Fig. 72, Plate XIV., afforded by calcite -or other uniaxial crystal in convergent polarised light. Artificial -double refraction has been produced in the glass by the strained -conditions, in a fashion concentrically symmetrical to the axis of the -cylinder, an interference figure being afforded symmetrical about the -axis of the cylinder as if it were an optic axis. - -The diamond crystallises in the cubic system, in octahedra, hexakis -octahedra, or hexakis tetrahedra, and should, therefore, theoretically -be without effect on polarised light. Yet it is rare to find a diamond -which does not show more or less colour in the dark field, owing to the -condition of strain in which it exists. It is notorious that the strain -is occasionally so great that a diamond explodes into powder shortly -after removal from its enveloping matrix of blue clay. The author, by -the great kindness of Sir William Crookes, was enabled to show on the -screen, both in a lecture at the Royal Society and in the Evening -Discourse to the British Association at Winnipeg, the images of ten -magnificent large diamonds, natural, perfectly formed crystals uncut and -unspoilt by the lapidary. They were mounted between two circular glass -plates of the usual 1⅞ inches diameter, the diamonds being attached by -balsam to one of them; each plate was held in a mahogany frame of 1⅝ -inches circular aperture, the two frames being then attached face to -face to form a single one, an enclosing cell, which could be placed on -the rotating stage as an object-slide for the projection polariscope. -The appearance of the diamonds on the screen in ordinary light is -reproduced in Fig. 82, Plate XVI., as well as is possible without their -natural colour, for while several of them are brilliantly colourless, -others are tinted, one being a bright green diamond. On producing the -dark field by crossing the analysing Nicol with respect to the -polariser, the darkness was dispelled by brilliant polarisation colours, -at once revealing the diamonds and outlining them clearly against the -dark background. On rotating the analyser the colours changed in the -usual manner of polarising objects, and bright colours were shown by all -the diamonds even when the Nicols were parallel. - -It is obvious, then, that both a transparent non-crystalline substance -such as glass, and a cubic crystal, must be free from strain in order -that it shall exhibit no colour in polarised light and, indeed, no -polarisation effects whatever, and behave as an isotropic substance. - -[Illustration: - - _PLATE XVI._ - - FIG. 82.—Ten Diamonds exhibiting Natural Faces, mounted for the - Lantern Polariscope, to show Polarisation Colours due to Internal - Strain. -] - -[Illustration: - - FIG. 121.—Doubly Refracting Liquid Crystals of Cholesteryl Acetate, - projected on the Screen in the Act of Growth (see p. 281). - - TWO FIGURES ILLUSTRATING THE HARDEST (DIAMOND) AND THE SOFTEST (LIQUID - CRYSTALS) OF CRYSTALS. -] - -The second special case to which attention may be called, that of a -plate of an ordinary uniaxial crystal such as calcite, cut -perpendicularly to the optic axis, is also obviously subject to the same -proviso, that the crystal must be free from strain in order to exhibit -the normal phenomena. Such a perfectly normal plate remains quite -obscure in the dark field in parallel light, producing neither colour -nor interference figure, even on rotation of the object stage with the -crystal, in its own plane. For the light traverses the crystal along the -optic axis, the axis of single refraction, and the vibrations occur with -equal velocity in all directions perpendicular to it. Hence there is no -division into two rays, one retarded behind the other on account of less -velocity of vibration, and therefore no interference colour. - -And now this leads us back to quartz, for this mineral is also uniaxial, -and we will investigate in the same manner in parallel polarised light -the plates of the mineral cut perpendicularly to the optic axis, which -have already been referred to in connection with the experiments -concerning the interference figures produced in convergent polarised -light. Suppose we take first the large plate of quartz 7.5 mm. thick and -over 2 inches in diameter. Placing it on the stage—instead of finding -the dark field to be unaffected by the introduction of the plate, and to -remain so on rotation of the latter in its own plane, as should -theoretically be the case if quartz were a normal uniaxial crystal, and -as calcite has been actually shown to do—we observe that it polarises in -brilliant colour, the whole hexagonal outline of the plate, clearly -focussed on the screen, being filled with an evenly brilliant violet -tint, the tint of passage, just as the central part of the interference -figure, within the innermost ring, had been coloured in the convergent -light experiment with the same plate. The colour changes with the -slightest rotation of either of the Nicols, passing into red for one -direction of rotation and into blue and green when the Nicol is rotated -in the other direction. The tint also alters when the section-plate is -rotated about its vertical diameter, by rotating the upper adjustable -part of the supporting column of the stage within its outer fixed -tubular column; this latter change is equivalent to a thickening of the -plate, the light beam having to traverse a longer path through the -quartz during such oblique setting of the plate. - -This colour is due to the same fact which produced colour in the central -part of the interference figure, namely, the optical activity of quartz, -the fact that the plane of vibration of a beam of plane polarised light -transmitted along the axis of quartz is rotated to the right hand or to -the left. The amount of this rotation is precisely equal, although -opposite in direction, for the two varieties of quartz, but the rotation -varies very considerably for different rays of the spectrum. It also -varies directly proportionally to the thickness of the plate. A plate -one millimetre thick cut perpendicularly to the axis rotates the plane -of polarisation for red hydrogen light (C of the spectrum) to the extent -of 17° 19′, for yellow D sodium light 21° 42′, and for greenish-blue F -hydrogen light 32° 46′. The rotation is a maximum for plates -perpendicular to the axis, and the effect is inappreciable in directions -at right angles thereto. It is clearly due to the oppositely spiral -winding of the regular-point-system of the crystal structure, round the -direction of the optic axis, the trigonal axis of symmetry of the -crystal, a structure which we have proved to be characteristic of quartz -by the beautiful experiments with the helical piles of mica plates, -absolutely reproducing the polarisation effects with quartz, as -described in the last chapter. - -The opposite optical rotation of the two varieties of quartz can be well -shown by constructing a “biquartz.” Two plates of equal thickness, -preferably either 7.5 mm. or 3.75 mm., are cut, one from a right-handed -and the other from a left-handed crystal, each exactly perpendicular to -the optic axis. The two edge-surfaces to be subsequently joined together -are also cut, ground and polished as true planes perpendicular to the -plate surfaces, and the two plates are then cemented together with -Canada balsam by these two prepared edge-surfaces, taking care that the -broad plate-surfaces of the two halves are absolutely continuous as if -the whole were a single parallel-surfaced plate of quartz. Such a -composite plate or “biquartz,” is one of the most useful aids to the -study of optical activity, being much used for enhancing the -sensitiveness of the determination of the angle of rotation. - -When the image of such a 7.5 mm. biquartz, mounted in the usual mahogany -frame and placed on the object stage of the projection polariscope, is -thrown on the screen—the Nicols being crossed for production of the dark -field, and the stage and crystal plate being strictly perpendicular to -the parallel beam of polarised light—the whole of the screen covered by -the image of the plate appears uniformly coloured with the violet tint -of passage. But the moment the analysing Nicol is rotated for a very few -degrees, one-half turns red and the other blue and then green. If the -Nicol be turned back again to the crossing position with the polariser, -and then rotated further in the opposite direction to the former -rotation, the appearances on the two sides of the sharply focussed fine -line of demarcation between the two halves are inverted, the side which -formerly turned red now becoming green, and _vice versa_. The two -varieties of quartz are thus oppositely affected, and it will be obvious -that the biquartz is a very delicate test for the exact crossing of a -pair of polarising prisms, or for the determination of the mutual -extinction of two rectangularly polarised beams of light in general. - -A very striking and beautiful mode of exhibiting this opposite and equal -rotation of the plane of polarisation by the two varieties of quartz may -next be described, an experiment which we owe to Prof. S. P. Thompson. A -composite plate of mica is constructed out of 24 sectors of 15° angle -each, the whole making up a complete circular plate. They are cemented -between two circular glass plates of the usual 1⅞ inch size, with -balsam; the sectors are laid down in succession on one of the plates -first, side by side, with the edge of every one in turn in close contact -with the edge of the next in order, so as to radiate from a common -centre. The second glass plate is only cemented after the arrangement -has been allowed to set for some days, when there is less risk of -disturbing the mounting of the sectors. The latter have all been cut -from the same film of mica, which has a thickness corresponding to a -retardation of one of the two rays produced by the double refraction of -the crystal behind the other equal to one and a half waves. Each sector -is so cut that the line bisecting the 15° angle is parallel to the line -joining the positions of emergence of the two optic axes of the crystal. - -[Illustration: - - FIG. 83.—A Disc _b_ of 24 Mica Sectors under Crossed Nicols, showing - Effects at _a_ and _c_ of Introduction of Left and Right-handed - Quartz Plates. -] - -On placing this wheel of mica on the polariscope stage, the Nicols being -crossed, the effect shown at _b_ in Fig. 83 is observed on the screen. -The four sectors 90° apart, the bisecting lines of which are vertical -and horizontal respectively, parallel to the vibration planes of the -Nicols, appear as a jet black cross; the sectors next to them appear -pale brown, and the next again a still paler delicate shade of sepia, -while the central diagonal ones of each quadrant, at 45° to the black -cross, are brilliantly white. - -On now introducing behind or in front of the stage a right-handed quartz -plate one millimetre thick, one of the pair of large ones described in -one of the convergent light experiments of the last chapter, the black -cross is observed to be deflected one sector to the right, as shown at -_c_ in Fig. 83; whereas when the left-handed companion plate is -introduced in like manner the cross moves over one sector to the left, -as indicated at _a_ in Fig. 83. The two quartz plates are mounted on the -same mahogany object frame, a specially long one with two large -apertures carrying the quartzes, so that first one and then the other -can be placed in or out of position, and when this is done rapidly the -movement of the cross from right to left and back again is very marked. - -Occasionally a natural biquartz is obtained, on cutting a plate out of a -crystal of quartz perpendicularly to the axis. For it is not uncommon to -find a crystal which, while apparently a single crystal, is really a -twin, the two right and left individuals being joined by an invisible -plane of contact, or “plane of composition” as it is called, so -beautifully have the two grown together. Figs. 84 and 85 show two kinds -of twins of quartz. The former consists of two obviously different -individuals, with the little _s_ and _x_ faces indicating right or -left-handedness clearly developed in an opposite manner. The crystal -shown in Fig. 85, however, appears to be a single individual, yet -differs from either a right-handed or a left-handed crystal in showing -the _s_ and _x_ faces developed on both right and left solid angles. It -is a case of complete interpenetration. - -In both cases the plane of twinning is parallel to the optic axis, and -to a pair of faces of the hexagonal prism of the second order, -perpendicular to a pair of the actual first order prism faces shown by -the crystal. They are examples of the well-known “Brazilian twinning” of -quartz, so called because many quartz crystals found in Brazil display -it. - -[Illustration: - - FIG. 84.—Pair of Brazilian Twins of Quartz. -] - -[Illustration: - - FIG. 85.—Completely Interpenetrated Brazilian Twins of Quartz. -] - -A natural biquartz of 3.75 millimetres thickness cut from such a crystal -as is shown in Fig. 85, the plate having a hexagonal outline just as if -the crystal were really a single one, may next be projected on the -screen. The Nicols being crossed, the outline of the crystal is seen -sharply defined, the whole area of the crystal being coloured a uniform -yellow, there being absolutely no trace of any dividing line. But the -moment one commences to turn the analysing Nicol different shades, -orange and green respectively, begin to develop on the two sides of the -line indicating the plane of composition of the twin, the hexagon being -divided by a diametral line joining two corners, which have been -arranged in mounting the plate in its carrier frame to be above one -another, so as to bring the line of composition vertical, as will be -clear from Fig. 86. On rotating the analyser further the difference is -still more marked, and we have blue on one side and orange red on the -other, developing still deeper into red and purple as the analyser -approaches the parallel position with respect to the polariser; when -this latter position is attained the transition violet tint is developed -evenly over the whole plate, and the dividing line has again -disappeared. - -[Illustration: - - FIG. 86.—A Natural Biquartz in Parallel Polarised Light. -] - -Another natural biquartz, also shown in the author’s lecture at -Winnipeg, introduces us to a new phenomenon. For when the Nicols are -crossed we observe a black band down the centre of the plate, marking -the line of division of the twins. When the analyser is rotated until it -is parallel to the polariser this black band changes to a white one, the -sequence of colours on the different sides of the band, that is, in each -half of the plate, being the same as just described. The effect with -crossed Nicols is more or less simulated in Fig. 87, Plate XVII., which -is a reproduction of a direct photograph of the screen picture. The -reason for this black band in the dark field, and for the white one in -the bright field, is that the two halves of the twin overlap at the -centre, the plane of junction of the two individual crystals being -oblique to the plate, instead of exactly perpendicular thereto as was -the case with the first natural biquartz. We are, in fact, beginning to -get the effect of two superposed wedges of quartz. - -[Illustration: - - _PLATE XVII._ - - FIG. 87.—Natural Biquartz exhibiting the Black Band (Nicols crossed) - at the Oblique Junction of the Right-handed and Left-handed Parts. -] - -[Illustration: - - FIG. 88.—Artificial Biquartz, the two parts being obliquely joined in - order to produce the Black Band. - - DIRECT PHOTOGRAPHS OF PICTURES PROJECTED ON THE SCREEN BY THE LANTERN - POLARISCOPE, USING PARALLEL LIGHT. -] - -When the obliquity is greater, or the crystal thicker, a white band -appears on each side of the black central one, the Nicols being crossed, -and when the thickness is as great as 6 to 7.5 mm. a spectrum band -appears on each side of the white one. - -That this obliquity of the surface of contact of the two intergrown -individuals (not the plane of twinning, which remains parallel to a pair -of faces of the hexagonal prism of the second order) is the true -explanation can be readily proved by reproducing the effect -artificially. A thick double plate of quartz is constructed, as shown in -Fig. 88, composed of two halves of respectively right-handed and -left-handed quartz, each 6 to 7 millimetres thick, and each of which has -had the edge-face of junction ground and polished obliquely at an angle -of 30° or so, and oppositely so, instead of perpendicularly to the -plates; the two halves are then cemented together in the usual manner -for a biquartz, with Canada balsam, in order to make a continuous plate. -On placing the plate of this construction possessed by the author on the -stage of the projection polariscope, the two halves exhibit on the -screen respectively brilliant red and green colour, with a vertical -central black band, and on each side of it first a white strip and then -a spectrum band, all the bands being parallel to each other, and the -whole effect being precisely what was observed with the thickest natural -biquartz. - -Thus, we have imitated the oblique junction of the twin parts of the -second and third biquartzes, and proved that this obliquity is the -reason for the phenomena of bands, the black band occupying the centre -where the two opposite rotations of the right and left quartz are -precisely neutralised. The dark field of the crossed Nicols consequently -prevails along this central strip, for the rotatory effect of the first -individual crystal on the light passing through it is exactly undone by -the subsequent passage of the rays through the other individual. On -either side of this neutral strip there is a little preponderance of -right-handed quartz on one side, and of left-handed quartz on the other, -and the usual effect of a thin plate of quartz is therefore seen, -namely, no colour but a little light, while further accretions of -thickness of the preponderating variety give all the colours of the -spectrum in turn, as with growing thicknesses of ordinary single quartz -plates, thus producing the spectrum band. - -The black band is also afforded when the plate is cut somewhat -obliquely, out of a twin crystal with a junction plane truly -perpendicular to the equatorial section, instead of cutting it truly -perpendicularly to the axis, the junction plane being then oblique to -the plate. The polarisation colours are not so strong, however, unless -the plate be made thicker. - -[Illustration: - - _PLATE XVIII._ - - FIG. 89.—Black Central Band and equidistant Spectrum Bands on each - side, afforded by Babinet’s Composite Plate of two Quartz Wedges, - one parallel and the other perpendicular to the Axis. (Direct - Photograph of Screen Picture as projected by Lantern Polariscope.) -] - -[Illustration: - - FIG. 94.—Section-plate of Amethyst, natural size, as seen directly in - Ordinary Light, showing Alternate Violet Sectors (see p. 223). -] - -This effect of a black band with flanking spectra is very similar to -that obtained, due to double refraction and not to optical activity, -when two thin wedges of quartz are cemented together to form a parallel -plate, one wedge being cut so that the optic axis is parallel to the -edge of the wedge, and the other with the optic axis perpendicular to -the edge. When such a composite plate of quartz, often known as a -Babinet plate from the name of its first constructor, is placed on the -stage of the polariscope, and rotated to the 45° position with respect -to the planes of vibration of the crossed Nicols, there is observed on -the screen a deep black band in the centre parallel to the edge of the -wedge, and a number of spectrum bands on each side, separated by white -equal interspaces, the rainbow coloured bands showing the orders of -Newton’s spectra. The effect, as seen on the screen, is reproduced -photographically in black and white in Fig. 89, Plate XVIII. - -These experiments lead us naturally to the study of a great variety of -quartz twins, involving some of the most beautiful and gorgeously -chromatic phenomena which it is possible to produce on the screen with -the projection polariscope. They will eventually bring us to the study -of amethyst quartz, in which the twinning is repeated so often that the -laminations of alternate right and left quartz are sometimes countless, -and almost approach molecular dimensions. - -The Brazilian twinning of quartz, parallel to a pair of faces of the -second order hexagonal prism {11̄20}, often occurs in a very erratic -manner, as regards the arrangement of the portions of the composite -crystal belonging to the two varieties, the surfaces of contact and -character of the interpenetration being frequently very irregular, and -often remarkably so. Thus Fig. 90, the upper figure of the coloured -frontispiece, gives some faint idea of the appearance presented on the -screen by a very beautiful quartz plate, one-half of which is entirely -composed of left-handed quartz, giving a rich even rose-red colour when -the Nicols are crossed, not very far from the violet transition tint, -the plate being nearly 7.5 mm. thick, while the other half consists of -an alternation of strips of right and left-handed quartz, joined -obliquely to the surface of the plate, the black band and its -accompanying white ones and spectrum bands being repeated two or three -times before the edge is reached. This is a very instructive case, for -it shows in this half of the plate, on a large scale, what occurs in -amethyst in a more minutely structural manner, the broad strips, the -sections of plates upwards of a quarter of an inch thick, of alternating -character becoming in amethyst thin lines, the sections of laminæ or -films of microscopic tenuity, their number being correspondingly -enormously increased. - -It may be interesting to state how this Fig. 90, and the lower Fig. 97 -of the frontispiece representing the projection on the screen of benzoic -acid in the act of crystallisation, were produced. The pictures on the -screen were directly photographed on the latest Lumière autochrome -plates, a transparency in the actual natural colours being thus obtained -in each case. These transparent colour-photographs were then used as -originals wherewith to reproduce the effects on paper by the most recent -improved three-colour photographic process. - -[Illustration: - - _PLATE XIX._ - - FIG. 91.—Sectorial 60° or 120° Intrusive Twinning of Right and - Left-handed Quartz, showing Ribbons with Central Black Band where - Oblique Overlapping occurs. -] - -[Illustration: - - FIG. 92.—Irregular Intrusive Twinning of Right and Left-handed Quartz. - - DIRECT PHOTOGRAPHS OF REMARKABLE SCREEN PICTURES AFFORDED IN PARALLEL - POLARISED LIGHT BY SECTIONS OF TWINS OF RIGHT AND LEFT QUARTZ. -] - -Two other typical cases of irregular quartz twinning may also with -advantage be demonstrated. The first is a plate in which there are -repeated 60° V-shaped or 120° wedge-shaped intrusions of one variety -into a greater mass of the other variety. The border of the V or 120° -wedge is composed of a ribbon, the outer edges of which are -spectrum-coloured and the central line of which is formed by the deep -black band, which is separated on each side from the spectra by a white -strip. Some idea of the beauty of this quartz plate, which was -generously lent to the author by Prof. S. P. Thompson, as projected on -the screen under crossed Nicols, may be gathered from Fig. 91, Plate -XIX., the upper homogeneous part of the plate being coloured a brilliant -green, and the lower part red. - -The second is an irregular interpenetration of one variety into the -other, in repeated V-shapes occupying the lower half of the image of the -plate as seen on the screen in the dark field of the projection -polariscope, like a range of sharp mountain peaks, the black bands being -so rapidly repeated as to be nearly continuous. These darker portions -thus appear to form the bulk of the mountains, while the upper untwinned -half of the crystal shows a clear and even sky blue; to make the -resemblance to a range of Alpine mountains even more complete, the wavy -line of demarcation between the twinned and non-twinned portions of the -plate is bordered by a white ribbon, of varying width, giving the -appearance of a snow-cap to each peak, which shows up clearly against -the blue sky. It will be obvious that this quartz plate affords an -altogether very beautiful series of phenomena in parallel polarised -light on the screen, for the colours change with every movement of the -analysing Nicol from the crossed position, the appearance for which has -just been described. Fig. 92, Plate XIX., gives only the faintest idea -of the beauty of the screen picture afforded by this section-plate. The -effect chosen as best for photographic reproduction purposes is one -afforded when the analysing Nicol is rotated somewhat away from the -crossed position with respect to the polariser. - -And now we arrive finally at amethyst quartz, three very beautiful -hexagonal plates of which—cut perpendicularly to the optic axis as usual -for quartzes intended to display optical activity, from an apparently -single hexagonal prism in each case—will be taken as typifying the -phenomena exhibited by this especially interesting variety of quartz on -the screen in parallel polarised light. The smaller one affords a screen -picture, with Nicols not quite crossed, such as is portrayed in Fig. 93, -Plate XX. We observe that the area of the hexagon is roughly divisible -into six 60°-sectors, and that alternate ones are uniformly coloured, -indicating that they belong to wholly right-handed or left-handed -quartz; whereas the other alternate sectors are most beautifully marked, -as if by line shading parallel or inclined at 30° to the edges of the -hexagon, by a considerable number of equally spaced dark or slate -coloured bands, close together but separated by white bands, with a -trace of spectrum colours along the middle of the latter. If we rotate -the analysing Nicol somewhat we can readily find a position, which is -not always that of crossed Nicols, for which these parallel bands of -laminar twinning are most clearly defined, as shown in the illustration, -the colours of the other sectors ever changing during the rotation. - -[Illustration: - - _PLATE XX._ - - FIG. 93.—Section-plate of Amethyst Quartz, showing Sectorial Repeated - Twinning of the Right and Left Varieties. -] - -[Illustration: - - FIG. 95.—Section-plate of large Amethyst Quartz Crystal, showing - relatively large Area of Sectorial Repeated Twinning of Right and - Left-handed Quartz (see p. 225). - - DIRECT PHOTOGRAPHS OF SCREEN PICTURES OF AMETHYST PROJECTED BY THE - LANTERN POLARISCOPE IN PARALLEL LIGHT. -] - -It is obvious that we have here to do with the same phenomenon as was -illustrated by the parallel bands shown on the large scale by the -section illustrated in Fig. 90 of the coloured frontispiece, the black, -white, and spectrum-coloured bands being simply repeated very many more -times in the same space, and in alternate sectors of the crystal. - -The twinning of amethyst in 60°-sectors is very characteristic of this -variety of quartz, and it is an interesting fact that the sectors which -show the laminar bands in polarised light often appear purple coloured -in ordinary light, the tint from which amethyst derives its name. This -is not necessarily or always so however, and the section just described -and illustrated in Fig. 93 appears quite colourless throughout on casual -inspection in ordinary light, in fact as a clear colourless hexagonal -section of ordinary simple quartz; a trace of the amethyst colour -becomes, however, apparent on closer examination when held obliquely, in -the sectors where the bands become visible in polarised light. - -The second plate of amethyst is a magnificent section 9 millimetres -thick and 2½ inches in diameter, of which alternate 60°-sectors are -deeply amethyst coloured, the tint being a pure violet of about the -wave-length of the hydrogen line near G of the spectrum. Moreover, even -to the naked eye when the specimen is held in the hand up to the light, -in certain positions the laminæ become visible as more deeply shaded -violet line markings. On placing it on the stage of the polariscope but -with the analysing Nicol removed, so as to observe the natural -appearance of the section in white light (for, although polarised by the -polarising Nicol, being unanalysed the section exhibits no polarisation -effects), these facts become clear to everyone in the room. The violet -staining of alternate sectors appears very deep, and traces of -lamination in the violet parts are just apparent on close scrutiny, the -other alternate sectors appearing colourless and unmarked except by a -few flaws almost always present in so large a section-plate of amethyst. -The natural appearance of this plate is shown in Fig. 94, Plate XVIII. -(facing page 218), as far as is possible photographically, the violet -sectors being clearly demarcated. - -On replacing the analysing Nicol the colourless sectors are seen to -polarise uniformly in brilliant colours, indicating a homogeneous -variety of quartz in each, either right or left-handed. Moreover, -whenever two of these naturally colourless parts touch each other, which -they do as the margin of the plate is approached, an irregular ribbon is -produced, composed of the black band in the centre, with first white and -then spectrum-coloured flanking strips on each side, the spectra forming -the edges of the ribbon. The violet sectors show the laminated twinning, -but, owing to the great thickness of this plate, in too complicated -(overlapping) a manner to be easily followed, a thinner plate being -required to show such fine laminations clearly. - -Finally, the third section is such a thinner plate, about 3.5 mm. thick -and nearly 1½ inches in diameter. This section of amethyst is probably -the most beautiful of all, for it not only shows the laminated twinning -to perfection, in three alternate 60°-sectors and in all six in the -middle part of the plate, but also these alternate sectors are -distinctly violet even to the eye when the specimen is held in the hand -against a white background; and the laminations are likewise also -clearly visible on holding the section obliquely up to the light. In -polarised light, either with crossed or parallel or anyway arranged -Nicols, the phenomena on the screen are of the most superb character. -The whole of the middle part of the plate appears made up of six -sectors, _all_ showing the fine laminar bands parallel to the edges of -the second order hexagonal prism {11̄20}, that is, at 30° to the edges -of the section, the crystal being a first order hexagonal prism {10̄10}. -Some idea of the arrangement will be afforded by Fig. 95, Plate XX. The -marginal parts develop into alternately right and left-handed sectors or -half-sectors, polarising in different and very brilliant colours, and -showing the ribbon bands at every junction. On rotating the analysing -Nicol the changes are remarkably beautiful, particularly for the -positions of the analyser when the laminar bands take on their deep -slate colour, with white and marginally spectral interstrips. The whole -phenomena, indeed, afforded by this plate of amethystine quartz, are the -most magnificent which the author has ever seen on the screen, in the -whole of his crystallographic experiences. - -The Brazilian twinning law of quartz, according to which the plane of -twinning is parallel to a pair of faces of the second order hexagonal -prism {11̄20}, appears capable of explaining all these varieties of -right and left-handed twins, the interpenetration of the intimate kind -shown in Fig. 85 (page 215) usually resulting in sectorial portions of -space being occupied by each kind, the surfaces of junction of -oppositely optically active parts being, however, very varied in their -distribution and character. Where they happen to be more or less -horizontal, a plate cut perpendicularly to the axis to include both -kinds would show Airy’s spirals in convergent polarised light, as may -readily be demonstrated by such a plate, one of several, in the author’s -collection. Where they are oblique, a plate cut at right angles to the -axis would, as we have seen experimentally, afford the black, white and -spectral ribbon bands in parallel polarised light. Where, however, the -mode of interpenetration is still more intimate, we have the rapidly -alternating laminæ of the two varieties, right and left-handed, building -up the beautiful structure of amethyst in thin layers. A section-plate -of such an intimate blending of the two varieties, cut as usual -perpendicular to the axis in order that any phenomena of optical -activity shall be exhibited at the maximum, affords no indication -whatever of optical rotation, the two varieties simply neutralising each -other’s effects, and the plate behaves as an ordinary uniaxial crystal, -affording in convergent polarised light a black cross like calcite, -complete to the centre. In parallel polarised light it shows of course -the laminated structure, but the tendency to remain dark under crossed -Nicols is shown by the fact that the tints exhibited by the laminations -are slates, greys, and even black, when the Nicols are crossed, the -delightful other colours only making their appearance when the analysing -Nicol is rotated. Thus the simple law of Brazilian twinning is quite -capable of explaining the whole of the phenomena exhibited by composite -crystals of the two varieties of quartz, and such an explanation is the -one accepted by von Groth, in the excellent description of quartz in the -last edition of his _Physikalische Krystallographie_. - -[Illustration: - - FIG. 96.—Plan of Amethyst Crystal. -] - -An interesting crystal of amethyst very similar to the third of those -just described, the one illustrated in Fig. 95, was described by Prof. -Judd in the year 1892 to the Mineralogical Society.[18] The plan of the -crystal is given in Fig. 96. The wedges marked _x_, _y_, _z_, are of a -pale yellow colour, as are also the three strips, sections of plates, -proceeding from the wedges and meeting at the centre _o_. The wedge _y_ -exhibits left-handed polarisation, and the wedge _z_ right-handed. The -large wedge _x_ is composite, the part marked _x__{r} being right-handed -and that marked _x__{l} left-handed. The surface of junction of the two -parts is not perpendicular to the plate, so where the two varieties -overlap, the part marked _x__{rl}, a ribbon band is shown in parallel -light and Airy’s spirals in convergent polarised light. The yellow parts -of the crystal exhibit ordinary rotatory polarisation colours, even -tints; but in the remaining sectors of the crystal, the lines of -division of which are indicated by the radial lines A, B, C, no trace of -circular polarisation is displayed, and the central part, where the -lamellæ are very well developed, gives the ordinary calcite-like -uniaxial interference figure. The more marginal portions, however, show -complicated interference figures, somewhat resembling those of biaxial -crystals, owing to irregular distribution of the two varieties of -quartz, and probable displacement of the optic axis by distortion. - -An ingenious theory of the formation of the lamellæ is put forward by -Prof. Judd in the same memoir. He had already shown that quartz is -endowed with planes of gliding, parallel to the rhombohedral faces, and -suggests that the lamellation is the result of the effect of high -pressure and possibly high temperature on the quartz crystal after its -formation. The lamellæ appear to be frequently parallel to the -rhombohedral terminal faces of the crystal, as if they were indeed glide -plane effects. It is quite conceivable that the gliding of layers of -molecules, which when permanent usually involves rotation and inversion -of the molecules, might result in alternately right and left structural -arrangements, and there is considerable evidence that the development of -the purple tint occurred subsequently to the growth of the crystal. It -is probably due to change in the state of oxidation of the trace of -manganese present as a minute impurity in the quartz crystal, and which -is concentrated between the lamellæ, just as the yellow tint is due to a -slight trace of iron (ferric) oxide. The theory is an interesting one, -and throws considerable light on the possible nature of intimate -lamellar twinning. - -One last experiment may now be referred to, the concluding experiment of -the Winnipeg lecture, and which is very reminiscent of the beautiful -slate colour of the lamellæ of amethyst. It is the actual -crystallisation, projected on the screen, of a thin film of melted -benzoic acid, which affords radiating closely packed long and narrow -crystals, shooting out on the screen from centres near the margin of the -field, very much like the individual crystals of repeatedly twinned -quartz in the beautiful amethyst crystal illustrated in Fig. 95. -Provided the film of melted benzoic acid be thin enough, the crystals -appear on the screen in parallel polarised light, under crossed Nicols, -tinted with the same beautiful shades of slate colour as amethyst, the -intermediate low-order tint between the black and the grey of Newton’s -first order spectrum. Some idea of the appearance on the screen is -afforded by Fig. 97, the lower of the two coloured figures in the -frontispiece. As in the case of Fig. 90, the screen picture was -photographed directly on a Lumière autochrome plate, and the -transparency in the actual colours thus obtained was employed as an -original wherewith to reproduce the picture on paper by the latest -three-colour photographic process. - -In carrying out the experiment a few of the flaky crystals of benzoic -acid are placed on one of the circular glass object plates of the -standard 1⅞-inch size for the projection polariscope; they are covered -by a second similar one, and the two plates are then held in a pair of -tongs and gently warmed over a small spirit lamp, or miniature Bunsen -lamp. As soon as the crystals have fused, and the melted substance is -evenly spread as a thin film between the two glass plates, the latter -are rapidly transferred to a special mahogany object frame, fitted with -a side slide to press the double-plate edge just sufficiently to hold it -in position in the frame, which is then at once placed on the rotating -stage of the polariscope. The screen appears quite dark at first, the -Nicols being crossed, but in a second or two as the slide cools the -benzoic acid begins to crystallise out at the sides, brilliant colours -and the deep greys being both developed, the former chiefly near the -edges of the crystals, rendering the crystallisation wonderfully -distinct and beautiful on the black background. Then long needle -crystals shoot out from various quarters one after another or -simultaneously, in lovely shades of slate or grey tinted with brilliant -colours at the margins and tips, the growing point cutting its way along -like a sharp brilliantly coloured arrowhead. Eventually an arch is -formed of such acicular crystals, radiating simultaneously from many -centres, gorgeously coloured in parts, but showing the yet more æsthetic -slates and greys in the main. Finally, the whole screen picture fills up -with a mass of interlacing yet ever distinct crystals, the last few to -crystallise in the centre usually doing so with a burst of especially -bright colour, as the thickness increases adequately for the double -refraction retardation to reach the more brilliant second order -spectrum, a concluding effect which evokes the emphatic delight of even -the most phlegmatic philosopher, inured to scenes of beauty in natural -phenomena. - -The series of experiments with quartz described in this and the previous -chapter, culminating with those revealing the alternate repetition of -extremely fine layers of right and left-handed quartz in amethyst, will, -it is hoped, have illustrated and rendered intelligible the important -structural principle of enantiomorphism or mirror-image symmetry. We -have only to imagine the layers to become thinner and thinner until we -approach ultimately the neighbourhood of the minute dimensions of the -chemical molecule, without as yet penetrating within the range of the -molecular forces; the two such oppositely constructed and intimately -blended structures, built up by atoms arranged oppositely screw-wise, -clockwise and anti-clockwise, will now form an ultramicroscopic mixture -of the two varieties in equal quantities, that is, in equal molecular -proportions. - -Such a structure will exhibit the symmetry of the system to which the -two individuals belong, but instead of only displaying that of the -enantiomorphous class of that system, possessing lower than the full -symmetry, as each variety does when crystallised alone, it will now -display the full holohedral symmetry of the system. That is, the -symmetry is enhanced by this intimate blending of the two complementary -enantiomorphous forms, the two together supplying all the possible -elements of symmetry of which the system is capable. Moreover, as we -have seen in the case of the lamellar portion of the amethyst crystals -represented in Figs. 95 and 96, there will now be no sign of optical -activity, for the two opposite rotations are equal and destroy each -other. - -Hence, such a compound crystal shows the holohedral symmetry of the -system, and is optically inactive. In such cases we are, in fact, -confronted with the phenomenon of pseudo-racemism, as defined in Chapter -XI. For we know that the two varieties are still present intact, -polarised light revealing them in the case of their grosser development -such as is found in amethyst, and the system of symmetry being clearly -the same, the forms developed being merely the sum of those of the two -individual varieties. - -Amethyst thus affords us a gross demonstration of the nature of -pseudo-racemism, and as such has proved an exceedingly illuminating -study. - -We can carry the process further, however, in imagination, until the two -differently helical molecules are themselves juxtaposed face to face, -right molecule to left molecule. When, however, this occurs, we have -entered into that most fascinatingly interesting region, the range of -molecular forces, a mysterious sphere of activities of which we are only -just beginning to learn something. Within this region of larger activity -the two oppositely constructed molecules are often known to combine -chemically to produce a molecular compound, just as potassium sulphate -molecules, for instance, will combine with those of magnesium sulphate -to form the well-known double salt. The double molecule now furnishes -the representative point of the space-lattice, in other words, a new -space-lattice is now erected, the units of which may be taken to be the -representative points of the double molecules. Such a space-lattice will -of necessity be of a totally different character to the old one -corresponding to the single molecule of either variety (for each variety -has the same space-lattice, the points, however, representing -differently, enantiomorphously, orientated atomic details). That is to -say, we shall have an entirely new kind of crystal produced, in all -probability belonging to a different crystal system. It is known as a -racemic compound, as described in Chapter XI. - -This is exactly what happens in the case of tartaric acid, the two -varieties, dextro or right-handed tartaric acid and lævo or left-handed -tartaric acid, not forming pseudo-racemic crystals of like but enhanced -(holohedral) symmetry, but a truly molecular compound, the well-known -inactive racemic acid, in which the phenomenon of “racemism” was first -discovered and from which it took its name. - -Now a molecular compound is notoriously regarded by chemists as a type -of chemical compound of low stability, molecular attraction or affinity -not being nearly so powerful as atomic affinity. Hence, under suitable -conditions it may be possible to induce the two component varieties to -crystallise out separately from the solution of the racemic compound. In -the case of racemic acid itself this does not readily happen, but in the -cases of certain of its metallic salts, sodium ammonium racemate, for -instance, specific conditions are known under which the two varieties of -crystals, right and left-handed respectively, may be separately -crystallised out from the solution, some of which conditions were -referred to in Chapter XI. Racemic acid itself, however, crystallises -quite differently to the two tartaric acids, namely, in triclinic -prismatic crystals. These are, in fact, absolutely different from the -monoclinic crystals of the dextro and lævo varieties of ordinary -tartaric acid, for racemic acid takes up also a molecule of water of -crystallisation on separating from its aqueous solution. There are -certain chemical differences also, due to the chemical union of the two -enantiomorphous molecules into a single double molecule, such, for -instance, as greater facility of reduction by hydriodic acid to succinic -acid. - -Thus our experiments with quartz have afforded us the means of acquiring -a clear idea of the nature of this most interesting type of crystal -structure which involves the principle of mirror-image symmetry. Racemic -acid and its similar structures, racemic compounds in general, are known -as “externally compensated” structures, the reflective principle here -acting externally to the single enantiomorphous molecule. It is but -another step, however, to imagine internal compensation of -enantiomorphous parts of a molecule, by mirror-image combination of such -parts, such as in all probability occurs in the case of the truly -inactive fourth variety of tartaric acid, in order to comprehend how the -principle enabled the 165 types of homogeneous structure involving this -kind of repetition to be arrived at, and thus, together with the 65 -regular point-systems already known, to afford us the complete set of -230 types of homogeneous structures possible to crystals. - - - - - CHAPTER XV - HOW A CRYSTAL GROWS FROM A SOLUTION. - - -One of the most deeply interesting aspects of a crystal, especially from -the point of view of the history of crystallographic investigation, -concerns the mysterious process of its growth from a solution (in a -solvent) of the substance composing it. The story of the elucidation, as -far as it has yet been accomplished, of the nature of crystallisation -from solution in water is one of the most romantic which the whole of -scientific progress can furnish. Again we are struck with the -parallelism between crystals and living objects. For just as the -discovery of bacteria, the infinitesimal germs of life, has given an -immense impetus to our knowledge of disease and been blessed with most -beneficent effects in combatting the ravages of the latter, so the -discovery that crystal-germs of most common crystallised substances, of -no larger size than bacteria, are floating about in our atmosphere, and -ready at any time to drop into our solutions and, if the latter are in -the proper receptive condition, to set them crystallising, is little -less marvellous, and has had as profound an effect on our knowledge of -the process of crystallisation. A true story, told to the Royal Society -the other day, may serve to illustrate the point. A new chemical -compound had been discovered, and at the time there could obviously be -no crystal-germs, minute crystallites of the dimensions of possibly only -a comparatively few chemical molecules, of this hitherto unknown -substance floating about in the air. It was found impossible to obtain -the deposition of crystals in the ordinary way, from solutions of the -substance in its ordinary solvent, although they were in the condition -of proper receptivity above referred to, on account of the absence of -such germs in the air. But later on, when the air of the laboratory had -become impregnated with such germs, on account of the daily handling of -the substance in the laboratory, no difficulty was any longer found in -obtaining good crystals quite readily from these solutions. - -We are at first inclined to wonder whether such extraordinary statements -can possibly be sober facts. Yet such is, indeed, the case, and it will -be very well worth devoting a chapter to the story of how we have at -length arrived at definite knowledge concerning the process of -crystallisation from the state of solution in water. For water is the -ordinary solvent from which we obtain our crystals, that is, such as are -prepared artificially in the laboratory. The laws which have been -discovered to hold for aqueous solutions are, however, equally -applicable to the cases where other solvents are used, such for instance -as the usual organic solvents like alcohol, ether, chloroform, and -benzene. - -The conditions under which crystallisation occurs from the liquid state, -or from solution of the substance in a solvent, have been accurately -determined experimentally by H. A. Miers,[19] and they bear out in the -main the predictions from theoretical considerations which were made by -Ostwald.[20] Taking first the case of crystallisation from solution, -there are two distinct curves representing the degree of solubility of -the solid substance and of supersolubility. The well-known ordinary -solubility curve is obtained by taking the temperature for abscissæ and -concentration for ordinates, so that any point on the curve indicates -the amount of the solid substance which the solvent can hold in solution -at that particular temperature. Now the fact that supersaturation may -occur has long been established, the phenomenon being of frequent -occurrence; and it is common knowledge that a supersaturated solution -may be preserved for a long time without crystals being deposited from -it, provided the liquid be maintained quietly at rest. Obviously, -therefore, this condition of supersaturation ought to be represented by -a second curve a few degrees lower as regards temperature than the -solubility curve, and its conditions were fairly fully predicted by -Ostwald, after collecting together and analysing the results of the -experiments of Gernez, Lecoq de Boisbaudran, J. M. Thomson, de Coppet, -Lefebvre, and Roozeboom. It was reserved for Miers, however, to discover -a means of experimentally tracing this curve, by observations of the -refractive index of the solution. The point at which the deposition of -crystals from the supersaturated solution occurs is immediately -indicated by a sudden change in the refraction of the liquid, the -refractive index attaining its maximum value at the temperature of -spontaneous crystallisation, and then dropping suddenly the moment the -crystals begin to fall. Moreover, the solution at the same time records -its own strength, for the refractive index varies directly as the amount -of salt dissolved. The determination of the strength of the solution at -the critical moment itself had previously proved an impossibility by -ordinary methods. - -Fig. 98 gives a general diagrammatic representation of Miers’ results -for a typical crystalline substance soluble in water. S is the ordinary -solubility curve, which may also be termed the “curve of crystallisation -by inoculation.” For as soon as the solution reaches this condition of -normal saturation it is liable to be caused to commence crystallising if -a germ crystal, that is, a miniature crystallite floating in the air as -dust, of the substance itself or of one isomorphous with it or capable -of forming parallel growths with it, fall into the solution from the -air. It has been a revelation to us that such minute crystallites of all -common substances are scattered broadcast in our atmosphere, and that -sooner or later one will introduce itself into any solution set to -crystallise which is not sealed up or placed in a vessel with a -filtering plug of cotton-wool in its neck or other aperture. - -[Illustration: - - FIG. 98.—Diagram illustrating the Conditions for Crystallisation from - Solution or the Liquid State. -] - -SS is the supersolubility curve, situated approximately 10° to the left -of the solubility curve as regards temperature, but about as much above -as regards concentration, so that the two curves usually run diagonally -and more or less parallel to each other across the diagram. This -supersolubility curve may be also called the “curve of spontaneous -crystallisation,” for it represents the conditions under which alone -crystals may begin to form without the initiating impulse of inoculation -by a germ-crystallite. On the suggestion of Ostwald it is also termed -the “metastable limit,” and the whole area between the solubility and -supersolubility curves is named the area of metastability, that which -represents the “metastable” condition of the solution. Within this area -the conditions are those for the start of crystallisation by -inoculation. The area beyond the supersolubility curve represents the -“labile” state, in which the conditions are those for spontaneous -crystallisation, inoculation being no longer necessary. These precise -results will, it is hoped, be quite clear with the aid of Fig. 98. - -Hence, when a cooling solution not quite saturated at the higher -starting temperature is stirred in an open vessel a slight shower of -crystals, started by inoculation, appears when the saturation point is -reached, which Miers calls a “metastable shower,” corresponding to the -ordinary solubility curve; the liquid then goes on cooling without -depositing the main bulk of the excess which that curve indicates ought -to be deposited, if it represent the whole truth. But when the -temperature of the supersolubility curve about 10° lower is reached, a -much more copious shower falls by spontaneous crystallisation, the -“labile shower.” - -In a closed vessel, such for instance as a glass tube sealed with the -aid of the blowpipe after the introduction of the solution, on cooling -after heating to a temperature superior to that of saturation, the first -shower never falls at all, no amount of shaking inducing the deposition -of crystals at the ordinary saturation point, proving that the slight -shower of the experiment in the open vessel is due to crystal-germs -introduced from the atmosphere. The second shower of crystals falls at -the lower temperature just as before, however, at the temperature of the -supersolubility curve, indicating that this shower is due in both cases -to spontaneous crystallisation. Solutions thus enclosed in sealed tubes, -to which inoculating dust crystals cannot have access, can never be made -to crystallise at any temperature higher than that given by the -supersolubility curve, however agitated, although they immediately do -crystallise, if shaken, as soon as that temperature is reached during -the cooling. If allowed to remain absolutely quiet, however, the -temperature may fall considerably lower before any crystallisation -occurs, the labile region being frequently well penetrated before this -happens. When crystallisation does supervene, the temperature usually -rises somewhat. After the labile shower has been deposited, the crystals -continue to grow steadily further, until the metastable region has been -traversed, and the saturation state is eventually reached, when final -equilibrium is produced. - -The proof that the crystals deposited in the metastable condition were -started by the advent of atmospheric germ crystals—that is, by -infinitesimal but perfectly structurally developed crystals, carried by -their very lightness like the particles of dust which are only revealed -in the path of a sunbeam as seen against a dark background—was afforded -by a series of experiments with a mixture of two rare organic chemical -preparations, salol (phenyl salicylate) a substance melting at 42.5°, -and betol (β-naphthol salicylate) another melting at 92°, which Miers -assumed were not likely to be present in ordinary air. The assumption -proved well grounded, and the first shower never fell at all in the -earlier experiments in which mixtures of these two substances were -allowed to cool in open vessels, from the state of fusion. But very soon -the air of the laboratory became impregnated with crystallites of both -substances, due to the very operations themselves being carried on in -contact with the air, and in the later experiments the first shower of -crystals did fall. The experiments were really designed to effect the -determination of the solubility curve for salol and betol in each other, -that is, the freezing-point curve of their mixtures, and the discovery -of the so-called “eutectic” point at which a mixture of constant -composition solidifies at a definite temperature. But incidentally the -experiments also served to establish similar laws for the production of -crystals from the liquefied state, by cooling below the melting point, -to those applying to crystallisation from solution. In the case of the -mixtures of the substances the one of lower melting point acted as a -solvent for the one of higher melting point, just as water does for -salt. Two curves corresponding to the ordinary freezing point and to the -limit of superfusion were established, analogous to the solubility and -supersolubility curves. Pure salol alone proved to crystallise -spontaneously at 33°, 9½° below its melting point, and the refractive -index attained a maximum for this temperature. Betol spontaneously -crystallised at 79°, 13° below its melting point. - -Two general cases of crystallisation are shown by the dotted curves ABCD -and ABE in Fig. 98. The first, represented by ABCD, is the case of a -supersaturated solution, made by adding the salt to hot water, being -allowed to cool slowly while stirred. The solution cools from A to B -without anything visibly happening, no crystal-germ falling into the -solution until B is reached, somewhere well within the metastable -region. When the germ has fallen in, however, crystals begin to appear -as a slight shower at B, and from B to C they continue to grow slowly. -On reaching the labile condition at C a cloud of crystals, the heavy -shower, is deposited, and the concentration falls rapidly to D on the -solubility curve, generally with slight rise of temperature. - -The second case is the important one employed by the author in the -investigations which will be found described in his “Crystalline -Structure and Chemical Constitution” (Macmillan & Co., 1910), for the -purpose of producing crystals of high perfection for goniometrical -investigation. The method can be confidently recommended as the one best -adapted to afford good measurable crystals, and is of quite general -application. The solution is made up so as to be in the metastable -condition, that is, only slightly supersaturated for the ordinary -temperatures. Eventually, while the solution is at rest in a protected -place, free from draughts or vibration, and after it has cooled to the -temperature of the air, a crystal-germ enters, followed probably by -others; each forms a centre of crystal growth, which proceeds very -slowly and deliberately, keeping pace with the evaporation in such a -manner that the labile condition is never reached. The natural result is -the production of very well-formed crystals bounded by excellent faces, -truly plane and free from striation or distortion. - -When the operation is arranged to occur during the night, as will -usually be the case, the solution being set out to crystallise in a -quiet and protected place on the previous afternoon or evening, the -slight fall of temperature during the night gently assists the process -and almost ensures a good crop of a few well-formed crystals large -enough for goniometrical purposes next morning. They should be removed -before the temperature begins to rise again with the advent of the sun, -dried with blotting paper and by air exposure for a short time, and -stored in a miniature bottle labelled with the name or formula of the -substance and the date of collection of the crop. In such cases the -labile state is never reached, and the course of the crystallisation is -represented by the curve BE. The whole conditions for the curve ABE, -however, would correspond to much lower temperatures, such as those -given at the foot of the diagram below the word “temperature,” rather -than to the upper row of temperature abscissæ suitable for the other -purposes of the diagram already referred to. Crystallisation might well -begin about 13° or 14°, as shown at B, and the liquid would cool a -couple of degrees or more during the night while crystallisation was -steadily proceeding, until equilibrium was reached at E on the -solubility curve. - -The diagram does not represent any substance in particular, but is a -perfectly general one, corresponding to the facts observed with most of -the very varied salts worked with by Miers and those of which the author -has had experience. The exact temperatures and concentrations will, of -course, differ for each substance. - -A beautiful experimental demonstration of crystallisation from the -metastable and labile conditions of solution respectively is afforded by -potassium bichromate, K_{2}Cr_{2}O_{7}. When deposited slowly from a -metastable solution under conditions of quietude, this salt is slowly -deposited in bright orange coloured and excellently formed crystals, -often of considerable size, belonging to the triclinic system of -symmetry; they are bounded by good pinakoidal (pairs of parallel) faces -intersecting in sharp edges. But when the crystallisation occurs from a -labile solution, being much more rapid, it takes the form of feathery or -arborescent branching skeletal growths, there being inadequate time for -the formation of well-developed crystals. - -Fig. 99, Plate XXI., is a photographic reproduction of well-formed -crystals of potassium bichromate, grown from a solution in the -metastable condition on a microscope slip, just as they are seen through -the microscope in the slow act of formation, employing a 1½ inch -objective. The crystallisation had been started by germ crystals of the -salt falling in from the air, after which the drop, placed within the -ring of hardened gold size on the slide, had been covered with a -cover-glass, under which the crystallisation had proceeded with -sufficient slowness to enable a successful photograph to be taken, when -the camera was subsequently attached above the vertically arranged -microscope. An upright micrographic apparatus had been designed by the -author specially for this photography of growing crystals, many of the -results of which are reproduced in this book. - -[Illustration: - - _PLATE XXI._ - - FIG. 99.—Potassium Bichromate slowly crystallising from a Metastable - Solution. -] - -[Illustration: - - FIG. 100.—Potassium Bichromate rapidly crystallised from a Labile - Solution. - - CHARACTERISTIC DIFFERENCE IN THE CRYSTALS DEPOSITED FROM METASTABLE - AND LABILE SOLUTIONS. -] - -Fig. 100 is the reproduction of another photograph taken under similar -conditions, but employing a hot and somewhat more concentrated solution -of potassium bichromate, and making the exposure at the moment when, in -the particular field chosen, a rapid labile crystallisation was just -completing itself, the rapidity of growth of the feathery skeletal -crystals having just become arrested. Indeed, the branches are -frequently terminated by small well-formed crystals, the rapid growth -having been succeeded by a final slow crystallisation where the solution -had discharged its labile excess and attained once more the metastable -condition. - -This experiment with potassium bichromate lends itself admirably to -lantern demonstration with the projection microscope. When the drop of -hot concentrated solution is first placed on the warmed microscope slip, -and the latter laid on the stage, nothing visible on the screen happens -for a minute or two, the solution becoming, however, more or less -rapidly cooled. But suddenly, the drop having cooled sufficiently to -bring the solution to the labile condition of supersaturation -corresponding to the conditions for spontaneous crystallisation -indicated by the supersolubility curve, arborescent or feathery growths -begin to shoot out from various points in the field, often near the -margin, and traverse the screen so rapidly that in a moment or two it is -filled with them. The crystallisation then slows down once more, the -labile shower of excess having become exhausted, and the terminations of -the branches and ramifications begin to develop into good little -crystals, which thus hang like fruit on a tree. The experiment is -rendered the more brilliant and beautiful by the bright orange colour of -the crystals. - -In Fig. 101, Plate XI., facing page 88, a reproduction of a photograph -of a similar crystallisation from a labile solution of ammonium chloride -is given. This salt is also particularly suitable for screen -demonstrations. The beautiful skeletal ramifications follow the axial -directions of the cubic axes, ammonium chloride crystallising in the -pentagonal-icositetrahedral class of the cubic system. Good crystals -may, however, be very slowly grown from metastable solutions, and they -usually exhibit as the principal forms the icositetrahedron -(predominating), cube, octahedron, rhombic dodecahedron, and the -class-distinguishing pentagonal icositetrahedron. The rapid growths by -spontaneous crystallisation of labile solutions, however, invariably -take the form of the rectangularly branching feathery crystals shown in -Fig. 101. - -Further light has been thrown on the act of crystallisation by another -most interesting research of Miers concerning “vicinal faces,”[21] such -as the three very low pyramid faces (forming a very flat triakis -octahedron) which often replace each octahedron face on a crystal of -alum which has been grown somewhat rapidly. The author has frequently -observed this phenomenon in the course of the numerous crystallisations -required for the investigation of the sulphates and selenates. It may be -described in general terms as the replacement of primary faces -possessing the simplest rational indices by faces having such high -indices that it is doubtful whether they ought really to be represented -by indices at all. The number of such vicinal faces which replace the -simple face depends on the symmetry of the crystal, to which, of course, -they conform. Thus, while three such vicinal faces replace an octahedral -face, and two replace the face of a tetragonal prism, the simple primary -prism face of a rhombic or monoclinic crystal would only be replaced by -one, which may have a deformation of as much as even 30′ from the -correct position of the prism face, and on either side of it. Indeed it -is possible for a whole succession of such vicinal faces to be developed -within the degree of arc which may in extreme cases separate the -limiting values on each side of the prism face, and such are often seen -and make up the well-known bundle of images afforded on the goniometer -by a bad face, a face which would cause the author at once to reject the -whole crystal for measurement purposes. One of the faces, even in cases -such as alum or a tetragonal crystal, where three or two might have -equal values as regards the symmetry, generally predominates, and -affords a very much more brilliant image of the goniometer signal than -the others in the bundle, so that an unwary observer might easily come -to the conclusion that this was the really valid image corresponding to -the octahedron face or to the simple primary prism face, or whatever -particular face was expected in the neighbourhood indicated by the -bundle of images. Obviously, however, it might only be one of three or -two equally valid faces of a vicinal form, which had grown -predominatingly during the last period of growth previous to removal -from the mother liquor. - -The explanation of this interesting phenomenon of the production of -vicinal faces is one intimately connected with the structure of -crystals, and it forms one of the strongest confirmations of the -correctness of the theory of crystal structure the basis of which is the -molecular space-lattice. Miers is in full agreement with the author in -emphasising the importance of the space-lattice formed by the “points” -representative of the molecules, and analogously chosen in the -molecules. He says: “Whatever structures may be necessary to account for -other features of crystals, there is little doubt that we are justified -in regarding their faces as the planes of a space-lattice.”[22] Now -Wulff,[23] who has contributed very considerably to our knowledge of the -nature of the act of crystallisation, has proved, from his own -investigations and those of Weyberg, carried out at his suggestion in -his laboratory at Warsaw,[24] that faces of greatest reticular density, -that is, those along which the points of the space-lattice are most -thickly strewn, are those which grow the most slowly, and therefore are -the best developed. This latter will be obvious on a little -consideration, for the faces of less reticular density which grow more, -tend in doing so to extend the boundaries of the faces of greatest -reticular density, and thus to enlarge those faces. Hence the usual -planes on a crystal must be those of high reticular density; and these -are such as are represented by the simplest indices, the faces most -dense of all in points being the primary ones. - -But it has been shown from the researches of Miers that vicinal faces -are often produced in preference to these simple index planes of high -density, and such vicinal faces, although the nearest (in angular -position) of all possible faces to those simple index planes, are -themselves of excessively low reticular density, so much so that if -represented by indices at all they can only be indicated by very high -numbers, not such as we are accustomed to consider as in keeping with -the simple spirit of the law of rational indices. Taking the example -worked out most fully by Miers, the octahedral crystals of alum, it is a -fact that the cubic faces of highest reticular density are those of the -cube itself, then come in order those of the rhombic dodecahedron and -those of the octahedron. Hence, the density of octahedral faces is very -high. But those of the very low triakis octahedron, which Miers finds to -replace the octahedron faces so frequently as vicinal faces, are of -excessively low reticular density. - -Miers explains the appearance of the latter instead of octahedral faces -by assuming that the supersaturated liquid in contact with the growing -crystal consists of the particles (molecules) of salt uniformly mingled -with those of water, the solvent, and that the act of crystallisation -consists of the escape of the water and solidification of the salt. -Consequently, the salt particles just before crystallisation cannot be -so dense as they are along primary planes of the crystal, as they are -separated by the water particles, which are presumably much more -numerous. Hence it is that they are not deposited along the planes of -high reticular density, but along vicinal planes of low density of -points. For instance, he shows that the shower of salt particles upon a -cube face would have to be so dense that there would be insufficient -room for the water particles. The density in a cube face is 114 times as -great as that in one of the vicinal planes observed. Now, 100 cubic -centimetres of solid alum weigh 172 grammes, and 100 c.c. of the -solution depositing crystals contain 9·74 grammes of alum. Thus the -density of the growing crystal of alum is nearly 18 times that of the -alum in the adjacent saturated solution. - -Consequently the deposition of the salt particles, in a moderately quick -crystallisation, when insufficient time is afforded for the deliberate -escape of the water particles and for the orderly rearrangement of the -salt particles, occurs along vicinal planes instead of along the primary -planes. For it must not be forgotten that whenever it has the -opportunity of coming into operation there is a directive molecular -force of some kind, which controls the operation of crystallisation, and -which undoubtedly attempts to cause, and given adequate time and scope -succeeds in causing, the production of faces of high reticular density, -the fundamentally important primary faces of lowest indices, and which -are often those along which cleavage occurs. Wulff emphasises this in -saying (_loc. cit._, p. 461): “Bei der Krystallisation orientiren sich -die Molekeln auf den Flächen des Krystalles ganz gleichförmig durch den -Einfluss der Richtkraft der Krystallisation.” The more rapidly the -crystallisation occurs, however, the less chance is there for this force -to attain its ultimate object. More will be said about this directive -force in the next chapter, after we have studied the remarkable “liquid -crystals” discovered by Lehmann. - -This highly interesting explanation of Miers, supported as it is by the -work of Wulff, and confirmed also in many respects by the observations -of the author, whose great aim throughout his investigations has been to -avoid the production of such vicinal faces, throws an important light on -the nature of the act of crystallisation. It renders the reason clear -why crystals which are very slowly grown from solutions only feebly -supersaturated and under conditions of absolute rest, protected from -either air currents or preventable earth tremors—conditions which the -author has taken quite exceptional pains to procure for the preparation -of the crystals used in the investigation of the sulphates and selenates -of the rhombic simple salt series and monoclinic double salt series—are -occasionally obtained quite free from any sign of such vicinal faces. -They are small perfect individuals exhibiting primarily the faces of -high reticular density, that is, the faces of the simple forms of low -rational indices; and these faces are absolutely plane, affording one -single brilliant image of the goniometer signal, which can be adjusted -with great precision to the cross-wires of the telescope. For the slower -the growth, the more time is afforded for the escape of the water -molecules, and for the salt molecules to deposit themselves as directed -by the molecular guiding force of crystallisation, along the planes of -high reticular density. In many of his experiments Miers expressly -stirred the solution, to prevent concentration currents, which had been -considered by Wulff of importance in the process, from coming into play -and causing unknown effects. Hence his experiments in which vicinal -faces were produced are not comparable with the author’s slow growths. - -The work of Miers assists in the proof that the constancy of angle to -within one or two minutes of arc is a real property of the crystals of a -substance. For previously the frequent presence of vicinal faces rather -than the simple forms of high reticular density, and which had been -mistaken for the latter, had caused Pfaff in 1878[25] and others to -conclude that variations from the true crystal angle amounting to as -much as 30′ were of common occurrence as the result of strain during -deposition. - -Brauns[26] in 1887 made a careful series of measurements of very good -octahedral crystals of lead nitrate, and found 13′ 20″ the largest -deviation of a good image from the theoretical position. He imputed it -to the action of gravity as a disturbing cause during deposition. The -researches of Miers have cleared away all this misconception, in proving -that the bright images referred to, taken for those of the simple -primary form, are not such at all, but vicinal faces of very low -reticular density. - - - - - CHAPTER XVI - LIQUID CRYSTALS. - - -We have seen in the foregoing pages that a crystal is usually a solid, -highly organised in a homogeneous manner, and, unless the symmetry be -developed to its highest extent, the crystal then belonging to the cubic -system, it is also in general anisotropic, that is, it exhibits double -refraction. Section-plates of it, more or less thin according to the -strength of the double refraction, exhibit colours in parallel polarised -light, and show the phenomenon of a single optic axis, or of two optic -axes, in convergent polarised light. Every variety of hardness, however, -is displayed, from that of the diamond down to that of a crystal as soft -as gypsum, and even softer. Moreover, many of the softer crystallised -substances develop the property of permitting one layer to glide over -another by gentle side pressure with a knife blade, when inserted in an -edge or face in an attempt to cut the crystal. Calcite and ice, for -instance, both possess such planes of gliding of the structural units -over one another in layers. There are also the border line cases of -crystals so soft as to be readily bent, and many well-known viscous -substances crystallisable only with great difficulty, some of which form -pliable crystals. - -But in the year 1876 Lehmann discovered a new property in an already -remarkable substance, iodide of silver, AgI, namely, that at -temperatures superior to 146° C. it can flow like a viscous liquid, -while exhibiting several of the properties which are characteristic of -crystals. Silver iodide is dimorphous, exhibiting a hexagonal form at -the ordinary temperature, which persists up to 146°. But during the -heating to the latter temperature a regularly accelerating diminution of -volume occurs, the feeble expansion in directions perpendicular to the -axis being overbalanced by a considerable contraction along the axis, -both quantities having been accurately measured so long ago as the year -1867 by Fizeau, by means of his delicate interference dilatometer. This -contraction, so unusual an occurrence with increase of temperature, -culminates at 146°, according to Mallard and Le Chatelier, in a sudden -change of condition into a cubic modification, accompanied by absorption -of heat. Now Lehmann, studying this cubic modification of silver iodide -under a microscope which he had devised—specially adapted for -observations at temperatures higher than the ordinary, by being supplied -with the means of heating the object under observation—found that it was -not only plastic, but actually a liquid. - -[Illustration: - - FIG. 102.—Lehmann’s Crystallisation Microscope. -] - -The form of Lehmann’s “Crystallisation Microscope,” as now constructed -by Zeiss, is shown in Fig. 102. Its essential features are that the -glass object-plate, which is somewhat wider than the usual microscope 3 -by 1 inches slip, is supported by little metallic columns at a height an -inch or more above the ordinary stage, and may be heated from below by a -miniature Bunsen burner, which is provided with a delicate graduated -gas-tap and is adjustable for its position, swinging in or out as -desired. The small Bunsen flame may be converted into a blowpipe flame -if necessary, an air-blast attachment to a mixing reservoir being -provided, to which the arm of the burner is hinged. Two cooling blasts, -connected with a gas-holder of air, are also provided, and are -adjustable to the most suitable symmetrical positions above the slide -for directing the cooling air on the part of the latter where the liquid -is situated. These arrangements enable the substance on the slide to be -rapidly or slowly heated or cooled at will. Electric connections are -also provided, in the event of the observer desiring to study the -behaviour of the liquid crystals under the influence of the electric -current. - -Considerably later, in the year 1889, the attention of Lehmann was -called by Reinitzer to another similarly singular substance, cholesteryl -benzoate, which appeared to consist of an aggregate of minute crystals -which flow as readily as oil, while preserving many of the characters of -crystals. - -In the next year, 1890, the substance para-azoxyphenetol, then recently -discovered by Gattermann, was observed by Lehmann to form a turbid -“melt” on fusion, which consisted of an aggregate of crystals flowing -with a mobility equal to that of water, and which take the form of -spherical drops showing a dark kernel inside, as shown at _a_ in Fig. -103, quite unlike a drop of ordinary liquid. The kernel disappears on -shaking, but reappears on coming to rest again. In polarised light the -drops show dichroism, that is two different colours in different parts -or directions, being divided into white and yellow parts, the yellow as -a pair of opposite approximately 60°-sectors, as indicated at _c_ in -Fig. 103. Under crossed Nicols they show a black cross, as represented -at _d_ in Fig. 103. - -Now obviously these drops are doubly refractive, and their whole optical -behaviour corresponds to the arrangement of the molecules in concentric -circles, such as that suggested at _b_ in Fig. 103. - -[Illustration: - - FIG. 103.—Liquid Crystals of Para-azoxy-phenetol arranged in Spherical - Drops. -] - -Another substance of like character, para-azoxy-anisol, was subsequently -found to behave similarly, and forms an excellent substance to use for -demonstration purposes. A reproduction of a photograph, kindly sent to -the author by Prof. Lehmann, of a slide of this substance is given in -Fig. 104, Plate XXII. It shows a characteristic field of such drops, -exhibiting white parts and yellow sectorial parts which photograph dark, -of para-azoxy-anisol mixed with a little para-azoxy-phenetol, oil and -resin (colophony), as seen under the polarising microscope with crossed -Nicols. - -The next and possibly most interesting step in this remarkable series of -discoveries was made by Lehmann himself in the year 1894. He alighted on -the fact that ammonium oleate, crystallised from solution in alcohol, -affords a splendid example of flowing crystals, which are sufficiently -large to enable their habits to be studied in detail. The individuals -are almost invisible in ordinary light, owing to the refractive index of -the crystals and of the mother liquor being approximately the same. But -in polarised light, using crossed Nicols, they are clearly revealed as -steep double pyramids with more or less rounded edges. Their section is -nearly circular in consequence, and they exhibit optical properties of a -uniaxial character, the optic axis being that of the double cone or -bipyramid. A characteristic individual is shown at _e_ in Fig. 105. When -two of these flowing crystals approach each other, as at _a_ in Fig. -105, they coalesce to form a larger single individual, as is indicated -in stages at _b_, _c_, and _d_ in the illustration. - -[Illustration: - - FIG. 105.—Liquid Crystals of Ammonium Oleate. -] - -When the cover-glass, under which they are growing on a microscope 3 by -1 inch slip, is moved to and fro so as to distort these remarkable -bodies, which we may well hesitate to call crystals, the singular effect -is produced of causing them all to become similarly orientated, for the -extinction directions follow the direction of the pressure. They at once -seek to regain their original form, however, on cessation of the -disturbance. A slide of the bipyramids under pressure is shown in Fig. -106. In the black extinguished portions of the field the flowing -crystals are flattened, according to Lehmann, and arranged so that the -optic axis is in all cases perpendicular to the tabular crystals and the -glass plates and parallel to the axis of the microscope. The black parts -are separated by oily strips, as shown in another slide under -considerable pressure, represented in Fig. 107, which are composed of -the tabular crystals standing on end, with their optic axes parallel to -the plates. These strips polarise the more brightly the more truly the -crystals stand perpendicularly to the plates. The two conditions are -shown diagrammatically at _a_ and _b_ in Fig. 108. - -[Illustration: - - FIG. 106.—Liquid Crystals of Ammonium Oleate under slight pressure. -] - -[Illustration: - - FIG. 107.—Ammonium Oleate under considerable pressure. -] - -[Illustration: - - FIG. 108.—Diagrammatic Representation of Arrangement of Molecules. -] - -Lehmann believes the explanation of these singular phenomena to be that -the “liquid crystals” of ammonium oleate are composed of piles or layers -of thin plates perpendicular to the optic axis. Disturbance detaches the -plates from their piled positions over one another, and sets them -parallel to the glass plate, except in places, the oily strips, where -the plates stand upright, perpendicularly to the micro-slip and -cover-glass. Lehmann, indeed, goes further, and asserts that the -molecules themselves are anisotropic, and probably take the form of -plates. - -[Illustration: - - FIG. 109.—A Crystal of Ammonium Oleate A, broken at B, each part - repairing itself at C, perfect again at D. -] - -An extremely interesting experimental observation of Lehmann’s with the -bipyramids of ammonium oleate is, that if one of them, for instance A in -Fig. 109, be broken into two parts, as at B, each part grows again and -completely repairs itself, becoming once more a perfect double pyramid, -as indicated in stages at C and D in the figure. - -[Illustration: - - FIG. 110. -] - -[Illustration: - - FIG. 111. - - Cruciform, Boomerang, and Arrow-head Twins of Ammonium Oleate. -] - -[Illustration: - - _PLATE XXII._ - - FIG. 104. - - Dichroic Crystal Drops of Para-azoxy-anisol. -] - -[Illustration: - - FIG. 112. - - Rectilinear Liquid Prisms of Para-azoxy-benzoic Acid. -] - -[Illustration: - - FIG. 115.—Tetragonal Astatic Magnet-system. -] - -[Illustration: - - FIG. 116.—Cubic Astatic Magnet-system. -] - -Twins of ammonium oleate are also shown in Figs. 110 and 111, the former -figure representing a twin of cruciform character, and the latter -exhibiting twins resembling a boomerang and an arrowhead respectively. - -This substance, ammonium oleate, thus appears to be one of the most -remarkable and interesting of all the bodies yet observed to afford -liquid crystals. Many other oleates produce liquid crystals also, but -the ammonium salt is by far the most striking, and very convincing as to -the reality of Lehmann’s discovery. - -Another substance of a different nature was discovered by Vorländer in -the year 1904, namely, the ethyl ester of para-azoxy-benzoic acid. A -characteristic microscope slide of it in ordinary light is shown in Fig. -112, Plate XXII., which is a reproduction of an actual photograph most -generously sent to the author by Prof. Lehmann. - -The individuals are described by Lehmann, who further studied the nature -of the substance, as almost perfectly rectilinear prisms with nearly -sharply defined basal plane end faces. A singular fact about this -substance is, that when two individuals approach each other they arrange -themselves parallel with a jerk, and then flow into each other, -producing a single larger liquid crystal, and often with such rapidity -that the eye can scarcely follow the movements. These coalescences -appear to be occurring all over the field at once, with the production -of larger and larger crystals. Indeed, Lehmann likens it to a struggle -between the innumerable individuals, in which the smaller ones are being -continually eaten up by the larger. - -Vorländer also prepared the ethyl ester of para-azoxy-cinnamic acid, and -Lehmann found it to be similarly interesting. The substance separates -from a solution in monobromonaphthalene in uniaxial prisms or -hemimorphic pyramids, the edges and solid angles of which are more or -less rounded, and which appear colourless in the direction of the axis -and yellow in all other directions. When pressed between the cover-glass -and the micro-slip on which the crystallisation is proceeding, -extinction of the light occurs throughout the whole mass when polarised -light is being employed and the Nicols are crossed. For throughout the -entire substance the particles—whether they are the molecules themselves -as Lehmann asserts or aggregations of them in the form of -ultramicroscopic crystals—arrange themselves with their optic axes (the -crystals being uniaxial) perpendicular to the cover-glass and -micro-slip, as in the case of ammonium oleate. Lehmann’s theory is that -the molecules themselves are tabular perpendicular to the axis, as in -the case just referred to, and that they are thus readily coerced by the -pressure of the flat cover-glass to take up positions parallel to it. - -Two further reproductions of photographs, taken in polarised light, of a -somewhat remarkable character, which have been placed at the author’s -disposal by the courtesy of Prof. Lehmann, are given in Figs. 113 and -114, Plate XXIII. Fig. 113 represents numerous doubly refractive and -dichroic strips marking the boundaries of elongated individual crystals -of the substance dibenzal benzidine, and affords a graphic idea of the -real character of the double refraction displayed by liquid crystals. - -[Illustration: - - _PLATE XXIII._ - - FIG. 113.—Elongated Liquid Crystals of Dibenzal Benzidine, showing - Double Refraction and Dichroism. -] - -[Illustration: - - FIG. 114.—Spherical Liquid Crystals of Para-azoxy-anisol, showing - Interference Colours under Crossed Nicols as the Effect of - Compression. -] - -Fig. 114 represents the effect of compression on para-azoxy-anisol, and -demonstrates very clearly the distribution of the interference colours -due to double refraction. - -We are thus face to face in these remarkable experiments with some new -facts concerning the nature of crystals. For we pass here into the -borderland between ordinary liquids—singly refractive and structureless, -in which the molecules are rolling over each other with every possible -orientation—and solid true crystals possessing homogeneous structure, -and the basis of which is a space-lattice arrangement of the chemical -molecules, determinative of the system of symmetry displayed. In this -wonderful borderland we certainly have had revealed to us, by the genius -and persistency of Lehmann, liquids which possess many of the attributes -of crystals, such as definite orientation of the ultimate particles, -double refraction, and optic axes. These are undoubtedly solid facts -which require to be faced. - -Whether Lehmann’s theory is to be accepted in full can only be decided -after much more investigation by several independent investigators. We -are now becoming familiar with the phenomena, as they have naturally -excited immense interest in all scientific circles, and demonstrations -of many of the experiments of Lehmann have been given in this country by -Dr. Miers, Prof. Pope, and others, and particularly by Messrs Zeiss, -with their new high temperature microscope, a description of the use of -which for the projection of liquid crystals on the screen will presently -be given. Prof. Lehmann himself has described the phenomena so clearly -and fully that it is quite easy for others to repeat his experiments, -and doubtless time would often be much better spent in doing so than in -criticising points of theory without observing the phenomena at first -hand. It frequently happens, in the inevitable march of scientific -progress, that striking new facts, such for instance as the discovery of -the composite nature of the chemical atom, are apt to cause either -alarm, even panic, as to cherished theories, or else unreasoning -scepticism. The happy mean between these two modes of receiving such -facts, the open philosophic mind, ever ready to widen the scope of the -horizon when a novel supposition is indubitably proved to be a real -fact, and to assimilate that truth into the theory, widening -correspondingly the scope of the latter if needful, is obviously the -ideal thing to cultivate, and one which eventually finds itself in -harmony with the authenticated final results of the new discoveries. It -usually happens that too sweeping conclusions are at first drawn from -such new facts, but time, with its further wealth of experience, -especially the accumulation of experimental _data_ which it brings in -its train, soon levels these down and relegates the facts to their -proper positions in the great scheme of natural knowledge. - -Lehmann’s view is that the ordinary effect of surface tension to cause -truly liquid particles to assume the spherical “drop” form is resisted -by a special force, which he terms “Gestaltungskraft,” and which we may -perhaps translate “Configuration-determining force.” This force he -considers is not identical with that of elasticity, but is that force by -virtue of which a “flowing crystal” continually seeks, while freely -swimming in the mother liquor or fused liquid, to take up its normal -configuration. Even if a spherical drop could be cut out of it, the -sphere would at once become a rod, prism, or pyramid or whatever the -normal configuration of the flowing crystals of the substance in -question might be. - -The much debated term “liquid crystal” has been given by Lehmann to the -normal configuration of each of the now considerable number of -substances which have been discovered to exhibit the phenomena of -flowing crystals. The latter appellation “flowing crystal,” which -Lehmann also uses, appears to the author to be in many ways more -suitable, however, and would avoid much of the criticism which has been -levelled at the term “liquid crystal.” - -As already indicated, Lehmann attributes the whole phenomena to a -fundamental cause, namely, anisotropy (optical dissimilarity in -different directions) of the molecules themselves, which he considers -must cause self-restoration of the structure after disturbance, a -process which he terms “spontaneous homœotropy.” He considers that it is -the molecular configuration-producing force, connected with the tabular -form of, and directionally differentiated distribution of energy and -force in, the single chemical molecules, which maintains the inner -structure of the flowing crystal in position. The polyhedral outward -form thus appears to be a necessary consequence of the inner structure, -on this basis that it is a force resident in the molecules themselves -which produces the structure. - -Now the revelation of new facts, as startling as those which are now -experimentally fully confirmed concerning flowing crystals, must -inevitably cause searching reflection as to whether the magnificent -geometrical work on the 230 homogeneous structures, and their -development in actual fact in the 32 classes of crystals, is to stand or -to be seriously affected. Again the author ventures to express the -opinion, that just what happened in regard to the historic differences -between the schools of Haüy and Mitscherlich, will in all probability -again occur, namely, both extreme views will be shown to depend more or -less on real facts, and other connecting facts will eventually be -revealed which will completely reconcile the two series with each other. -In the author’s opinion, the geometrical work will stand, as the grand -generalisation it really is. But it will be interpreted in the future -without the somewhat arbitrary assumptions which have more or less -accompanied it. From these it will be freed, and then rise purified and -elevated to its real dominating position in regard to crystal -morphology. - -Lehmann, with the natural enthusiasm of the discoverer of one of the -most remarkable facts for which the last few decades have been famous, -may have carried his theory too far, and particularly in that part of -his work, to which the author has not hitherto referred, in which he -describes certain phenomena of flowing crystals as akin to the movement -of living organisms such as bacteria, and thus brought even some of the -sound facts under the criticism of the sceptic more than might otherwise -have been the case. He may also have made his theory far more -revolutionary than is essential. But the one incontrovertible thing -stands out plainly, namely, that the “flowing crystals” with which he -has made us acquainted are an indubitable experimental fact. Flowing -crystals are produced, however, by a relatively few substances of very -complex molecular constitution, involving a large number of atoms in the -molecule; they are mostly compounds of carbon, and in number possibly -not one per cent. of the innumerable substances known to produce -ordinary solid crystals. That the theory of crystal structure can -eventually be made to include these few remarkable substances is highly -probable, when many more facts have been accumulated. - -Lehmann would appear to have made one point very clear, which at once -removes an objection long felt by the author to the theory of crystal -structure as it stands at present, namely, that the chemical molecule is -endowed with a directive orientative force, which is certainly concerned -in crystallisation. To assume, as has been done, just because it is not -necessary from the point of view of the geometrician in developing his -possible homogeneous structures, that no directive force is operative in -crystallisation, and that all is a mere question of the most convenient -mechanical packing of the molecules, is, in the author’s opinion, going -beyond what the experimental facts justify. If Lehmann’s discovery of -flowing crystals does nothing more than return to the molecule the -property always hitherto attributed to it, of possessing in itself some -directive force by reason of which it arranges itself homogeneously by -mutual accommodation with its similarly endowed fellow molecules, when -its motion in the liquid state has been sufficiently arrested by its -approach to its fellows within the range of molecular action (four or -five molecular diameters), either by cooling or the falling out of -previously separating solvent molecules, it will have achieved a notable -thing. - -What does occur at the moment of crystallisation is at the present time -one of the most interesting unsolved questions in crystallography, and -one calling most urgently for solution. Attention was directed to the -problem in the last chapter, in connection with the suggestive work of -Miers on vicinal faces. It was shown that it was only when the directive -force had time to come properly into operation that the primary faces of -fundamental importance were produced, and that when the crystallisation -was rapid vicinal faces formed instead. Lehmann believes that a single -kind of chemical molecule is only capable of producing a single specific -space-lattice, and that polymorphism is due to alteration of the -molecules themselves at the critical temperature of transformation. He -showed so far back as 1872 that this limit could be actually observed -under the microscope, as a definite line of demarcation between the two -varieties as the temperature fell, one side of the field attaining the -critical temperature slightly before the other, and the defining line -between the two kinds thus travelling over the field. Internal friction -did not appear to Lehmann to enter into the question at all, as he -considered it would have done if a rearrangement of the molecules were -the sole cause of the change. The molecules themselves, he states, must -have been undergoing change, and such rearrangement of them as occurred -must have been due to that fact. - -[Illustration: - - _PLATE XXIV._ - - FIG. 117.—Arrangement of Astatic Magnet-systems in a Plane. -] - -[Illustration: - - FIG. 118.—Arrangement of Astatic Magnet-systems in Space. -] - -Lehmann suggests a very interesting explanation of the molecular -orientative force of configuration, namely, that it is due to the action -of the electronic corpuscles (forming the elementary atoms) rotating in -the molecule. For the molecules of flowing crystals behave like freely -suspended astatic systems of magnets, which are constantly setting -themselves, even while moving about, in a crystalline space-lattice. He -suggests that the molecules are really magnets the poles of which -mutually attract and repel one another; that two equal magnetic -molecules are arranged alongside with opposite poles against each other, -thus mutually binding each other, or that four horse-shoe magnets may be -arranged with opposite poles together, in a tetragonal astatic system, -as shown in Fig. 115, Plate XXII. The latter may be grouped in space in -a cubic astatic system, as represented in Fig. 116 on the same Plate; -while Figs. 117 and 118, Plate XXIV., are further suggestive of how a -homogeneous structure of such astatically distributed molecules can be -built up, Fig. 117 representing a single plane of them, and Fig. 118 the -complete arrangement in space. - -An astatic system of molecules of this nature would have lost all power -of attraction by a magnet, and the fact would thus be accounted for that -no striking crystallographic results have ever attended experiments on -crystallisation in a magnetic field. Astatic systems, however, such as -that shown in Fig. 115, would certainly arrange themselves in -space-lattices. For such parallel arrangements would, in general, -involve differences in different directions, with regard both to -internal friction and to the power of thermal expansion and of such -regular dilatational or other deformational changes as we know are -provoked by different physical conditions of environment. These -differences would naturally, in turn, give rise to external polyhedral -form. - -Lehmann then goes on to point out that either electric currents or -mechanically moved quantities of electricity, such as moving negative -electronic corpuscles, can give rise to just such magnetic effects, and -he suggests that these corpuscles are the true cause. He supposes that -the directive forces result in astatic combinations which find their -equilibrium when the latter have taken up their positions at the eight -corners of a cube or other elementary parallelepipedon of one of the -fourteen possible space-lattices, the positive atoms being encircled -spiral-wise by the negative electronic corpuscles in alternately -opposite directions. Such parallelepipeda would seek homogeneous -repetition by virtue of the fact of the corners exhibiting alternating -polarity. - -These theoretical ideas of Lehmann have naturally called forth much -discussion, criticism, and scepticism. But, so far, his experimental -facts have been fully substantiated by further investigation. Much more -practical work is urgently required, however, before the subject can be -considered as laid on a secure foundation. So much may be said, however, -that it is clear that we must concede the existence of a directive force -of crystallisation, and not be led by the pure geometry of the subject -of crystal structure to ignore facts of such interest and undoubted -importance as have been brought into prominence by the remarkable work -of Lehmann. - -A further interesting contribution has recently been made by -Vorländer[27] to the facts regarding the relationship between chemical -constitution and the formation of liquid crystals. It must have already -struck the reader that most of the substances which exhibit liquid -crystals are composed of a large number of chemical atoms, being either -long-chain compounds of the fatty acids or complex derivatives of the -hydrocarbon benzene, C_{6}H_{6}; also that many of the latter are “para” -compounds, that is, derivatives in which the substitution groups are -inserted in the benzene ring of six carbon atoms in the “para” position, -which is that at the opposite corner of the hexagon to the carbon atom -to which a substitution group has already been attached. This renders -the para compounds the most extended in a straight line of all the -benzene derivatives. Now Vorländer finds that a particularly favourable -condition for the production of liquid crystals is a linear structure of -the molecule. As the para substitution products of benzene derivatives -possess this elongated structure, many of them exhibit the development -of liquid crystals. The more linearly extended the structure becomes, -that is, the longer the straight chain of atoms is, the more favourable -become the conditions. The advent of a third substitution group, -however, which would have the effect of producing a kink in the chain, -or of bending it, appears to destroy the possibility of the production -of liquid crystals. This interesting observation may afford the key to -many of the extraordinary phenomena of liquid crystals which have been -described, and is undoubtedly one of prime importance. Further -favourable conditions for the formation of liquid crystals, according to -Vorländer,[28] are the aromatic character, and the presence of the -doubly-linked carbon and nitrogen groups C:C, C:N, and N:N, which are -usually so rich in energy. - -The idea of the formation of a specific crystalline homogeneous -structure, merely because the mechanical fitting-in of the molecules -occurs with the minimum of trouble or maximum of ease for this -particular type of all the 230 possible types, is certainly not -applicable to the case of Lehmann’s liquid crystals. With this, -moreover, is also connected the question of softness or hardness of -crystals, which was referred to at the opening of this chapter. For the -so-called liquid crystals are extreme cases of softness, and yet in -these cases the molecules must still be arranged in accordance with the -internal structure of a crystal, either parallel or enantiomorphously -definitely orientated with respect to each other, for otherwise it is -not possible to account for the optical properties resembling the -orientated ones of a crystal. Yet the condition being that of a liquid, -the molecules must be able readily to pass and roll over each other, and -hence cannot be at the close quarters where mere “fitting-in” comes into -play. - -Again, as has been pointed out earlier, many soft crystals, even such as -calcite, which are only relatively soft, attaining the position of as -much as four in the scale of hardness, readily exhibit the property of -being deformable upon glide-planes. The molecules in these cases have -been shown to undergo a movement which has two components, a -transference and a rotation, a fact which has been thoroughly -substantiated by optical investigations of the parts of the crystal -concerned before and after gliding. There cannot, therefore, have been -merely “fitting-in” of the molecules, but their orientated positions -must have been determined and maintained by the organising force, which -is probably purely physical and not chemical, but is nevertheless the -cause of crystallisation; it draws the molecules within a certain range -of each other, corresponding to and dependent upon the temperature, -causes or enables them to arrange themselves in the marshalled order of -the particular type among the 230 possible arrangements, and keeps them -at the same time from approaching nearer to each other than within these -prescribed limits corresponding to the temperature. It is doubtless -within these limits that gliding can occur parallel to such planes as -leave the molecules most room for the purpose, and which are directions -of least resistance. - -[Illustration: - - FIG. 119.—Lehmann’s Crystallisation Microscope arranged for - Projection. -] - -Connected with this important question is the principle enunciated by -Bravais, as a result of his discovery of the space-lattice, that -cleavage occurs most readily parallel to those net-planes of the -space-lattice which are most densely strewn with points. The force just -referred to, whether we term it cohesion or otherwise, is obviously at a -maximum within such a plane, and at a minimum in the perpendicular -direction where the points are further off from each other. Moreover, it -has been fairly well proved also, from the experiments of Wulff, -described in the last chapter, that the direction or directions of -maximum cohesion are those of slowest growth of the crystal; so that -faces parallel to those directions become relatively more extended owing -to the more rapid growth of other faces on their boundaries, and thus -become the most largely developed and confer the “habit” on the crystal. -All these are facts so important as evidences of a controlling force at -work in crystallisation, that a _purely_ geometrical theory of the -formation of crystals which would make “facility of fitting-in” of the -molecular particles its chief tenet, obviously does not tell us the -complete story. Hence the author desires to utter a warning against -going too far with the pure geometry of the subject. The geometricians -have done a grand work in providing us with the thoroughly well -established 230 types of homogeneous structures, as a full and final -explanation of the 32 classes of crystals, and so far their results are -wholly and unreservedly acceptable. - -The phenomena of “liquid crystals” lend themselves admirably to screen -demonstration, for which purpose an excellent improved form of the -crystallisation microscope of Lehmann, shown in Fig. 119, is constructed -by Zeiss, and its actual use in the projection, with the aid of the -well-known Zeiss electric lantern, but specially fitted for the purpose, -is shown in Fig. 120. - -A magnification of 600–700 diameters on the screen is very suitable, -employing a Zeiss 8–millimetre objective without eyepiece. This -objective affords directly a magnification of 30 diameters. For ordinary -eye observation an eyepiece magnifying 6–8 times is added, thus -affording to the eye a magnification of about 200 diameters. - -[Illustration: - - FIG. 120.—Zeiss Apparatus for the Projection of Liquid Crystals. -] - -The lantern is supplied with a self-feeding electric arc lamp, ensuring -a steady light. A collective lens of extra light-gathering power is -fitted in front as condenser, and from it proceeds a light-tight tube -provided with a water cell to filter out most of the heat rays which -accompany the light. The electric lantern with Brockie-Pell or Oliver -self-feeding arc lamp, shown in Figs. 71 and 79 (pages 186 and 202), is -also equally suitable, and with the water cell, and parallelising -concave lens removed from the large Nicol polariser, affords a parallel -beam of the same character as the Zeiss apparatus. The microscope stands -on a sole plate provided with levelling screws, and is naturally -employed in the vertical position for such work with fused substances. A -mirror inclined at 45° at the foot of the microscope directs the -parallelised rays from the optical lantern through the microscope, and -another above the optical tube reflects them to the screen. - -The heating apparatus consists of a form of miniature Bunsen burner -fitted with blowpipe blast, the respective pressures of gas and air -being regulated by means of two taps with graduated arcs for obtaining -greater delicacy of adjustment. The tabular plate seen to the left in -Fig. 119 is the graduated semi-circle of the two taps; below it is seen -the cylindrical mixing chamber for gas and air, in the event of the -necessity for using the Bunsen as a blowpipe. There are two separate -attachments for indiarubber tubes to this cylinder, conveying -respectively gas and air. Above the object stage a double air-blast is -fitted, each tube of which is hinged with a universal joint, so that it -can be readily adjusted to any desired position on either side of and -above the glass plate (supported on little metallic uprights) on which -the experiment is being conducted. A polarising Nicol prism and an -analysing Nicol, both constructed in a manner which protects them from -the effects of heat more effectually than is the case with the usual -form, are provided for obtaining the projections in polarised light. The -objective and analysing Nicol, as well as the substage condenser, are -also specially protected from injury by heat, by being surrounded with a -water jacket, supplied with running water, and a disc-like screen just -above the objective assists in deflecting the heat rays from the optical -tube and its Bertrand lens and other usual fittings. The miniature -Bunsen flame is usually brought about an inch below the object-plate, -and the size of the flame can be regulated with the utmost precision, so -that a fairly constant temperature can be obtained for a considerable -time. With the aid of the blowpipe air-blast temperatures up to 700° C. -can be safely employed. - -The microscope shown in position on the projection apparatus in Fig. 120 -is a still more recent form introduced by Zeiss, embodying several -further conveniences and improvements. - -The following substances lend themselves particularly well to projection -purposes. Para-azoxy-anisol with resin, which exhibits the phenomenon of -rotating drops; cholesteryl acetate, which affords a fine example of -spherical liquid crystals; paraazoxy-phenetol with resin, which gives -beautiful interference colours; and the acetyl ester of -para-azoxy-benzoic acid with resin, which shows the uniting of crystals -to form larger and larger individuals. - -Perhaps the most interesting and beautiful of all is cholesteryl -acetate, a characteristic field of which is shown in Fig. 121 on Plate -XVI., facing page 208. It is interesting that on this Plate XVI. there -are represented the very hardest and the softest of crystals, namely, -diamonds and liquid crystals. In order to obtain the finest effect the -heating and cooling should be carried out very slowly. The little Bunsen -burner, with a very minute flame, is first placed under the slide, and -allowed to act until the substance melts and forms a clear liquid. The -gas jet is then removed and the air-blasts, both of which are -simultaneously actuated when the tap controlling them is turned, are -very gently brought into operation, one on each side of the centre of -the slide, there being a good working distance of a quarter of an inch -or more between the slide and the objective. The cooling is thus brought -about very slowly. The Nicols should be crossed, and at this time the -field is quite dark, the liquid substance being at this temperature -(well above 114.5° the ordinary melting point) an ordinary singly -refractive liquid. - -As soon as the temperature has become reduced to that at which the -particular modification of cholesteryl acetate is produced which forms -liquid crystals, spots of light make their appearance at various points -in the field, and each expands into a beautiful circular and more or -less coloured disc marked by a rectangular sectorial black cross, which -latter is well shown in the illustration (Fig. 121). These beautiful -apparitions continue to occur, and each to expand to a certain size, -which is rarely exceeded, until the whole field becomes filled with the -wheels or crossed discs, the general effect very much in some respects -resembling that afforded by a slide of the well-known polarising -substance salicine. These discs, however, are liquid, being spherical -drops, of the structure already described and illustrated in Fig. 103, -and that this is so is at once made apparent on touching the cover-glass -with a pen-knife or other hard pointed substance, which immediately -causes them to become distorted. They recover instantly their shape -again when the pressure is removed. When the cooling, moreover, has -proceeded still further, there is a sudden change, and acicular solid -crystals shoot over the screen, tinted with all the colours of the -spectrum, until the field is full of them, the ordinary solid -modification of the substance having then been produced. The experiment -may be repeated with the same specimen of the substance, mounted on the -same slide, covered with the usual thin cover-glass, time after time for -months, at reasonable intervals. - -In concluding this chapter it may be mentioned that absolute proof of -the double refraction of the liquid crystals of several different -substances, derivatives of cinnamic acid, has been afforded during the -year 1910. For direct measurements have been carried out by two -independent investigators, Dorn and Stumpf, of the two refractive -indices corresponding to the ordinary and extraordinary rays in each -case, the crystals being uniaxial. - - - - - CHAPTER XVII - THE CHEMICAL SIGNIFICANCE OF CRYSTALLOGRAPHY. THE THEORY OF POPE AND - BARLOW—CONCLUSION. - - -Nothing in connection with the subject of crystallography is more -surprising than the neglect and apathy with which it has been for long -treated by the chemical world. That crystalline structure is intimately -related to chemical constitution will have been made abundantly plain -during the course of this book. Yet in spite of the great work of -Mitscherlich, essentially a chemist, and of a large amount of striking -work which has been steadily accumulating during the last thirty years, -with results of vital importance to chemistry, it is only at the -eleventh hour that chemists are really awakening to the vast -significance which crystal structure has for them. - -The explanation undoubtedly is, that the long interregnum of conflicting -investigations, doubt, and controversy, which followed the work of Haüy -and Mitscherlich, and preceded the beginning of really accurate and -painstaking investigation of an organised and systematic character, had -caused chemists to regard with more or less of indifference the work of -the crystallographers. Added to this we must remember that the subject -of crystallography has hitherto been taught, when taught at all, merely -as an appanage of mineralogy, although the pure chemical substances -which crystallise well infinitely outnumber the naturally occurring -minerals, and the results afforded by them frequently possess a much -greater value by reason of the purity of the substances and their more -definite chemical constitution. Also the mathematical and geometrical -side has usually been unduly emphasised, and carried on in lectures -without any practical goniometrical work at all. Moreover, the current -text-books have often proved forbiddingly full of calculations and -formulæ, and of the obsolete and unenticing symbols of Naumann. - -At last we have come to see that the subject is one of fascinating -interest when its study is commenced in a practical manner from the -beginning, armed from the very first lesson with the goniometer. The -crystal itself is then our main and highly interesting study; its -exterior form unravels itself in a most delightfully simple manner when -we follow the arrangement of its faces in zones on the goniometer -itself; and its symmetry becomes immediately patent to our eyes in all -ordinary simple cases, when we construct for ourselves its plan in a -stereographic projection, drawn at first in freehand while still at the -goniometer. The calculations also become perfectly simple when we have -learnt that only the simplest of the very easy formulæ of spherical -trigonometry are required, and which a knowledge of only elementary -plane trigonometry enables us to apply. Aided by a few very helpful -rules, such as those of Napier for calculating right-angled spherical -triangles, and the rule of the anharmonic ratio of four poles in a -zone—which, when the positions of three crystal faces of the zone are -known, at once enables us to calculate the situation of any fourth face -of the zone—we have at once a stock-in-trade which carries us over all -difficulties in the way of calculation, and relegates this side of the -work to an altogether subordinate position, although accuracy in -carrying it out is, of course, absolutely essential and even vital. - -Especial interest has recently been attracted on the part of chemists to -the bearing of crystallography on their science by a remarkable theory -which has been advanced by Pope and Barlow, connecting the internal -structure of crystals with chemical valency, the power which the atom of -a chemical element possesses of combining with the atoms of other -elements, and which is generally expressed by the number of atoms of a -monad element, such as hydrogen or chlorine, with which it is capable of -combining chemically. Thus the electro-positive metal potassium is said -to have monad valency because it is capable of combining with one atom -of the electro-negative element chlorine to form the salt potassium -chloride, KCl; calcium possesses dyad valency because it can unite with -two atoms of chlorine to form calcium chloride, CaCl_{2}; aluminium is -triadic as it can combine with three atoms of chlorine producing -aluminium chloride, AlCl_{3}; carbon is a tetrad because it can take up -four chlorine atoms, forming carbon tetrachloride, CCl_{4}, and -phosphorus a pentad as it can fix five with production of phosphorus -pentachloride, PCl_{5}; while sulphur is a hexad because it can take up -as many as six atoms of chlorine, forming sulphur hexachloride, SCl_{6}. - -Occasion has already been taken in Chapter XI. to refer to the able work -of Prof. Pope with regard to optically active carbon compounds, and in -Chapter IX. the important contribution of Mr Barlow to the completion of -the theory of the homogeneous partitioning of space has likewise been -discussed. In collaboration these two investigators have now propounded -a theory which connects the chemical and geometrical sides of -crystallography in a somewhat startling manner, which has naturally -aroused very considerable discussion, and which, whether right or wrong, -cannot fail to have the best results in attracting investigators to the -subject. - -Starting from the facts which have now been laid down in this book as -having been firmly established by the most careful measurement and -experimental investigation—notably the constancy of the interfacial -angles of the crystals of the same substance, the fixed positions of the -atoms or their spheres of influence in the molecule and in the crystal, -and the arrangement of the molecules in space-lattices and the atoms in -point-systems—Pope and Barlow assume that valency, the expression of the -relative combining power of the chemical elements, is a question of the -size of the sphere of influence of the atom of an element, and that the -relative sizes of such spheres of influence determine the modes in which -they can be packed, that is, the nature of the homogeneous crystal -structure which they can build up. The theory consequently renders the -chemical phenomenon of valency and the physical phenomenon of -crystalline form mutually interdependent. - -It will thus be apparent that the essence of their conception is that -the chemical molecule may be considered as made up of a number of -spheres corresponding to, and representing, the spheres of influence of -the atoms composing it, and that the volume of each sphere is roughly -proportional to the valency of the atom which it represents. They then -assume that the sum of the valencies of the atoms present in the -molecule may be substituted for the molecular volume, and the quantity -thus arrived at is termed by them the “valency volume.” By using the -valency volume instead of the molecular volume in the author’s formulæ -for calculating the molecular distance ratios, which have been shown in -Chapter X. to afford the relative dimensions of the unit cell of the -space-lattice, and the distances of separation of the centres of gravity -of the molecules from each other along the directions of the crystal -axes, Pope and Barlow arrive at new ratios, which they term equivalence -parameters. By the use of these latter they have attempted to account -for the crystalline structure of a number of substances—chiefly organic -compounds, in the investigation of which Prof. Pope has proved so -adept—which are connected morphotropically in the manner described in -Chapter VIII., and of others which are still less intimately connected. - -Unfortunately, the equivalence parameters do not make clear the -relationships in an isomorphous series, as do the molecular distance -ratios; for they are, from their very nature and mode of derivation, -almost identical for all the members of an isomorphous series, the -valencies of the interchangeable elements being the same. The molecular -distance ratios have also the great advantage of being derived from the -three measurements which have now been brought to the highest pitch of -experimental accuracy, namely, atomic weight determinations, density -determinations by the Retgers immersion method, and goniometrical and -physical measurements (optical and thermal) with instruments now -available of the utmost refinement. Structural constants thus derived -are obviously of especial value. It would thus appear that the theory -requires modification so as to take account of the experimentally proved -regular increase in volume and in the directional dimensions of the -structural unit cell of the crystal space-lattice, when one element of -the same family group and of the same valency is interchanged for -another. Indeed, as the theory stands at present it entirely ignores and -fails to offer any explanation of the highly important physical property -of density, specific gravity. That this physical constant, and the -equally important constant molecular volume, derived by dividing the -molecular weight by the density, possess a real and significant meaning -in isomorphous series, formed by the interchange of elements of the same -family group, has been clearly proved in Chapter X. This fact is, -indeed, so obvious that any further development of the theory must of -necessity take account of it. - -A precise statement of their conception has recently been made by Prof. -Pope in an excellent Report on the Progress of Crystallography, issued -by the Chemical Society early in the year 1909. He states that they -(Pope and Barlow) “regard the whole of the volume occupied by a -crystalline structure as partitioned out into polyhedra, which lie -packed together in such a manner as to fill the whole of that volume -without interstices. The polyhedra can be so selected that each -represents the habitat of one component atom of the material, and are -termed the spheres of atomic influence of the constituent atoms. Up to -this point no assumption is made other than that clearly indicated by -the result of crystallographic measurements, namely, that each atom -present in a crystalline structure exerts a distinct morphological -effect—or, what is the same thing, appropriates a certain definite -volume. The assumption is next made that the crystalline structure, -which is resolvable into individual molecules and ultimately into -individual atoms, exists as such by reason of equilibrium set up between -opposing attractive and repulsive forces operating between the component -atoms, and that this equilibrium results in the polyhedra representing -the spheres of atomic influence assuming shapes which are as nearly as -possible spherical.... The polyhedra thus arrived at may be regarded as -derived by compression of a close-packed assemblage of deformable, -incompressible elastic spheres,[29] the compression sufficing for the -practical extinction of the interstitial space. When such an assemblage -is released from pressure it is evident that in place of polyhedra, the -shapes of which approximate as closely as possible to the spherical, -closely packed spheres are presented; the distances between the sphere -centres can be substantially in the same ratios as the distances between -the centres of the corresponding polyhedra in the unexpanded mass, and -the equilibrium condition of maximum sphericity of the polyhedra will be -presented in the expanded mass of spheres by the existence of the -maximum number of contacts between spheres. The whole method of treating -the primary assumption thus resolves itself into finding close-packed -assemblages of spheres of various sizes representing by their relative -volumes the spheres of influence of the component atoms of any -particular crystalline structure.” - -Some very interesting evidence of the validity of their fundamental -assumption of spheres of influence of the component atoms as the -ultimate structural units is brought forward. They show that there are -two modes of closely packing equal spheres, which give rise respectively -to a cubic and a hexagonal crystal structure, the latter having a -specific axial ratio of the vertical to the three equal equatorial -horizontal axes; and that the chemical elements which are solids and -crystallise, and the structural units of which can naturally be assumed -to be equal spheres, being those of the similar atoms of the same -chemical element, do practically all crystallise either in the cubic -system or in the hexagonal system with the specific axial ratio -indicated by them. The theory as it concerns chemical valency is -obviously not affected by these interesting facts, as the spheres of -influence present are those of the identically similar atoms of the same -element. But the theory has received considerable support from the -results of the investigation of a number of carbon compounds, chiefly -derivatives of benzene. - -Thus, in spite of the very wise decision of Barlow at the time of -developing his theory of the homogeneous partitioning of space, to keep -quite clear of attributing shape to the structural unit atoms or -molecules, and to consider them as points, he, in common with the other -contributors to that splendid geometrical work, appears driven to -consider the question of shape when, in collaboration with his chemical -colleague, he endeavours to apply his geometrical results to the -practical problems of chemistry. It may be inevitable that we cannot get -away from the idea of shape of the fundamental structural units. Yet the -moment we do admit the idea, and begin to talk of polyhedra, or even of -spheres, in close or any other packing, we enter the debatable land, -concerning which the experimental evidence is as yet but shadowy and -liable to many interpretations. Hence it is that we have the -parallelohedra of Von Fedorow, having volumes proportional to the -molecular volumes, the more general plane-faced cell of fourteen faces, -the tetrakaidecahedron of Lord Kelvin and its deformed derivatives, and -now the polyhedra of Pope and Barlow. The more indefinite -“Fundamentalbereich” of Schönflies appears to be left behind, and we -have embarked on a definite course of attributing shape to the component -atoms or their regions of influence in the crystal structure. Von -Fedorow has developed his particular theory in a very masterly manner, -and with the aid of it professes, and with very considerable success in -many cases, to determine the correct mode of setting up a crystal for -truly comparative descriptive purposes, and has derived therefrom a -remarkable method of crystallochemical analysis. - -Moreover, there is yet another view, that of Sollas,[30] that the -packing of the molecules is a more open one altogether, a view to which -he has been guided by consideration of the molecular volume. Sollas has -offered some remarkable explanations of crystal structure, notably in -the case of the dimorphous forms of silver iodide. The abnormal -contraction which occurs on heating this interesting substance, and its -sudden transformation at 146° from the ordinary hexagonal into a cubic -modification as discovered by Lehmann, appear capable of very clear -explanation on the basis of his theory. According to this theory of -Sollas the volumes of the spheres of influence of the atoms of the -different elements of the same family group, such as those of the group -of alkali metals or those of the halogens, chlorine, bromine, and -iodine, vary progressively in a manner which is dependent on the atomic -volumes of the elements, which have a real comparative significance when -the elements belong to the same family group. - -It is probable that there is a considerable substratum of truth behind -these various apparently conflicting views, and what is now required is -that the germ of real fact shall be winnowed from the husk of fallacious -speculation, just as occurred in the happy recent settlement of the old -issue between Haüy and Mitscherlich. As in that case, moreover, it will -be experimental work of superlative accuracy which can alone offer the -desirable evidence on which a satisfactory arbitration can be founded. - -It is thus obvious that we have now arrived at a stage in the history of -crystallography when more experimental data, and many more measurements -of the most carefully conducted character, on pure materials and -excellently developed crystals, are most urgently needed, in order to -decide these important, indeed fundamental, questions, the present state -of which the author has endeavoured to present with judicial -impartiality. When one looks around, and sees the almost complete lack -of opportunities for the training of investigators in this rapidly -growing branch of science, the importance of which to chemistry and -physics is increasing every day, while the field is ripe for the -harvesters, one is inclined to feel depressed with the thought of the -opportunities which are being lost. Our country has, in this science at -any rate, a fine record, having with few breaks led the van of progress -from the time of Wollaston, the inventor in the year 1809 of the -reflecting goniometer, and of Miller, the originator of our method of -describing crystals and the pioneer of accurate experimental work, down -to the present day. It may be, also, that our country’s reputation is -safe at this moment. But it is in the hands of a band of investigators -so small, and often of the private and not professional nature, carrying -on the work for sheer love of it and deep interest in it, that the -wonder is that so much has been done, and it is the provision for -carrying on our national tradition of leadership in crystallography in -the future that is a matter for the deepest concern. - -If the perusal of this book should have awakened sufficient interest in -the minds of some of its readers to prompt them to offer themselves as -recruits to this small band of investigators, and especially if it -should have inspired the zeal and enthusiasm of a few young students -looking around for a promising and fascinating field of work, and, -finally, if it should prove to be of assistance in obtaining the means -of training such recruits with the help of the best and most accurate -experimental apparatus which can be obtained, the author’s main objects -in writing it will have been attained. - - - - - INDEX - - - Airy’s spirals, 197 - - Alpine snow-field, 49 - - Alum, ammonium iron, 43; - cæsium, 42; - potash, 5, 16, 17, 77; - vicinal faces of potash, 248, 251 - - Amethyst quartz, 222–229; - as example of pseudo-racemism, 231, 232 - - Ammonium chloride labile crystallisation, 248; - iron alum, 43; - NH_{4} group, and its isomorphism with alkalies, 82, 83, 131; - oleate, liquid crystals of, 259–263; - oleate, twins of, 262 - - Ampère’s researches on ammonia, 83 - - Anatase, crystal of, 37 - - Antimony oxide, dimorphism of, 87, 88 - - Apatite, crystal of, 37 - - Aragonite, biaxial interference figure of, 189, 190 - - Armstrong and Pope on sobrerol, 153 - - Arsenic oxide, crystals of, 5, 88 - - Astatic systems of molecular magnets of Lehmann, 271, 272 - - Asymmetric carbon atom, 144, 145 - - Axes and axial planes of crystals, 51, 55, 56 - - Axial ratios, 129 - - - Babinet’s double-wedge quartz plate, 219 - - Bacteria, destruction of one enantiomorphous form by, 149 - - Barium nitrate, 152 - - Barker, researches on perchlorates, 123 - - Barlow, discovery of remainder of 230 point-systems, 118, 119, 140; - and Pope’s theory, 285–292 - - Bartolinus, Erasmus, 17 - - Benzoic acid, screen experiment on crystallisation of, 229–231 - - Bergmann and Gahn’s laws of cleavage, 18 - - Berzelius and Mitscherlich, atomic weights and isomorphism, 80, 82, 83, - 85 - - Beudant’s researches on the vitriols, 75, 76 - - Biaxial crystals, 60°-prisms and refractive indices of, 162, 163, 184; - optic axes of, 185; - optic axial angle of, 191 - - Biot’s researches on tartaric and racemic acids, 143 - - Biquartze, natural and artificial, 181, 211, 214, 216, 217 - - Black band of quartz twins, 216–218 - - Boisbaudran, Lecoq de, 238 - - Boyle, Robert, 17 - - Brauns, 254 - - Bravais, 114; - space-lattices, 114 - - Brookite, 91 - - - Cæsalpinus, 15 - - Cæsium alum, 42 - - Calcite, crystals of, 10–13, 38; - amount of double refraction of, 174; - in rock sections, 177, 178; - uniaxial interference figure of, 189, 190; - plate perpendicular to axis of, 209; - refractive indices of, 174; - 60°-prism of, 165, 168 - - Calcium carbonate, three habits of crystals of, 11 - - Calcium dextro-glycerate, optical activity and crystal form of, 155–160 - - Carangeot’s contact goniometer, 18–20 - - Carbonate, calcium, 11, 87; - potassium sodium, 44 - - Carbonates of alkaline earths, 74 - - Carbon, dimorphism of, 137 - - Carbon dioxide, liquid in quartz cavities, 46 - - Cavendish, 23 - - Chemical significance of crystallography, 73, 283, 284; - valency and crystalline form, 285–291 - - Cholesteryl benzoate, liquid crystals of, 258; - acetate, spherical liquid crystals of, 281, 282 - - Chromate, potassium, 84 - - Chromates and manganates, isomorphism with sulphates, 84 - - Classes, the 32 crystal, 6, 33 - - Cleavage, 17; - and glide-planes, 275, 276 - - Cobalt sulphate, 78 - - Conditions for growth of crystals, 240, 244, 245 - - Constancy of crystal angles, 6, 13, 14, 17, 23, 132 - - Convergent light experiments, 186, 188 - - Copper sulphate, 40, 76, 78 - - Coppet, de, 238 - - Crookes, Sir William, 138, 208 - - Crossed-axial-plane dispersion of optic axes, 89, 94, 95; - of ethyl triphenyl pyrrholone, 106, 107, 108; - of gypsum, 89 - - Crystal, definition of, 4; - germs and their influence on crystallisation, 236, 237 - - Crystals, modes of formation of, 4, 5 - - Cube and its perfection of symmetry, 38; - axes of, 52 - - Cubic system, 37 - - Cyanide, potassium cadmium, crystals of, 42 - - - Dalton, 25 - - Dark field of polariscope, 188, 202 - - Davy, Sir Humphry, researches on ammonia, 82, 83 - - Delafosse and morphotropy, 101 - - Deville and Troost’s researches on tantalum chloride, 84, 85 - - Diamond, 8, 137, 138, 207, 208 - - Dibenzal benzidine, liquid crystals of, 264 - - Digonal axis of symmetry, 36 - - Dimensions of structural parallelepipeda, 129 - - Dimorphism, 79, 87; - of antimony oxide, 87, 88; - of carbonate of lime, 79, 87; - of mercuric iodide, 97; - of sodium dihydrogen phosphate, 80, 81; - of sulphur, 86; - of vitriols, 79 - - Directive molecular force in crystallisation, 139, 269, 274–277 - - Dog-tooth spar, 10–12 - - Double refraction, interference colours due to, 176, 177; - measure of, 169, 170; - of biaxial crystals, 184; - of uniaxial crystals, 183 - - Double sulphates and selenates, 35, 79, 121, 127, 128, 132 - - Dulong and Petit’s law, 84 - - - Electronic corpuscles, constituents of atoms, 113, 124 - - Elements of a crystal, 68, 69 - - Enantiomorphism and optical activity, 140; - 11 classes showing, 150 - - Epsom salts, 76, 78 - - Ethyl triphenyl pyrrholone, 105, 106 - - Eutropic series, definition of, 132 - - External molecular compensation, 134, 234 - - Extinction directions, 204, 205 - - - Fedorow, von, discovery of remainder of 230 point-systems, 118, 119, - 140; - theory of, 291, 292 - - Ferricyanide, potassium, 44 - - Ferrocyanide, potassium, 43 - - Ferrous sulphate, 76, 78 - - Fletcher, indicatrix of, 184 - - Fluorspar, single refraction of, 206 - - Form, definition of a, 11, 60, 61 - - Frankenheim on morphotropy, 101; - discovery of space-lattices, 114 - - Frankland and Frew, 155 - - Fuchs, von, researches on sulphates of barium, strontium, and lead, 75, - 77 - - Fuess reflecting goniometer, 64–66 - - Fundamentalbereich of Schönflies, 112, 113 - - - Gattermann, 258 - - Gay-Lussac on alums, 77; - on ammonia, 83; - on racemic acid, 142 - - Gernez, 238 - - Gessner, 14 - - Gmelin, researches on racemic acid, 142 - - Goniometer, contact, 19, 20; - reflecting, 64–66 - - Graphite, 137 - - Groth, von, morphotropic researches, 98, 102–104 - - Growth of a crystal, 1; - from solution, 237–254 - - Guglielmini, researches on crystal structure, 17 - - Gypsum (selenite), cleavage of, 203; - crystals of, 14; - extinction directions of, 204; - 60°-prism, experiment with, 163, 168; - twins of in polarised light, 205, 206 - - - Habit of crystals, 12, 13 - - Hardness of crystals, 255, 274 - - Hatchett’s discovery of columbium, 85 - - Haüy, 22; - and Mitscherlich, 70, 75, 77, 88, 132; - fundamental forms, 22, 23; - law of constancy of form, 23; - lattices, 29; - law of rational indices, 24, 29, 30; - modernisation of theories of, 30–32; - molécules intégrantes, soustractives, and élémentaires, 25–28; - structural units, 24 - - Hemihedral classes of crystals, 34 - - Hexagonal axes of symmetry, 36; - prism, 11; - system, 36, 53 - - Hexakis octahedron, 39; - indices of, 61 - - Hjortdahl and morphotropy, 101 - - Holohedral classes of crystals, 34 - - Homogeneity, 6, 16; - definition of, 114 - - Homogeneous structures, the 230 types of, 6, 111 - - Hooke, Robert, 16 - - Huyghens, 17; - discovery of laws of double refraction, 17; - investigation of calcite, 17 - - - Ice, crystalline form of, 47 - - Iceland spar, discovery of, 17; - rhombohedron of, 10–12 - - Inactive tartaric acid, 144 - - Inactivity, true optical, 235 - - Indicatrix of Fletcher, 184 - - Indices of crystal faces, 11 - - Intercepts on crystal axes, 58 - - Interference colours due to double refraction, 177; - figures of biaxial and uniaxial crystals, 189–191 - - Internal structure of crystals, 15, 111–120 - - Iodide of mercury, dimorphism of, 97 - - Isomerism, chemical and physical, 142, 143 - - Isomorphism, Mitscherlich’s conferment of the term, 81; - doctrine of, 81, 82; - limitations of, 85; - recent clearer definition of, 121–132 - - Isotropic crystals, 206 - - - John of Berlin’s discovery of racemic acid, 142 - - - Kipping and Pope, definition of racemism and pseudo-racemism, 153, 154 - - Kopp, 100 - - Kundt’s powder, 149 - - - Labile solutions, 241 - - Laurent and Nickle’s organic researches, 99 - - Lavoisier, 23 - - Law of rational indices, 24, 50, 57, 59 - - Le Bel and van t’Hoff’s explanation of optical activity, 151 - - Le Blanc, researches on alums, 77 - - Lehmann, researches on liquid crystals, 256–282 - - Liquid crystals, 255–282; - and polymorphism, 138, 139; - list of substances forming, 280, 281 - - - Magnesium sulphate, 76, 78 - - Mallard and Le Chatelier on silver iodide, 256 - - Manganate, potassium, 96 - - Manganates, 84, 96 - - Manganese sulphate, 78 - - Marignac and isomorphism of tantalum and niobium compounds, 84; - and morphotropy, 101 - - Mercury iodide, dimorphism of, 97 - - Metastable solutions, 240 - - Methyl triphenyl pyrrholone, 105, 106 - - Mica-sectors plate for testing sign of optical rotation, 212, 213 - - Microscope, Lehmann’s crystallisation, 256, 257, 276–280 - - Miers, H. A., researches on crystallisation, 238–243; - on red silver ores, 109; - on vicinal faces, 248–254 - - Millerian indices, 57 - - Mirror-image symmetry, 118, 119, 134, 135; - illustrated by quartz, 231 - - Mitscherlich, experiment with gypsum, 90–94; - work of, 70–97 - - Mixed crystals, 77, 86 - - Molecular compound, racemic acid a, 150 - - Molecular volume and distance ratios, 129, 130 - - Molecule, individuality and directive force of, 139, 269 - - Monochromatic illuminator, 192, 193 - - Monoclinic system, 39; - axes and axial planes of, 53, 54 - - Morphotropy, 98–104 - - Muthmann, researches on permanganates, 123 - - - Naphthalene tetrachloride, 99 - - Newton’s seven orders of spectra, 177 - - Nickel sulphate, 76, 78 - - Nicol prism, 174, 175, 187 - - Nitrobenzenes, von Groth’s researches on, 104 - - Noble, Sir Andrew, experiments on liquefaction of carbon, 138 - - - Optical activity and mirror-image symmetry, 141; - antipodes, characters of, 152; - characters of crystals, 7 - - Optically active classes of crystals, 150, 151 - - Optic axes of biaxial crystals, 185; - axis of uniaxial crystals, 165; - axial angle, 191; - axial angle of ethyl triphenyl pyrrholone for different wave-lengths, - 107, 191; - axial angle of gypsum at different temperatures, 90–94, 191 - - Ostwald’s predictions of crystallisation phenomena, 238, 240 - - Oxides of arsenic and antimony, isodimorphism of, 88 - - - Para-azoxy-anisol, liquid crystals of, 259, 265; - -benzoic acid, 263; - -cinnamic acid, 263; - -phenetol, 258, 259 - - Parametral form, 56 - - Pasteur’s law, 155; - research on morphotropy of tartrates, 100; - research on tartaric and racemic acids, 142–150 - - Penfield’s diagram of spherical projection, 61, 62 - - _Penicillium glaucum_, destruction of dextro component of racemic acid, - 148 - - Perchlorates and permanganates, isomorphism of, 84, 96 - - Pfaff, 254 - - Phases, different solid, 137 - - Phenol and resorcinol, von Groth’s researches on, 103 - - Phosphate, ammonium magnesium, 43; - sodium dihydrogen, dimorphism of, 80, 81 - - Phosphates and arsenates, isomorphism of, 73, 74 - - Phosphorus, dimorphism of, 138 - - Photomicrographs of growing crystals, mode of obtaining, 41 - - Pistor’s goniometer, 89 - - Planeness of crystal faces, 6, 7 - - Polarisation colours due to optical activity, 179 - - Polarisation, rectangular, of spectra from doubly refractive prisms, - 164 - - Polariscope, the, 187, 188 - - Polymorphism, 133–137 - - Pope and Barlow, theory of, 285–291 - - Positive and negative uniaxial and biaxial crystals, 166 - - Potassium bichromate, metastable and labile crystallisation of, 246, - 247; - manganate, 96; - nickel sulphate, 35; - selenate, 96; - sulphate, 13 - - Priestley, 23 - - Primitive form of Romé de l’Isle, 20, 27 - - Projection polariscope, for convergent light, 92, 186, 188, 189; - for parallel light, 201, 202 - - Progressive change of crystal angles in isomorphous series, 125; - of double refraction, 126; - of molecular distance ratios, 130, 131; - of position of optical ellipsoid, 128 - - Propyl triphenyl pyrrholone, 106, 108 - - Proustite, 109, 110 - - Pseudo-racemism, 153, 154 - - Pyrargyrite, 109, 110 - - Pyroelectrical properties of crystals, 149 - - - Quartz, 170; - crystalline form of, 171, 172; - crystals in rock sections, polarisation colours of, 175–178; - crystals, liquid cavities in, 45; - crystals on sand grains, 2, 3; - double refraction of, 174; - interference figure in convergent polarised light, 194–196; - optical activity of, 173, 179–181, 210; - plates, preparation of for polariscope, 181; - polarisation colours of due to optical activity, 209, 210; - refractive indices of, 174; - screw point-systems of, 151; - 60°-prism experiment with, 165, 168; - Steno’s research on, 16; - twinning of, 215, 216, 219–221, 225; - two varieties of as examples of mirror-image symmetry, 171 - - - Racemate, sodium ammonium, Pasteur’s researches on, 147, 148 - - Racemic acid, 142–146, 233, 234 - - Racemic forms and racemism, 150, 153, 233 - - Rammelsberg and morphotropy, 101 - - Rational indices, 24, 50, 59, 116 - - Reflection of light by crystal faces, 8 - - Refractive index, meaning of, 167 - - Reinitzer, 258 - - Reusch’s artificial quartzes, 198, 199 - - Rhombic system, 39; - axes and axial planes of, 52, 54 - - Rhombohedron and its axes, 10, 11, 54, 55, 172 - - Rings and brushes, optic axial, 190, 191 - - Rock-salt, cube of in quartz cavity, 45; - 60°-prism experiment with, 162, 168 - - Romé de l’Isle, 18–20; - researches on alums, 77 - - Roozeboom, 238 - - Rotation of plane of polarisation by quartz of two varieties and - different thickness, 180, 210, 211 - - Royal Institution experiment with diamonds, 8 - - - Sal-ammoniac, 82, 83 - - Salol and betol, Miers’ researches on, 242, 243 - - Sand grains with quartz crystals, 3 - - Scalenohedron of calcite, 11 - - Scheele’s discovery of tartaric acid, 142 - - Schlippe’s salt, 45 - - Schönflies, discovery of remainder of 230 point-systems, 118, 119, 140 - - Seebeck, researches on ammonia, 82, 83 - - Selenates, isomorphous with sulphates, 96; - of alkalies, 121 - - Selenic acid, 95 - - Selenite, polarisation colours of films of, 203 - - Selenium, discovery by Berzelius, 96 - - Sella’s warning against hasty generalisation, 102 - - Senarmontite, 88 - - Silver iodide, 256, 292 - - Single refraction of cubic crystals, 162, 183 - - Snow crystals, 49 - - Sobrerol, 153 - - Sodium chlorate, 151; - sulphantimoniate, 45 - - Sohncke, regular point-systems, 117; - mirror-image molecular arrangement, 141; - two point-systems of quartz, 151 - - Sollas, crystal structure, theory of, 292 - - Solubility and supersolubility, 238–241; - curves of, 240 - - Solutions, optical activity of, 151; - metastable and labile, 240, 241 - - Space-lattices, 50, 114–116; - triclinic illustration of, 115 - - Specific gravity, importance of determinations of, 129 - - Spheres of influence of atoms, 113 - - Steno, 16 - - Stereographic projection, 34, 62, 67; - of topaz, 68; - of double sulphates, 35 - - Stereometric arrangement of atoms in molecule, 124, 136 - - Story Maskelyne, 34 - - Strain, polarisation colours of glass and diamond due to, 207, 208 - - Structural units of crystals, 24, 111–113 - - Sulphantimoniate of sodium, 45 - - Sulphate, ammonium magnesium, 44 - - Sulphates of alkalies, 121; - of barium, strontium and lead, 74 - - Sulphur, dimorphism of, 86, 87, 137; - monoclinic form of, 4, 86 - - Symbol of a face or form, 57 - - Symmetry, axes and planes of, 34, 36, 41, 55; - elements of, 34 - - Systems, the crystal, 6, 7, 33 - - - Tartaric acid, 142–148; - dextro, 143, 146–148; - lævo, 144, 146–148; - pyro-electrical properties of, 149 - - Tartrate, hydrogen potassium, 43 - - Tetartohedral classes of crystals, 34 - - Tetragonal system, 37; - axes and planes of, 36, 52 - - Thallium, relation of to alkali metals, 131 - - Thénard and Gay-Lussac, research on ammonia, 83 - - Thomson, J. M., 238 - - Thomson, Sir J. J., discovery of composition of atoms, 112 - - Topaz, crystal of, 40; - stereographic projection of, 68 - - Transition tint, 180 - - Triclinic bipyramid, 56; - system, 40; - axes and planes of, 53, 54 - - Trigonal system, 37; - axes and planes of, 37, 53, 55 - - Triphenyl pyrrholone derivatives, 105–108 - - Triple tartrate of sodium, potassium, and ammonium, 91 - - Tutton, fixed positions of atoms in crystals, 122–124; - law of progression of crystal properties in isomorphous series, - 121–123, 129; - nature of structural units, 134, 135; - research on calcium dextro-glycerate, 157, 160; - researches on simple and double sulphates and selenates, 121; - researches on triphenyl pyrrholones, 105–108 - - - Uniaxial crystals, optic axis of, 165; - 60°-prisms of, 164; - two refractive indices of, 166, 183 - - - Valentinite, 88 - - Vanadium family group of elements, 85 - - Vauquelin, researches on alums, 77 - - Vicinal faces, 248–254 - - Vitriols, the, 15, 76, 78 - - Von Lang, 23, 34 - - Vorländer, 263; - nature of molecules forming liquid crystals, 273 - - - Water, exceptional thermal dilatation of, 47 - - Water flowers in ice, 47 - - Water of crystallisation, 76, 78, 79 - - Wave-length of most luminous part of spectrum, 179 - - Werner’s fundamental form, 20 - - Westfeld, 18 - - Weyberg, 250 - - White of higher orders, 178 - - Wollaston’s reflecting goniometer, 63, 81; - work on carbonates and sulphates of barium, strontium and lead, 74, - 75 - - Wulff, 250, 252, 253 - - - Zinc sulphate, 76, 78 - - Zone of crystal faces, 63; - circle, 63 - - - - - PRINTED BY - TURNBULL AND SPEARS, - EDINBURGH - ------ - -Footnote 1: - - See page 57 for explanation of indices. - -Footnote 2: - - _Proc. Roy. Soc._, 1908, A, 81, 40. - -Footnote 3: - - _Comptes Rendus_, 1842, 15, 350, and 1845, 20, 357. - -Footnote 4: - - _Comptes Rendus_, 1848, 27, 611, and 1849, 29, 339. - -Footnote 5: - - _Jahresbericht_, 1849, 19. - -Footnote 6: - - _Comptes Rendus_, 1848, 26, 535. - -Footnote 7: - - _Journ. für Prakt. Chemie._, 1865, 94, 286. - -Footnote 8: - - _Mem. R. Accad. di Torino_, 2A, 17, 337, and 20, 355. - -Footnote 9: - - _Pogg. Ann._, 141, 31. - -Footnote 10: - - _Mineralogical Magazine_, 1888, 8, 37. - -Footnote 11: - - _Journ. Chem. Soc._, 1896, 69, 507. - -Footnote 12: - - _Ann. de Chim. et Phys._, 1848, 24, 28 and 38; also 1850, 28, 56; - _Comptes Rendus_, 1848, 26, 535; also 1849, 29, 297; also 1850, 31, - 480; also 1853, 37, 162, and 1858, 46, 615. - -Footnote 13: - - _Journ. Chem. Soc._, 1891, 315. - -Footnote 14: - - _Ibid._, 1897, 989. - -Footnote 15: - - _Journ. Chem. Soc._, 1891, 59, 233. - -Footnote 16: - - _Ibid._, 1891, 59, 96. - -Footnote 17: - - _Phil. Trans. Roy. Soc._, 1895, A, 185, 913. - -Footnote 18: - - _Mineralogical Magazine_, 1892, 10, 123. - -Footnote 19: - - _Journ. Chem. Soc._, 1906, 89, 413; _Proc. Roy. Soc., A_, 1907, 79, - 322. - -Footnote 20: - - “Lehrbuch der Allgemeinen Chemice,” vol. 2, part 2, p. 780. - -Footnote 21: - - _Phil. Trans._, 1903, A, 202, 459. - -Footnote 22: - - _Phil. Trans._, 1903, A, 202, 519. - -Footnote 23: - - _Zeitschr. für Kryst._, 1901, 34, 449. - -Footnote 24: - - _Loc. cit._, p. 531. - -Footnote 25: - - _Sitzungsber. d. Physik. Med. Soc._, Erlangen, 1878, 10, 59. - -Footnote 26: - - _Neues Jahrbuch_, 1887, 138. - -Footnote 27: - - _Ber. der deutsch. Chem. Ges._, 1907, 40, 1970. - -Footnote 28: - - _Zeitschr. f. Phys. Chemie._, 1907, 57, 357. - -Footnote 29: - - This apparent contradiction in terms the author takes to mean that, - whatever compression is produced by pressure on one part of the - sphere, is counterbalanced by a corresponding protuberance produced in - the part not confined under pressure, the total volume being - incompressible. - -Footnote 30: - - _Proc. Roy. Soc._, 1898, 63, 270, 286, and 296; 1901, 67, 493. - ------------------------------------------------------------------------- - - - - - TRANSCRIBER’S NOTES - - - 1. Silently corrected obvious typographical errors and variations in - spelling. - 2. Retained archaic, non-standard, and uncertain spellings as printed. - 3. Re-indexed footnotes using numbers and collected together at the end - of the last chapter. - 4. Enclosed italics font in _underscores_. - 5. Denoted superscripts by a caret before a single superscript - character or a series of superscripted characters enclosed in - curly braces, e.g. M^r. or M^{ister}. - 6. 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