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CRYSTALS AND THE FINE-STRUCTURE 
 OF MATTER 
 
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CRYSTALS AND THE 
 FINE-STRUCTURE 
 OF MATTER 
 
 FRIEDRICH RINNE 
 
 PROFESSOR OF MINERALOGY IN THE UNIVERSITY OF LEIPSIC 
 
 TRANSLATED BY 
 
 WALTER 5S. STILES 
 
 WITH A DRAWING BY A, DURER, AND PORTRAITS OF THE LEADING INVESTIGATORS 
 IN THE STUDY OF FINE-STRUCTURE AND 202 FIGURES 
 
 NEW YORK 
 E. P. DUTTON AND COMPANY 
 PUBLISHERS 
 
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 PREFACE 
 
 to the scientific publica yearago. The book 
 
 set out to consider experiments on the fine- 
 structure of matter from a new and essentially 
 crystallographic standpoint, corresponding to its 
 title ‘‘Crystals as types of the fine-structure of 
 matter.”’ 
 
 The friendly communications I have received, the 
 reviews in the press, and the fact that translations 
 of the book have been prepared, will testify to the 
 good reception given to the work by the public. 
 The demand for the first edition being so great, I 
 have thought it well to extend the work consider- 
 ably for the new issue. 
 
 Historical details are introduced, the treatment 
 of crystallography is amplified, tabular summaries 
 and sections on the atom domain and _ stereo- 
 chemical axes are added. Instructive cases of poly- 
 morphism are also described. In addition, the book 
 contains many new explanatory figures, and portraits 
 of some leading scientists. By these changes the 
 work is not, I think, increased in size by more than 
 half of its previous bulk, without at the same time 
 being enriched by many glimpses into the interesting 
 relations of the science of fine-structure. 
 
 The need for a treatment of the subject easily 
 
 Vv 
 
 a oer first edition of this work was submitted 
 
 67055 
 
"i CRYSTALS AND MATTER | 
 
 understood by the general reader is, however, con- 
 stantly observed. For those more closely interested 
 a list of text-books on crystals has been included. 
 An index is appended. 
 
 I hope that this new and much enlarged edition, 
 the preparation of which has been a labour of love 
 for me, will help, as did its predecessor, both by its 
 text and diagrams, in placing in the right light the 
 physical, or, rather, the natural philosophical side of 
 crystallographic science, and in acquiring new friends | 
 for the very fertile study of the fine-structure of 
 matter. 
 
 To these preliminary remarks I add my thanks 
 for the advice offered to me by the reviewers, whose 
 suggestions have been willingly followed, together 
 with the expression of my gratitude to my assistants, 
 Dr. K. H. Scheumann and Dr. E. Schiebold, for 
 friendly help in the editing of this new edition. The 
 Saxon Academy of Sciences facilitated the publica- 
 tion of the book by the loan of a large number of 
 printing blocks. 
 
 INSTITUTE OF MINERALOGY AND PETROGRAPHY 
 IN THE UNIVERSITY OF LEIPSIC 
 IN THE SPRING OF 1922 
 
 NOTE 
 
 THE translator wishes to tender his best thanks to 
 Professor Rinne for very kindly reading through the 
 proofs, and offering many helpful suggestions. Most 
 of these have been adopted and have materially 
 assisted in the work of translation. 
 
 W. OAS bes 
 
CONTENTS 
 
 CHAP. PAGE 
 PREFACE . ; : : ; é ; : : Vv 
 
 I. INTRODUCTION . : . ; ; : I 
 II. THE STUDY OF FINE-STRUCTURE (LEPTOLOGY) . : 5 
 III. CRYSTALLOGRAPHY AND LEPTOLOGY 6 
 
 t, Historical,.p.6. 2. The Laue Effect, PA LD ise 
 M. v. Laue, p. 13. 4. Further X-ray methods, 
 p.16. 5. X-ray results in the domain of chemis- 
 try, p. 20. 6. Crystals as stereochemical types, 
 p. 25. 7. Outlines of general crystallographic 
 morphology, p. 28. 8. Extension of the mani- 
 fold of crystallographic types by twin formation, 
 Pp. 35: 
 
 IV. FINE-STRUCTURAL UNITY OF MATTER ‘ vty) es (8) 
 
 1. Fine-structure of amorphous bodies Ged 
 with that of crystals, p. 39. 2. Physical investi- 
 gations on the general forms of atoms and mole- 
 cules, p. 43. 3. Atom domains, p. 46. 4. Differ- 
 ence between the structure of individual leptons 
 and crystals, p. 49. 
 
 V. THE GENERAL CHARACTERISTICS OF THE FINE-STRUCTURE 
 OF MATTER ; . ‘ ; : ; ad eo 
 
 y. Crystals, p. 54. 2. Gases and liquids, p. 59. 
 
 3. General character of the fine-structure, p. 60. 
 VI. THE SERIES OF TRANSFORMATIONS OF MATTER . 61 
 
 1. States of aggregation ; gases, liquids and Henna 
 
 crystals, crystals, p. 61. 2. Discontinuities of 
 
 lower order in theseries. Polymorphism. Enan- 
 
 tiomorphy, p. 69. 3. Review and unified con- 
 
 ception, p. 76. 
 
 VII. GENERAL TECTONIC ARRANGEMENT IN THE FINE-STRUC- 
 TURE OF CRYSTALS : 78 
 
 1. Analysis of een emia! Ties p. 8, 
 
 2. Valency, p. 80. 3. Molecules in the crystal. 
 
 Structural groups (leptyles}. Lattice types, 
 
 p. 87. 
 VIII. ASSOCIATION OF THE FINE-STRUCTURAL PARTICLES IN 
 MIXED CRYSTALS AND OUTGROWTHS ON CRYSTALS . 97 
 
 1. Transitions between chemical combination and 
 physical mixture. Mixed crystals, p. 97. 2. 
 Physico-chemical significance of the mixed 
 crystal, p. 98. 3. Out-growths with substances 
 not isomorphous, p. 102. 
 
 b vii 
 
Vill 
 
 CHAP. 
 
 IX. 
 
 Xi, 
 
 XII. 
 
 >, bee 
 
 XIV. 
 
 XV. 
 
 CRYSTALS AND MATTER 
 
 MORPHOTROPY . : : : : , : 
 1. Historical, p. 105. 2. Stereochemical axes, 
 p. 106. 3.:Morphotropic constructions, p. 107. 
 
 4. Examples from the mineral world, p. 111. 
 
 . ISOTYPY 
 
 1. Crystal Pans Eyota pastes rae Sryeees foun 
 as stability types, p. 115. 2. Fine-structure 
 groups as stability types, p. 116. 
 
 CRYSTAL GROWTH AND SOLUTION : ; ; 
 
 1, Pure crystal growth, p. 121. 2. Mixed crystal 
 growth, p. 128. 3. Collective crystallisation, 
 p.131. 4. Crystal solution, p.133. 5.Summary, 
 Pp. 137. 
 
 CHEMICAL ACTIONS ON CRYSTALS 
 
 1. Anisotropy of chemical actions on crystals, D- 139. 
 2. Anisotropic chemical reactions of molecules, 
 p-140. 3. Structural rigidity of electrons, atoms, 
 molecules, and crystals, p. 141. 4. Crystallo- 
 graphic chemical changes. Undermining and 
 reconstruction, p. 143. 5. Resistance to mechani- 
 cal disruption and chemical attack, p. 154. 
 
 AN ATTEMPT TO FORM SOME IDEA OF THE COURSE OF 
 CHEMICAL REACTIONS FROM OBSERVATIONS ON 
 CRYSTALS 
 
 1. Chemical SreMeay notions! Dp. 158. | 2. 
 chemical processes and discontinuous reactions. 
 Mass action and catalysts. Heat as a catalyst, 
 p.160. 3. Crystallographic indicators of chemical 
 processes, p. 163. 
 
 ANALOGY OF THE MORPHOLOGICAL ACTION OF PHYSICAL 
 AND CHEMICAL FIELDS ON CRYSTALS 
 
 1. Thermal influences on the crystal form, p. 170. 
 2. Chemical influences on the crystal form, p. 174. 
 3. Comparison of the thermal and chemical in- 
 fluences on the crystal form, p. 176. 
 
 CRYSTAL PHYSIOLOGY AND THE CLASSIFICATION OF 
 ATOMS. 
 
 1. einen uel gat Uheaioieey of the cndat cated p. 
 180. Electrons, atoms, and molecules, p. 181. 
 3: Kiam types, p. 182. 4. Atom sub-types, 
 p.185. 5. Elements, p.185. 6. Normal mixture 
 and separation of isotopes, p. 186. 
 
 CONCLUSION 
 
 INDEX 
 
 PAGE 
 105 
 
 II5 
 
 I21I 
 
 139 
 
 158 
 
 170 
 
 180 
 
 187 
 IQI 
 
LIST OF PLATES 
 
 Dr. W. C. RONTGEN . 
 
 MELANCHOLY 
 From an Ferrero b by Apert Diirer. 
 
 RENE Just Haty 
 P. v. GROTH 
 
 A. SCHOENFLIES 
 E, v. FEDOROW 
 M. v. LAUE 
 
 FIGS. 15 AND 16 
 P. DEBYE 
 
 P. SCHERRER 
 
 Sirk WILLIAM HENRY BRAGG 
 From a Photograph by Elliott & Fry, Ltd. 
 
 WILLIAM LAWRENCE BRAGG 
 From a Photograph by Elliott & Fry, Ltd. 
 
 G. v. TSCHERMAK 
 
 Fias, 50, 51 AND 52. 
 Fic, 86 
 
 Fics. 163 AND 164 
 Figs, 165, 166 AND 167 
 
 Frontispiece 
 
 TO FACE PAGE 
 I 
 
 22 
 
 26 
 38 
 66 
 140 
 
 144 
 
= OF Te 
 we “Ualvexaite uF wis 
 ih! 
 
 ™ pec i 
 
/ 
 } 
 
 MELANCHOLY 
 
 FROM AN ENGRAVING BY ALBERT DURER 
 
CRYSTALS AND THE 
 FINE-STRUCTURE OF MATTER 
 
 I. INTRODUCTION 
 
 HE reader sees before him a reproduction of a 
 drawing by the great German master, Albert 
 Diirer, whose art has here represented the 
 problems of natural science and technical practice, 
 together with the simple means available in his time 
 for their elucidation. Among the symbols in the 
 picture we observe in a very prominent place a 
 gigantic crystal, which is surely an indication that 
 Diirer saw a scientific problem of the first importance 
 in the solution of the riddle of this regular form. 
 
 Over the whole scene rests a gloomy air of 
 speculative resignation, the Faust-like expression of 
 the feeling that we, ultimately and in spite of fervent 
 endeavour, “nothing can know.” The artist has 
 called his picture “‘ Melancholy.” 
 
 A Durer of our own day would certainly have 
 drawn more hopefully. It would seem as though the 
 darkness enveloping the great mysteries of nature 
 were lifting a little. By studying crystals and using 
 them in other investigations much light is thrown on 
 the ultimate significance of matter and on the nature 
 of the forces which, acting from one minute particle 
 
 to another, result in the coherence of the universe. 
 of . 
 
2 CRYSTALS AND MATTER 
 
 Crystals are proving themselves more and more 
 the ideal substances of physics and chemistry. Even 
 before the last great discoveries in this branch of 
 knowledge the late Woldemar Voigt, Professor of 
 Theoretical Physics at G6ttingen, who had a unique 
 knowledge of this subject, drew attention to the 
 exceptional regularity of the crystalline parts of 
 matter in a fine simile, which is quoted below : 
 
 ‘Imagine a couple of hundred picked violin 
 players in a large room, all playing the same piece 
 of music on faultless instruments, but beginning 
 simultaneously at widely different places, and start- 
 ing afresh each time they reach the end. The finest 
 ear would be unable to recognise the piece actually 
 played in this uniform medley of sound. Now such 
 music is presented to us by the molecules of gaseous, 
 liquid, and ordinary solid bodies. A crystal, on the 
 other hand, corresponds to the orchestra described 
 above when it is guided as a whole by one able 
 conductor, so that all eyes hang on his slightest 
 gesture, and all hands play the same bar. In this 
 way the melody and rhythm of the piece presented 
 become completely effective, the number of the per- 
 formers not hindering but intensifying the result.”’ 
 
 This picture makes it clear how crystals can 
 present a large series of phenomena which are absent 
 in other bodies, and that, in them, certain character- 
 istics are developed in wonderful variety and elegance, 
 which elsewhere occur only as monotonous mean 
 values. This fact is briefly referred to in Voigt’s 
 text-book at the end of the chapter on the esthetic 
 side of crystal science as follows :— 
 
 ‘In my opinion the music of physical laws in no 
 other branch achieves harmonies so full and rich as in 
 
INTRODUCTION 3 
 
 crystal physics.’ The full truth of his view has been 
 strikingly proved in the last ten years; it could not 
 have been brought to notice in more impressive 
 fashion than by the results of the researches initiated 
 by M. v. Laue on the action of X-ray impulses in 
 crystals. The fine-structure of crystals acts like 
 a grating diffracting the X-rays. In this “ Laue 
 effect,’’ which will be discussed later, crystalline sub- 
 stances are seen to be the best ordered materials. 
 It is easily understood that, since these experimental 
 demonstrations of the constitution of crystals, the 
 investigations not only of mineralogists but also of 
 physicists and chemists have been especially con- 
 centrated on the crystal form. Further magnificent 
 results have been obtained which are extensions of 
 the discovery of M. v. Laue and his fellow-workers, 
 W. Friedrich and P. Knipping. These matters cer- 
 tainly merit the widest dissemination in scientific 
 circles. Many expositions with this object have 
 already appeared. Besides taking a share in the 
 work of research, I have frequently done my best 
 to promote the’ same end by means of notes in 
 journals and papers. 
 
 The present small work has, in addition, a 
 wider purpose. In a treatment which aims at 
 being comprehensible, as far as possible, to general 
 readers, while at the same time offering much new 
 material to fellow-students, the attempt is also made 
 to deduce from the properties of crystals the main 
 characteristics of the fine-structural constitution of 
 matter. 
 
 With this in view the title has been slightly altered 
 from that of the first edition. The aim of this book 
 is thus to show that crystals exemplify not only the 
 
4 CRYSTALS AND MATTER 
 
 morphological but also the physical and chemical 
 constitution of matter. The crystal is therefore 
 treated from the standpoint of its fine-structure, 
 and a discussion of amorphous bodies (for example 
 gases and liquids) is included. 
 
II. THE STUDY OF FINE-STRUCTURE 
 (LEPTOLOGY) 
 
 study, or leptology, because the customary term 
 
 stereochemistry does not cover completely the 
 field of the inquiry intended. In stereochemistry 
 we study the form and arrangement of the particles 
 comprising various substances in order to explain 
 thereby their chemical properties. Alongside this 
 science, another has arisen, which may justly be 
 termed stereophysics, and this also is concerned with 
 the constitution and association of the particles of 
 matter, dealing especially with their movements and 
 physical properties (e.g. crystal optics). Finally, a 
 third allied subject is included, namely, crystal 
 structure, or the study of the fine-structure of 
 crystalline bodies from the point of view of their 
 geometrical relations. 
 
 Thus from the trunk of Greek Atomic Theory, 
 which is more than two thousand years old, three 
 branches of knowledge have sprung forth and blos- 
 somed—stereochemistry, stereophysics, and crystal 
 structure. Their intimate relations to one another 
 justify a name to include them all. If now the 
 particles of matter, the electrons, atoms, ions, and 
 molecules which constitute gases, liquids, and crys- 
 tals are termed collectively fine-structure-particles or 
 leptons (from Xemros, fine, delicate), the name sug- 
 gested above, fine-structure-study or leptology indi- 
 cates’ precisely the end in view. 
 
 5 
 
 I INTRODUCE the expression fine-structure- 
 
Ill. CRYSTALLOGRAPHY AND LEPIOUCOGGS 
 
 HISTORICAL 
 
 EARLY two thousand five hundred years 
 Nig. in ancient Greece, where Babylonian, 
 
 Persian, and Egyptian wisdom was _ asso- 
 ciated with the Greek genius, philosophical theories 
 as to the fine-structure of matter were extant. 
 These theories were due to Democritus and to his 
 friend Leucippus, and also to Epicurus, who lived 
 some hundred years later. Their writings have not 
 come down to us; their ideas, however, are pre- 
 served in the poem “ De rerum natura ’’ of Lucretius 
 (96-55 B.C.). 
 
 According to this they assumed all matter, 
 considered in its finest state, to have a granular 
 structure, and to consist invariably of aggregates 
 containing large numbers of atoms. These very 
 small particles are, in their opinion, extended but 
 indivisible, unchangeable, and for different sub- 
 stances, of different shape, magnitude, and weight. 
 They oscillate and move about at random. Between 
 them is empty space. 
 
 Under the influence of Aristotle this theory 
 retreated into the background, and it found no 
 favour in intellectual circles of the Middle Ages 
 until scientists, such as P. Gassendi (1592-1655), 
 and the great physicist, I. Newton (1642-1726), 
 endowed the old fundamental idea with a new 
 
 validity. As a consequence, closer relations between 
 6 
 
CRYSTALLOGRAPHY, LEPTOLOGY 7 
 
 leptology and crystallography were soon estab- 
 
 lished. 
 
 In this connection Ch. Huygens (1629-1695) 
 
 was able to explain the form of calc-spar, its cleavage, 
 and the variation of its hardness and double refraction 
 with direction, by assuming a regular packing of 
 (a conception which 
 
 spheroidal particles 
 Barlow and _ Pope 
 also employed suc- 
 cessfully). Later, 
 
 the principal founder 
 of scientific crystallo- 
 graphy, Abbé René 
 Just Haiiy 
 
 (1743- 
 
 Fic. 1.—Fine-structure of calc-spar 
 according to Ch. Huygens. 
 
 later 
 
 ‘ 
 fa 
 
 Cy 
 ) 
 | 
 rl 
 @ 
 | 
 | 
 a 
 se 
 oe 
 
 \a 
 
 (usannme 
 iy 
 
 ne 
 ate 
 
 ue 
 
 Fic, 2. —Crystalline fine-structure according to 
 Re ye tiauy: 
 
 1822,) endeavoured to establish his expositions on 
 
 a fine-structural basis. 
 
 +) 
 
 “additive molecules 
 
 He thought of crystals as 
 constructed of, for the most 
 
 part, contiguous building stones, the form of which 
 determined the cleavage, and he deduced by means 
 of “ decrescence”’ (i.e. regular, step-like reduction in 
 the size of molecular plates on a nucleus) the crystal- 
 
 line forms from such ‘ 
 
 ordinary fineness of 
 
 ‘Primitive Bodies.’’ 
 
 The extra- 
 the staircase structure renders 
 
 the boundary planes perfectly smooth to the eye, 
 and even to the most sensitive optical tests. 
 
8 CRYSTALS AND MATTER 
 
 An experimental law of crystallography, that 
 of the “‘ simple rational axial sections of the crystal 
 faces,’ 1 found in the above its obvious explanation. 
 We understand by this law that in the external form 
 
 coc eoc 
 Fic. 93, Fic. 4. 
 Illustration of the law of simple rational axial sections of crystal faces. 
 
 of crystals, surfaces arbitrarily placed do not occur, 
 but only those which stand in a definite relation to 
 one another. In contrast to the freedom of an archi- 
 tect who, in building a house with the roof D, as 
 
 Fic. 5.—The relationship of the faces of an aragonite crystal on three inter- 
 secting axes. 
 
 shown in Fig. 3, can at will give a small or large 
 value to its angle of inclination, in crystal structure 
 (Fig. 4) only certain slopes are possible, such as those 
 
 1Commonly known as the “ Law of Rational Indices.” 
 
RENE JUST HAUY 
 Born 28th February, 1743, Died 3rd June, 1822 
 
sé 
 
 ,. , a H ‘ 
 
 a 
 
 Wisivchou: oe acumen - 
 
 4 » ae a 
 
CRYSTALLOGRAPHY, LEPTOLOGY 9 
 
 from 6 to c, from 6 to $c, 2c, 2c, cc, or, in other words, 
 according to simple rational ratios. Similar remarks 
 hold for the other external faces of the crystal with 
 respect to the points of section on the spacial axes 
 a, b,c. If the units of these are marked out, say, 
 for aragonite (Fig. 5), by means of the axial points of 
 the face O, = a: 6:c in which ratio the length of 6 
 is set equal to unity, and each of the other surfaces 
 be imagined displaced parallel to itself till it passes 
 through the end point 10, then experimentally all 
 such surfaces cut the axes a and 
 cin the manner discussed above. 
 For example, the face 6 cuts 
 these axes In a:b: oc; m,, 
 Meer COC Dy, 1M cod. UC. 
 Sritieed..0 2 2c; and'so.on: pas 
 
 A consideration of Fig. 6 a@étetetdteretseey2 7 
 renders this law of crystallo- Crys Bas 
 graphy, which restricts in a ee nat RAR TEL 
 far-reaching fashion the exter- 
 nal form of crystals, immediately intelligible accord- 
 ing to the ideas of R. J. Haity. 
 
 Against the great advantage of the agreement 
 between the Hatiy theory and crystallographic prac- 
 tice several considerations, especially those of a 
 physical nature, must be advanced. The compres- 
 sibility of the crystals, and their volume changes, 
 with change of temperature, make their structure 
 according to the Hatiy scheme improbable. How- 
 ever, these difficulties could easily be got over, as the 
 founder of the theory himself pointed out, if the 
 closely-packed ‘‘ additive molecules ”’ are replaced at 
 their centres by particles freely oscillating, which 
 mutually maintain each other in this arrangement. 
 
10 CRYSTALS AND MATTER 
 
 Thus we arrive at the idea of space-lattices as systems 
 of particles, arranged periodically in three dimensions 
 (Fig. 7). oe 
 Such a system is formed by the points of inter-- 
 section of three families of planes, each family 
 
 Fic. Uae HOPE En) Fic, 8.—Molecular lattice according 
 to A. Bravais. 
 consisting of parallel planes, the distance between 
 consecutive members being constant. According to 
 A. Bravais (1811-1863), the space-lattice particles 
 consist of chemical molecules, the symmetry of which 
 is determined, by) ices 
 stallographic considerations 
 (Fig. 8). The Munich phy- 
 sicist, L. Sohncke (1842- 
 1897), and his mineralogical 
 colleague, P. v. Groth, on 
 the other hand, replace the 
 Bravais molecular lattice by 
 what they call point sys- 
 tems, which are atom lat- 
 ie scree em seeorling tices placed regularly one 
 inside the other, as indicated 
 in Fig. 9, where the scheme includes only two types 
 of atom. 
 At the same time, the necessary geometrical 
 
CRYSTALLOGRAPHY, LEPTOLOGY 11 
 
 representation of all space groupings possible in 
 accordance with crystallographic laws was carried 
 out. A. Schdénflies? has given in a classical treatise 
 the whole system of space-lattice arrangements. The 
 same work has also been accomplished by the Russian 
 investigator, E. v. Fedorow. 
 
 ee CP ECL: 
 
 The year 1912 came as the great harvest year in 
 the physics of space-lattice ideas. This led to the 
 ever-memorable researches instituted by M. v. Laue, 
 and carried out jointly with P. Knipping and 
 W. Friedrich, in Munich, on the use of crystals as 
 diffraction gratings for X-rays. A polychromatic 
 impulse of the radiation is split up by diffraction 
 at the particles of 
 the crystal grating 
 into a spectrum. of 
 monochromatic rays. 
 These, received on a 
 photographic plate, 
 give rise after de- 
 velopment to a figure 
 symbolic of the Fic. 10.—The ‘“‘ Laue Effect.”’ 
 atomic arrangement, 
 in the form of a Laue diagram (Fig. 10). This diffrac- 
 tion effect can be considered ? formally as a reflexion 
 
 ? 
 
 1A. Schénflies, “‘ Kristallsysteme and Kristallstruistur,”’ 1891. 
 
 2 Thus each point in a Laue diagram is to be regarded as the 
 point of impact of X-rays which have been reflected at a structure 
 plane in the crystal. Fig. 12 makes this clear for two planes, E,F, 
 and E,E,. The primary ray P is reflected at E,E, to R,, and at 
 E,E, to R,. R, and R, are thus the points of impact of secondary 
 rays on the photographic plate. Rhythmic arrangements of struc- 
 ture planes are emphasised in the radiogram by corresponding 
 
12 CRYSTALS AND MATTER 
 
 of the incident rays at the planes of atoms in the 
 crystal, in which case, however, a reflected ray is 
 obtained only if the condition n\ = 27 sin a is 
 fulfilled? (Fig. 11). This aspect was especially em- 
 phasised by W. H. and 
 W. L. Bragg. 
 
 P 
 
 Fic, 11.—Reflexion of X-rays. Fic. 12.—Diagram showing re- 
 flexion of an X-ray at two struc- 
 tural planes of a crystal. 
 
 In this way the space-lattice idea of crystallo- 
 graphy became the starting-point of an extraordinary 
 development of physical science ; for not only the 
 nature of X-rays as wave phenomena, but also the 
 actuality of the atom was experimentally proved, 
 
 repetitions of reflected rays of the same intensity ; the Laue diagram 
 of beryl (Fig. 14) is an especially beautiful example of this. 
 
 The customary arrangement of the points on elliptical curves is 
 easily explained with Fig. 13. In this S,s, is the primary ray, which 
 is reflected along Ks, by a plane cutting the plane of the paper 
 at right angles and in the line Kz. If this plane be rotated about 
 Kz, the reflected ray traces out a conical surface with Kz as axis. 
 The receiving photographic plate PP cuts this cone in an ellipse 
 S,\S,. For other inclinations of the line Kz, parabolas, hyperbolas, 
 or straight lines are obtained for the series of reflexions from those 
 planes which run parallel to some one direction (corresponding to 
 Kz) ; or which, as the crystallographer says, lie in a “‘ zone.”’ 
 
 1n=I1, 2, 3...2XA= wave lengths. v» = distance between 
 reflecting planes. a = glancing angle. ‘The path difference for the 
 two rays I and 2 is obviously equal to wu, thus it equals 2y sin a 
 (Fig. 11). 
 
E. v. FEDOROW 
 
me La f 
 
 oun THE. LORNA 
 = JF HE 
 | -HakeaL Lay Of i - ‘ais 
 
CRYSTALLOGRAPHY, LEPTOLOGY 13 
 
 once and for all, by the Laue effect. The existence 
 of atoms is as certain now as that of the macrocosm 
 of the starry heavens. Laue’s experiment may with 
 
 Fic. 13.—Formation of zone curves in Laue diagram. 
 
 justice be described as a solemn deposition of nature 
 concerning its most intimate structure. 
 
 M. v. LAUE 
 
 Under these circumstances it will be of interest 
 to the reader to hear something more of the young 
 German scientist and his celebrated researches, which 
 opened up an unlimited field for investigation in 
 the fine-structure of matter by the application of 
 X-rays. 
 
 M. v. Laue was born on the gth October, 1879, 
 at Pfaffendorf, near Coblenz. While a student in 
 Berlin he received the greatest encouragement in his 
 scientific work from the creator of the quantum 
 theory, M. Planck, whose assistant he was from 
 
14 CRYSTALS AND MATTER 
 
 1906 to 1909. When v. Laue removed to Munich in 
 1909, stimulated by the work of Réntgen and the keen 
 
 Fic. 14.—Laue diagram of the end surface of a crystal or beryl. Photographed 
 by F. Rinne. 
 
 interest of Sommerfeld in X- and y-rays, his attention 
 
 was directed to the question of the nature of these 
 
 rays. Moreover, it chanced that at Munich Uni-— 
 
surface 
 
y =e ’ : p 
 i a i 1 j , : : ; i i 
 { -* i is 
 7 {ow ; — rh { a boii} *y oe 
 ; = 2 % > . 7 
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 A i >, é =i es el wwe 
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 7 - ' : x ae, rye wie a e -_ 
 ! = =. fa ; 
 Ps : : = 4 a i k : eal 3 
 
 . 
 ie? 
 q * 3 a - if - 
 
 7 3 
 FT 7 ) : 7 4 ® ie ¢ ' ~~ 
 » ‘ P " 
 
CRYSTALLOGRAPHY, LEPTOLOGY 15 
 
 versity, P. v. Groth, in his lectures and writings, was 
 emphasising the space-lattice idea, and this ultimately 
 took its place in the circle of ideas of the physicists 
 there. In this way it happened that, when v. Laue 
 was visited in February, 1912, by his colleague, 
 P. P. Ewald, they discussed the studies of the latter 
 as to the relations of long electro-magnetic waves 
 in a space-lattice, and the question occurred as to 
 the action of such waves, which are short compared 
 with the dimensions of the lattice. Laue’s optical 
 knowledge told him that grating spectra must arise ; 
 he expected interference phenomena in the passage of 
 X-rays through crystals, and mentioned this to Ewald. 
 Copper vitriol served as the material for the first 
 experiment, large regular pieces of this being easily 
 obtained. Friedrich and Knipping left the direction 
 of the incident beam to chance. On the photographic 
 plate behind the crystal there appeared in the first 
 experiment the expected grating spectra. In the 
 continuation of the work, Friedrich and Knipping 
 investigated - regular crystals having the highest 
 possible symmetry, such as zinc-blende, and illu- 
 minated them in the direction of the crystallographic 
 axes, with the beautiful results submitted to the 
 reader in the original figures 15 and 16. 
 
 By applying here the results for the ordinary and 
 the crossed grating the theory of the phenomena was 
 immediately obtained, and Professor Sommerfeld was 
 able, on the 8th June, 1912, to lay before the Munich 
 Academy the joint work of Friedrich, Knipping, and 
 v. Laue on the interference of X-rays. 
 
 In 1912 M. v. Laue obtained the Professorship 
 of Theoretical Physics at Zurich University. From 
 1914 to 1919 he acted as Professor in the same 
 
16 CRYSTALS AND MATTER 
 
 subject at Frankfurt University ; a summons to 
 Berlin University brought him to his present position. 
 A further recognition was accorded him in the Nobel 
 Prize for physics for the year IgI4. 
 
 FURTHER METHODS IN X-RAY WORK ON CRYSTALS 
 
 It is to a number of investigators, happily an in- 
 creasing number, that we owe extensions of the method 
 of the fundamental research of Laue, Friedrich, and 
 Knipping. W. H. and W. L. Bragg‘ used plates cut © 
 
 Fic. 17.—Rotation spectrogram of adularia. After E. Schiebold. 
 
 in known directions from crystals, and rotated then 
 on a spectrometer about an axis lying in the surface 
 of the specimen. As soon as the angle between the 
 plane of the specimen and one of the incident X-rays 
 satisfied the equation m\ = 27 sin a reflexion occurred. 
 The reflected ray lying in the plane of incidence was 
 detected by means of a cylindrical ionisation chamber, 
 and its inclination determined. The quantity of 
 
 1W. H. and W. L. Bragg, “ X-Rays and Crystal Structure.” 
 
CRYSTALLOGRAPHY, LEPTOLOGY 17 
 
 ionisation served aS a measure of the intensity of the 
 reflected radiation. 
 
 Other rotation methods with photographic deter- 
 mination of the direction of the reflected rays have 
 been elaborated, especially by H. Seeman and E. 
 Schiebold. 
 
 Interesting researches on substances possessing a 
 fibrous or flaky structure, such as occur in nature, 
 in plants, and animals, or can be prepared by drawing 
 or pressing metals, have been carried out by members 
 of the Research Institute in the Chemistry of Fibrines 
 at Dahlem, near Berlin; among these, Polanyi and 
 his collaborators, Becker, Herzog, and Jancke, may 
 be specially mentioned.! 
 
 1 Substances built up of fibres or flakes, which lie with all their 
 fibre axes or flake normals parallel, but otherwise indifferently, give 
 on the passage of X-rays perpendicular to this direction an X-ray 
 effect similar to that of a crystal plate which is rotated about an 
 axis passing through it. The series of reflexions, one after the other, 
 produced by the latter, as a result of the rotation, are shown by the 
 pack of fibres or flakes simultaneously. All fibres or flakes the struc- 
 ture planes of which satisfy the equation n\ = 2y sin a give rise to 
 reflected rays. As was shown by H. Seeman and E. Schiebold, in 
 particular for rotating plates, and by M. Polanyi for fibrous sub- 
 stances and stretched metals, we get in this way characteristic 
 diagrams. The reader more closely interested is referred to Fig. 18. 
 In this PSt denotes the primary ray. It is reflected at surfaces 
 whose normals N,, N,, etc., corresponding to various positions of 
 rotation about the rotation axis DA (or the axis of the fibres, as 
 the case may be), are shown in the figure. The incident ray, the 
 normal to the surface, and the reflected ray, lie in every case in one 
 plane, e.g. the ray S,, corresponding to N3 liesin the plane E,. If 
 monochromatic radiation be used and only reflexions of a given order 
 be considered (i.e. if we assume definite values for \ and ” in m\ = 2r 
 sin a), then reflected rays arise which pass through the intersections 
 of the a-circle with the great circles corresponding to various planes 
 of incidence. Take, for example, N = 3. In this case the inter- 
 sections on the circle E, are, on the sphere, S,S’; ; on the photographic 
 plate PhPl, S,S’,. For other values of ”, and therefore of a, other 
 
 2 
 
18 CRYSTALS AND MATTER 
 
 Of the greatest importance is a method emanating 
 from P. Debye and P. Scherrer, in Géttingen, which 
 
 ‘ 
 
 , ‘ 
 ‘ 
 i) 
 
 Fic, 18.—Explanation of the rotation spectrogram. After E. Schiebold, 
 
 Fic. 19.—Reproduction of Fic. 20.—Reflexion cone of X-rays obtained 
 a Polanyi diagram. with crystal powder. 
 
 renders the investigator independent of the possession 
 
 of oriented plates or of crystals with any regularity 
 at all. 
 
 reflexions will arise. Taking all the intersections into consideration, 
 they lie on the curve similar to a lemniscate, shown in the figure. 
 In general, of course, there are four reflected rays; in special 
 positions of the reflecting planes, only two. Compare S, and S’,, S; 
 and S’,, in Fig. 18, also Fig. 19. 
 
CRYSTALLOGRAPHY, LEPTOLOGY 19 
 
 These scientists employed powders of the finest 
 crystalline particles, such as can be obtained by 
 precipitation from solution or by continued pul- 
 verisation. In such a case the structure planes re- 
 flecting the X-rays lie at random in all directions. 
 Those of them, however, which are inclined to the 
 primary rays at the glancing angle a, satisfying the 
 equation m\ = 2rvsina, give rise to a reflected ray ; 
 and since such positions occur all round the primary 
 ray, a cone of rays is produced, as P. Debye and P. 
 Scherrer showed, instead of the | 
 single reflected ray obtained in 
 the Laue diagram. This effect is 
 that which would be obtained if 
 
 Fic. 21.—Camera for the Debye- Fic, 22.—Unrolled film with Debye- 
 Scherrer method. Scherrer diagram. 
 
 a Laue diagram were rotated about its central normal. 
 
 Since monochromatic light is employed, the hollow 
 
 cones of reflected rays obtained are thin, and cut 
 
 the interposed photographic plate in separate circles 
 
 (Fig. 20). 
 
 In order to make the distance travelled by all the 
 reflected rays to the receiving surface the same, and 
 to facilitate photographic registration, a cylindrical 
 camera, with the specimen as a small roll at its 
 centre, is employed, following the suggestion of 
 P. Debye and P. Scherrer. A film is placed round 
 the inner wall of the camera; after exposure and 
 photographic development this is unrolled, examined, 
 and measured (Figs. 21 and 22). Independently of 
 Debye and Scherrer, the American investigator Hull, 
 
20 CRYSTALS AND MATTER 
 
 has devised a similar method. W. H. Bragg has also 
 combined the Debye-Scherrer with his spectrometric 
 ionisation method. , 
 
 With respect to the manipulation of the experi- 
 mental results, the reader interested further is 
 referred to the more detailed works on the subject. 
 
 Phot +late (elevation) 
 
 GRC AD 
 
 topor N20) 
 1 
 | | Phot. 4 late (see tun 
 
 A Phot. sil (cylinder) ——~ 
 ee 
 a 
 
 Incident 
 Leam 
 
 Fic. 23.—Construction of the Debye-Scherrer diagram of powdered 
 graphite. 
 
 X-RAY RESULTS IN THE DOMAIN OF CHEMISTRY 
 
 The X-ray spectra of H. Moseley (1888-1915) 
 provided chemistry with the natural series of the 
 
 1W. H. and W. L. Bragg, “‘ X-Rays and Crystal Structure.” 
 E. Marx, ‘‘ Handbuch der Radiologie, Band 5 (Kathodenstrahlen und 
 Rontgenstrahlen).’”’ P. Niggli, ‘‘Geometrische Kristallographie des 
 Diskontinuums.”’ F. Rinne, “ Einfuhrung in die Kristallographische 
 Formenlehre sowie Anleitung zu Kristallographisch-optischen und 
 roéntgenographischen Untersuchungen,”’ 5th edition, published by 
 Dr. M. Jaenecke, Leipsic. Works by P. P. Ewald and E. Schiebold 
 on X-rays and Crystals are in course of preparation. 
 
DR? PP. DEBYE 
 
 Professor of Physics in the University of 
 Zurich 
 
 DR. P. SCHERRER 
 
 Professor of Physics in the University of 
 Zurich 
 
THE LIBRARY 
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CRYSTALLOGRAPHY, LEPTOLOGY 21 
 
 ¢ 
 
 elements according to their ‘‘ atomic numbers,” an 
 arrangement to which there are no exceptions up to 
 the present time. In L. Meyer’s and D. J. Mendeléeff’s 
 “Natural System of the Elements,’ where they are 
 set down in the order of increasing atomic weight, 
 the arrangement being broken at certain points and 
 so split up into series, there were several very awk- 
 ward anomalies. Contrary to the guiding principle 
 of the arrangement according to increasing weight, 
 the properties of argon necessitated its being placed 
 before potassium in spite of its higher atomic weight. 
 For similar reasons cobalt came before nickel, and 
 tellurium before iodine. This anomaly has now been 
 removed. The spectrometric diffraction of X-rays, 
 using crystal plates as gratings, with anticathodes of 
 the various substances, proved that the elements are 
 ordered by their X-ray spectra in complete accord- 
 ance with the corrected Meyer-Mendeléeff table. The 
 square root of the frequency of the spectral lines is 
 a linear function of the atomic number in the sys- 
 tem. Accurate determinations in particular by the 
 Lund physicist, Manne Siegbahn, have definitely fixed 
 this extremely important scientific result, “the 
 atomic number,’’ which constantly characterises the 
 elements. In addition, these X-ray investigations. 
 rendered possible determinations of exact stereo- 
 chemical formule by a physical method. W. H. and 
 W. L. Bragg have led the way in this with remarkable 
 ingenuity. They measured the 7 values of crystals, 
 for example, of fluor-spar, for various directions, 
 such as perpendicular to the cubic, rhombododeca- 
 hedral and octahedral surfaces, and were able to 
 conclude therefrom, with the help of the intensity 
 relations of the appropriate spectra, the positions of 
 
22 CRYSTALS AND MATTER > 
 
 the atoms. Later the fine-structural constitutions of 
 rock-salt, fluor-spar, zinc-blende, diamond, calc-spar, 
 and other important crystal types were elucidated, 
 not only as regards general structure, but including 
 the absolute magnitudes of their 7-values. 
 
 Distinguished scientists, including P. Debye, 
 P. Scherrer, P. Vegard, A. W. Hull, R. W.oWyekaas 
 and L. W. Mackeehan, carried on the investigation 
 of such atomic point lattices. 
 
 Fic. 24.—Stereograms. Elementary cells in the fine-structure of crystals.} 
 a. Fine-structure of copper, gold, silver, gold and aluminium. 4, Fine 
 structure of rock-salt. c¢. Fine-structure of fluor-spar. d. Fine-structure 
 of zinc-blende. ¢. Fine-structure of the diamond. /f. Fine-structure of 
 iron pyrites. 
 
 1 Some idea of the minuteness of crystalline lattice structure—of 
 the diamond, for example—is obtained by a comparison such as the 
 following : Imagine cubical boxes of 1 metre each side arranged one 
 behind the other in a straight line from Berlin to Cairo. Now reduce 
 this enormous length of nearly 3000 kilometres to 1 millimetre; a 
 measure of the actual size of the elementary cells is obtained in the 
 proportionally reduced boxes. There are about 2,837,000 of these 
 to the millimetre, and a similar number of carbon atoms lie on 1 milli- 
 metre of the edge of a diamond cube (Fig. 24e). Ina cubic millimetre 
 of the gem there are 178 trillion carbon atoms. 
 
 In considering the question further, it is noticed that in Fig. 24a the 
 
SIR WILLIAM HENRY BRAGG 
 
 Professor of Physics in the University of London 
 
 WILLIAM LAWRENCE BRAGG 
 
 Professor of Physics in the University 
 of Manchester 
 
 Li) t 
 
CRYSTALLOGRAPHY, LEPTOLOGY 23 
 
 If, however, it were immediately assumed that 
 already a comprehensive idea of the microcosm 
 
 copper atoms are to be thought of as being at the corners and at the 
 centres of the sides of an elementary cube. Its length of side a is 
 equal to 3°61 X Io ~° cm., i.e. 0:000,000,0361 cm. 
 
 For silver this length amounts to 4:06, for gold 4:07, and for 
 aluminium 4:07, all multiplied by 10o-% cm. The sodium atoms in 
 rock-salt NaCl are arranged in exactly similar fashion (Fig. 245). 
 The corresponding chlorine atoms occur at the mid-points of the 
 Sigeseand atithe centre of the cube; @='5:8 X 10-°% cm. In 
 Fig. 24c the calcium atoms of fluor-spar CaF, are placed in the same 
 way as the copper and sodium atoms in Fig. 24a and 6 respectively. 
 Now divide up the elementary cell into eight smaller cells by three 
 planes, parallel to the sides and passing through the centre, and 
 instal in each a fluorine atom. These atoms, owing to their positions 
 at the centres of the cells, form a small cube inside the larger one ; 
 a= 5°44 X I0—8, 
 
 For zinc-blende ZnS (Fig. 24d) the metal is arranged as in the 
 previous types, and like fluor-spar, the cubic structure is divided 
 into eight compartments ; in this case only every other one contains 
 a sulphur atom. The value of a for the zinc sulphide cube is 
 5°4 Xx 10—5 cm. The structure of the diamond (Fig. 24e) is ob- 
 tained from that of zinc-blende, if both the zinc and sulphur atoms 
 of the latter are replaced by carbon atoms; a@ = 3°53 X to—& 
 cm. Finally, iron pyrites FeS, (Fig. 24f) reproduces the earlier 
 arrangements in the positions of the metallic atoms. Its sulphur 
 particles set themselves on the diagonals of the eight smaller cells 
 mentioned above, each one being about a quarter of the diagonal 
 length from either the cube edge or the cube centre. Taken together 
 the sulphur atoms form a rhombohedron, as Fig. 24f indicates. If 
 the corners of this were moved along to the mid-points of the 
 diagonals, the sulphur atoms would reproduce the arrangement of 
 the fluorine atoms in fluor-spar CaF, (Fig. 24c) ; @ = 5°37 X 107° cm. 
 
 It is necessary that the joining lines shown in the figure should 
 be constantly kept in view. Moreover, it must be remembered that 
 each atom represented by a point is itself a kind of planetary system 
 with central body and satellites. 
 
 Regarded in this way it is more than ever obvious that we have 
 here a marvellous microcosmic system ; the particles hover in the 
 crystal space like stars in the heavens, all mutually supporting one 
 another in their regular arrangement, which we have learnt to 
 measure accurately to a ten-millionth of a millimetre. 
 
24 CRYSTALS AND MATTER 
 
 of the crystalline world had been obtained, our 
 jubilation would be premature. Only a compara- 
 tively small number of crystal stereochemical for- 
 mule are available to-day, ten years after the first 
 Laue research on the subject. This is to be attri- 
 buted to the unsettled state of the times, and, in 
 addition, to the indirect nature of the experiments.! 
 Indirect methods must be used which, in complicated 
 cases of atomic arrangement, are extremely difficult 
 mathematically. 
 
 This applies especially to the case of the com- 
 pounds of organic chemistry.? 
 
 1The atomic world is not disclosed by the microscope. As 
 E. Abbé showed, the microscope is, in a certain sense, blind to objects 
 smaller than about -o005 mm. (5 X 107-°cm.). The wave length of 
 light visible to the eye, or of light such as can be used in photography, 
 is coarse compared with the fineness of the leptons. The atoms are 
 a thousand times smaller (only about 10~—* cm.), and can therefore 
 no more act on light waves, which are large in comparison, than a leaf 
 can influence the waves on which it floats. The wave lengths of X-rays, 
 however (10~° to 10~* cm.), correspond well with such minuteness. 
 
 A leptoscope using these fine waves has, however, not yet become 
 possible, for no substance is known which would serve optically as 
 a lens. We thus must rely on the original diffraction effect as in the 
 case of the ultramicroscope with ordinary light. Probably at some 
 time an image of the fine-structure will be obtained with X-rays 
 by making use of the reflecting power of the structure planes in the 
 crystal. M. Wolfke called attention to the possibility of separating 
 in practice, as is done in theory, the formation of the image into 
 the production of, first, a diffraction figure, and then, by further 
 diffraction, the actual image. The production of the first diffraction 
 figure would be assigned to the X-rays ; it would be photographed 
 and then transformed by repeated diffraction, using ordinary light, 
 into the image of the sub-microscopic object. With regard to several 
 restricting conditions, those interested might read the ‘“‘ Physikalische 
 Zeitschrift,’ vol. xxi, p. 495, 1920. The suggestion has not, so far, 
 matured sufficiently to give practical results. 
 
 * Recently W. H. Bragg, and also K. Becker and W. Jancke, 
 have made some very welcome contributions to our knowledge of 
 
CRYSTALLOGRAPHY, LEPTOLOGY 25 
 
 CRYSTALS AS STEREOCHEMICAL TYPES 
 
 This being the case, it is obvious that, in the 
 difficult work of establishing stereochemical formule, 
 any external assistance is weleome. Crystal mor- 
 phology makes its appearance in such a capacity. 
 When carefully considered, this subject, in a certain 
 sense, 1s merely macrostereochemistry. Of course, 
 we must not conclude that a crystal is an enor- 
 mous molecule similar to sub-microscopic molecules. 
 That would be a false conception, being inconsistent 
 with two actual properties of molecules, namely, 
 
 Fic. 25.—Variety of forms of calc-spar. 
 
 constant form and constant weight. In this respect 
 we may compare the scheme for the benzene ring 
 with the protean multiplicity of calc-spar (Fig. 25), 
 and the exclusiveness of the molecule with the 
 capacity of the crystal to grow and so to increase 
 its magnitude and weight. A crystal form, however, 
 is definitely characterised as a stereochemical symbol 
 in the sense that it is a visible, and therefore easily 
 examined, sample of the leptonic structure Its 
 
 such substances. The dimensions of the elementary cells, and the 
 number of molecules in each, are now known for indigo, anthracene, 
 urea, succinic acid, hydroquinone, anthraquinone, naphthalene, and 
 many other organic compounds, although the precise positions of 
 the atoms are still uncertain, 
 
26 CRYSTALS AND MATTER 
 
 principal surfaces represent series of planes through 
 the leptocosm occupied by a network of atoms, and 
 Fig. 26 shows clearly that in this way surfaces with 
 rational axial sections arise; for the planes densely 
 
 Fic. 26.—A plane in the space-lattice showing edges and surfaces. 
 
 packed with matter form stable boundaries. The 
 principal edges of a crystal indicate the directions of 
 rigidly set lines of atoms, and its morphological 
 symmetry is symbolic of the arrangement of its 
 fine-structure particles. 
 
 Fic. 27.—Rock-salt. Fine-structure Fic. 28.—Mirror symmetry of 
 of the crystal faces and edges. Circles pyroxene. 
 represent Na, dots Cl. 
 
 The explanatory Fig. 27, which represents the 
 known fine-structure of rock-salt, enables this to be 
 clearly understood. In this way substances not yet 
 investigated will be in great part determined from 
 their external forms alone, and thus macrostereo- 
 
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CRYSTALLOGRAPHY, LEPTOLOGY 27 
 
 chemistry will play the part which was emphasised 
 as early as the year 1903 by G. v. Tschermak. 
 
 In this sense Fig. 28, for example, gives us hints as 
 to the structure of pyroxene, which 1s still unknown. 
 Fig. 29, in the same way, provides definite indications 
 as to the structure of that remarkable silicious 
 material, quartz, for it exhibits in the crystal form, 
 and therefore, we may assume, in its fine-structure 
 (not yet experimentally investigated), not mirror, but 
 only rotational symmetry, and serves as a macroscopic 
 symbol of the famous conclusions of L. Pasteur con- 
 cerning the “ asymmetrical’”’ molecule. In the crystal 
 
 Mirror plane 
 
 Fic. 29.—Right- and left-handed quartz. 
 
 forms of quartz, actual right- and left-handed struc- 
 tures are visibly and clearly characterised, and the 
 compensated racemic variety also exists; this is 
 shown in certain twin formations in which right- and 
 left-handed quartz are combined in regular fashion. 
 
 In addition, the remarkable property possessed by 
 most crystals of splitting along definite planes affords 
 an indication of the arrangement of the particles ; it 
 may, of course, be assumed that the particles in the 
 surface formed by a cleavage lie close to one another, 
 each bound firmly to its neighbours. Perpendicular 
 
 1 This is illustrated by imagining the two forms of Fig. 29 dis- 
 placed parallel into one another. 
 
vik CRYSTALS AND MATTER 
 
 to the cleavage plane, as in a direction of weaker 
 cohesion, greater distances prevail, and in conse- 
 quence splitting along such planes is possible. 
 
 OUTLINES OF. GENERAL CRYSTALLOGRAPHIC 
 MORPHOLOGY 
 
 I. THE RELATIONS OF THE GROSS STRUCTURE 
 
 In the way indicated above, the study of crystal 
 morphology arises as an introduction to the subject 
 of stereochemistry of the solid’ state. [t= isjiam 
 this way, a part of chemistry. Every student of 
 chemistry must, therefore, find himself immersed in 
 a specialised subject at the outset. After a greatly 
 changed cultivation of the crystallographic soil, much 
 superfluous undergrowth of names and derivations 
 having been rooted up and simple methods of 
 development followed, the wanderer who picks his 
 way carefully has no very special troubles to fear in 
 this region ; indeed, it may be said that one is now 
 able to go forth in this province, as in a well-tended 
 garden, with artistic enjoyment. 
 
 As an example of such a guiding plan in ae 
 ornamentation of the inorganic world, so important 
 in stereochemistry, the sans scheme will be 
 briefly developed. 
 
 The idea of deriving the raultisiticns of ei. 
 graphic forms from five types all mutually connected, 
 which may be called primitive forms, has already 
 been brought forward by G. v. [schermak, and fol- 
 lowed in his teaching. The fundamental rules of 
 crystallographic symmetry and, therefore, of crystal- 
 structural types, are embodied in these primitive 
 forms, for they are fundamental in the varied mor- 
 
CRYSTALLOGRAPHY, LEPTOLOGY 29 
 
 phology of crystals, being exemplified by the so-called 
 centre of symmetry, axes of symmetry, and sym- 
 metry planes. If the crystal structure possesses a 
 centre of symmetry, to every boundary plane there 
 belongs an equivalent parallel opposite surface, and 
 thus lines through the centre of the crystal cut the 
 external surface in two corresponding points. The 
 symmetry axes indicate, in a certain sense, the 
 rhythm in which similar structural particles grouped 
 about a given direction repeat themselves in different 
 positions ; thus if, say, the axis is senary, as in a six- 
 sided prism, then this will appear to an observer, 
 after a rotation of 60° about the vertical, just as 
 
 Fic. 30.—The five primitive forms of aun pedion, pinacoid, 
 sphenoid, doma, prisma. 
 
 before. A symmetry plane divides a body into 
 
 halves, which have the appearance of an object 
 
 and its mirror image (Fig. 28, p. 26). 
 
 The primitive form # (the pedion, Fig. 30) repre- 
 sents a surface standing alone, i.e. a form devoid of 
 symmetry ; #7 (the pinacoid), with a surface and 
 a similar parallel surface, is the embodiment of 
 the principle of centre symmetry ; s (the sphenoid), 
 with one surface and another flap-lke surface, 
 represents the fundamental idea of rotation in binary 
 rhythm ; d (the doma) represents the principle of 
 reflexion ; and finally, m (the prisma), the combina- 
 tion d +s, d+ 1, or s + #1, which all give the 
 same m. From these five types, to some extent 
 representative of five structural rules in the inorganic 
 
30 CRYSTALS AND MATTER 
 
 world, all the multiplicity of the remaining macro- 
 stereochemistry may be derived by an application of 
 the ideas of A. Schénflies as cases of rhythmical 
 repetition of the primitive forms, according to the 
 numbers 2, 3, 4, and 6, about a principal direction 
 in the structure. Thus the ternary rhythm must 
 appear in a whirl form and dominated by the octant 
 (Fig. 31). It is fundamental that the rhythms may 
 be developed in a simple (gyric) fashion, or by 
 
 4 
 N v 
 Fic. 31.—Simple and isometric (octant) FIG. 32.—Gyric and gyroidal 
 
 whirl structure. rhythms. 
 
 rotation and reflexion together (gyroidal) (Fig. 32). 
 Nevertheless, this development is very convenient, as 
 the combined procedure leads to only four new forms. 
 
 In the following table the thirty-two crystal classes 
 are summarised as types of the crystallographic, 
 and, therefore, stereochemical arrangements. These 
 classes were known already in the time of J. Ch. F. 
 Hessel (1831), and have all been discovered in 
 crystalline materials, except in the case of 34. They 
 are to some extent the architectural styles in 
 
a 
 
 fr vat See oS Ly al 8. 
 S3| 38) 3§ ao| 33 | 38 
 KUE} aS | meg “EE | Ste] ges 
 reed | RC) 4 Sag opel ob ics a | age 
 ea ca} oa a A} ey 
 Primitive forms: Triclinic and 
 monoclinic system ™m co rms 
 Binary rhythm of the primitive 
 forms: Rhombic system . 2d 2m — — 
 Ternary rhythm of the oS 
 forms: Trigonal system 3a 3m 3p 35 
 Tetrad rhythm of the rene 
 forms: Tetragonal system 4d | 4m 4p 4s 
 
 Senary rhythm of the primitive 
 
 CRYSTALLOGRAPHY, LEPTOLOGY 31 
 
 TABLE OF THE THIRTY-TWO CRYSTAL CLASSES. 
 
 II. 
 
 I. Gyric Development. Gyroidal 
 Development, 
 
 Structural Elements. 
 
 forms : Hexagonal system 6d 6m — — 
 Ternary rhythm of the primitive 
 
 forms with octants: Isometric 
 
 system ta zm — 
 
 The rows are series of the same rhythm, the columns those of 
 the same primitive forms. 2p and 2, being identical with m and s, 
 are bracketed, and are only included in the table for the sake of 
 completeness. _ 
 
 The symbols of the classes are to be read, for example, as 
 three p, three pi, and so or; 3 bar p, 3 bar s, etc.: written out in 
 full they are, say, for the ternary series, ehgayte Pian trigyric 
 pinacoidal, trigyric sphenoidal, trigyric domatic, trigyric prismatic ; 
 sodium periodate, dolomite, quartz, tourmaline, calc-spar, are 
 examples of these. The series is concluded with trigyroidal pedial 
 (no example yet known) and trigyroidal sphenoidal (benitoite). For 
 the isometric rhythm the development is characterised as isometric 
 pedial, etc. 
 
 Further details in F.:Rinne’s ‘ Einfuhrung in die Kristallo- 
 graphisch-optischen sowie réntgenographischen Untersuchungen, ” 
 4th edition, Leipsic ; published by Dr. Jaenecke. 
 
 the crystal realm. A cursory examination of the 
 table shows the simplicity of the relations running 
 through it. 
 
32 CRYSTALS AND MATTER 
 
 2. REPRODUCTION OF CRYSTALLOGRAPHIC FORMS 
 IN PROJECTION 
 
 It is especially convenient in considering the 
 matters treated above to turn from pictures of the 
 crystallographic configurations in perspective and 
 to symbolise the formal relations in projection. In 
 Fig. 33 this is done for the primitive forms. To 
 render the figure clear, it is noticed that an X in the 
 diagram represents a surface in the upper portion of | 
 a crystal which has been cut in half by a horizontal 
 
 FIG. 33..—Projection diagrams of the five primitive forms of crystallography. 
 
 Fic. 34.—Projection diagrams of the senary rhythm of the primitive forms. 
 
 plane ; a circle represents an under surface. Such a 
 “figurative point ’’ in the centre of the projection 
 denotes a surface parallel to the paper, whilst one on 
 the circumference represents a plane perpendicular 
 to the paper, meeting the circle in the point marked. 
 Finally, the points between the centre and the cir- 
 cumference are symbols of surfaces placed obliquely. 
 
 It is now easy to represent the effect of a rhythmic 
 repetition of the five primitive forms in projection. 
 In Fig. 34 this is done for a senary rhythm, using 
 oblique surfaces as the general case. The result gives 
 
CRYSTALLOGRAPHY, LEPTOLOGY 33 
 
 the five simple whirl elements of the hexagonal 
 system. A development in accordance with the 
 principle of rotation-with-reflexion gives nothing new 
 in the hexagonal series. In a similar manner we 
 obtain in the other systems general examples of the 
 external crystallographic forms, and at the same 
 time indications as to the arrangement of the particles 
 in the fine-structure. 
 
 3. FINE-STRUCTURE RELATIONS OF GENERAL 
 CRYSTALLOGRAPHIC MORPHOLOGY 
 
 For the purposes of leptology, this grouping in 
 crystal classes is further subdivided into a large 
 
 er Refi exton- 
 Simple Simple with- 
 yoluion atin peheniGn translation 
 Trigonal Trigonal Mirror plane. Plane of reflexion- 
 rotational serew with-translation 
 axis axis 
 
 Fic. 35, a-d.—Simple rotation and translation-with-rotation (screw form). 
 Simple reflexion and translation-with-reflexion. 
 number of space groupings, thus completing the 
 classification. As new variations of the symmetry 
 elements, we have to consider the following: transla- 
 tion-with-rotation and translation-with-reflexion, that 
 is to say, the resultant motions arising from rotation 
 or reflexion, together with translation. The result in 
 the former case is a screw form. Fig. 35) serves as 
 an example of such a screw axis, and Fig. 35d shows 
 a translation with reflexion. 
 For the rest, it is recognised as a result of the 
 3 
 
34 CRYSTALS AND MATTER 
 
 space-lattice principle, that for every crystal struc- 
 ture, considered from a stereochemical standpoint, all 
 
 Fic. 36.—Asymmetric fine- Fic. 37.—Fine-structure with 
 structure. symmetry centre. 
 
 Fic, 38.—Fine-structure with Fic. 39.—Fine-structure with arrange- 
 binary arrangement. ment on a binary screw axis. 
 
 Fic. 40.—Fine-structure with Fic, 41.—Fine-structure with trans- 
 reflexion. lation-with-reflexion. 
 
 Fics. 36-41.—Simple stereograms of crystallographic fine-structure. 
 
 the symmetry elements, and therefore all the fine- 
 structural particles, occur in the crystal in periodic 
 
CRYSTALLOGRAPHY, LEPTOLOGY 35 
 
 sequence, as Figs. 36-41 show for the simplest cases 
 of crystal structure. In short, allowing for all the 
 crystallographic possibilities, there are 230 types of 
 crystallographic fine-structure, and consequently a 
 similar number in the stereochemistry of the solid 
 state. Every crystallographic substance is consti- 
 tuted according to one of these schemes of A. Schén- 
 flies and E. Fedorow, and its specific character in the 
 material world is expressed in the absolute measure 
 of the periodicity in the point system, and in the 
 magnitude of its angles. 
 
 EXTENSION OF THE MANIFOLD OF CRYSTALLO- 
 GRAPHIC TYPES BY TWIN FORMATION 
 
 Reviewing the fine-structural relations, the thirty- 
 two classes with their subdivisions into 230 space 
 groupings present themselves as the embodiment of 
 the principle of association. Each crystal unit is the 
 model of a structural style containing certain elements 
 of this aggregate. Taken together they constitute 
 a complete system of forms based on Hatiy’s funda- 
 mental law (p. 8), a system to which mathematical 
 thought has nothing to add. 
 
 It is thus a surprising thing to observe that nature 
 in numberless cases pushes this principle of associa- 
 tion still further, for she takes as unit the whole 
 crystal and combines it in regular fashion with its 
 like to give so-called twinnings. It was G. v. Tscher- 
 mak again who pointed out the parallelism between 
 form development from elements, such as the five 
 primitive forms and this higher association into 
 twins. If a certain crystal lacks a centre of sym- 
 metry, as, for example, the “‘ hemimorphic succinic 
 iodimide ’”’ shown in Fig. 42, this is occasionally 
 
36 CRYSTALS AND MATTER 
 
 remedied in nature by a regular growing together of 
 two individuals. The complex (Fig. 43) is centre 
 symmetrical. In the same way, twin formation with 
 
 FIG. 42.— FIG. 43.— FIG. 44.— 
 Hemimorphic crystal of Centre symmetrical twin Right-handed 
 succinic iodimide. of succinic iodimide. quartz. 
 
 respect to a symmetry axis is accompanied by an 
 increase in the symmetry. If, for example, a right- 
 handed quartz (Fig. 44) exhibits the symmetry 
 elements 3s, then its association, often occurring as 
 
 Fic. 45.—Left-handed quartz. Fic. 46.—Right- and left-handed 
 
 quartz, twinned. 
 
 an ingrowth with a left-handed quartz in a twinning, 
 ‘as in Fig. 46, is of the type 3m. Thus there is 
 an increase in the symmetry from the sphenoidal s 
 form to the prismatic m form (p. 31). On the other 
 
CRYSTALLOGRAPHY, LEPTOLOGY 37 
 
 hand, a regular twin formation of two right- or left- 
 handed quartz crystals (Fig. 47) has the symmetry 
 6s. Such regular non-parallel combinations can be 
 constructed by rotating one crystal with respect to 
 the other about a fixed axis (in the example of 
 Fig. 47, 60° about the whirl axis). A combination 
 along a twinning plane z, after application of a rotation 
 of 180° about the normal to z, acts similarly as illus- 
 trated by gypsum (Figs. 48 and 49). 
 
 Fic. 47.— Fic. 48.—Gypsum. Fic. 49.— 
 Two left-handed quartz, Gypsum twinning. 
 twinned. 
 
 The symmetry is thus augmented by a mirror 
 plane, and the type 2d replaces that of m. 
 
 Such extraordinarily widespread phenomena in 
 nature are certainly of considerable general signific- 
 ance. This consists primarily in the recognition that 
 a visibly high symmetry may in reality represent an 
 ageregation of parts of lower symmetry, even if in 
 the limit the complexity is no longer recognisable 
 by the eye owing to fineness of the parts. Actually 
 there are many gradations of such mimesy from 
 macroscopically definite twinnings (such as pseudo- 
 hexagonal chrysoberyl) (Fig. 50) to the finest struc- 
 tures arising from “‘ polysynthetic’’ repetition of 
 
38 CRYSTALS AND MATTER 
 
 very thin lamellz, shown, for example, by microcline, 
 the fineness of which verges on the sub-microscopic. 
 
 We may recognise, therefore, in twin formation 
 an attempt towards higher symmetry than that 
 possessed by the individual crystal; in addition, 
 however, the relations also indicate that the twin 
 grouping, in contrast to the ideal parallelism in the 
 structure of a simple crystal, presents a less compact 
 fitting-in of the particles with each other. The space-. 
 lattices of the parts of the twin do not pass con- 
 tinuously one into the other. In this respect it is 
 of further interest that, according to the ideas .of 
 Ch. Friedel, fine-structural parallelopipeds, at least 
 of higher orders, may be constructed which satisfy, 
 or nearly so, the condition of parallelism. The 
 diagram of Fig. 52 makes this clear. 
 
Fic, 50.—Chrysoberyl sextet as an Fic. 51.—Fine twinning of microcline 
 example of mimesy (micrograph) 
 
 ie 4 A | 
 
 i Wy 
 
 fi cite il Hf 
 Aas Ff ease 
 
 Fic. 52.—Fine-structure of a twinning 
 
.s Tile LIBRARY. 
 OF THE 
 URIVERSITY oF jLLINOES 
 
IV. FINE-STRUCTURAL UNITY OF MATTER 
 
 FINE-STRUCTURE OF AMORPHOUS BODIES COMPARED 
 With FHAT OF CRYSTALS 
 
 were to leave the fine-structural relations of 
 
 crystals, which have chiefly interested us up to 
 now, without attempting to connect them up with 
 general leptology which concerns the physicist and 
 chemist, in their study of the numerous examples of 
 amorphous bodies. Actually, crystals and amor- 
 phous bodies are very closely related regarding their 
 fine-structure. The name of the latter class hardly 
 accords with the facts of leptology. The ordinary 
 chemical formule and ideas concerning the mor- 
 phology of atoms of gases and liquids, indicate 
 clearly a definite anisotropic structure. In this way, 
 chemical formule, such as, for example, 
 
 / CoH, N(CHs)2 ; 
 NaCl; (NH,) Cl; C,H, C 
 
 I; would be contrary to scientific principle if we 
 
 a 
 PG, N(CH,)2 
 
 NH, | NH, 
 
 Cl Ni Cra bis roa CO rs, 
 
 NH; | NH; 
 NH, 
 and the diagrams of Fig. 53 represent fine-structural 
 schemes: To F. M. Jaeger must be accorded the 
 special merit of having first recognised and enlarged 
 39 
 
40 CRYSTALS AND MATTER 
 
 upon the symmetry relations of the individuals of 
 amorphous bodies in his very fine work, “ Lectures 
 on the Principle of Symmetry” (1917). Figures such 
 as 53, 54, and many others of atoms, ions, and mole- 
 cules show, without further remark, a regularity in the 
 form of the individual leptons, a regularity a 
 
 O : 
 
 Electron H: Ion 
 ca. 10-13 cm. ca. 10-16 em. OR ee Atom ca. 0°55 X 10-8 em. 
 
 nae 
 
 A, Molecule Crystal 
 
 Fic. 53.—The leptonic series. Models of an electron, atom, ion, molecule, and 
 a crystal. 
 
 d Inosite t Inosite 
 
 ef 
 eH 
 °OH Symmetry plane 
 
 o86 
 
 Fic. 54.—Examples of symmetrical molecules. 
 
 referred to by chemist and physicist. Considerations 
 of symmetry, whether it be total absence of sym- 
 metry, centre symmetry, rhythmic architecture in the 
 sense of gyric or gyroidal repetition, or, finally, mirror 
 symmetry, are of importance here as in crystals. 
 
 In fact, we may assert with P. Debye and 
 P. Scherrer that, allowing for the difference of scale, 
 
UNITY OF MATTER 41 
 
 as regards general structure there is no essential 
 difference “‘ between a crystal and a chemical mole- 
 cule, for both have the characteristic property of 
 containing atoms regularly arranged.” 
 
 With respect to this intimate relationship, it 
 may be added that between the leptonic forms of a 
 liquid or gas from which crystals are separating and 
 the crystals themselves there is a definite morpho- 
 logical connection. To each crystal type will belong 
 certain characteristic preliminary forms occurring in 
 crystallisation. 
 
 This fundamental and general relationship of the 
 fine-structures of atoms, molecules, and crystals may 
 also be applied to the arrangement of the negative 
 electrons forming the external shell of the atom, 
 which swarm of peripheral corpuscles gives to the 
 atomic complex its “shape.’’ We speak of the 
 symmetry of atoms as of crystals, meaning by that 
 the symmetry of the external electron arrangement. 
 The tetrahedral form of a carbon atom, for example, 
 is indicated by four electrons at the corner of a 
 tetrahedron. Following A. Johnsen, we presuppose 
 a minimum symmetry of the atom in crystallo- 
 graphy, and this is determined by crystallographic 
 symmetry relations. The minimum symmetry for 
 a given kind of atom changes with the symmetry 
 conditions prevailing at the place where the particle 
 occurs in the fine-structure. The carbon atom of 
 the diamond has to fulfil other symmetry relations 
 than that of graphite. In consequence, the arrange- 
 ment of the electrons in these two cases must be 
 different. That this is so is shown distinctly by their 
 different optical properties. Thus we may compare 
 the transparency of the diamond, on the one hand, 
 
42 CRYSTALS AND MATTER 
 
 with the blackness of graphite, on the other. But 
 even in the same crystal the electron arrangements 
 of similar atoms may differ. In the diamond all the 
 carbon atoms are identical ; in graphite two varieties 
 must be postulated to satisfy symmetry considera- 
 tions. Recognising this, the mutual influence of 
 different atoms and ions in the crystal is to be borne 
 in mind, for their forms must depend on their neigh- 
 bours for the time being. The action of a fluorine 
 ion on the fine-structure of a neighbouring sodium ion 
 is different from that of chlorine, bromine, or iodine 
 ions, and these again are affected differently by the 
 electron orbits of sodium potassium, rubidium, and 
 caesium. Moreover, temperature itself must be recog- 
 nised as a factor tending to determine the structure. 
 The electron arrangement in free atoms or mole- 
 cules will not be essentially different from that in 
 the crystal. Indeed, recently the physicists M. Born, 
 H. Landé, and others, following crystallography, 
 speak of cubic and other polyhedral atoms. The 
 diagrams in Figs. 55-57 are drawn in accordance 
 with this general idea. In particular, the known 
 periodicity of the series of the elements with respect 
 to the number eight, suggests as a probable distri- 
 bution a surface-centred octahedral arrangement for 
 elements with eight external electrons. By increasing 
 the volume, that is, by adding further electron shells 
 for the elements of higher atomic number, other 
 stable arrangements, partly of a crystallographic 
 nature, but approximating to an isotropic distri- 
 bution, will arise; such configurations are treated 
 in detail in a recent treatise by H. Tertsch.t Fin- 
 ally, the electron swarm becomes less stable as 
 
 1Compare p. 84 and p. 181. 
 
UNITY OF MATTER 43 
 
 the charge on the nucleus is increased, so that the 
 series of possible elements on our earth ends with 
 uranium, a spontaneously 
 disintegrating atom with 
 ninety-two external elec- 
 trons. 7 
 
 © Nucleus 
 electron 
 
 ata 
 
 e Nucleus with two electrons 
 © Hlectron 
 
 Fic. 57. 
 
 Fics. 55-57.—Schemes for electron groupings in atoms, molecules, and 
 crystals. Examples: carbon atom, methane molecule, diamond. 
 
 PHYSICAL INVESTIGATIONS ON THE GENERAL FORMS 
 OF ATOMS AND MOLECULES 
 
 Fitting in with our ideas on the analogy 
 between crystals and individual leptons, a similar 
 correspondence in the behaviour of crystalline and 
 amorphous substances of the same chemical com- 
 position is shown in certain physical processes. We 
 find that the curves obtained by Cl. Schaefer and 
 M. Schubert for the reflexion of short ultra-red 
 
44. CRYSTALS AND MATTER 
 
 waves by quartz and opal completely correspond, a 
 clear indication of the fine-structural similarity of the 
 SiO, particles, in spite of the difference in the gross 
 forms, quartz showing the characteristic space- 
 lattice structure lacking in the opal. In their action 
 on X-rays, too, amorphous and crystalline materials 
 behave in much the same 
 general fashion. An ex- 
 tremely fine crystal pow- 
 
 | der approximates for this 
 4 --f---4 purpose to the amorphous 
 Oe eee material with which the 
 
 2,4 6) 8 ORR 8 ee eey complete crystal 4S een 
 ultra-red for quarts and opal. After linked up through inter- 
 Cl. Schaefer and M. Schubert. mediate forms. Accord- 
 ing to Debye and Scherrer, the subdivision in soot 
 actually extends to complexes of only thirty atoms of 
 carbon; but these still give the normal X-ray effect 
 of crystals. 
 
 With amorphous substances circular shadings 
 round the point of impact of the primary ray are 
 obtained, although, on account of the small number 
 of components in the kinetic unit, the considerable 
 internal heat motions, and the irregular state of 
 aggregation! of the particles, only one or two weak 
 and diffuse rings appear on the plate. It seems to 
 me, however, that all this points to a definite regular 
 form of the particle. Fig. 59 illustrates the produc- 
 tion of such an effect, exaggerated for the sake of 
 clearness, with the imaginary case of a molecule 
 with cubic arrangements of its atoms.’ 
 
 i 
 SCC Neel 
 
 1 Also occasionally, as in glass, owing to the interaction of the 
 different types of molecules. 
 
 * For readers interested in crystallography, diagrammatic deriva- 
 tions of the lines in the Debye-Scherrer diagram are shown in Fig. 59, 
 
UNITY OF MATTER 45 
 
 NotE.—The conception of colloidal matter is not 
 synonymous with that of amorphous substances. 
 Its characteristic 1s a medley of particles of magni- 
 tude 10-* to 107-7 cm., which thus lie between 
 leptonic dimensions (about 10~® cm.) and micro- 
 
 Ses Phot. plate section) 
 Phot. film (cylinder) 
 
 iP 
 
 | 
 : 
 oe Se ae 
 
 Incident Beam Incident Beam Incident Beam 
 
 geass 
 meas, 
 
 Fic. 59.—Debye-Scherrer diagram. Example of a cubic molecule shown 
 schematically. 
 
 scopic visbility (about 10-5 cm.); they may be 
 either crystalline or amorphous. The colloidal solu- 
 tions of gold and silver, for example, previously re- 
 garded as amorphous, show, according to P. Scherrer, 
 obtained from the principal structure planes in the imaginary cube 
 molecule. These are obtained by applying the principle of reflexion, 
 
 but it must be remembered that the fundamental phenomenon is 
 one of diffraction. 
 
- 
 
 46 CRYSTALS AND MATTER 
 
 the typical phenomena of crystalline materials (Fig. 
 60, upper diagram). Silica gel also gives a crystal 
 effect, but at the same time acts an amorphous sub- 
 
 Fic. 60.—Debye-Scherrer diagrams of colloidal silver and silica gel. After 
 P oeherrer. 
 
 stance (Fig. 60, lower diagram). In this preparation 
 
 we have to deal with an amorphous gel containing 
 
 small SiO, crystals disseminated in it. 
 
 ATOMIC DOMAINS 
 
 Stereograms of crystals, such as that of Fig. 61, 
 which shows the elementary cell of metallic sodium 
 in representing the atoms as mere points, give no 
 indications as to their solidarity.t It is possible, 
 however, to endow such symbols of the crystal 
 structure with a dynamical basis, and, in fact, this 
 is done by describing about each atom a domain, to 
 represent a portion of space which it claims for 
 itself and keeps free from other atoms. Under the 
 action of attractive and repulsive forces the atoms 
 in the structure mutually maintain each other in an 
 equilibrium arrangement, which is characterised in 
 the close packing of these atom domains, which are 
 to the first approximation spherical. The radius of 
 the domain is given simply as half the least distance 
 between the atoms in a stereogram similar, say, to 
 Fig. 61. Thus Fig. 62 shows the domain picture for 
 the sodium of the previous stereogram. 
 
 In this case the radius of the domain of each atom 
 
 1 In the scheme for diamond (p. 43) this is indicated. 
 
UNITY OF MATTER ~— 47 
 
 amounts to 1°86 x 10~*® cm. Passing from one 
 material to another as from sodium to sodium chloride 
 (Fig. 63), thus comparing simple 
 substances with compounds, we 
 arrive finally at a table of magni- 
 tudes for the atomic domains, a 
 table obtained by W. L. Bragg, who ~ 
 
 first carried out systematic experi- Ti 
 
 ments on these lines. Fic, 61.—Stereogram 
 
 P. Niggli has given a similar  °% scum cyst. 
 
 scheme. Further applications in crystallography are 
 due to G. Aminoff, and the author has also expressed 
 
 Fic. 62.—Atom domains in stereo- Fic. 63.—Atom domains in stereogram 
 gram of sodium crystal. of rock-salt. 
 
 his views on these very promising relations, both in 
 
 his lectures and in occasional publications. Some 
 
 new determinations may here be put before the 
 
 reader for his guidance (compare also p. 107). 
 Diameters of crystallographic atom domains in 
 
 10 ® cm. units :— 
 
 E35 ane ees-OOulbiger e. ”.|-2-69 \ = Pe Me is ty ine, 1+26 
 eas. epenine are vg. “Sct Cla Ae ae ot Rea a . 2°00 
 | he eae ort 9 45 3°90.) Bre Jae Ae ANC. pea 
 Bpre" may Bart a e436}. ee eg me owe so 
 Cac. » 5°04 
 
 That the atomic domain depends on the nature of 
 the surrounding atoms and the electric charge, has 
 
48 CRYSTALS AND MATTER 
 
 already been emphasised by K. Fajans, H. Grimm, 
 and W.L. Bragg. The alkali metals, for example, 
 show somewhat larger values than their salt ions 
 tabulated above. We have, in fact, Li, 3:02; Na, 
 B7 ER NS ASO WARD M4 NOS Ue 1on 
 
 The transference of these numbers obtained in the 
 case of the crystal to the individual leptons, is a 
 
 Oo) 02) 
 Co ee 
 
 Brides + dno coco 
 RTT Co 
 
 "290 ‘aekie. eee 
 Fic. 64.—Molecular domains Fic. 65.—Molecular domains of 
 OF 0 guNor CO. CO)., iH PEIN 35 70 
 
 procedure in perfect accord with the views adopted 
 in this book. In accordance with the considerations 
 on p. 39 et seq., the chance of error is not so great as 
 might at first appear. An investigation of this point 
 is contained in the Figs. 64 and 65, and the values 
 given agree very well with those arrived at from de- 
 terminations of the mean free paths of the molecules. 
 
 Molecular Diameter. 
 Crystallographic Data. 5 : : 
 y Hed Landolt and Bornstein’s Nernst. Theoretical 
 
 Tables. Chemistry. 
 O, ued §2) eek De ems ae 00 107~*% cm. | 2:00, 4.) a0ssee 
 N, os 2600, rOp Bem. S80 400) NO 3 cms ao 7 eee 
 GO, 2s BOO rs TO Orca 2-9 a. 1G Sem eo) ee 
 Cl, eo 4240-S°> 2107 cm.)| 48) | SIO om ara A 
 
 The mathematical treatment of the question leads 
 to the most difficult branch of mathematics, the 
 
UNITY OF MATTER 49 
 
 problem of several bodies. Fora long time, however, 
 approximations must suffice. Nevertheless, thanks 
 to the efforts of M. Born, A. Landé, K. Fajans, 
 F. Madelung, H. Thirring, and others, methods of 
 approach to the end in view have been laid down, e.g. 
 with respect to the attractive and repulsive action 
 of the ions. The attraction here is put inversely as 
 the square of the distance apart of the ion centres, 
 whilst the potential of the repulsive forces (corre- 
 sponding for the alkali halides to the compressibility 
 of the crystal) involves a higher (5-9) power of the 
 distance. This will naturally depend on the size and 
 nature of the particles, and in view of the anisotropy 
 of the sphere of action, may vary with direction. 
 
 Although the working out of exact mathematical 
 theories of the fine-structure of matter must be left 
 to the future, a glance at the spacial crystal schemes 
 at once suggests the diagnosis and interpretation of 
 many peculiar properties of substances. 
 
 DIFFERENCE BETWEEN THE STRUCTURE OF 
 INDIVIDUAL LEPTONS AND CRYSTALS 
 
 The chief difference in the construction of crystal- 
 line bodies and the individuals of amorphous sub- 
 stances lies in the restriction of the rhythm. In the 
 fine-structure of crystals, owing to the fact that 
 each structural unit is joined to its neighbours, 
 this rhythm exhibits a three-dimensional periodicity. 
 Such space-lattice structure is clearly only possible 
 when the repetition is according to the numbers 
 2, 3, 4, and 6, or with no repetition at all. Pentad, 
 septad, and compound axes of higher period are here 
 theoretically excluded, nor are they found in practice. 
 
 4 
 
50 CRYSTALS AND MATTER 
 
 A boundary surface not containing rifts or gaps is 
 known to be impossible with such polyhedra (Fig. 66) + 
 We thus have here the essential principle in the 
 restriction of the crystalline forms to the thirty-two 
 classes of page 31, and thus to 230 space groupings. 
 
 Crystallographic Rhythms 
 
 FECOOHH 3 
 
 Digyral Trigyral Jetragyral Aterogyraf 
 
 Non-crystallographic rhythms possible in molecules 
 
 Fic. 66.—Diagrams showing crystallographic and two non-crystallographic 
 rhythms (pentad and septad). 
 
 2 ted 2 cag 
 Fic. 67.—Irrational axial section for eight-fold rhythm. 
 
 Further, this fine-structural limitation is in agreement 
 with Haitiy’s crystallographic law of simple rational 
 axial sections. A crystal rhythm corresponding to 
 the number 8 (Fig. 67), cannot arise, for such a 
 
 1F, A. Wilfing has already pointed out the importance of this 
 fact in crystallography. 
 
 a 
 
UNITY OF MATTER 51 
 
 regular octagon would give an axial section of 
 2°4142 ... Similar relations hold good when repeti- 
 tion occurs according to any number other than 
 2, 3, 4, or 6, from which it also follows that of the 
 “yegular’”’ polyhedra of mathematics, the cube, 
 tetrahedron, octahedron, and icosahedron, only the 
 
 BOY 
 
 Fig. 68. Fic. 70. Fic. 7I. 
 
 Fics. 68, 70, 71.—The mathematical regular polyhedra of crystallography. 
 first three are represented in the inorganic realm of 
 nature. The Figs. 68, 70-72 should show the reader 
 that the first three fall within the above restriction 
 in the rhythmic arrangement of their surfaces, whilst 
 the icosahedron, the non-crystallographic five-fold 
 repetition of which is clearly shown in Fig. 72, does 
 
 Fic. 72.—The non-crystallographic regular icosahedron. 
 
 not. Such inadmissible rhythms are absent from the 
 macroscopic crystal form, and also from the fine- 
 structure. With regard to the latter, the Laue dia- 
 grams are unimpeachable witnesses (Fig. 73) ; in the 
 structure of free atoms and molecules, both crys- 
 tallographic and non-crystallographic rhythms may 
 equally well occur. 
 
 NoTE.—Fic. 69, which appeared erroneously in the original, has been 
 deleted. 
 
52 CRYSTALS AND MATTER 
 
 e eee . om. a nie Pe . . 
 ee a H 
 2 ae & or o Sea a : 
 . wey Ay Os Se e e e ie A ee y i Py 
 e Ye Pe ee a2? © ® ®@ e 5 . aOR oi" 6 age e 2 sree 
 bo dK) *e, ® °@e- 5 ‘°°? “ ris 4 e Ae O°8e" 0. : e 
 a e 6 ‘ ie? eco Ps ae) or or Py eg es a ee ° 
 oe f is . . @ «= %6 
 eee oe ae st sees See 
 ® e. @ e of e ~ “2 Cn ) e-.° e 
 ee Orne Yee ° ° ve. ce Na 
 . * ee © e ee ei iY ete G e Par e nies © i. ae 
 ° Ay ae sad oS ® e° e seek at eer ae’: a 
 ee 4h ote g co's & e ee. . Xe * ae $ ee, = x 
 ® @ Cae, ett te a Ai 
 e ® Seg? e Vee, ene Noy ee : 
 e eel we By ° @ e a9 Sb ars ® =e 2 wee . 
 76°. hg 29g 28 - ° pite* os) . eee Ce0-6 <7 Yea oe 
 @ a Si\e°8 Se e res 5 acti sie eee e wee 
 e e& eo? ° e ® oF. e - oO se: AO ae 
 e oS @ee SP a! e e e See ae 5 at 
 ° a 3 ; 
 ee, 2 ae labes ; 
 ° e e o%8 c & « : - : a 
 e e F 
 = b 
 : y ; . % . 
 e hd a ; 
 i xs e. “ 
 wae, ? ; e e. . 
 x ; ad 5 ° ° e @ ° 
 ° e e e ° te eo ®@ «6 . 
 ee .- @& e e e . ; : 
 hese Ms ry} Pig e° . e e e corie Ab e @ Rn 
 s . : . e OF , m 
 rey ADRS e ~* ® oe? - 3 we ky) ; ay, 
 ° ate . or ae? de ° n A ? 
 eee ein Want | ana) oe e € ; 
 e we ee Ayo e e e ° e e 
 3 a A . e ; e . e .° e, z 
 ? S e. ret s . ‘ : : 
 ° -@ ° ts ‘ e” ~° ®@ 
 ' 3 Were : ae 6 +3 3s e, e ° 
 om 6 ~® e e oe ‘ @ e Q a e : 
 ee ° ee ® e e@ 6¢ e ae 5 as 4 Pd ‘ / 
 ¥ e Chk ert PuAP ar peer) v3 - e e . 
 e aa) ore e were Z ° . e ) e 3 
 e mh pee e e, . ( ses , >| 
 se : - . ° 
 i! Pas ° e ang 7 ; 3 2 
 e e e e 
 e e 6 e . . 
 ° e 
 
 rj oo ’ 
 
 e*e. ,e° Sey ote ey °@, Jee 
 eee 
 
 O10 see Pov ." 9. 8e® 2 Shia y 
 
 : 
 
 Fic, 73.—Laue diagrams with repetition according to the numbers 1, 2, 3, 4, 
 
 and 6. After photographs by F. Rinne. a. Cyanite. b. Sanidine. 
 c. Calc-spar. d. Rock-salt. e. Beryl. 
 
UNITY OF MATTER 53 
 DIAGRAMS 
 
 Binary type. Rhombic system. 
 Ternary type. Trigonal system. 
 Tetrad type. Tetragonal system. 
 
 Senary type. Hexagonal system. 
 
 Fic. 74.—Whirl forms of the crystal types deduced from the primitive forms. 
 
V. THE GENERAL CHARACTERISTICS OF 
 THE FINE-STRUCTURE OF MATTER 
 
 FINE-STRUCTURAL CHARACTER OF CRYSTALS 
 
 F, as we have emphasised, the difference in the 
 
 architectural rhythms of the structures of indi- 
 
 vidual leptons and crystals is merely a matter of 
 detail, then the question arises as to what constitutes 
 the prevailing characteristic in the fine-structure of 
 all substances. 
 
 Taking crystals as types, we find that they ex- 
 hibit very definitely two peculiar qualities-—firstly, 
 change of properties with direction (anisotropy) ; 
 and, secondly, stability (isostasy). 
 
 A. CHANGE OF PROPERTIES WITH DIRECTION IN 
 CRYSTALS 
 
 This anisotropy makes its appearance in the mor- 
 phology of the crystal, in the regular arrangement 
 of the external faces. Rock-salt, for example, may 
 develop in a certain direction a surface, say, that 
 of the cube; and this development repeats itself 
 in a definite series of isolated directions, to which 
 the edges and angles of the structure conform. 
 
 The diamond is provided with an octahedral 
 form in a similar way. As to the fine-structure, 
 Figs. 75-78 represent with remarkable clearness the 
 various modes of formation of certain principal 
 
 54 
 
FINE-STRUCTURE OF MATTER 55 
 
 surfaces of the minerals, showing the anisotropy of 
 the structure. 
 | The change of physical properties with direction 
 is very obvious in crystals showing cleavage. Thus 
 for rock-salt there are three directions, perpendicular 
 to the cube surfaces, in which the cohesion of the 
 crystal is a minimum. The resistance to splitting 
 in these directions is only one-third of that in the 
 direction of the cube diagonal. The cleavage planes 
 are, therefore, regularly oriented surfaces of maxi- 
 mum brittleness. In like manner, many crystals 
 show particular planes of maximum plasticity. These 
 
 Fic. 75. Fic. 76. Fic. 77. Fic. 78. 
 
 Fics. 75-78.—Structure of the diamond parallel to surfaces of the cube, rhombic 
 dodecahedron, pyramidal cube, and ‘octahedron. 
 are surfaces in which internal displacements may 
 easily occur; with ice they arise as planes parallel 
 to the surface of the ice floe. Hardness is also a 
 directional property in crystalline materials. The re- 
 sistance to disrupture, which shows itself as hardness, 
 is often different in different directions of the crystal ; 
 thus it is a familiar fact to diamond workers that the 
 cube surfaces of the gem are more difficult to prepare 
 by polishing than the octahedron surfaces. Garnet, 
 too, is harder on the cube surfaces than on the 
 octahedron and rhombic dodecahedron, according 
 to the researches of P. J. Holmquist on polishing. 
 Even on the same surface of a crystal the hardness, 
 
56 CRYSTALS AND MATTER 
 
 as measured by scratching, varies with the orientation 
 of the scratch made by the test needle. Cyanite is 
 a classical example of this. 
 
 Further, demonstrations of this change of pro- 
 perties with direction are given, often very strikingly, 
 by optical tests. On passing ordinary daylight 
 through the mineral cordierite such extremes as 
 these arise in the absorption of light ; a preparation 
 in a certain direction appears dark blue; that in 
 another direction, yellow; and in a third, grey. | 
 Moreover, the customary idea of constant wave 
 
 Anisotropy of Crystals 
 (Regular change of properties with direction] 
 
 oS 
 : 
 ee 2 
 ae A | right blue | P= 
 eZ a8 
 yx 
 Anisotropy Anisotropy of Anisotropy of Anisotropy of Anisotropy of 
 of form Cohesion, for Lightabsorption Heat Conduction Ghemical Reaction 
 
 Example: Rock-salt. example: Cleavage. Example: Cordierite. Example: Gypsum. Example. Tourmaline. 
 Example: Mica. 
 
 Fic. 79.—Demonstrations of the anisotropy of crystals. 
 
 length as a measure of the velocity of propagation 
 of the light fails for all non-isometric crystals. For 
 such substances the value of X for a given colour 
 changes with the direction of the light, and wave 
 length curves can be drawn showing the variation 
 diagrammatically. 
 
 Variation of thermal properties with direction 
 may also, in many crystals, be very clearly demon- 
 strated. Fig. 79d indicates the heat conduction in 
 a cleaved plate of gypsum. A thin layer of wax has 
 been deposited on the gypsum plate, and the prepara- 
 tion heated from a central point with a hot wire. 
 
FINE-STRUCTURE OF MATTER 57 
 
 The wax is melted to various extents in different 
 directions corresponding to the heat conduction in 
 gypsum ; the perimeter of the figure formed by the 
 melted wax is indicative of the heat conduction in 
 the crystal beneath. 
 
 Even in chemical actions a definite change of 
 properties of materials with direction is unmistak- 
 able. Thus, for calc-spar, varying intensity in the 
 reaction . CaCO, + 2HCl = CaCl, + H,O + CO, is 
 clearly shown by the different amounts of CO, 
 liberated under the same conditions from the different 
 . faces of the mineral. Hence resistance of calc-spar 
 to attack by hydrochloric acid varies with direction. 
 The differences which arise are surprisingly large. 
 According to O. Miigge, quartz is attacked by fluoric 
 acid 150 times more easily in the direction of 
 the whirl axis than in the direction perpendicular 
 thereto. 
 
 It is of considerable significance that in crystal- 
 line materials “ simple vectorial’ directional differ- 
 ences can occur. The tetrahedron is a morphological 
 example of this; it lacks a centre of symmetry. 
 
 A chemical example is depicted in Fig. 79e. The 
 diagram represents a tourmaline sphere which has 
 been transformed by caustic potash to a bee-hive 
 shaped body, a clear indication that the chemical 
 reaction between the silicate and its corrodent occurs 
 much more rapidly in the direction from below to 
 above than from above to below. 
 
 It is, therefore, a characteristic of crystalline 
 materials evidenced by numerous experiments of 
 morphological, physical, and chemical nature that 
 they exhibit different properties in different direc- 
 tions. 
 
58 CRYSTALS AND MATTER 
 
 Corresponding to this the sphere of action of 
 crystals is also anisotropic, as growth phenomena in 
 particular show. A crystalline sphere grows to a 
 body with edges, corners, and plane faces. 
 
 By SITARILIIM OF CRYSTALS 
 
 Associated with the characteristic of anisotropy 
 in the crystal is the second general property, internal 
 equilibrium or isostasy ; the whole constitution of — 
 the crystal as a stable form exemplifies this. A. Nold 
 
 Benzene molecule 
 
 Fic. 80.—Morphological anisotropy of molecules. 
 
 has especially taken this into account in his work 
 on the crystal structure of the diamond. However, 
 in the fine-structure of crystals the question is one 
 of dynamical equilibrium, that is to say, kinetic 
 stability or isodynamostasy.* 
 
 1 As a parallel to this, there is the isostasy of geology, from which 
 science the name isostasy is here borrowed. The earth, however, is 
 not in isostatic, but in isodynamic equilibrium. Its rotation in- 
 volves equatorial bulging, and as a result of this there arise fissures 
 tending to prevent such an adjustment ; these, as the boundaries of 
 continents, and rifts within them, run from north-east to south-west, 
 south-east to north-west, and meridionally. 
 
FINE-STRUCTURE OF MATTER 59 
 
 FINE-STRUCTURAL CHARACTER OF GASES AND 
 LIQUIDS 
 The general characteristics in the construction 
 of the individual leptons (electrons, ions, atoms, 
 and molecules) cannot be different. The graphical 
 schemes and formule given for the structure of atoms 
 and molecules indicate morphological anisotropy ; 
 
 as 
 
 Benzene drop 
 
 Fic. 81.—Pseudoisotropy of an aggregate of molecules (benzene drop). 
 
 but their morphological, as well as their chemical and 
 physical anisotropy, will be entirely obscured by the 
 irregular arrangement of the particles and will thus 
 be transformed into an isotropy by averaging. Mor- 
 phologically, this is illustrated by the spherical form 
 of free gases (e.g. the earth’s atmosphere) and in the 
 drop shape of liquids. The general stereophysical 
 conception of the electrons, atoms, and molecules as 
 kinetic units at once indicates their isodynamostatic 
 character. 
 
60 CRYSTALS AND MATTER 
 
 GENERAL CHARACTER OF THE FINE-STRUCTURE OF 
 MATTER 
 
 According to the above, the fine-structural ar- 
 rangements of every substance represent anisotropic 
 stability forms. 
 
 Their structure will be conditioned by attracting 
 and repelling forces. The particular arrangement of 
 the particles of an aggregate is always the result of 
 a complex action of all its particles on one another; 
 it is not characterised by some kind of linear or 
 curvilinear force threads. 
 
 In addition, the aggregation units, whether atoms, ~ 
 ions, molecules, or crystals, also act on each other 
 when in close proximity. There arises, besides the 
 endoleptonic field of force effective in the individual 
 structures, an interstitial field depending on the 
 reciprocal relations of the component substances, and 
 which, therefore, is not constant for each type of 
 atom, but is a function of the nature and arrange- 
 ment of the neighbouring ones. What is observed 
 in chemical processes is a consequence of this inter- 
 action, whether it be a change in arrangement, asso- 
 ciation of previously separated similar or dissimilar 
 particles to higher units, disruption or substitution, 
 or whether the action be partly more physical or 
 definitely chemical. The setting up of a physical 
 field, a change of temperature in particular, or a 
 change of pressure, may initiate similar processes 
 or modify them. In other words, all physical and 
 chemical actions of substances proceed in accordance 
 with these ideas. 
 
VI. THE SERIES OF TRANSFORMATIONS OF 
 MATTER 
 
 See OULDS LIQUID CRYSTALS) CRYSTATS 
 
 HE broadest survey of the general physical 
 relations of fine-structural aggregates under 
 
 the influence of attractive and repulsive forces 
 anisotropically directed, is afforded by a considera- 
 tion of the changes of state which all substances pass 
 through when their physical conditions are altered. 
 By changing the temperature of a substance, that is, 
 by speeding-up or retarding the motions of its fine- 
 structural particles, it is possible, as is well known, to 
 pass the substance through a long series of meta- 
 morphoses extending through the gaseous, liquid, 
 and solid crystalline states. Thus H,O, for example, 
 traverses the states, water vapour = liquid = ice, 
 and on the basis of the mechanical theory of heat, 
 we are in a position to form a very clear picture of 
 these changes. 
 
 In the gaseous state there is very considerable 
 leptonic unrest. With individual motions of the 
 velocity of a bullet the particles speed hither and 
 thither, although only travelling minute distances 
 in straight paths; colliding, rebounding, thrusting 
 aside, each molecule wins for itself a portion of space 
 the size of which is the same for all gases. That the 
 same number of material particles of any type take 
 
 up the same space is the essence of Avogadro’s 
 61 
 
62 CRYSTALS AND MATTER 
 
 Hypothesis. J. Loschmidt (1865) was able, in 
 furtherance of this hypothesis, to estimate the num- 
 ber of particles per c.c. Ato° C. and one atmosphere 
 pressure there are 27:6 trillion ; truly a dense popu- 
 lation of space, although, however, it should again 
 be emphasised that from particle to particle a space 
 of average dimensions thirty to forty times the size of 
 one particle must be assumed. The inter-connection 
 of the individual atoms or molecules by force fields . 
 is therefore very slight in gases; the resistance to 
 mechanical subdivision of masses of gas, in a certain 
 sense the hardness, is correspondingly small. More- 
 over, the disperseness depends on the pressure in 
 accordance with Mariotte’s law, and on the tempera- 
 ture, an alteration of 1° C. involving a change in 
 volume of = for any gas whatever. A permanent 
 arrangement of the particles, say about some instan- 
 taneous centre of their motion, is prevented by their 
 rapid movement and diffuse distribution, for they 
 roam about passing from one place to another by 
 irregular diffusion. 
 
 In liquids, on account of the smaller distances 
 of the particles, a field loosely binding the molecules 
 is present, as well as the endoleptonic forces, and 
 with this comes increased resistance of the mass 
 to subdivision, as the viscosity indicates. Indeed, 
 internal friction may often increase to considerable 
 hardness, as, for example, in glass, which is to be 
 regarded in a physical chemical sense as a “ rigid 
 liquid.’”’ Silica glass in its cohesion stands only a 
 little way behind quartz. For all liquids, the hard- 
 ness varies with the temperature. Warm water 
 flows through a funnel much faster than cold, owing 
 to there being a large diminution of the internal 
 
TRANSFORMATIONS OF MATTER — 63 
 
 cohesion of the particles, while glass on heating ap- 
 proximates to an ordinary liquid. 
 
 In general, a permanent arrangement of the par- 
 ticles does not arise in liquids. The case is different 
 for the so-called liquid crystals of O. Lehmann. 
 There the molecules arrange themselves in the inter- 
 stitial field of force more or less regularly oriented 
 with respect to each other, very often with one 
 direction parallel, so that the anisotropy inherent in 
 each one is shown by the single optical axis (as in 
 a crystal with a simple whirl axis). Three-dimen- 
 sional periodicity of arrangement, however, is lacking 
 in this microcosm. Thus Huckel, on investigating 
 such substances with X-rays by the Debye-Scherrer 
 method, did not obtain crystal diffraction patterns. 
 On the contrary, there appeared only the indistinct 
 interference ring shown by amorphous _bodies.} 
 These so-called liquid crystals are not then strictly 
 crystals, but rather intermediate stages to true 
 crystals, and as such “ penecrystals,’’ as one might 
 call them, they are of very great interest in science. 
 Their discovery and preparation by O. Lehmann and 
 D. Vorlander, in particular, is one of the finest 
 achievements of science. 
 
 Still further connections and transitional stages 
 between the structures of individual leptons and 
 typical crystals would possibly, if not probably, 
 be established if large molecules containing many 
 atoms, such as those of albumen and starch, with 
 
 1 In spite of this result, it would be of interest to investigate the 
 effect of passing the rays perpendicular to the common direction of 
 the molecules. Quite possibly interference diagrams would be 
 obtained of the type shown by fibres and flakes, although less 
 distinctly than for these substances. 
 
64. CRYSTALS AND MATTER 
 
 numerous similar groups in their structure,! were 
 capable of existing in space-lattice arrangement 
 wholly or in part. They would then give a Debye- 
 Scherrer crystal effect, contrasting, possibly only 
 
 Fic. 84. 
 Fics. 82-84.—Schemes for the gaseous, normal fluid, and liquid crystal states. 
 
 in the matter of degree, with ordinary molecules, 
 which, owing to the small number of similar struc- 
 tural groups in the kinetic unit, do not furnish a 
 space-lattice arrangement. [Let us assign, in this 
 
 1 Probably, too, atoms containing many electrons. 
 
TRANSFORMATIONS OF MATTER 65 
 
 connection, at least eight equal valued particles 
 to an elementary parallelopiped. Substances with 
 formule such as Cys9H729Niig5¢Oo43 (Serum-albumen), 
 CrssHicosNigsOoisFeS; (Dog’s Hemoglobin), gliadin 
 with thirty-eight molecular radicals of glutaminic 
 acid, and similar complexes, should possess enough 
 similar groups for crystalline structure in the mole- 
 cule; aggregates of only about thirty atoms of 
 carbon in graphite still show space-lattice character 
 
 Fic. 85.—Scheme for the crystal state. 
 
 according to Debye and Scherrer, as has already 
 been mentioned. 
 
 The results of the experiments of P. Scherrer, 
 R. O. Hertzog, and W. Jancke on cotton, cellulose, 
 starch, etc., for which the crystal effect with X-rays 
 was observed, certainly deserve further consideration 
 from the above point of view as being evidence for 
 the fine-structural nature of “crystalline molecules.”’ 
 
 On crystallisation the already anisotropic particles 
 arrange themselves in regular fashion into a space- 
 lattice ; by this the formation of external surfaces 
 in accordance with Hatiy’s Law (p. 8) is rendered 
 
 5 
 
66 CRYSTALS AND MATTER 
 
 possible. Thus, in the external ornamentation, we 
 have a reliable criterion for the crystalline nature of 
 a substance. A fine-structural medley of individual 
 leptons gives under the uniform action of the surface 
 forces, a spherical form to the lepton complex 
 (Fig. 81, p. 59), a form, moreover, not foreign to 
 crystalline materials, especially when very small 
 masses are considered. Fig. 86, showing drop-shaped 
 globulites, slightly curved longulites, pearl-like mar- 
 garites, in strings, and hair-like trichites, gives some’ 
 elegant examples from the mineral world. With 
 larger crystallisations of benzophenone and ice, for 
 example, R. Nacken has also obtained circular crystal 
 forms by using special methods of cooling. 
 
 The great disperseness of gases is enormously 
 diminished on crystallisation; in liquids, too, in 
 general, the same thing occurs.!. In this connection 
 some figures for sodium and the diamond, as two 
 extreme types, will be of interest to the reader. 
 
 Number of atoms per c.c. Ratio. 
 1. Gas at boiling-point (820° C.) . ‘ : A 5°5 * Io" x 
 2. Liquid at boiling-point . ; - 19,500 x 1038 355 
 3. Liquid at solidifying- point, 97: 6° 5 ‘ -') 24,500 *hiGr mn 
 4. Crystal at solidifying-point 2 ‘ : . 25,000 x 10!8 f 13 
 
 In solid sodium, with its body-centred space- 
 lattice (only two atoms in the elementary cell) and 
 its large a-value for the cube edge (4:3 x 10> 8 cm.), 
 we are dealing with a soft metal. The condensation 
 from gas to fluid is considerable; that for fluid to 
 crystal, small. 
 
 With the hard, compact diamond (eight atoms in 
 the elementary cell, 4 = 3-53 x I0~ ° cm); aaimuen 
 
 1Exceptions to this, as in the case of H,O, for example, are 
 
 explained by assuming molecular variations in the liquid when the 
 crystallisation point is reached. 
 
Fic. 86.—Globular and curvilinear crystals of microscopic 
 dimensions 
 
is A, =! ie 
 ¢ i a ¢ i 
 ek, y ha 
 te ? 
 ' ; are ; 
 a 7 4 
 ey ; 
 1 bn 
 ) 
 7 wt 
 iy 
 | 
 f ; 
 ' . 
 4 , 
 eh 
 4 ‘] 
 We TRE LIBRARY 
 ; 7 at 9 Lee 
 : Wr een 
 ~~ gurbeagert OF TRLIRBES 
 : . 
 j . 
 ) 
 i 
 } e | 
 | ‘ ’ 
 ! @ 
 ; 
 : : 
 ya! 
 ii a ee es 
 : . | F, 7 x in’ ie 
 = ; ce ake fe ‘ 
 Is me ; io, “aN te 
 K, at 34 ‘ c ue 
 eee ae 
 ra ea! aie : ee 
 ON ae 
 
TRANSFORMATIONS OF MATTER 67 
 
 more extensive condensation occurs on crystallisa- 
 tion; I c.c. of this gem contains 180-000 trillion: 
 carbon atoms, compared with 1-3 trillion in carbon 
 vapour at 5500° C. Naturally, with such close 
 packing of the particles of matter, the ree be- 
 tween them becomes enormous. 
 
 A contraction of the substance in crystallisation 
 until the external surfaces of the atoms touch is not, 
 however, to be assumed, in view of the known possi- 
 bilities of diffusion in the crystal. With change of 
 temperature zeolites absorb and expel H,O through 
 their siliceous substance, liquids such as carbon 
 disulphide can pass in and out of dehydrated chaba- 
 site, gold atoms penetrate lead to a perceptible extent 
 in a short time, and irregular structures in isomor- 
 phous mixtures of metals or salts adjust themselves 
 on tempering by wandering of the particles in the 
 solid crystal. The atoms must, therefore, in these 
 cases be “ able to pass by one another.’’ The con- 
 ception of “‘ close packing of spheres,’’ so useful in 
 crystallography, must not therefore be understood 
 as an actual contiguity of the material atoms. As 
 has already been stated on page 46, the matter here 
 is one of a subdivision of space, dependent to some 
 extent on the external conditions, into “spheres of 
 influence ’’ around the atoms. 
 
 In correspondence with these relations the kine- 
 matic conditions alter as the series of metamorphoses 
 of matter is passed through. While the motion of 
 the individual leptons in the ideal gas state of 
 matter, neglecting collisions of the particles, is prac- 
 tically independent of the surroundings, this free- 
 dom of path is restricted in liquids by the reciprocal 
 force fields, and the liberty of the structural units 
 
68 CRYSTALS AND MATTER 
 
 of crystals amounts merely to a tenth of the distance 
 between the atoms. Lower temperatures naturally 
 signify here a slowing down and limitation of the 
 motion of the structural groups, and we may imagine 
 that at very low temperatures the particles in a 
 certain sense ‘freeze hard.’’ But even at the 
 absolute zero the energy of intra-atomic motion still 
 remains ; the frictionless agitation of the structural 
 units of the atom determines the general constitution 
 of matter, and isyeternal. 
 
 The types of fine-structural state are therefore 
 easily distinguished. Differences in the motion, in 
 the distances apart, and in the mutual interaction of 
 the particles, determine the constitution of these 
 states under discussion, and thus nothing is more 
 natural than that changes of state should modify 
 the structure of the particles. If this does not occur 
 to the extent of an actual change in the chemical 
 character of the substance on transition from the 
 gaseous to the fluid and solid states, the identity of 
 the molecules in the different states of aggregation 
 cannot, after what has been said, be admitted: the 
 particles of gases and liquids are changed in passing 
 through the series of metamorphoses of matter,} 
 
 Further, in the opinion of physical chemists, the 
 state of affairs is complicated still more in the gaseous 
 
 1In this connection the physical condition of a substance is 
 occasionally indicated in the abbreviated symbol for its chemical 
 nature as a printed formula. Just as the charges of the ions are 
 expressed by * and ’, or + and —, the gaseous, liquid, amorphous, 
 and solid crystalline states are denoted by the symbols }, ~, - +, and 
 —— (the usual crystallographic sign) respectively. For example: 
 
 H,0, H,O, H,O; and. SjO,. for silica glass. In chemical equations, 
 
 aT, oo 
 too, such as MgCO, = MgO + CO,, the actual physical condition of 
 the substances is shown immediately. 
 
TRANSFORMATIONS OF MATTER 69 
 
 and fluid phases by equilibria between different types 
 of molecule. 
 
 A. Smits follows up this idea in his theory of 
 polymorphism, even to the extent of relating the 
 process of crystallisation of substances without 
 decomposition to an internal equilibrium between 
 different types of molecule. 
 
 DISCONTINUITIES OF LOWER ORDER IN THE TRANS- 
 FORMATION SERIES. POLYMORPHISM. ENANTIO- 
 MORPHY. 
 
 Besides the reverse changes from the gaseous to 
 the liquid state,1 and from these to the crystalline, 
 
 Series of metamorphoses of matter 
 
 Total energy 
 
 Temperature decreasing ———» ~973° 
 
 Fic.£87. —Subdivision of the series of metamorphoses. 
 
 within each of these states (and also for liquid 
 crystals) less drastic discontinuous transitions are 
 possible. The relations of crystals whose greater 
 capacity for changes of internal structure is obvious 
 in their frequent polymorphism? are here again 
 typical. 
 
 1 A transition which can be accomplished continuously by appli- 
 
 cation of a certain pressure and temperature. 
 2 Termed allotropy for elements. 
 
70 CRYSTALS AND MATTER 
 
 Examination characterises these crystal modifi- 
 cations as forms of the same substances in a chemical 
 sense, which differ from each other in their energy 
 content. In the fine-structure this is expressed as 
 a variation in the arrangement of the component 
 particles, that is, in the architecture ; the stabilities 
 are different, and tend, therefore, to cause trans- 
 formation of one modification into another. Transi- 
 tion occurs if the difference is so considerable that 
 the internal resistance to structural change can be 
 overcome. 
 
 This may be brought about by a correspondingly 
 large change in the external conditions, generally 
 the temperature or pressure, and sometimes both ; 
 from among many cases of this we mention, as 
 instructive examples, the abrupt change of borazite } 
 and of a-quartz= f-quartz at 575° (Figs. 88 and 
 89), or the transitions of ice. According to a diagram 
 worked out by G. Tammann, ice passes through five 
 different forms with increase of pressure at — 30° C. 
 Occasionally the catalytic influence of a chemical 
 field, that is, the intimate proximity of a certain 
 substance, brings about a transition which would not 
 
 1 The investigation of this mineral gives very elegant physical 
 chemical demonstrations of fine-structure if a borazite plate is viewed 
 between crossed nicols and its temperature raised. Owing to the 
 considerable double refraction it transmits bright polarisation colours. 
 On exceeding 265° the isotropy of the a-borazite spreads out from a 
 point like a dark curtain over the previously radiant field, fluctuating 
 with each small variation of temperature, and finally enveloping the 
 whole in deep shadow. On cooling, the curtain rolls back and the 
 structural particles are restored to their old rhombic equilibrium 
 arrangement. In other cases change of modification is immediately 
 evidenced optically by a colour change. Red mercuric iodide is a 
 striking example of this. On heating above 126° it changes from 
 its tetragonal equilibrium arrangement to a yellow rhombic form. 
 
TRANSFORMATIONS OF MATTER 71 
 
 otherwise occur. H. E. Boeke and the author have 
 shown that the iron sulphide of magnetic pyrites 
 serves aS an example of this; transformation only 
 
 Hexagonal Tsometrical 
 a-Quartz a-borazite 
 575° ————_—__—__ | 265° 
 Trigonal Orthotrimetrical 
 
 (rhombical) 
 Cy 
 
 B-Quartz 
 
 Fic. 88.—Homoomeric modification of quartz and borazite. 
 
 Fic. 89.—a@ and 4. Laue diagrams of 8-quartz and a-quartz. After F. Rinne. 
 
 takes place in the presence of a little carbon or excess 
 of iron. 
 
 After a change in the fine-structure a new equili- 
 brium arrangement is in every case established, 
 
72 CRYSTALS AND MATTER 
 
 which need not, however, represent the most stable 
 configuration under the existing circumstances, but, 
 on the contrary, may be, in accordance with Ostwald’s 
 step-rule, that which in energy content is nearest the 
 original state. 
 
 For polytype modifications such as those of car- 
 borundum, discovered by H. Baumhauer, H. Espig’s 
 experiments, carried out in my institute, have shown 
 that equally large elementary cells contain the same 
 number of molecules (24). 
 
 These modifications cannot differ, therefore, in 
 specific weight; they will, nevertheless, require 
 different amounts of energy to change their molecular 
 
 cei <a > 
 7 ee NEES vo ee 
 
 Carborundum Carborundum Carborunduin 
 Type I Type IT Type LIT 
 
 Trigonal Hexagonal 
 
 Fic. 90.—Polytype modifications of carborundum. 
 
 motions, i.e. their specific heats are different. <A 
 change of one carborundum modification into another 
 is not known. Such changes of structure are to be 
 observed, however, in other cases, for generally the 
 modifications are structures with definitely unequal 
 stabilities, except at the transition point. 
 
 If the difference in the fine-structure is not too 
 great the modifications are homdomeric, that is to 
 say, transformable one into the other without break- 
 ing up the whole structure. This is the case for the 
 quartz referred to, and also for borazite and leucite. 
 With these minerals it is easy, by heating one modifi- 
 cation above the limiting temperature to change it 
 over into the other, and, on reversing the process, 
 
TRANSFORMATIONS OF MATTER 73 
 
 by lowering the temperature to return to the first 
 modification as often as desired. The solidity of 
 the crystal structure does not preclude exact ex- 
 periment, and Laue diagrams give a picture of 
 this “dance of the leptons’’ in the change from 
 B- to a-quartz. In this case the ternary arrange- 
 ment of the particles becomes, as though at some 
 mysterious word of command, a senary arrangement, 
 as Figs. 89a and 0b show, a truly wonderful glimpse 
 of a fine-structural sanctuary of nature. A similar 
 case arises with iron, and here the investigations of 
 A. Westgren have disclosed the relations of the fine- 
 structure. Four modifications 
 
 of iron are known—a, f, y, | 
 and 6 irons. On heating to | 
 above 769° the a passes into 
 
 Ppomemvanicry, at -oo6" this "0 ¢ 
 gives the Y modification, and Fic, g1.--- Stereograms of iron 
 finally an increase of tempera- aie eet So a 
 ture to 1401° leaves the metal in the 8 state; the 
 transitions in every case are indicated by thermal 
 changes. 
 
 The transformation a= 8 is very clearly shown. 
 Here at 769° lies the upper limit of the technically 
 important property of ferromagnetism (capacity for in- 
 tense magnetisation) possessed by the metal. In their 
 fine-structure a and 8 irons are closely allied, differing, 
 however, from the y metal; Westgren was able to 
 _establish that the former have body-centred lattices, 
 whilst that of y-iron is surface-centred (Fig. 91). 
 The transition from one variety of iron into another 
 takes place quite smoothly. On the other hand, 
 with carbon tetrabromide for example (Fig. 93), 
 clearly marked fractures arise on slowly heating or 
 
74 CRYSTALS AND MATTER 
 
 cooling, and even more so in the transition from 
 diamond to graphite, an example of allomerism 
 which P. Debye and P. Scherrer have made classical. 
 
 Planes of the cube, rhombic dodecahedron, a pyramidal 
 
 Planes of the cube, rhombic dodecahedron, a pyramidal cube, 
 
 cube and the octahedron. 
 and the octahedron. 
 
 Fic. 92a.—Atom domains of a-iron.! 
 Fic. 924.—Atom domains of y-iron. 
 
 In diamond the carbon particles are arranged in 
 aliphatic tetrahedral grouping, but in graphite they 
 are arranged in the aromatic ring form which is 
 
 1 The lower scales in Figs. 92a and 6 relate to depth distances. 
 
TRANSFORMATIONS OF MATTER 75 
 
 a very different arrangement, and one requiring 
 much more space from ring to ring. Graphite is 
 in a certain sense an elongated diamond (Fig. 94). 
 
 Fic. 93.—Change of modification of carbon tetrabromide, the external form 
 remaining intact but showing internal mosaic formations. 
 
 The restoration of the atoms to the diamond arrange- 
 ment has not yet been accomplished. 
 In conclusion, the remarkable enantiomorphy of 
 
 Diamond © 
 a-250-10-8com 
 €=306-10-8om c -5 97: 10-5an 
 @Ce7123 aca F203 
 
 Fic. 94.—Allomeric modifications of carbon (diamond and graphite). 
 
 crystals, as shown by quartz and cane sugar, for 
 example, should also be mentioned. 
 
 The two varieties which call for consideration 
 are differentiated as right- and left-handed; as 
 
76 CRYSTALS AND MATTER 
 
 regards their macro- and micro-stereochemistry, each 
 variety taken by itself is without mirror symmetry, 
 but the two together are symmetrical one to the 
 other, as the halves of the human body. Their 
 stability to changes of external conditions is exactly 
 the same. They are obtained mixed together indis- 
 criminately, and are not convertible one into the 
 other. 
 
 Clearly then the modifications of a crystalline 
 material like the isomers of liquids and gases are 
 chemically different substances, although, at the 
 same time, the divergences in their material relations 
 
 , ly right left 
 Cane sugar 
 
 Fic. 95.—Enantiomorphy of cane sugar. 
 
 are, for the most part, not very large. In the case 
 of carbon, however, they are not small (graphite is 
 oxidisable to graphitic acid, diamond is not), nor 
 with calcium carbonate are they inconsiderable 
 (boiling with cobalt nitrate solution colours aragonite 
 violet, but calc-spar blue) ; quartz, finally, scarcely 
 shows them at all. 
 
 REVIEW AND UNIFIED CONCEPTION OF THE 
 STATES OF MATTER 
 A survey of the whole series of metamorphoses 
 from the gas to the last modification in the tempera- 
 ture scale presents a picture of discontinuous energy 
 changes, and between these discontinuous changes 
 
TRANSFORMATIONS OF MATTER 77 
 
 steady variations within each state, which possibly 
 represent energy exchanges in minute jumps. Ac- 
 cording to that idea, large energy quanta are com- 
 posed of small quanta, which for the observed 
 continuous changes follow each other in a steady 
 series, whilst for the sudden discontinuous changes 
 they rush over a steep fall. Structurally, in the 
 former case one is concerned with fine leptonic jumps, 
 in the latter with motions of larger leptonic complexes. 
 Regarding the fine-structural changes of sub- 
 stances in this way, we arrive at a unified conception 
 of the states in which matter is presented to us. 
 Starting from the absolute zero and continuing to 
 the highest temperature obtainable, matter passes 
 through a series in which the transition points 
 (whether these be boundaries between the solid, 
 crystalline, liquid, or gaseous states, or terminating 
 points of sub-states within these) represent obstacles 
 on a constantly rising road, which must be overcome 
 by especially large expenditure of energy. The 
 substance deviates more and more from the character 
 it possesses by virtue of its own particular constitu- 
 tion. In ordinary experiments carried out at room 
 temperature, that is, at about 290° absolute, we must 
 in consequence assume considerable restrictive in- 
 fluences at work on the crystal, resulting in modifica- 
 tions of its proper constitution.!. The investigation of 
 matter at very low temperatures eliminates, as much 
 as possible, the enormous influence of the surround- 
 ing circumstances on the primitive structure. Thus, 
 of all the states of aggregation, the solid crystalline 
 approximates most nearly to that of ideal matter. 
 
 1 Of these, temperature is the most active; after that, pressure 
 and chemical action are the most important. 
 
VII. GENERAL TECTONIC ARRANGEMENTS 
 IN THE FINE-STRUCTURE OF CRYS TAS 
 
 ANALYSIS OF STEREOCHEMICAL FORMULA 
 
 structural researches from a chemical stand- 
 point suggests many questions of general 
 importance. 
 
 First, it is an obvious conclusion that the sug- , 
 gested schemes express the analytical constitution 
 of the substances they represent. It is, of course, 
 necessary to argue correctly from the fine-structure. 
 For example, the diagram for the stereochemical 
 formula of fluor-spar CaF, (Fig. 96a) contains in the 
 tectonic unit there depicted 14Ca and 8F atoms, 
 leading apparently to the incorrect formula Ca,,F, ; 
 but it must be remembered that each Ca at a corner 
 of the figure is shared by eight cubes, since the 
 fine-structure is repeated indefinitely, often on all 
 sides. Consequently, each of these only represents 
 an eighth of a Ca. Similarly, a Ca on a surface of 
 the cube (as on a wall separating two cubes) must be 
 counted as $Ca, whilst the 8F atoms in the inside 
 of the figure must be reckoned in full. We have, 
 therefore, (8.4.+6.4)Ca and 8F, or Ca,F, corre- 
 sponding to the actual CaF,. In the same way, the 
 reader will easily decipher the stereogram of iron 
 carbide Fe,C, and the formula of ice H,O (Figs. 
 96d and c). 
 
 A CONSIDERATION of the results of fine- 
 
 78 
 
TECTONIC ARRANGEMENTS 79 
 
 If, according to this, a stoichiometrical balance 
 for a portion of the crystalline body surrounded by 
 others is recognised, it seems unmistakable that the 
 crystal as a whole does not conform to the require- 
 ments of the law of simple multiple proportions, if 
 its space-lattice structure extends to the external 
 surfaces. In this case the particles lacking neigh- 
 bours would not balance analytically, and there would 
 be an excess of one kind. The edge of the elementary 
 cell of fluor-spar (shown in Fig. 96a) is 5:44 x 1078 
 cm.long. A cube of the mineral of 5-44 x I07? cm. 
 
 Fic. 96.—Stereochemical formule. a. Fluor-spar CaF,. After W. H. and 
 W. L. Bragg. 6. Iron carbide Fe,C. After A. Westgren. c¢. Ice H,O. 
 After G. Aminoff. 
 
 side which is thus of colloidal dimensions would 
 give, in accordance with the above scheme, the ratio 
 Ca: F = 36:51 : 63:49, instead of the theoretical value 
 33°33 : 66:66. By increasing the size of the cube the 
 deviation from the ideal ratio would gradually become 
 smaller, and, finally, analytically indetectable. Since, 
 however, it is unlikely that for crystals the law of 
 simple multiple proportions does not hold good, there 
 only remains the assumption that the regular space- 
 lattice arrangement of the particles fails in the 
 surface zone. 
 
80 CRYSTALS AND MATTER 
 
 VALENCY 
 
 No less important chemically is the question as 
 to how valency is exhibited in crystals. We may 
 expect, therefore, that the study of the specially 
 regular crystalline materials will lead to a better 
 understanding of the controversial nature of valency. 
 For gases and liquids the only rules of valency 
 which have actually been experimentally confirmed — 
 are restricted mainly to the specification of how 
 
 many atoms have found a place 
 
 round another atom under de- 
 
 finite chemical and physical 
 
 conditions. With regard to the 
 
 fine-structure, it is of interest 
 
 to note that, in substances with 
 
 simple valency, it is rare that 
 
 ZE. more than eight atoms are united 
 
 to one of another (as the series 
 
 FNa; F,Mg; B,Al; 2, Sip eye 
 
 B.S; —; F,0Os shows), and that 
 
 these combining numbers for a 
 
 Fic. 97.—Net ofasimple given atom depend on the tem- 
 
 crystal and its twin form. erature, ie. on the motion of the 
 
 particles. Both of these are clearly structural 
 characteristics permitting of obvious explanation. 
 
 As to the spatial arrangement of the particles, 
 no definite result is arrived at in terms of valency, 
 although it appears quite natural in CH, for us to 
 place the 4H satellites round the central carbon atom 
 in similar positions on the corners or sides of a 
 tetrahedron (compare Fig. 56, p. 43). In particular, 
 it is entirely a formal characterisation if valency or 
 co-ordination bonds are represented as single directed 
 lines of force between the atoms and molecules. In 
 
TECTONIC ARRANGEMENTS 81 
 
 Fig. 97 it is clearly seen that such bonds between 
 similar atoms at the common surface of a twin have 
 different directions from those in the simple crystal. 
 We must, however, note that such tensors have only 
 the character of pedagogical simplification, and not 
 that of reality, just as the customary representation 
 of chemical formule on paper is, to a certain extent, | 
 merely a simplified projection of very complicated 
 spacial arrangements. The “structural chemical ”’ 
 mode of representation in a plane, and the drawing 
 in of directed bonds, are but symbolisations, super- 
 ficial and linear respectively. Plane arrangements 
 of the atoms or of their centres of motion are not, 
 of course, excluded; for benzene, in fact, such an 
 arrangement is very probable. In this example we 
 have a limiting case in which matter is practically 
 reduced to a plane. A methylation of benzene leads 
 at once to a definitely three-dimensional molecular 
 structure.t. In general terms, the constellation of the 
 particles must be thought of as a complex action of 
 the anisotropic fields of force, gradually weakening, 
 which always surround the particles. As is well 
 known, especially in inorganic chemistry, we cannot 
 get on without introducing some spacial co-ordination 
 action besides valency ; A. Werner has discussed this 
 very ingeniously. Organic chemistry, owing to the 
 special nature of the nucleus of all its compounds, 
 carbon, for which the valency and co-ordination 
 
 1 This does not lessen the great value in teaching the notions 
 of directed singular valency and co-ordination forces, and their 
 representation as lines in plane formule. It is an indication of the 
 considerable practical utility of this handy method of representation 
 that such a formulation of the certainly very complicated interaction 
 of the oscillating particles generally guides us safely through the 
 
 innumerable phenomena of chemistry. 
 6 
 
’ 
 
 89 CRYSTALS AND MATTER 
 
 numbers are the same, might easily attempt to dis- 
 pense with such conceptions. 
 
 In contrast to these uncertainties in the case of 
 amorphous materials, X-ray experiments on crystals 
 have disclosed not only how many particles at any 
 time surround one atom, but, in addition, details as to 
 the spacial arrangement of the centres of the particles 
 in terms of centimetre co-ordinates. Thus it is 
 known definitely that, in the case of diamond, each 
 | C atom is placed at the 
 centre of a tetrahedron, 
 at the corners of which 
 are four others, each 
 1°53 X 10~° cm. from the 
 centre; in rock-salt each 
 Na atom is surrounded at 
 a distance of 2:81 x 1078 
 cm. by 6 Cl atoms on the 
 normals to the surface of 
 a cube, and each Cl simi- 
 larly by 6 Na’s.1. In addi- 
 tion, spacial representation of the atomic domains 
 gives us a Clear insight into the dynamical relations ; 
 in Fig. 98 this is done for sodium chloride. 
 
 All the other rules of valency lead us here to 
 somewhat insecure ground. It is true that in crystal 
 stereochemical figures connecting links may be looked 
 upon as valency tensors, and it is in this way seen 
 that in the diamond the tetravalency of carbon is 
 clearly shown graphically. In other cases, a val- 
 ency distribution is symbolised, as for rock-salt in 
 i values, and in similar ways for other substances. It 
 must not be forgotten, however, that the methods 
 
 Fic. 98.—Axial arrangement of the 
 atom domains in rock-salt. 
 
 ‘See also Fig. 96, p. 79, and Fig. 24d, p. 22. 
 
TECTONIC ARRANGEMENTS 83 
 
 discussed above are only graphical aids, and merely 
 summarise diagrammatically the field of force. 
 
 In this connection it must be ascertained whether 
 any further knowledge of valency relations is available 
 from a consideration of the crystallographic evidence 
 of a definite “form ’’ for the structural particles : 
 and here we are helped by ideas of H. J. van’t Hoff, 
 who assumed that it is the definite form of the atoms 
 which involves directional maxima in the tendency 
 to aggregate. These maxima therefore determine 
 the fine-structural distribution possible, and therefore 
 the valency. 
 
 Fic. 99.—Valency schemes for a and 4 Rock-salt ; ¢, calc-spar. 
 
 Thus a carbon atom, owing to the tetrahedral 
 arrangement of its external electrons, presents a 
 specially stable distribution for four hydrogen atoms 
 at the corners of a tetrahedron rotated with respect 
 to the carbon tetrahedron, so that the H atoms lie 
 over its four faces (Fig. 56, p. 43). The normal 
 quadrivalency of carbon arises then primarily as the 
 result of a stable arrangement in the field of force, 
 and kinetic relations are therefore of importance here. 
 
 This anisotropy in the shape of the atom will exist 
 also in the atomic domain, concerning which refer- 
 ence is made on page 46; in these domains, however, 
 the corners and edges will be rounded off. 
 
84. CRYSTALS AND MATTER 
 
 A zero valency arises if a stability already high 
 permits of no further additions. The relations of 
 substances possessing the character of rare gases 
 may, following the ideas of Bohr, W. Kossel, and in 
 particular H. Tertsch, be regarded in this way. 
 After filling an electron zone with 8, again 8, then 18 
 (= 6+ 12), again 18, and finally 32 (= 24 + 8) 
 electrons corresponding to the periods in the system 
 of the elements,’ a shield so stable from outside in- 
 fluence is constructed that a fine-structural deforma- ° 
 tion and chemical change is rendered difficult to a 
 high degree. Thus zero valency is obtained in each 
 case. Figs. 100-103 represent these stable electronic 
 arrangements diagrammatically in crystallographic 
 form. Helium (with two negative electrons) has 
 probably pinacoidal stability (Fig. 30, p. 29).2 The 
 other numbers, 8, 1% =6+12, 32 =8-4+ 24, are in 
 accordance with the crystallographic motives of the 
 cube (six surfaces, eight corners), the rhombic dode- 
 cahedron (twelve surfaces over the edges of the cube), 
 and the pyramidal cube (twenty-four surfaces over 
 the six of the cube). 
 
 In the construction of molecules analogous cases 
 are not improbable. That tetrahedral grouping, as 
 
 17 H, 2 He; 1 Li, 2 Be, 3 B, 4.C, 5'N, 6 O, 7 F, 3 Nee 
 2 Mg, 3 Al; 4 Si, 5 P,'6'S;4 Cl, 8 Ar; 2K, 2 Ca, 3'8cua eee 
 6.Cr, 7 Mn; 8 Fe, 9 Co, to Ni, 11 -Cu, 12° Zn, 13 Gay 4) ee 
 16 Se, 17 Br,.18 Kr; 1 Rb, 2 Sr, 3-Y, 4 21, 55ND) 0))iGw eee 
 9 Rh, 10. Pd, 11 Ag, 12 Cd, 13 In, 14 Sn, 15 Sb, 16 Te, 170} p ee 
 t Cs, 2 Ba, 3-19 ‘the rare earths, 20 W, 21-22 Os,'23 If, 24° Pte aera 
 26 Hg, 27 Tl, 28 Pb, 29 Bi, 30 Po, 31-32 Em; t-2 Ra, 3 Ace 
 5 Br, 6 U, 
 
 * The total electrons in the negative shell may probably be 
 arranged for the rare gases in the following way: He2; Neonz2 + 8; 
 Ar 2+8+8; Kr 2+8+6+12+8; X=2+8+46412 
 +6+12+8; Em=2+84+8+6+412+6+4 12 + 244 8. 
 
TECTONIC ARRANGEMENTS 85 
 
 Fia. 100. Fic, 101. 
 
 Fic, 103. 
 
 Fics. 100-103.—Crystallographic schemes for zero valency due to perstable 
 electron arrangements and uranium as the final term in the atom series. 
 Rare gases: Shell with 8 electrons for neon and argon; shell with 6 + 12= 18 
 electrons for krypton and xenon; shell with 24 + 8 = 32 electrons for 
 emanium ; uranium as the final member of the electron aggregates. 
 
86 CRYSTALS AND MATTER 
 
 is to be assumed for CH,, with its great regularity, 
 favours to a certain extent chemical stability is shown 
 definitely by the properties of methane. 
 
 To substances which require only small variations 
 in their external electron shells to attain the external 
 structural balance of a rare gas, we must, on the other 
 hand, readily assign high chemical affinity. The 
 left and right-hand neighbours of the rare gases in 
 the systems of the elements are of this type; the 
 alkalies and halogens. Their tendency to combine 
 one with the other is great, and according to 
 W. Kossel, this is brought about through the loss of 
 an electron by the alkali and by a gain of one by the 
 halogen. 
 
 The following small table of the names and 
 numbers of electrons in the atoms renders this clear: } 
 
 Halogens. Rare Gases. Alkalies. 
 I 2 4 
 Hydrogen. Helium Lithium. 
 9 IO II 
 Fluorine. Neon. Sodium. 
 17 18 19 
 Chlorine. Argon Potassium. 
 an 36 ou 
 Bromine. Krypton Rubidium. 
 ey 54 55 
 Iodine. Xenon. Caesium. 
 
 In a similar manner it will be readily assumed 
 that molecules, too, with the great internal tension 
 which must arise if the stability is not fully estab- 
 lished, tend to like transformations. 
 
 Moreover, in connection with these assumptions 
 as to the mode of action of valency, the occasionally 
 
 1In view of the analogy between Li H and the halides, H is 
 included in this comparison. 
 
TECTONIC ARRANGEMENTS 87 
 
 observable changes of valency in changes of crystal 
 modification, i.e. in stereochemical structural rear- 
 rangements, should be noticed. In the diamond the 
 quadrivalency of carbon is expressed as four equally 
 large maxima in the arrangement. In graphite 
 structure, on the other hand, it is seen that a 4 x I 
 is differentiated into a 3 + 1 valency. This change 
 in the form of the field of force is, to some extent, the 
 first step towards a diminution of valency ; it forms 
 an intermediate stage to a tri- or divalency of carbon. 
 Since it may well be assumed that, 
 in the process of changing from dia- 
 mond to graphite, a variation of 
 atomic form by rearrangement of 
 the electrons takes place (which the 
 optical differences of clear diamond 
 and black graphite, besides con- 
 siderations of crystallographic sym- 
 Mereyemindicate),* then it 1s quite 
 obvious that a change in the shape 00: (oh viee” 
 of the atoms goes hand in hand 
 
 with a change in their method of arrangement. 
 The idea that morphological changes themselves 
 occur under the influence of alterations in the ma- 
 terial surroundings, and of alterations of tempera- 
 ture affecting the internal motions, harmonises in 
 the best possible way with the experimental result 
 that the valency of the substance may depend on the 
 kind of matter with which it is in valency connection, 
 and also on the temperature. 
 
 MOLECULES IN THE CRYSTAL STRUCTURE 
 
 It will appear of great importance to the chemist 
 in dealing with these crystallographic stereochemical 
 
88 CRYSTALS AND MATTER 
 
 formule to investigate the positions of the mole- 
 cules which come under his consideration in the 
 chemistry of gases and liquids. Occasionally it 
 happens that the molecular scheme appears in the 
 elementary cell, as, for instance, those in Fig. 24, 
 page 22. That has, however, as little foundation 
 as the representation of the macroscopic crystal as a 
 molecular unit ; for we are concerned in both cases 
 with more or less arbitrary sections from the fine- 
 structure. For zinc sulphide, Fig. 105 might just as - 
 
 Fic. 105.—A cube section. Fic. 106.—A rhombic dodeca- 
 hedral section. 
 
 Stereochemical formula for zinc-blende ZnS. 
 
 well be proposed as Fig. 106, both of them are con- 
 sistent with the ratio Zn:S =1:1. It appears from 
 this that the discussion of the appropriate architec- 
 tural relations must be attempted in some other way. 
 
 In particular, it must be remembered that the 
 chemical properties of a substance are characterised 
 above all in the organisation of its general architec- 
 ture. As we study an example of architectural art, 
 such as a cathedral, not only as an architectural 
 unit, but recognising its morphological subdivision, 
 so it is an important chemical problem to investigate 
 crystals in the same way. The customary method 
 
TECTONIC ARRANGEMENTS 89 
 
 of writing formule as Hg, Cl’, Na~ Cl’, (NH,)° Cl’, 
 eee). , NH(CH,).1) C,H, (COOH) (NH) accords 
 with this. In such symbols the chemical indi- 
 viduality is emphasised, as well as the appropriate 
 tectonic organisation, as exhibited in reactions. 
 Hence special efforts must be made to find out the 
 inner groupings in the architecture 
 of crystals. 
 
 On the other hand, however, it 
 must be added that the kinetic 
 units in the erection of higher 
 chemical complexes, as in the ag- 
 gregation of atoms to molecules, 
 and so to crystal units, may 
 become indistinguishable in the 
 fine-structure, or may even be dis- 
 solved altogether. This is the case, 
 for example, in the formation of a 
 molecule of H, from its atoms. 
 It cannot be ascertained in the 
 scheme shown which of the two 
 electrons belonged, originally, to a 
 given nucleus. The earlier indi- _ Fics. 107 and 108.— 
 
 : 4 ’ Models of the hydrogen 
 vidualities are in such extreme atom and the hydrogen 
 cases completely destroyed. passa 
 
 Therefore, we are not to restrict ourselves, as a 
 matter of course, to a rigorously uniform representa- 
 tion of the fine-structure of the crystal, but must 
 expect numerous gradations in view of the great 
 abundance of chemical conditions which exist in it. 
 Naturally, in formal mathematical representation, an 
 arrangement in an atomic space-lattice is always 
 possible by joining up the similar and similarly 
 situated atoms into a lattice unit. This procedure, 
 
90 CRYSTALS AND MATTER 
 
 however, invests the crystal structure with practically 
 no physical chemical properties. The presentation 
 of a purely atomistic structure of the crystal, or its 
 comprehensive characterisation with co-ordination 
 bonds, is obviously too general. 
 
 Considering the results already obtained in crystal 
 leptology with a view to a physical chemical inter- 
 pretation, the desired organisation into radicals is, 
 in fact, often definitely shown in the architecture. 
 The Braggs, as originators of fine-structural schemes, ~ 
 have pointed out the dumbbell-like connection of the 
 two sulphur atoms in iron pyrites, and, in somewhat 
 greater detail, P. Niggli has expressed the opinion 
 that the representation of the complete crystal as a 
 purely atomistic space-lattice complex certainly may 
 always be carried out formally, but that many inter- 
 connections between certain atoms forming structural 
 groups (as P. Niggli called them) stand out clearly 
 on a merely architectural consideration of the 
 schemes. These structural groups I have designated 
 geometrical radicals or leptyles in adaptation to 
 chemical ideas. Finally, groups of a molecular 
 nature are occasionally unmistakable in the crystal 
 arrangement. Such leptyles occur in iron pyrites 
 with its doublet S,, in calc-spar with the ion 
 CO;, and in rutile and anatase with the molecular 
 TiO, complex. <A fine example of leptyle grouping 
 has been investigated by Dickinson (‘‘ Journal of 
 the American Chemical Society,” vol. xliv., 1922, 
 p. 287). Moreover, for organic substances with ring 
 formation, such a fine-structural grouping must be 
 assumed according to P. v. Groth, who formerly 
 brought into prominence the atomistic crystal struc- 
 ture. As regards cyclic chains of atoms so funda- 
 
TECTONIC ARRANGEMENTS 91 
 
 mental chemically, which can with certainty be 
 attributed to the molecules of benzene, naphthalene, 
 and other organic substances, it is more than probable 
 that these chains persist in crystallisation. Crystal- 
 lised organic compounds will often be molecular 
 
 Fic. 109.—Physical chemical system of molecules split up geometrically into 
 six different space-lattices. 
 
 aggregates loosely knit into space-lattices. This con- 
 ception W. H. Bragg has taken as fundamental in 
 his present researches on organic compounds. 
 
 In the same connection considerations of cleavage 
 can be brought forward. According to E. Schiebold, 
 
 @ 
 lron pyrites 72,5; Rutile 77Q; 
 
 @ Fe oS Cy e7 
 Fic. 110.—The leptyles S, in Iron pyrites FeS,, TiO, in rutile, CO, in cale-spar. 
 
 such planes are to be regarded as plane rifts in the 
 fine-structure, as it is accepted that these surfaces of 
 separation do not cut through any strong chemical 
 bond. With graphite this is clearly shown, as 
 cleavage along the principal plane in the fine-structure 
 
92 CRYSTALS AND MATTER 
 
 separates out to some extent packs of ring radicals 
 with their 3 + 1 valency by cutting through the 
 extenuated connecting links. According to A. John- 
 sen, twin cleavage with separate motion of the 
 structural groups Ca and CO, demonstrates this 
 same relation. We are concerned here with a very 
 remarkable property of calc- | 
 spar which, on the applica- 
 tion of a lateral pressure, 
 permits portions of the whole 
 structure to slide to a position 
 symmetrical with the station- 
 ary part (Fig. 112). 
 
 Fic, 111.—Structural groups in 
 graphite. 
 
 Oo 2 SY 6 8 10 12 14 1616 20 
 
 Fic. 113.—Analogous selective reflexion in the ultra-red of the hydrated sulphates 
 of Mg —- K, Co— K, Zn — K. After Cl. Schaefer and M. Schubert. 
 
 In this so-called simple displacement or twin 
 slipping, the Ca atoms on one side, the CO; complexes 
 of the fine-structure on the other, shift apart each as 
 a complete whole. 
 
 The researches of Cl. Schaefer on the reflexion of 
 ultra-red radiation by crystalline salts and the con- 
 
TECTONIC ARRANGEMENTS 93 
 
 stant effect, there observed, of the groups NO,, SOu,, 
 SeO,, CrO,, and H,O, also illustrate the idea under 
 discussion, as occasionally in these cases definite 
 radicals can be separated out partly by chemical 
 treatment of the crystal structure. 
 
 In short, there is no lack of well-supported 
 - opinions that the crystal lattices are organised to a 
 varying degree in natural complexes, more or less 
 simple, up to the molecular type. With this in view, 
 A. Reis has classified crystal lattices from a chemical 
 standpoint, and has emphasised, following the lead of 
 W. Nernst, F. Haber, M. Born, and others, the special 
 nature of salts as ionic structures. 
 
 Thus neutral atom lattices, atom ion lattices, 
 neutral and charged radical lattices, as well as mole- 
 cular lattices, are to be differentiated, to which series 
 I should like to add one more, the mixed lattice. 
 Calc-spar Ca: CO,;, and zeolites such as heulandite,} 
 are examples of mixed lattices. The H,O contained 
 in the latter mineral is set so loosely in the fine- 
 structure that it evaporates from the crystal on 
 the temperature being raised, until an equilibrium 
 content is reached. On lowering the temperature 
 the H,O is again taken up from the surroundings 
 without the remaining structural complex being 
 damaged or in any way broken up, as is shown 
 by the Laue diagrams. That indicates great struc- 
 tural independence of the water particles. They are 
 
 1Indeed, fundamentally all lattices with isotopic substances 
 (p. 185) have already a mixed character. The isotopes of a given 
 atomic variety, say, Cl,;, Cl,;, and Cl,., are, of course, to be 
 thought of as separated in space and replaceable one by another 
 (compare Fig. 119, p. 98). 
 
94 CRYSTALS AND MATTER 
 
 to be considered as regularly arranged structural 
 particles only loosely coupled to the silicate space- 
 lattice. 
 
 Complete kinetic units in the sense of individual 
 leptons in ideal gases are themselves quite ineffective 
 in the isodynamostasy of the whole; for the leptyles, 
 however, this naturally is not so. Like all structural 
 particles, they occur in the crystal in complex 
 structural connexion with those above and below. 
 The kinetic unit for crystalline materials is the © 
 whole crystal. In this sense it replaces as a new 
 
 UTI AS 
 Ves 
 XX 
 
 WN a 
 
 Pigart4, Fig. 125. 
 
 Fics. 114 and 115.—Schemes for atom and radical lattices. 
 
 compound the individual molecule of gases and 
 liquids. Within the crystal, affinity tensors, in the 
 sense of valency theory, pass in and out from one 
 structural particle to another throughout the whole 
 structure ; the crystal is in this respect a leptoblast. 
 With reference to the cohesion prevailing through the 
 whole system, P. Pfeiffer has made the relations clear 
 in the designation ‘‘ molecular linkage.”’ 
 
 The cohesion of the structural groups arising as a 
 result of force fields, in general, very strong, prevents 
 the disrupture which is possible for liquids and gases 
 on the application of mechanical force. This cohesion 
 
TECTONIC ARRANGEMENTS 95 
 
 enables a crystalline material more or less to with- 
 stand fracture, and renders crystals, in consequence, 
 correspondingly hard materials. 
 
 ho 
 
 5 
 & 
 RY 
 
 Yeths, yooh, 
 
 2 
 
 Fic. 116.—Scheme for the molecular lattice of an aromatic compound C,H,Xop 
 (compare also Fig. 85, p. 65, and Fig. 109, p. 91). 
 
 The chemical facts alone indicate how the inner 
 structure is organised in general, yet surprising 
 
 Fic, 117.—Mixed lattice (calc-spar). 
 
 results sometimes arise. This is shown by a re- 
 search in my institute by Lotte Berndt on olivine, 
 to which is ascribed the formula of an orthosilicate 
 
96 CRYSTALS AND MATTER 
 
 Mg,(SiO,). According to these experiments a corre- 
 sponding organisation into the structural groups of 
 Mg and SiO, leptyles (charged) is to be expected. 
 The symmetry of the space grouping proves, however, 
 that SiO, leptyles do not arise in the fine-structure. 
 On the contrary, the atoms in the olivine must be 
 
 < 
 
 / 
 | 
 
 We 
 \ 
 
 SF 
 = 
 
 \ 
 
 = 
 
 SS 
 Ro 
 re 
 
 Fic. 118.—Leptoblastic structure for diamond. 
 
 separated into MgO- and Si0,-groups. Presumably 
 this will be shown by reflexion experiments in the 
 ultra-red, page 92. More comprehensive experiments 
 on silica salts, as to their nature as ortho-, meta-, or 
 polysilicates, would be of the greatest interest in 
 crystal stereochemistry. 
 
VIII. ASSOCIATION OF THE FINE-STRUC- 
 TURAL PARTICLES IN MIXED CRYSTALS 
 AND OUT-GROWTHS ON CRYSTALS 
 
 TRANSITIONS BETWEEN CHEMICAL COMBINATION 
 AND PHYSICAL MIXTURE. MIXED CRYSTALS 
 
 dealing with the structure of crystals that they 
 
 permit, to a greater or less extent, indiscriminate 
 heterogeneity of their constituents without injury 
 to their stability. Such is the case for isomorphous 
 mixtures. We find here certain atoms or groups 
 of atoms replaced by others in the fine-structure of 
 the crystals. It is certain that crystals with isotopic 
 constituents (p. 185) must be reckoned as isomor- 
 phous mixtures. In NaCl crystals, for example, 
 three kinds of chlorine atoms are to be assumed, 
 Cl,; (i.e. with the atomic weight 35), Cl,,, and 
 perhaps Cl,,. Although these atomic sub-types 
 have different masses they do not differ chemically. 
 Such isotopic mixed crystals represent the ideal case 
 of isomorphous substitution.t In addition, in the 
 interchanges of isomorphous mixed crystals, we are 
 concerned with substituents quite analogous chemic- 
 ally ; in the mixtures NaCl and NaBr, Ba(NO,;), and 
 Pb(NO,),, KCl and K(CN), for example, the Cl and 
 
 1 Owing to the uniform intermixture of the chlorine isotopes on 
 the earth, one has presumably in rock-salt always 77 per cent. Cl,,, 
 and 23 per cent. Cl;,, with traces of Cl,). Theoretically, however, 
 any arbitrary mixture is possible. 
 
 " 97 
 
 [< is an especially interesting point to note in 
 
98 CRYSTALS AND MATTER 
 
 Br, Ba and Pb atoms respectively displace each other 
 in the fine-structure. This interchange may also 
 occur with chemical radicals such as NH,, which 
 replaces K, or CN, which is interchangeable, with 
 Br. Similarly, the groups’ SO,, S5éO;,*CrOj@aae 
 the members of many other series may be substi- 
 tuted one for another. 
 
 With such complex space-lattices, it is not uncom- 
 mon for optical anomalies! (or in the isometric sys- 
 
 Fic. 119.—Scheme for the isomorphous Fic. 120.—Scheme for the isomor- 
 mixture of NaCl,,, NaCls,, NaClyo. phous mixture of NaCl and NaBr. 
 
 tem double refraction in the place of the isotropy 
 possessed by the pure components) to indicate, accord- 
 ing to G. Tammann, unequal apportionment of the 
 substituents. For barium lead nitrate the abnormal 
 phenomena disappear with time, even at ordinary 
 temperatures or on heating; they are not found, 
 therefore, in mixtures which, like pyrop in eruptive 
 rocks, crystallised at higher temperature. 
 
 PHYSICAL CHEMICAL SIGNIFICANCE OF THE MIXED 
 CRYSTAL 
 
 As regards the X-ray effect, the fine-structural 
 heterogeneity in question is not evidenced experi- 
 mentally in diffraction, which is an indication that in 
 isomorphous mixtures the substitution is, as assumed 
 above, complete. If in mixed crystals there existed 
 intergrowing layers of first one and then the other 
 
 +R. Erauns has given from the results of very many actual 
 observations an excellent summary of these anomalies. 
 
FINE-STRUCTURAL PARTICLES — 99 
 
 substance, such as very thin sheets of KBr and KCl 
 growing together to give the mixed crystal (KBr 
 and KCl), then each component would naturally give 
 its own interference pattern in the Debye-Scherrer dia- 
 gram. This, however, is not the case. On the other 
 hand, Hadding found such double patterns for Nak 
 felspar, showing that here one has a sub-microscopic 
 mechanical mixture of the two felspar varieties, and 
 nota simple interchange of sodium and potassium ions. 
 
 How are we, therefore, to think of this newly im- 
 planted portion of the fine-structure in combination 
 with the whole? These matters lead to the con- 
 sideration of a fundamental question in chemistry, 
 namely, the relation between chemical compound 
 and physical mixture. 
 
 If the existence of mutual coupling by the action 
 of forces effective only very near the atom is to be 
 emphasised, then isomorphous mixtures must come 
 under the headings of chemical compounds. All 
 atoms in a mixed crystal stand certainly in definite 
 chemical connexion one with another. Moreover, no 
 reliable line of demarcation between compounds and 
 mixtures can be laid down in passing from one pure 
 crystal through the series of mixtures to the other 
 pure substance. In the unmixed calcium carbonate 
 of calc-spar, for example, a chemical compound in 
 every sense exists. If now a manganese atom occu- 
 pies one of the numerous corresponding places not 
 occupied by a calcium atom, then certainly the 
 constitution of the whole as a chemical compound 
 is not abruptly destroyed. Thus, by repeated sub- 
 stitution of atom for atom by correspondingly small 
 quanta in the crystal, one may pass right to the other 
 end of the series CaCO; — MnCOQ,. 
 
100 CRYSTALS AND MATTER 
 
 It is impossible to draw a line between chemical 
 combination and mixture. The case of alteration in 
 the structure of zeolithic isomorphous mixtures will 
 be similarly judged. In these silicates more and 
 more Ca, for example, can be replaced by Na, if 
 a solution of NaCl is allowed to act on a crystal con- 
 taining calcium. In accordance with the laws of 
 mass action, a crystallographic chemical change in 
 the structure takes place, which presents in its 
 various stages a complete series of isomorphous 
 mixtures from the Ca to the Na, mineral. In con- 
 sequence, the structural parts must be in chemical 
 combination with one another. 
 
 Although then isomorphous mixtures in the 
 regular chemical coupling of their structural parts 
 suggest chemical compounds, still on the other hand 
 they do not accord with the second characteristic of 
 such compounds. As H. Kopp has already empha- 
 sized, the distinguishing feature of a chemical com- 
 pound is constancy of atomic proportions, in spite of 
 very great variation of the original conditions. 
 
 Typical molar mixtures, the constitution of which 
 follows change of the external conditions in a con- 
 tinuous fashion, show the opposite effect. Evidently 
 this is the case for isomorphous mixtures. If, for 
 example, the KBr content of a mixed solution of 
 KBr and KCl be steadily increased, the composition 
 of the mixed crystals (K(Cl, Br)) separating varies 
 continuously in correspondence. 
 
 Meanwhile we recognize immediately, in these 
 isomorphous mixtures, that the difference in ques- 
 tion is bridged over in the fact that the continuous 
 variations of their chemical constitution are actually 
 at the same time discontinuous. It is only because 
 
FINE-STRUCTURAL PARTICLES | 101 
 
 of the atomistic, and therefore analytically very 
 small, discontinuous changes in that large leptonic 
 unit, the crystal, that practical continuity obtains. In 
 this it will be readily assumed that those intermediate 
 stages, for which the substituents stand in simple 
 stoichiometrical ratio to one another, and for which 
 a uniformity of distribution in the space-lattice is to 
 be anticipated, will stand out somewhat from the 
 others in, say, their optical or chemical properties. 
 The researches of G. Tammann on isomorphous 
 mixtures point to this conclusion. 
 
 We may venture to conjecture that for these 
 ratios the linkages in the force field are strengthened. 
 
 Mixed crystals then have a specially important 
 physico-chemical interest, owing to their explicit 
 character as intermediate stages between chemical 
 compounds and physical mixtures. 
 
 The inner meaning of this is that chemical and 
 physical methods of association are not so essentially 
 different as has been assumed for some time in 
 chemistry. 
 
 Not uncommonly, a claim for the sharpest separa- 
 tion of the two conceptions is made. Many, especi- 
 ally old chemists and physicists, have, however, 
 signified their acceptance of the same conception 
 as is arrived at here. Of these may be named 
 am bocrhave, C. L..Berthollet, J: B. Biot, J. C. Pog- 
 gendorff, H. Kopp, Guldberg, and Waage, D. Men- 
 deléef, H. J. van’t Hoff, and many others, and as 
 W. Nernst, the master of modern chemistry, expresses 
 it shortly in his text-book of theoretical chemistry, 
 “the differences between physical mixture and 
 chemical combination are really matters of degree, 
 and we find in nature all gradations between the two.”’ 
 
102 CRYSTALS AND MATTER 
 
 OUTGROWTHS WITH SUBSTANCES NOT ISOMORPHOUS 
 
 This being so, these ideas may be extended 
 even further to include the observed outgrowths 
 according to definite laws which occur for various 
 substances, e.g. rutile (TiO,) and hematite (Fe,O,), 
 or potassium iodide and the complicated silicate, 
 mica. When the particles are near in a leptonic 
 sense, physico-chemi- 
 cal reciprocal actions 
 come into play which 
 may lead to coupling 
 up. It is convenient 
 following F. Grandjean 
 to speak of a “‘ champ 
 moléculaire de contact.”’ 
 As is known, crystallis- 
 ing potassium iodide 
 deposits on mica regu- 
 larly oriented with 
 respect to its mica sub- 
 Ke Aa Tell ccvatals showitie 
 
 FIG. 121.—Crystallisation of NaCland ‘ 
 ar pee aa and separating tahedron surfaces, +4 
 contrast to the cubic 
 form of freely formed individual crystals. 
 
 Apart from these macroscopic compounds, a 
 series of phenomena may be obtained in macroscopic 
 and microscopic experiments, which extend the idea 
 of the formation of oriented crystallographic aggre- 
 gates from the colloidal state down to the greatest 
 dispersivity, i.e. to molecules. 
 
 Thus chemical and physical modes of coupling of 
 the particles converge in their essentials to the same 
 
FINE-STRUCTURAL PARTICLES 103 
 
 thing. Strong and weak linkages are connected by 
 intermediate cases, which can, even in the same 
 substance, be followed through as gradations from 
 the one extreme of chemical combination to the other 
 of evident indifference. Certainly, in this generally 
 important connection the frequent occurrence of a 
 variation of the isomorphous miscibility with tem- 
 perature is worthy of notice. NaCl and KCl, for 
 example, at high temperatures, as they are obtained 
 by crystallisation from the molten state, occur in the 
 space lattice in arbitrary proportions, Na, K and 
 
 i 1EN 
 SZ; 
 oy) 
 
 Fic. Rare an of Fic. 123.—Parallel outgrowth 
 rutile on iron glance. of KCl and NaCl. 
 
 As crystallographic evidence of the weakening force field, and as symbolic 
 forms of adsorption compounds (p. 130). 
 Cl particles being then linked up in some regular 
 fashion. On lowering the temperature this power of 
 coupling up internally, possessed by the NaCl-KCl 
 crystal fine-structure decreases steadily. The mix- 
 ture separates out, and finally NaCl and KCI particles 
 lie in contact with, but independently of, each other 
 in a physical mixture.t The reciprocal action is not, 
 however, entirely extinct at lower temperatures. 
 On a freshly cleaved substratum of rock-salt a con- 
 centrated KCl solution deposits small cubic sylvine 
 
 1JIn Fig. 121 these relations are set out in a diagram for which 
 we are indebted to the studies of Kurnakow, Shemtschushny, and 
 Nacken. A. Smits gives also the complete system 
 KC] — NaCl — H,O. 
 
104 CRYSTALS AND MATTER 
 
 crystals in parallel orientation to the substratum, 
 which is a definite indication that an actual chemical 
 field is set up between the KCl and NaCl, under the 
 influence of which the KCl particles arrange them- 
 selves parallel to the NaCl space-lattice. In the re- 
 verse case, also, NaCl will deposit on sylvine oriented 
 parallel to the KCl space-lattice. 
 
 In view of these examples, an important role 
 must certainly be ascribed to temperature in deter- 
 mining the capacity of aggregation in solution of the 
 molecules. 
 
IX. MORPHOTROPY 
 
 HISTORICAL 
 
 HE particular type of fine-structural constel- 
 lation in the molecule or crystal, depends 
 
 naturally, in the first place, on the nature of 
 the components. It is, therefore, of great interest to 
 compare in this respect various substances with each 
 other, and the most likely method of attack appears 
 to be by chemical substitution. 
 
 On these lines P. v. Groth has founded the science 
 of morphotropy. By comparing crystalline forms, 
 he succeeded in showing how the replacement of 
 Hieby. (Ol); NO,, NH,, CH,, Cl, in benzene (C,H) 
 manifests itself in changes of corresponding angles, 
 and ultimately of the symmetry. It became possible 
 to determine the morphological value of atoms and 
 radicals, and by multiplying examples to lay down 
 general rules in this respect. | 
 
 Isomorphism, with its interchanges of allied sub- 
 stances, say of potassium by rubidium in the sul- 
 phate, presents itself here as isomorphotropy. Sub- 
 sequently, as the result of work by W. Muthmann 
 and F. Becke in particular, it was possible to inter- 
 pret these morphological values with respect to 
 definite physico-chemical masses such as the mol. 
 The axial ratio a:b:c, with a different unit of 
 length 6 = 1, for every crystal, was now replaced by 
 
 the cm. values of the “ topic axes,’’ so that absolute 
 105 
 
106 CRYSTALS AND MATTER 
 
 comparison could be carried out.. Further, the axial 
 lengths of the elementary cell can be employed as 
 cell axes, and, finally, those of molecular and atomic 
 regions as leptonic axes; this is actually done in a 
 table which follows. The morphotropic variations of 
 the fine-structure have been termed by A. Johnsen, 
 topotropy. 
 
 A brief statement such as the following gives a 
 general idea of these important relations. 
 
 STEREOCHEMICAL AXES 
 
 Using the Loschmidt number N = 6:06 x I0*? as 
 the estimated number of molecules in the mol (gram 
 molecule) ! of any substance, it is possible to calculate 
 the number of molecules in any other mass, and 
 further, by a knowledge of the density) mass in 
 grams per c.c.), the number in an arbitrary unit 
 of space. Of these space units the elementary cell 
 (a term due to P. v. Groth) interests us in fine- 
 structure work; the reader is already acquainted 
 with it in Fig. 24, page 22. Its dimensions are 
 determined by X-ray methods. The number and 
 also the weight of the molecules belonging to it can 
 be easily calculated theoretically. 
 
 The known cell volume, and the number u of 
 molecules composing it, give immediately the portion 
 of space appertaining to each of the m molecules of 
 Preece 
 
 1 By this is meant the number representing the molecular weight 
 in grams of the substance. For NaCl the components Na and Cl of 
 atomic weights 23:0 and 35:4é respectively, it amounts to 23 + 35-46 
 
 * The weight of a mol Gy is connected with the volume of a 
 mol Vm and the density S by the equation Gy = VmS. The mol 
 axes are obtained in cm. by a simple calculation. In .crystallo- 
 
MORPHOTROPY 107 
 
 Nothing now stands in the way of numerical 
 comparisons of the fine-structure of crystallographic 
 and chemically similar substances. 
 
 As an example of such an investigation, the table 
 for the closely related isometric salts of KCl, KBr, 
 and KI is here given. Sodium chloride, NaCl, and 
 the cyanide, KCN, are also included. In structure, 
 all these are of the sodium chloride type (p. 22). 
 
 Mol. Cell (cube.) Molecular Domain. a3 
 
 sO 
 
 Specific oe: 
 sere At ase. | Volume | Weight | Axes | Volume | Weight | “%°S | Volume | Weight] Axes | 56 
 1O-=: LO-— ame no 
 
 C. Cm). 8 ann ord as ae eben Canin et Cal 53 
 
 KCi 1-990 | 37°64 | 74°56| 3°346| 247-72] 492-96/ 6°280| 61°93 | 123°24 | 3-956] 4 
 KBr 2°756 | 43°19 | 119:02| 3°508 | 285°52| 786:92| 6°585 | 71°38 | 196°73| 4-148] 4 
 KI 3°134 | 52°97 |16 6-02 | 3°756 | 350-24 | 1097°64| 7°049 | 87°56 | 274°41 | 4°441| 4 
 KCN 1-546 | 42°13 | 65:11 | 3-480 | 279-73] 432°30| 6°54 | 66°93 | 108-08] 4°117| 4 
 NaCl | 2°173 | 26:90 | 58-46) 2-996| 177°58| 386°52|5°628| 44°47 | 96°63/3°543) 4 
 
 In the series KCl, KBr, KI, with the same cation 
 K+ and varying anion, the continuously increasing 
 appropriation of space by the compounds, both as 
 regards volume and axial lengths, is very definitely 
 indicated, as is also the fine-structural equivalence 
 of the group (CN) and the bromine ion. Sodium and 
 potassium show their morphological action in a 
 similar way. 
 
 MORPHOTROPIC CONSTRUCTIONS 
 
 ¢ 
 
 Making use of the results for “atomic domains ”’ 
 (p. 46), fine-structures can often be predicted with 
 
 graphy they are termed “‘ topic axes’”’ by F. Becke and W. Muthmann. 
 Denoting the Loschmidt number by N, we have A ae Vir (Vi, = 
 
 ~ 
 
 leptonic volume, i.e. volume of one molecule), and placing N molecules 
 in the cell Z, Vz =7.V{. 
 
108 CRYSTALS AND MATTER 
 
 surprisingly close approximation to the actual 
 measurements. In a certain sense the spherical 
 domains lie at hand as bricks to be built-up ina 
 definite structural scheme. To take an example, the 
 number d,, = 3°51 obtained as a mean from various 
 calcium compounds, and the value d, = 1:26 from 
 MgO lead to the cubic elementary cell constructed in 
 the NaCl type, of CaO with an edge, a = 3:51 + 
 1°20.= 4:77 X 10-8 cm.; X-ray value= 4°38 X 10m ae 
 
 Further, let us suppose that we are dealing with — 
 the still unknown fine-structure of rubidium chloride. 
 The NaCl type is here to be expected with an elemen- 
 tary cell of side 6-59 x I0~8 cm. corresponding to 
 the assumptions dp, = 4:47 + do, = 2-12 X 107 
 which will be approximately true. 
 
 The simple relation holding for the atomic do- 
 mains of allied substances makes it possible to in- 
 vestigate, either graphically or mathematically, new 
 types of atoms with regard to their atomic domains. 
 In this way the series Me ++ = 2:99; Cat? ==3:5am 
 ort + = 3:96; Batt = 4°36 leads to) thesragmams 
 ion Ra++ = 4:90. Taking the domain diameter of 
 OQ- ~ = 1:26, we arrive at the elementary Ccelma 
 radium oxide Rat+t+O-~—, built in the NaCl 
 pattern with presumably a cube edge of length 
 rl, pec O we cas 
 
 Even more complicated constructions may be 
 carried out giving satisfactory agreement with X-ray 
 and goniometric investigations. 
 
 In this connection the trigonal crystalline salt 
 CsCl . ICl may serve as anexample. In Fig. 124 the 
 spherical atom domains of the elements named are 
 piled up one on the other in close packing. Their 
 dimensions (4d,, = 2°52 + dq = 2:12 + d, = 2°93 
 
MORPHOTROPY 109 
 
 + dq = 2°12 + $de, = 2°52) give the result 12-21 
 hero, °* cm. X-ray research leads to the same 
 number 12:2, certainly a good, if somewhat for- 
 tuitous agreement, as regards accuracy. If with 
 this close packing of spheres two material axial lines 
 are constructed, the Cs ions touching those of Cl 
 and I, the arrangement shown in Fig. 124 is obtained 
 with an angle e = 39° 44’, which corresponds to an 
 angle 41° 57’ in the goniometric measurements of 
 the crystal. We might well rest content with this 
 satisfactory agreement, but the fact is of crystallo- 
 graphic interest that a very small elongation of the 
 caesium domain in this trigonal, and therefore 
 geometric rotational structure, gives the angle 
 e = 42°, which the crystal actually exhibits according 
 to goniometric observations. Fig. 125 illustrates 
 this point. 
 
 In Fig. 126 NaF .HF, as an example morpho- 
 tropic to CsCl. ICl, is similarly constructed. Here 
 also it is easy, by a slight alteration of the hydrogen, 
 to build up a crystal form of the compound verified 
 almost exactly by X-ray researches. 
 
 Finally, reversing the argument, it appears pos- 
 sible also to test a substance of known molecular 
 volume with respect to its structural type. The 
 case of the caesium salts CsCl, CsBr, CsI, may here 
 serve asanexample. Their molecular volumes (mole- 
 cular weight divided by specific weight) are, according 
 to a table in P. Groth’s “ Chemischer Kristallo- 
 graphie,” 41°80, 47:40, 56°85 respectively. If these 
 molecular volumes are calculated for a body-centred 
 elementary cell, the results come with convincing 
 Seetoximation, tor) CsCl = 42756), (CsBri = 48-72, 
 CsI = 58-70, indicating the existence of this type 
 
110 CRYSTALS AND MATTER 
 
 of structure, while a construction in the NaCl scheme 
 gives the widely deviating values 55:28, 63°28, 76-25. 
 
 Fic. 124.— FIG. 125.— 
 Construction of trigonal CsCl. ICI Construction of trigonal CsCl . ICI 
 from spherical atom domains. using rotational ellipsoids for Cs. 
 
 Fic. 126.—Construction of NaF .HF 
 from atom domains. 
 
 Moreover, recent X-ray experiments of Davey have 
 
 shown that the former assumption is correct. 
 Strictly, it must be supposed, according to 
 
 K. Fajans and H. Grimm, that the space dominated 
 
MORPHOTROPY 1 
 
 by an ion, say K+, will be affected by its particular 
 Mech poursy apo itay Clasbrevor lode husiion no 
 constituent in a compound is the atomic domain 
 always exactly the same.! For the most part, how- 
 ever, the relations are comparatively simple, inasmuch 
 as it is possible, according to W. Biltz, F. A. Hen- 
 glein, and E. Schiebold, to obtain graphically definite 
 linear connections between the magnitudes concerned 
 (p. 184). In short, we are on a very promising road 
 leading to a knowledge of the activities of the par- 
 ticles in the crystal microcosm, and consequently to 
 the prediction of the properties of crystals. 
 
 EXAMPLES OF FINE-STRUCTURAL MORPHOTROPY 
 FROM THE MINERAL WORLD 
 
 For the formulation of general laws every material 
 is of the same value. At the same time, for those 
 interested in natural science, there is a special attrac- 
 tion in investigating the constitution and the laws of 
 association of the substances which make up the 
 structural material of the earth. Moreover, the fre- 
 quently large size of such crystals is a favourable 
 factor in the investigation; without the naturally 
 occurring forms of quartz and calc-spar fundamental 
 physical and chemical phenomena, such as double 
 refraction, circular polarisation, etc., would certainly 
 not have been discovered so soon. 
 
 In the study of fine-structure, rock-salt, zinc- 
 blende, lead-glance, calc-spar, magnetite, and several 
 
 1 Hydrogen appears to be particularly sensitive in this respect. 
 G. Aminoff estimated from the crystal structure of Mn(OH), and 
 ice fyy = 1°15 X 10-5 cm. and 1:12 -++10—-% cm. respectively. 
 Vegard obtained for NH,Cl and NH,Br, 0-83 and 0-84 x 1078 cm. 
 
 respectively, whilst, again, NH,I and another modification of 
 NH,Br gave rH = 1°12, or 0-99 x 108 cm. 
 
112 CRYSTALS AND MATTER 
 
 more substances, have served for investigations 
 which have become classical, whilst there remain 
 in the collections of the mineralogist many likely 
 materials for fine-structural investigation, such as 
 quartz, the felspars, mica, augite, hornblende, gyp- 
 sum and anhydrite, apatite, cinnabar, and numerous 
 other mineralogical forms besides. 
 
 With a view to adding to these examples from the 
 mineral world, particularly in a morphological con- 
 nection, I proposed to one of my students, Herr 
 M. Mechling, that he should undertake, under the 
 direction of my assistant, E. Schiebold, and myself, 
 
 @) 
 Fic. 127.—Topotropy for FeS, (iron pyrites) and CoAsS (cobaltite). 
 
 a study of cobaltite, CoAsS, so that the results might 
 be applied in a topotropical connection with reference 
 to iron pyrites FeSs. 
 
 The structure scheme for iron pyrites, according 
 to investigations of W. H. Bragg and W. L. Bragg, 
 is shown in Fig. 127: Fe at the corners and on the 
 surface centres of the elementary cube containing 
 four molecules, and of side 5:40 x 10-8 cm.; S, as 
 a double atom placed centre symmetrically on the 
 ternary axis. Considered topotropically, Co appears 
 in the place of Fe, and an As appears for one S of the 
 doublet in this equilibrium system, deforming it and 
 giving rise to another stable arrangement. It is 
 
MORPHOTROPY 113 
 
 found that the length of the cube side is only very 
 slightly changed, namely, from 5-40 to 5-66 x Ico 78 
 cm., the remaining S is displaced from its position on 
 the body diagonal of the cube, and, at the same time, 
 the substituted As does not take the position of the 
 displaced S atom. The centre symmetry is lost 
 on account of the leptonic inequality of As and $ 
 in the new arrangement of the atoms. Cobaltite 
 structure belongs not to the pinacoidal group, as does 
 iron pyrites, but to the pedial group. All mirror 
 symmetry in the fine-structure has vanished.! 
 
 For the unusually numerous isomorphous mix- 
 tures in the mineral world it is further possible not 
 only to compare topotropically the extreme substances 
 of the sequence of mixtures, but also to follow out 
 the process of change, step by step, in the series. 
 With this in view, Lotte Berndt, under the direction 
 of Dr. Schiebold and myself, has worked on a par- 
 ticular case. The mineral investigated was olivine. 
 The elementary structural cell of this Mg,Si0, sub- 
 stance (so-called forsterite) is a rectangular parallel- 
 opiped. It is of considerable interest to see how 
 this elementary cell changes when iron atoms are 
 introduced for one part of the Mg in Mg,SiO,, by 
 which process a so-called isomorphous mixture (Mg, 
 Fe).SiO, (chrysolite) is obtained, and _ resulting, 
 finally, in Fe,SiO,. The topotropic effect is brought 
 out in the following table : ? 
 
 1 According to later research on etch figures by H. Schneider- 
 hdhn, there occurs for cobaltite a rhombic transformation modifica- 
 tion with very tine twinned lamelle. The consequent mimetic 
 isometric structure gives rise in the Laue diagram to an effect of 
 correspondingly high symmetry. 
 
 * In this az, tz, cz denote the edge lengths of the elementary cell. 
 
 8 
 
114 CRYSTALS AND MATTER 
 
 az bz cz in 1o—8 cm. 
 Forsterite Mg,SiO, 4°74 10‘19 5:97 
 Chrysolite (Mg, Fe),Si0, with about 
 7 per cent. FeO . ; . 4°84 10°40 6:10 
 Fayalite Fe,SiO, . ; : ~. 4°99 10°89 6°31 
 
 It is indeed a triumph of physics that, thanks to 
 X-rays, the absolute values of such small displace- 
 ments in the microcosm can be measured. 
 
a WEARS OME SARA 
 
 UnvolaliorolLeniny CRYSTAL CLASSES, CRYSTAL 
 BORNMOCAS es LA DLULLY “ly PES 
 
 HE fine-structure of atoms, molecules, and 
 crystals may justly be compared with the 
 astronomical megacosm. In both cases the 
 question is one of statico-dynamical stability sys- 
 tems. It is immediately obvious, then, that as 
 certain structural arrangements recur in star systems, 
 so also types. arise in minute structure. 
 
 In the visible crystal world such isotypes are 
 given in the crystal systems (syngonies) and crystal 
 classes. 
 
 The groupings in the table of page 31 must, in 
 this way, be thought of as stability schemes, and it 
 is probable that they could 
 be calculated as a conse- 
 quence of attracting and 
 repelling forces. In this 
 way new light is thrown 
 on the crystallographic 
 ideas stated at the be- 
 ginning of this book, for 
 there we had to deal with 
 cases of isodynamostasy on Fic. 128.—Isotypic form 
 a large scale. Further, the Be toate 
 considerations may be extended to the specific gonio- 
 metric ratios within the subdivisions. With this in 
 
 115 
 
116 CRYSTALS AND MATTER 
 
 view I have noticed the remarkable analogy in the 
 crystal forms of very different substances, especially 
 those of simple chemical constitution, such as the ele- 
 ments, oxides, sulphides, halides, etc., in which certain 
 architectural characteristics are repeated. For these 
 the stability form of the complete isometric system 
 is required; for trigonal substances the arsenic, 
 quartz, and graphite type, as well as that of 
 hexagonal magnesium. These crystallographic form 
 groups stand in close morphological relation, as the 
 plan in Fig. 128, page 115, indicates diagrammatic- 
 ally. 
 
 FINE-STRUCTURE GROUPS AS STABILITY TYPES 
 
 Furthermore, by means of X-ray investigation of 
 crystals, there is an opportunity of examining the case ' 
 of isotypy with reference to the fine-structure. In 
 
 1As examples of special isotypy the following hexagonal 
 
 substances (magnesium type) are tabulated :— 
 End surface : 
 pyramidal 
 I : 1°6391 62707 
 I: 1°5802 > 61° 17. 
 IT: 11-6554 627 235 
 1 31'6288 62° 0” 
 1 21-6216 61° 54’ 
 I : 1°6305 62°25 
 I : 1-6006 61° 35’ 
 Greenockite, Cds , I : 1:6218 61° 54’ 
 Magnetic pyrites, FeS Biles ove Biot Lay 62° 19’ 
 Covellite, CuS . : ; S atiek Sono O12 At 
 
 I 
 
 I 
 
 I 
 
 I 
 
 I 
 
 I 
 
 I 
 
 I 
 
 I 
 
 a:c 
 
 Magnesium, Mg 
 Beryllium, Be . 
 Cadmium, Cd : 
 Irodosmine, (Ir, Os) . 
 Zinc oxide, ZnO 
 Beryllium oxide, BeO 
 Wurtzite, ZnS . 
 
 Arsenic-nickel, NiAs” : 16389 62°90" 
 Antimony-nickel, NiSb Ee7 220 63° 18’ 
 Silver iodide, on > 1°6392 62° 9’ 
 Ice, H,O a oF hy 61° 50’ 
 Tridymite, SiO, : : 16530 62°\2%0 
 Cadmium iodide, Cals > 1°5940 OL 29) 
 Lead iodide, Pbl, > 1°6758 62° 40’ 
 Carborundum, CSi : > 1-6324 Nae a 
 Copper glance (pseudo-hexagonal), Cu,S : 16707 02” 30. 
 Chrysoberyl (pseudo-hexagonal) 
 
 BeOA1LO;  . . : : <) Pt Oat 61° 55’ 
 
ISOTYPY 117 
 
 fact, the correctness of the idea becomes more and 
 more obvious, for certain structural schemes occur in 
 remarkable abundance, and in peculiarly close archi- 
 tectural relation to one another. For the examples 
 mentioned up to now, the case of a surface-centred 
 elementary cell with its tetrahedral placing of the 
 atoms in the form of the isometric type for elements 
 and simple compounds, occurs very frequently in- 
 deed. This corresponds to a very stable style of 
 structure, which still stands out prominently on 
 trigonal deformation. On closer consideration of 
 the elementary cells (Fig. 129 and 24, p. 22), the 
 
 Fic. 129.—Lattice type of rock-salt and its trigonal deformation to the calc-spar 
 type. 
 
 pre-eminent importance of the tetrahedral grouping 
 will be easily recognised as parts of the sections there 
 depicted, not only for diamond and zinc-blende, for 
 example, but also for copper, rock-salt, fluor-spar, 
 and in the deformed calc-spar. In Fig. 130 tetra- 
 hedral types of structure are reproduced to show 
 their special character. 
 
 In view of the isotypic agreement in the stability 
 of the isometric diamond and zinc-blende fine- 
 structure, it was of great interest to see if this equili- 
 brium form transmits itself to any extent to the 
 hexagonal type, a form I have ascribed to mag- 
 nesium, and to which carborundum consisting of C 
 
118 CRYSTALS AND MATTER 
 
 and Si also belongs. 
 
 The attractive problem of 
 investigating the fine-structure of this carbide by 
 
 means of the delicate and yet so powerful agency of 
 
 3, 
 Fic. 130.—Examples of tetrahedral structure: Copper, rock-salt, diamond, 
 zinc-blende, iron pyrites, calc-spar. 
 
 . 
 6 
 ome 
 xm 
 
 + y,". ¥ 
 
 rs 
 
 ae.’ 
 
 Fic, 131. Fie. 132. 
 Fics. 131 and 132.—Family relationship between the Laue diagrams of a 
 
 diamond twinning (plate parallel to the octahedral surface) and of carborun- 
 dum (plate parallel to the end surface). 
 
 X-rays was undertaken and carried out with great 
 success by H. Espig, under the direction of Dr. E. 
 Schiebold and myself, on type II. of carborundum. 
 
ISOTYPY 119 
 
 I had already earlier pointed out, in a comparison 
 of the Laue diagrams of diamond and silicon carbide, 
 the great resemblance of these spectral symbols of 
 the fine-structure ; this is brought out well enough 
 in Figs. 131 and 132. As the crosses in the figure 
 for carborundum show, it contains all the reflexions 
 
 Fic 133.—Fine-structure of modification II. of carborundum. C, dark; Si, 
 light circles. 
 
 of the diamond. The detailed X-ray study of 
 H. Espig very clearly indicated the correctness of 
 the assumption of a structural affinity. The tetra- 
 hedral diamond structure is involved to some extent 
 as a component of carborundum. Its carbon atoms 
 constitute goniometrically a form nearly identical 
 
120 CRYSTALS AND MATTER 
 
 with those of diamond. In this way, one of the two 
 (for the diamond, equal valued) carbon families is 
 reproduced here, the Si atoms of the carborundum 
 replacing the second tetrahedral group of the dia- 
 mond. Here, however, an important change in 
 configuration occurs, in that this type of atom is 
 arranged not in tetrahedra, but in slender trigonal 
 pyramids, which are set with the apex of one in the 
 body of another. Moreover, the rearrangement of 
 the C and Si stars in this complicated heaven is also 
 shown in the variation of the side length of the car- 
 bon tetrahedron ; it amounts in 
 the diamond to 2°5 x 1078, and 
 - In carborundum to 3°I x Io~7 8 
 cm. Thus both the diamond and 
 Fic. 134.—Tetrahedral fine. Carborundum belong, crystallo- 
 
 structure of hexagonal zinc 
 
 oxide. Large atom domain graphically, to a special type, 
 (d= 2:64)Zn, small, (¢=1'26)0. and it is now definitely recog- 
 nised that they are in consequence intimately con- 
 nected fine-structurally. 
 
 The same thing applies to hexagonal zinc oxide, 
 which W. L. Bragg has investigated. According to 
 him its oxygen atoms form everywhere the middle 
 points of tetrahedra whose corners are occupied by 
 zinc atoms. Hexagonal zinc oxide stands, therefore, 
 in close relation to isometric zinc sulphide, and also 
 to the diamond.! According to G. Aminoff, an 
 analogous case in connection with isotypy arises for 
 Mg (OH), and H,O. In its formal fine-structure, 
 Mg(OH), can be regarded as H,O.OH, structure, 
 for which one H, has been replaced by a magnesium 
 atom and spacial condensation has occurred. 
 
 1Isometric ZnS shows in its Zn and S tetrahedral groups the 
 two tetahedra of the diamond (compare Fig. 24d and 24e, p. 22). 
 
XI. CRYSTAL GROWTH AND SOLUTION 
 
 HE conception of the crystal as the stability 
 form of attractive and repulsive anisotropic 
 
 forces, involves the assertion of its reaction 
 to external physical or chemical changes in order to 
 accommodate itself to the new conditions. 
 
 In this sense we may consider the extensibility 
 and compressibility of crystalline materials, and 
 many other physical properties, under the influence 
 of temperature or pressure change. The capacity 
 of crystals to react to their surroundings stands out 
 as specially obvious in the easily observable pheno- 
 mena of growth and solution. These may therefore 
 be briefly considered here. 
 
 PURE CRYSTAL GROWTH 
 
 Every crystal has grown from a tiny nucleus, and 
 in this process of enlargement by addition of succes- 
 sive shells of parallel placed particles, the anisotropic 
 character of the substance is very clearly shown. 
 In this way we get, in general, not a sphere, but a 
 faceted body, indicating that the nucleus grows with 
 different velocities along different directions, in the 
 form of a growth pyramid; the crystal is thereby 
 divided up genetically, as F. Becke first emphasised. 
 Directions of similar growth recur at intervals. 
 
 According to the fundamental work of A. Johnsen, 
 
 which was extended by R. Gross and others, the curve 
 I2I 
 
122 CRYSTALS AND MATTER 
 
 of growth velocity, which is obtained by drawing pro- 
 portional vectors from some fixed point in accordance 
 with this growth anisotropy of 
 the crystal, shows maxima and 
 minima, and, moreover, very 
 sharp gradients. Null or 
 
 _ Fic. 135.—Growth by deposi- Fic. 136.—Growth pyramids. 
 tion of successive shells (example 
 of quartz). 
 
 {219 
 
 Fic. 137.—Intensity curve for the growth of rock-salt. 
 
 infinite extremes do not occur. Normal to the minima 
 directions, the large crystal surfaces naturally unfold 
 
CRYSTAL GROWTH AND SOLUTION 123 
 
 themselves, since these remain near 
 the origin. To them is to be as- 
 cribed a lower surface energy com- 
 pared with the faster-growing sur- 
 faces. 
 
 F. Haber, F. Paneth, P. Niggli, 
 and others have represented the 
 outer zone of the crystal as un- 
 saturated with respect to valency. 
 It attempts, therefore, to reach equil- 
 ibrium by addition of new particles. 
 Harmony with the outside is not 
 attained, for a new surface is there, 
 and the growth still continues. ey hie 
 
 P. Niggli made a considerable of growth. 
 
 ©. @, O' @: 0 @€ 0 .¢ CO. €@:. 0 6. 0 
 
 e©oe 0 © 0 & 0 @ 0 @ 0 Ox 
 (ei0 
 
 o-Altoms B=CL 
 e=/iltoms A-Na 
 
 On @ ON. 07 0) 10-20 OY On 6 
 eye 0 ® 0 © 0 @ 0 @ 
 
 De O— O: (Oia Os. O-- 0-20 40:0 
 DN 
 
 <3 0 60" - 0: 80; 50-0 ...6- 0 
 
 Fic. 139.—Surface zone of unsaturated valency for rock-salt according to 
 P. Niggli. 
 
124 CRYSTALS AND MATTER 
 
 advance in this matter when he assumed the growth 
 velocity proportional to the thickness A of the as- 
 sumed unsaturated crystal layer, a magnitude differ- 
 ent for different surfaces. In this way he arrived 
 at theoretical growth intensity curves, which, with 
 lead-glance and rock-salt (Fig. 139), for example, 
 agree very well with our ideas derived from the 
 morphology of these minerals. 
 
 Valeton has developed some very illuminating 
 views on the growth of crystals, with special reference 
 to the structure of rock-salt from sodium and chlorine 
 ions. The deposition of new particles depends, he 
 
 FIG, 140. Fic. 141. Fic. 142. 
 
 Fics. 140-142.—Ion groupings on the cube and octahedron surfaces of rock- 
 salt. After Valeton. 
 
 says, on the fine-structural nature of the surface. 
 The chess-board aggregation in the cube planes of 
 positive Na and negative Cl atoms is very clearly 
 differentiated from the uniform octahedral surfaces 
 containing either sodium or chlorine (Figs. 140-142. 
 See. also, Hig. 27, p. 26): . The vformer ares 
 favourable to the retention of an impinging ion of 
 the solution than the latter. An ion which is to be 
 retained on a cube surface must strike almost exactly 
 in the middle of a field, whilst for the octahedral 
 surface it depends on the sign of the ion, there being, 
 on the whole, a 50 per cent. chance. The cube will 
 therefore grow more slowly than the octahedron, and 
 
CRYSTAL GROWTH AND SOLUTION 125 
 
 will, in consequence, predominate in rock-salt. An 
 important point to be noticed in this connection is 
 that whilst with the Niggli idea cube the octahedron 
 and rhombic dodecahedron surfaces do not differ in 
 the thickness of their unsaturated zones (zero in each 
 
 aV2= 7292 
 2 Qa =9 ] 23 9 #4, 
 
 Fic. 145a. Fic. 1450, 
 
 Fics. 143-145.—Surface structure of the cube, rhombic dodecahedron and 
 octahedron surfaces of rock-salt, indicating the arrangement of the atom 
 domains. (The second scale relates to depth distances.) 
 
 case), on Valeton’s views there arises the possibility of 
 growth differences for these surfaces. In this respect 
 the great dissimilarity of the surfaces may be indi- 
 cated in greater detail by Figs. 143-145. These show 
 fine-structural aspects of such surfaces, indicating not 
 
126 CRYSTALS AND MATTER 
 
 merely the external plane but also the region beneath 
 by representation of the atomic domains. The action 
 of the attractive forces, which presumably are capable 
 of making their influence felt over several periods of 
 the fine-structure from the inside, outwards, will for 
 a cube surface be screened off by packs of only two 
 layers by the covering of the atomic domains.! 
 Matters are otherwise for the octahedron, or even 
 for the pyramidal cube surfaces, as Figs. 144 and 
 145 demonstrate. 
 
 Certainly these fine-structural differences with 
 respect to the region below the surface, together 
 with the net density (as the measure of the number 
 of particles belonging to the layer), are of immediate 
 significance as obvious explanations of the variation 
 of growth with direction in the crystal. Further, it 
 must be remembered in all these considerations that 
 in crystal growth we are concerned with the action 
 of a complicated chemical field between the crystal 
 and its surroundings, otherwise there would be no 
 possibility of explaining the great influence on the 
 crystallisation of the other things present in the 
 solution, which can give to sodium chloride, in one 
 case, the form of a cube; and in another case, that 
 of an octahedron or a pyramidal cube (Fig. 146).? 
 
 1 Here, as in other cases as for diamond (p. 55), and still toa 
 considerable extent in zinc-blende, canals as penetrating rifts in 
 the structure persist in these constructions which traverse the 
 crystal in certain directions, forming a check pattern. Their signifi- 
 cance, taken together, as a porosity of the crystal, remains to be 
 investigated in another place. 
 
 ? We might assume for this that the surface of the crystal is 
 chemically compensated by a special arrangement of the particles 
 there, and under the influence of the solution immediately rearranges 
 itself (compare p. 79). 
 
CRYSTAL GROWTH AND SOLUTION 127 
 
 In consequence, such additions to the solution 
 must possess the power of altering the factors con- 
 ditioning growth velocity at the crystal surfaces. 
 For example, the growth normal to the octahedron 
 surface may be retarded, and this external form 
 imparted to the crystal. Such, more or less, effective 
 protection of the surface will be explained by the 
 adsorption of the other substance present in solution, 
 an explanation proposed by R. Marc. Experiment 
 very definitely supports his hypothesis of a retarda- 
 tion of growth by adsorbed substances. With 
 calcium sulphate, for example, on addition of merely 
 
 a b c 
 
 Fic. 146.—Rock-salt crystal: a, from pure solution; 8, from solution containing 
 glycocoll; c, from solution containing formamide. 
 
 a trace of a certain dye, which is absorbed by the 
 crystal, we get, in contrast to the well-formed crys- 
 tals obtained from pure solution, an irregular mass 
 of thin sheets. When the crystal surface is satu- 
 rated adsorptively it loses the power of acting as 
 a nucleus. Such crystals may be shaken up in 
 supersaturated solution for days without disturbing 
 the supersaturation. 
 
 The particular form of a crystal constitutes a 
 morphological symbol of the equilibrium of the 
 balanced force fields which arise between its own 
 substance and the materials around. 
 
128 CRYSTALS AND MATTER 
 
 MIXED CRYSTAL GROWTH 
 
 For the analytical chemist crystallisation and 
 crystal growth have a special importance. Crystal- 
 lisation is for him a process of molecular selection, 
 inasmuch as one kind of matter separates cleanly in 
 crystal nuclei, on which more of the same substance 
 is then deposited. This process brings to mind ina 
 general way the ability of organisms in an abundance 
 of material, say, of a solution, to apply it to the 
 crowth of a certain one. 
 
 Of considerable general interest in physical 
 chemistry, and of significance to the analytical 
 chemist, is a circumstance which may diminish the 
 value of crystallisation as a means of purification. 
 If, in the chemical field between crystal and solution, 
 besides the substance chemically identical with the 
 crystal, there exist also others, which, although 
 different, are chemically analogous to it, as, for 
 example, besides the chloride of potassium, those of 
 Rb or (NH,), then interchange of atoms or radicals 
 may occur. 
 
 This has been known as an experimental fact 
 since the researches of J. N. Fuchs and F. E. Mit- 
 scherlich, and later, in the light of space-lattice 
 theory, has become very clearly understood, fine- 
 structurally.1. The case here is one of the growth of 
 isomorphous mixtures (p. 97). Moreover, different 
 mixtures, or the pure extremes of substances so 
 related, are able to build themselves up into layers. 
 A splendid example of this is shown in Fig. 148 for 
 zonal tourmaline. | 
 
 The inclusion of such alternative structural units 
 in the growth can naturally only take place in so far 
 
 1 See Fig. 147. 
 
CRYSTAL GROWTH AND SOLUTION 129 
 
 as the stability of the whole is not imperilled. Often 
 that is not at all the case for extensive isomor- 
 phous mixtures as for felspar, olivine, and other 
 minerals. In other cases, warpings of the form or 
 optical anomalies point to discordance in the structure 
 of such heterogeneous lattices. There exists also a 
 certain danger with respect to the architectural 
 cohesion, that on change of the temperature the 
 alternating particles may lose more and more their 
 structural equivalence by the swinging round of 
 their electrons, i.e. by change of form. A pair of 
 
 FIG. 147 .—Isomorphous alternation Fic. 148.—Isomorphous stratification 
 of Cl and Br in the space-lattice. Example: tourmaline. 
 
 substances such as Na and K is simply incapable of 
 this alternating incorporation in the growth, unless 
 the atomic structures are first in a similar condition. 
 This leptonic isomorphism depends on the physico- 
 chemical factors, temperature, and pressure, as well 
 as on the action of the surrounding material, these 
 being the effective influences on the fine-structure. 
 In this respect the marked change in the miscibility 
 with temperature of NaCl and KCl already men- 
 tioned on page 103, is of interest. 
 
 Especially worthy of note in these phenomena 
 of regular incorporation in mixed growth are those 
 2 
 
130 CRYSTALS AND MATTER 
 
 cases in which occasionally a substance quite foreign 
 is incorporated. Many minerals bound together 
 macroscopically in regular fashion, such as cyanite, 
 staurolite, rutile, and iron-glance exhibit the relation- 
 ship (Fig. 122, p. 103). For very small dimensions 
 of the definitely oriented guest material, and for more 
 extensive growth of the surrounding host, a kind of 
 colloidal solid solution with regular packing of the 
 substances, but without any stoichiometrical ratio 
 one to another, is arrived at as, 
 for example, in “ adsorption com- 
 pounds,” “occlusion,” and the 
 like. 
 
 As to the forces of cohesion 
 between the crystal particles and 
 such foreign bodies, no deviation 
 will be made from the hypothesis 
 of electrical coupling. This hy- 
 Ser ieee pothesis has continually increas- 
 oe paint eee) ing support, as, for instance, in 
 
 connection with the flocculation 
 of colloidal substances. There again it is possible to 
 employ crystals as types with actual visual observa- 
 tion. Electrically bound charges may, in fact, be 
 immediately observed for pyro-electric crystals like 
 quartz, which on change of temperature becomes 
 electrified positively and negatively respectively on 
 its alternate column edges (Fig. 149)... To show the 
 distribution of the oppositely charged regions in such 
 cases, A. Kundt employs a mixture of sulphur and 
 red lead which has become electrified by being blown 
 out of a suitable sprinkling apparatus. The positive 
 red lead sticks to the negative parts of the crystal, 
 coating them red, the negative sulphur deposits on 
 
CRYSTAL GROWTH AND SOLUTION 131 
 
 the positive regions, which consequently appear 
 yellow. In this way an adsorption coupling of 
 crystallographic nature can be immediately detected 
 for quartz. 
 
 COLLECTIVE CRYSTALLISATION 
 
 It is further of great physico-chemical interest 
 that forces tending to form aggregates exist not only 
 between crystal and solution, but also between 
 crystal and crystal. In fact, it is certainly one of the 
 most remarkable phenomena of the crystal world 
 that in spite of the rigidity of the material there is 
 this tendency to collect together to larger individuals, 
 1.e. the space-lattices of neighbouring crystals tend to 
 set themselves parallel and to link up; this process, 
 which I have studied in various cases, I have termed 
 “collective crystallisation,’ which indicates _ its 
 nature. The effect is, in view of the coalescence of 
 small particles to form large ones, a “ coarsening of 
 the grain.’”’ The process may be demonstrated in a 
 few minutes by heating up cast steel. Every chemist 
 is acquainted with it in the much-used platinum 
 crucible, and the technical ighting workman, in fine 
 tungsten filaments, the innumerable small crystal 
 individuals of which, on glowing, are turned round 
 into single crystals which may be a metre in length. 
 Tungsten trioxide also is convenient for showing the 
 phenomena, and calc-spar too, if care is taken that in 
 the intense heating the CO, is not allowed to escape, 
 as is the case if the experiment is carried out in 
 a carbonic acid bomb. The marmorisation of dense 
 limestone in contact and regional metamorphosis is 
 also a case of this collective crystallisation.! 
 
 1 What happened here was a transformation of ordinary dense 
 limestone into marble by molten masses, perhaps of granite type, 
 
132 CRYSTALS AND MATTER 
 
 In such cases of closely packed, new-grown struc- 
 tures (as Figs. 150) and 151) show), the individuals 
 
 Fic. 150.—Collective crystallisation for iron: a@, Martin steel; 6, the same 
 Martin steel glowed in the furnace. 
 
 Dare 
 iors, s 
 
 ROM 
 on; 
 
 ‘ z “ 
 XK MG SI 
 At fi ; 
 Late tb) sed! S (ado 
 WS Coe IS Kea 
 Soars Keg 
 oa iret 
 dew AAS 
 
 Fic. 151.—Collective crystallisation of calc-spar: a, limestone; 4, marble. 
 
 hinder each other in the production of regular 
 crystallographic forms. If, on the other hand, they 
 
 which, arising from the depths of the earth, stuck fast in the stony 
 crust, and then, by reactive action of the gases and liquids liber- 
 ated, and the high temperature, extensively transformed the matter 
 around. Here, at the “ regions of contact,’’ collective crystallisation 
 played a great part, and, in addition, many substances finely dis- 
 seminated in the original material became associated into large 
 crystals of, perhaps, graphite, andalusite, augite, garnet, etc., so 
 that here again collective crystallisation was effective. 
 
CRYSTAL GROWTH AND SOLUTION 133 
 
 swim around freely in a liquid, association by collec- 
 tive crystallisation, with new development of crystal 
 surfaces, can proceed. This is the case for the 
 highly interesting combination of contiguous ammo- 
 nium oleate crystals described by O. Lehmann. 
 Elongated pyramidal individuals coalesce to a struc- 
 tural unit more or less definitely crystallographic in 
 contour, or, at least, to a group of crystals with their 
 
 Fic. 152.—Collective crystallisation for ammonium oleate. After 
 O. Lehmann. 
 principal axes parallel (Fig. 152). Moreover, it may 
 probably be assumed that in every crystallisation, to 
 begin with, numerous sub-microscopic crystals arise 
 which form by aggregation visible crystal nuclei in 
 the above manner. 
 
 CRYSTAL SOLUTION 
 
 Corresponding to many of the observations on 
 crystal growth, a knowledge of crystal solution has 
 lately been developed to a very gratifying extent. 
 The observations here indicate that the principles of 
 the phenomena are closely connected. It appears of 
 use in dealing with the solution of crystals, as in the 
 discussions on their growth, to employ the very clear 
 picturisation of the atomic domain stereograms. In 
 
134. CRYSTALS AND MATTER 
 
 this way we see immediately, as the examples of 
 Figs. 143-5, page 125, show, the anisotropy which 
 occurs both in solution and growth. It becomes clear 
 that the displacement of the surface, by the attack 
 of the solvent on the crystal structure, takes place 
 with varying rapidity, depending on the direction of 
 attack. Research definitely corroborates this state- 
 ment, to which the pretty phenomena of etch figures 
 bear immediate witness. These are formed in large 
 numbers, and mostly of microscopic dimensions, on 
 crystal surfaces attacked by a solvent, and represent, 
 in the form of cavities or eminences with regular 
 
 Fic. 153, a, 6, c.—Etch figures and the corresponding light figures on a side 
 surface of gypsum, together with figures for the same mineral dehydrated. 
 
 edge and surface boundaries the symmetry of the 
 crystal faces on which they occur (Fig. 153a). On 
 examining all the results so obtained, the structural 
 style of the whole crystal body becomes evident. 
 With the reflexion goniometer corresponding light 
 figures may be measured (Fig. 153b). In both figures 
 the binary character of the surface is brought out. 
 Good examples of solution anisotropy are very neatly 
 shown by plates cut into circles, an example of which 
 is shown in Fig. 154. It represents an initially 
 circular plate of gypsum, which on immersion in 
 water has changed to a pointed figure of elliptic 
 periphery, owing to the anisotropy of the solution 
 velocity. The relations of polished crystal bodies of 
 
CRYSTAL GROWTH AND SOLUTION 135 
 
 regular, and in particular, cubic form, or of crystal 
 spheres, are known in even greater detail. Growth 
 and solution velocity mutually correspond, the repre- 
 sentative vectors rising and falling together. The 
 directions of rapid growth are those of rapid solution, 
 and vice versa. During the solution of a crystal a 
 struggle takes place between the anisotropic velocities 
 of solution. The surfaces, which quickly approach 
 to the mid-point of the crystal, and which are those 
 having large solution velocities in the direction of 
 
 © > 
 Uy Lo y oh i y / 
 100 ox UY Ny, 
 nM 
 Yi il S 
 403 
 
 Fic. 154.—Anisotropy of crystal Fic. 155.—Reproduction of a per- 
 solution. Example: Initially circular manently conformal solution form. 
 plate of gypsum in dissolution. Example: Anhydrite in sulphuric acid. 
 
 their normals, will gain more and more area, sup- 
 pressing the slower ones which will probably be 
 present at the start, and will finally form the 
 boundary planes of the residual body. The scheme 
 of Fig. 155 depicts such a final form for anhydrite. 
 In the same connexion several figures from a research 
 of W. Schnorr on rock-salt may be mentioned. 
 Specimens of the mineral which are initially cubic 
 (Fig. 156) assume the forms shown in Figs. 157, 158, 
 159, 160, when immersed in an unsaturated solution 
 of sodium chloride containing urea. At first there 
 
136 CRYSTALS AND MATTER 
 
 are formed, on the cube edges, surfaces of the pyra- 
 midal cube, which suppress the original form, but 
 are then suppressed by those of the icositetrahedron. 
 The icositetrahedron so produced is quite unaltered 
 in shape on further solution. A stability form as the 
 expression of the equilibrium attained is arrived at 
 in this way. Under constant conditions, during solu- 
 tion, as also during growth, the velocity of displace- 
 ment of a surface is found to be always the same. 
 
 Fic. 158. Fic. 159. . Fig. 160. 
 
 FIGs. 156-160.—Solution process for a rock-salt cube in an unsaturated solution 
 of sodium chloride containing urea. After W. Schnorr. 
 
 An analogy between growth and solution also 
 arises in this way, since, in both cases, the question 
 is one of definite reciprocal action between crystal 
 and its environment, depending on the nature of 
 the other substances present, 1.e. on the chemical 
 field. The morphological action of an anisotropic 
 solution depends not only on the crystal but also 
 on the particular solvent. This may be very clearly 
 shown for anhydrite, which W. Burckhardt has in- 
 vestigated on this point at my suggestion. Accord- 
 ing to whether sulphuric acid, nitric acid, water, 
 
CRYSTAL GROWTH AND SOLUTION 187 
 
 or salt solution was employed, different solution forms 
 were obtained from a cubic cleave of the rhombic 
 mineral (Fig. 161). 
 
 The symmetry of the solution body is naturally 
 the same in all states of its formation and for every 
 solvent. It is identical with that of the growth 
 figure. From the phenomena of solution the par- 
 ticular class of the thirty-two crystal classes to which 
 a substance belongs may be determined. 
 
 ie oO 
 
 (100} i{010} {001} {100}; a {100} ;{010};{ o01}i{0.1.10} {100}; {010};{007} ;{so%} 
 is d 
 
 Fic. 161.—Variation of the solution form for different solvents. Example : 
 Anhydrite. a, Tri-pinacoidal initial cleave ; 4, solution in sulphuric acid ; 
 ¢, in nitric acid or water ; d, in salt solution. 
 
 SUMMARY OF CRYSTAL GROWTH AND SOLUTION 
 
 In a general survey of the processes of crystal 
 growth and solution it must be considered that the 
 question is, broadly, one of the displacement of the 
 boundary between solid crystalline and liquid (in 
 certain cases, gaSeous) masses which are exerting 
 forces on each other in opposite directions. In 
 growth the attractive forces of the crystal prepon- 
 derate ; in solution those of the surrounding matter 
 are greatest. The external surface is correspondingly 
 displaced. All circumstances which diminish the 
 electrical connection of the crystal particles act in 
 the direction of solution, and vice versa. Water, 
 with its high dielectric constant, and consequent 
 extensive diminution of the electric attraction, brings 
 many crystals to destruction. 
 
138 CRYSTALS AND MATTER 
 
 Such crystals dissolve as soon as the conditions 
 of temperature, i.e. of the motion of the particles, 
 and the modifying influence of other substances in 
 the solution, permit a displacement of the surface 
 inwards as the result of the opposing forces acting in 
 this direction. For insoluble substances the internal 
 cohesion preponderates over physico-chemical forces, 
 tending to break up the structure. 
 
XII, CHEMICAL ACTIONS ON CRYSTALS 
 
 ANISOTROPY OF CHEMICAL ACTIONS ON CRYSTALS 
 
 and chemical action are dealt with separately, 
 
 no fundamental contrast in the processes is 
 implied. On the contrary, they are very closely 
 related. Thus the reaction CaCO, + 2HCl = CaCl, 
 + H,O + CO, for calc-spar manifests its regular 
 anisotropy not only in pretty microscopic etch 
 figures (and also light figures) 
 (Fig. 162), but also in the varying nea So 
 amounts of CO, developed in unit 
 time on the morphologically dif- 
 ferent surfaces. The same thing 
 applies for the reaction body. Fig. 
 163 gives a beautiful example of ae 
 this. Fig. 164 depicts such a jy 162.—Rtch figures on 
 case indicating the variation with  “le-spar crystal. 
 change of the corrosive agent employed ; it refers to 
 the decomposition of a tourmaline by caustic potash 
 and hydrofluoric acid, which was recently studied in 
 my institute by Lotte Kulazewski. The reactions 
 with the silico-borate in question are especially inter- 
 esting, as, owing to the trigyric domatic character of 
 the mineral, they proceed vectorially (differently in 
 the upper and lower halves of the tourmaline sphere), 
 as a comparison of the diagrams Figs. 164a and 6 and 
 c and d indicates. Figs. 164a and c represent the 
 
 139 
 
 A LTHOUGH the phenomena of growth, solution 
 
 a 
 aN 
 m4 
 ie | 
 
 ~ferrr--ccn 
 
140 CRYSTALS AND MATTER. 
 
 upper, Figs. 6 and d the lower hemispheres with 
 respect to the light reflexion of the reaction body ; 
 both hemispheres are depicted up to beyond the 
 equator. On allowing the reaction to proceed 
 further, the vectorial character of the chemical 
 process is realised morphologically with extraor- 
 dinary effect, in that the sphere becomes more and 
 more flattened on one side, and finally is transformed 
 
 into a dome-shaped body, which has already been 
 
 illustrated in Fig. 79¢, page 56. In cases of centre 
 symmetrical structure the chemical action is ten- 
 sorial, i.e. equal in any one direction, and the 
 corresponding opposite direction. 
 
 ANISOTROPIC CHEMICAL REACTIONS OF MOLECULES 
 
 For individual molecules in gases or liquids the 
 morphological principle of chemical reaction cannot 
 differ from that for crystals. Indeed, the funda- 
 mental conception of the stereochemistry of chemical 
 reactions, such as the localisation of the replace- 
 ment of one or more hydrogen atoms in the benzene 
 molecule by Cl, NH,, CH;, or other radicals,! corre- 
 sponds to this assertion. The process of chemical 
 action, according to this, is always definitely aniso- 
 tropic, and the action for molecules only differs from 
 that for crystals in the suppression of a restricting 
 rhythm in the structure, and consequently in the 
 chemical relations. A centre or mirror symmetrical 
 arrangement, or a “ plurality’ of places liable to 
 attack may naturally, however, play a réle in chemical 
 operations with molecules, inasmuch as completely 
 similar topical relations (such as the two (NH,) groups 
 
 1 For the mechanics of such processes, compare page 158. 
 
Fic. 163.—Reaction body of a sphere of topaz after treatment with caustic 
 petash. After MW. Eichler 
 
 Fic. 164.—Reaction anisotropy with ternary rhythm obtained by treating a 
 tourmaline sphere with 1 (left-hand figure), caustic potash ; a, top surface; 
 6, under surface. 2 (right-hand figure). hydrofluoric acid ; ¢, top surface ; 
 d, under surface of the sphere. (Vectorial anisotropy of chemical reaction.) 
 
 After Ch. Kulaszewskt 
 
CHEMICAL ACTIONS ON CRYSTALS 141 
 
 in (NH,), CO; molecule) may show in similar reactions 
 their equal fine-structural and equal chemical rank in 
 the complex. 
 
 STRUCTURAL RIGIDITY OF ELECTRONS, ATOMS, 
 MOLECULES AND CRYSTALS 
 
 According to the above, an estimate of the 
 chemical nature of substances and of their changes 
 turns on the investigation of fine-structure and its 
 variations. This can only be carried out completely 
 by reference to the stages in the graduated series 
 which extends from the simplest form of matter, 
 from electrons to atoms and molecules, and finally 
 to crystals. 
 
 With respect to the possibility of initiation of 
 chemical reaction and its continuance, the general 
 structural rigidity of the particles is the question 
 of prime importance, inasmuch as the resistance 
 to chemical change is dependent on this factor. As 
 the particular circumstances vary considerably, we 
 are interested here in the general state of affairs, 
 that is, whether, for the purpose of the present con- 
 sideration, the graduated series from electron to 
 crystal is arranged in accordance with the structural 
 rigidity. An examination of the series from the 
 lowest to the highest member indicates unmistak- 
 ably an increasing fine-structural complication, and 
 parallel with this, a decrease in the rigidity of the 
 various types. 
 
 The architecture of the electron has remained up 
 to now imperturbable. Such forms are, we may 
 say, fortresses still untaken. 
 
 Practically the same thing holds for the atom as 
 for the electron, as far as the central portion, the 
 
142 CRYSTALS AND MATTER 
 
 nucleus is concerned. A variation of this inner 
 structure, the main mass of the atomic system, is 
 only possible by enormous expenditure of energy. 
 As we know, Rutherford has managed to cause disin- 
 
 tegration of nitrogen atoms N into 2H+ and 3Hett 
 by powerful bombardment with Het* particles. 
 Only one out of every 100,000 shots resulted in 
 an actual collision. Although, therefore, the nucleus 
 
 of the atom is not indivisible, it is a structure ex- — 
 
 tremely difficult to split up. 
 
 For the outer shell the case is otherwise. As 
 regards this external zone, the atomic system does 
 not, in point of fact, accord with its name. On the 
 contrary, the atom shell is quite easy to split up, 
 and is thus ‘‘eutomic.’’ Since the acceptance of 
 Sv. v. Arrhenius’ conception of ions, and following 
 that, the fine-structural explanation of their forma- 
 tion, as a splitting off of electrons from the outer 
 sphere or an insertion of the same, the matter has 
 become one of easily effected superficial variations. 
 The idea of structural variations in the atom is made 
 use of when light emission and absorption are ex- 
 plained as the transposition of an electron from one 
 stable path to another, with consequent loss or gain 
 of energy quanta. In such cases quite small quanti- 
 ties of energy render possible very real changes in the 
 outer sphere of the atom. 
 
 It would be superfluous here to go into details 
 concerning the disruption and aggregation of mole- 
 cules and their transformation by substitution ; this 
 is, in fact, the main topic of chemistry. The incon- 
 ceivable abundance of these phenomena shows how 
 such changes can take place, with varying energy 
 
CHEMICAL ACTIONS ON CRYSTALS 143 
 
 exchanges, which, in general, occur with relatively 
 greater facility than in a physical field (e.g. by a 
 rise of temperature or through the action of matter). 
 
 Passing through the structural series to the final 
 and most complicated forms, namely, crystals, which 
 are specially considered here, the relations point in 
 the same direction of readier variation in the archi- 
 tecture. With crystals it is quite usual for the struc- 
 ture to collapse on application of physical or chemical 
 influences, and for the matter to assume a lower 
 structural type. For instance, in the action of HCl 
 on calc-spar, the space-lattice complex of the latter 
 is broken up with the formation of free molecules, 
 CaleH.,O and CO,. 
 
 CRYSTALLOGRAPHIC CHEMICAL CHANGES IN THE 
 STRUCTURE. UNDERMINING AND RECONSTRUCTION 
 
 In less frequent cases, however, chemical changes 
 occur in crystalline materials without destruction of 
 the crystal structure, i.e. with preservation of the 
 high crystalline status of the substance. We are 
 then concerned with a particularly important case 
 of the so-called topochemical reactions of V. Kohl- 
 schiitter, that is, with a structural undermining. 
 Further, it is occasionally possible to reverse the 
 process or to substitute something else for the sub- 
 stance removed, and thus to transform the crystal 
 structure without at the same time destroying the 
 crystalline character of the material. But naturally, 
 these are extreme cases which are linked up by inter- 
 mediate examples of more or less drastic disturbance 
 with the other extreme of complete destruction of 
 the crystalline form during chemical reaction. 
 
 It is profitable to compare a process of ideal 
 
144 CRYSTALS AND - MATTER 
 
 crystallographic chemical undermining with the par- 
 tial destruction of a framework structure from which 
 the filling between the beams has been removed. 
 On account of this, the structure becomes less com- 
 pact without, however, collapsing, and the principal 
 structural lines are stid maintained. The simile may 
 be readily extended to a reconstruction. The beams, 
 although remaining in position, may, in the partial 
 disruption, be damaged to some extent by splitting. 
 and transverse fracture, and as structural particles 
 in an undermining or reconstruction may become dis- 
 placed, the stability of the whole may be impaired and 
 finally lost. 
 
 In the light of space-lattice theory, it may be 
 assumed for the ideal case of undermining that from 
 a point system such as the one of Fig. 9, page Io, 
 one space-lattice is removed without the remainder 
 collapsing, although it may be deformed to a new 
 equilibrium arrangement. In reconstruction compen- 
 sation is made for the removed lattice. We must, 
 however, expect many gradations of the above, while 
 still more complicated rearrangements and dismem- 
 berments even to the dissolution of the molecule, may 
 occur. 
 
 The structural residue which remains after glow- 
 ing the natural fluorine-containing cerium didymium 
 lanthanum calcium carbonate (so-called parisite) is, 
 according to G. Aminoff, surprisingly well preserved, 
 so that it responds tolerably well to the searching test 
 of the Laue diagram. Figs. 165-166 give a diagram- 
 matic representation of this interesting case. 
 
 A more or less extensive dehydration may be 
 effected for minerals of the zeolite group without the 
 destruction of the crystal form. It is known that 
 
Fic. 165. Fic. 166. 
 
 Fics. 165, 166.—Laue diagrams of parisite (synchysite) and metaparisite 
 After G. Aminoff 
 
 2.01) 
 
 Bic. 
 
 167.—Onptics of heulandite and metaheulandite 
 
eS eee Sc Te RR ce ; vs 
 oats SPT Mee a 
 UUvERSITY ge HLS © i 
 
 tn ‘ 
 
 “1 
 
CHEMICAL ACTIONS ON CRYSTALS 145 
 
 in heulandite, for example, the loss of water takes 
 place practically continuously, and that sometimes 
 an equilibrium between the hydrosilicate and its sur- 
 roundings is established. Interesting parallel pheno- 
 mena to this are the optical relations of the mineral. 
 Like the hands on a clock, the extinction directions 
 move round indicating the water content of the sub- 
 stance. Here then is a good opportunity to study a 
 chemical equilibrium by an optical method (e.g. by 
 polarised light). Observations by O. Weigel and 
 K. H. Scheumann confirm quantitatively that there 
 is, in fact, a very exact parallelism between the 
 chemical composition and the optics of the material, 
 for their variations accord precisely. The crystalline 
 nature of heulandite remains undisturbed for this 
 variation to and fro of the chemical composition, at 
 least as regards the first stages of dehydration, as is 
 proved by the Laue diagrams. The continuation of 
 the process leads to more drastic deformations. In 
 the removal of water from this zeolite, so long as 
 about three mols remain, we are concerned, not 
 with a process very vital to the architectural sta- 
 bility, but rather with the removal of a constituent 
 which is only loosely, although regularly coupled 
 to the silicate space-lattice. This is in complete 
 agreement with the view that heulandite is mor- 
 phologically closely allied to its felspar anhydride. 
 The diagrams of Fig. 170 show the marked analogy 
 in the appropriate angle relations. The magni- 
 tudes of the axial ratios indicate the same thing ; 
 but, in addition, the morphological influence of the 
 H,O becomes apparent in the length of the “db” 
 axis. 
 
 In this particular we find a considerable mor- 
 IO 
 
146 CRYSTALS AND MATTER 
 
 — ee eee 
 
 114"}0"' 
 
 ee ow) a oe we es) on ee inane) Sp) Oe eee 
 
 Fic. 169. 
 
 Fics, 168 and 169.—Laue diagrams of heulandite and metaheulandite. 
 
CHEMICAL ACTIONS ON CRYSTALS 147 
 
 phological difference in the structures of felspar and 
 heulandite :-— 
 Sanidine i Re eM es Gee a eae it arta halig Meek ay Rie 
 Heulandite GC AON es Te 2) Ore Ba Od 
 
 Among zeolites scolecite is of very great interest. 
 Its change into metascolecite leads, just as in a 
 change of modification, to an actual transformation 
 of the silicate structure, indicating that here the 
 water content is certainly of fundamental importance 
 to the whole. Although we may use the methods 
 
 Heulandite 
 
 Sanidine a 
 id along the (oro! . 
 
 along the (o10) 
 
 Fic. 170.—Comparison of sanidine and Fic. 171.—Laue diagram of 
 heulandite. metascolecite. 
 
 of ordinary and X-ray optics, which prove the 
 persistence of a space-lattice arrangement in meta- 
 scolecite, the phenomena may also be conveniently 
 demonstrated pyroelectrically. With reference to 
 Fig. 172, I found, in fact, that on dehydration of 
 scolecite the front and side surfaces become inter- 
 changed as regards the symmetry relations. I found 
 a case of structural undermining of exceptional 
 chemical simplicity in brucite, the natural trigonal 
 Mg(OH),. At about 400° expulsion of the water 
 begins, which wanders out from the point system 
 
148 CRYSTALS AND MATTER 
 
 by diffusion and evaporates from the crystal to the 
 outside, the effect increasing with increasing tem- 
 perature, till finally MgO as a pseudomorph of 
 Mg(OH), is left. A comparison shows that the 
 crystal optics of a trigonal body, although weakened, 
 
 Fic. 172.—-Pyroelectric effect for scolecite and metascolecite (sprinkled with 
 sulphur and red lead). 
 
 still remain with reversal of the double refraction, 
 
 the directions of the optical axes being unchanged 
 
 (Fig. 173). The more sensitive X-ray tests show, 
 
 on the other hand, that the change has not occurred 
 
 without deformation. While brucite gives a Laue 
 
 Fic. 173.—Optics of brucite (MgO . H,O) and metabrucite (MgO). 
 
 diagram of mere points, I found for MgO, obtained 
 from brucite, a star-like X-ray figure (Fig. 174-5), a 
 fact also mentioned by G. Aminoff. It indicates 
 regular bending such as can be produced with the 
 same effect on mica, rock-salt, and other substances. 
 In addition, inner variations of the structure occur 
 
CHEMICAL ACTIONS ON CRYSTALS 149 
 
 on the expulsion of the H,O. The high tempera- 
 ture appropriate to the undermining is favourable 
 to this. O. Paul informs me that he actually 
 obtained with glowed metabrucite the Debye- 
 Scherrer diagram of periclase, that is, of the iso- 
 metric form of MgO, which according to him and 
 Gerlach is given on heating magnesite, MgCO,, to a 
 red heat. 
 
 Such phenomena are transitional to those for 
 which very drastic rearrangements in the fine- 
 
 Fic. 174. Fic. 175. 
 
 Fics. 174-175.—Laue diagrams of brucite and metabrucite. (X-ray star figure.) 
 
 structure occur in the topochemical reaction of dehy- 
 dration, as in the change of gypsum CaSO,2H,0O to 
 the so-called subhydrate CaSO,4H,O, and then to 
 the anhydride CaSO,. The constitutional difference 
 between anhydrous and dihydrol calcium sulphate 
 is at once indicated by the macrostereochemistry of 
 the crystal form. Gypsum is monoclinic, while anhy- 
 drite is rhombic, which is quite a different type of 
 structure (Figs. 176-177). 
 
 This variation in the function of water in crystal 
 structure is also shown in physical chemical diagrams, 
 
150 CRYSTALS AND MATTER 
 
 such as are depicted in Fig. 178. In hydrated barium 
 chloride the H,O is an essential fine-structural con- 
 stituent. On heating the crystal it 1s expelled in 
 quanta (to some extent in large fine-structural, and 
 
 200 
 
 (60 
 
 ! 
 | 
 | 
 i 
 : | 
 
 i 
 Heulandite | 
 ‘ Ca Silz Digs On 7 7a5.5/1,0 
 
 l 
 
 | 
 ! 
 | 
 ‘ 
 | 
 | 
 
 7emperature 
 
 | 
 | | 
 | | 
 { 
 
 ‘ Mols N;O—» ” “ 
 
 Brucite 
 Mg 0-120 
 
 Fic. 176. 
 
 a 
 Fic, 177. reosrernt kl 
 
 Fics. 176, 177.—Gypsum Fic. 178.—Diagrams for the dehydration of 
 (CaSo,2H,O) and anhydrite heulandite (after O. Weigel), brucite (after O. 
 (CaSo,). Westphal), and hydrated barium chloride. 
 
 Temperature 
 
 Ea LIne 
 
 Dihydrated barium chloride 
 
 Jemperature 
 
 thus chemical, aggregates) at ‘boiling points ”’ 
 corresponding to the bends in the curve of Fig. 178. 
 Brucite shows one segment corresponding to evapora- 
 tion, and the researches of O. Weigel and K. H. 
 Scheumann indicate that for heulandite the curve is, 
 
CHEMICAL ACTIONS ON CRYSTALS 151 
 
 at least to begin with, nearly a straight ascending line, 
 which, according to O. Weigel, is of interest in that it 
 shows singular points for the simple stoichiometrical 
 ratios of silicate and water. At these positions evapo- 
 ration is checked by momentary strengthening of the 
 bonds between water and silicate. 
 
 The undermining of crystals may, however, take 
 place to an extent much greater than is represented 
 by the removal of a relatively small part of the con- 
 stituents which water usually represents. The calcium 
 aluminium hydrosilicate CaOAl,036S10,¢ . 5°5 aq. of 
 heulandite, for example, may be reduced fine-structur- 
 
 Fic. 179.—a-6—Form and optics of heulandite and its silicon dioxide; 
 c-d—form and optics of desmine and its silicon dioxide. 
 ally to SiO... The entire filling of basic constituents 
 is then removed, the result being just as though the 
 skeleton of some silicious plant had been prepared by 
 burning the organic wrapping. The relict of the zeo- 
 lite so obtained still shows (especially after glowing, 
 probably under the influence of collective crystal- 
 lisation) definite optical agreement with the original 
 substance, the hard rigid pseudomorph of which it 
 represents. If, starting from desmine zeolite, all 
 the basic constituents are simultaneously withdrawn 
 with hydrochloric acid, a SiO, optically analogous 
 to the original desmine is obtained. Thus the same 
 chemical substance SiO, appears here to have a 
 
152 CRYSTALS AND MATTER 
 
 varying structure depending on its previous history ; 
 in the one case it is a heulandite, in the other a 
 desmine residue. 
 
 We may experiment in the same way with dark 
 mica (biotite) and break it down to a very soft SiO, 
 in flakes, similar to biotite, which, as Si0O,-metabio- 
 tite, is similar optically to mica. ‘The X-ray experi- 
 ments show, however, that in such extensive under- 
 mining of the structure considerable disturbances in 
 a leptonic sense have occurred. X-ray diagrams are 
 no longer obtainable for the residual silica of the 
 zeolites and mica. This is also the case for so-called 
 koenenite a 3MgO. Al,O; . 2MgCl, .6H,O, which may 
 be reduced to Al,Os, in very soft flakes corresponding 
 to the form of the original crystal. Doubtless, in 
 such cases, there occurs extensive devastation of the 
 inner architecture within the external frame of the 
 structure, which still stands. Using ordinary light, 
 this view is not supported to the same extent as 
 with the sensitive X-rays, which fail to give regular 
 reflexion because of the increased agitation of the 
 atoms in the structure, following the rise in tem- 
 perature.t Moreover, weak double refraction may 
 accompany needle, flake, or prism structure. In 
 other cases, however, the weakening of the structure 
 makes itself, ultimately, macroscopically evident. 
 For scolecite and olivine, for example, one obtains as 
 a residue of the chemical action a silica gel no longer 
 coherent. The loosening of the structure has then 
 
 1A roughness of the surface reduces the reflexion of ordinary 
 light. The work of E. Wagner shows that in the same way the 
 capacity to reflect X-rays is diminished by the deviation from 
 planeness which arises in the planes of the space-lattice, owing to 
 the increased motion of the particles following a rise in temperature. 
 
CHEMICAL ACTIONS ON CRYSTALS 153 
 
 become so great that a spontaneous disintegration 
 ensues. Here then is an interesting series of sub- 
 stances which in their general construction and 
 fine-structure form bridges between crystalline, 
 amorphous solid, and, finally, on the attainment of 
 the greatest dispersion, fluid materials. 
 
 A crystallographic reconstruction corresponding 
 to a substitution in molecular chemistry may be 
 
 ee, 
 
 Pseudomorph of SiO, Silica gel from 
 Srom desmine scolecite 
 
 Cie gi 
 23. e) ae: ‘ 
 aietae 
 ; 
 
 oe 
 
 Fic. 180.—(Left-hand figure) : Desmine and its silicon dioxide. (Right-hand 
 figure) : Scolecite and its silicon dioxide. 
 
 P><P<P> 
 
 Fic. 181.—Reconstruction of chabasite: 1, chabasite; 2, metachabasite with 
 carbon disulphide : 3, metachabasite with ethyl alcohol. 
 
 easily effected for zeolites, either by replacing water 
 or by exchanging, more or less extensively, the Ca 
 for Na by the action of a Na-salt solution. When 
 water has been removed, other substances, such as 
 carbon bisulphide, alcohol, etc., may also be intro- 
 duced. In every case the crystalline nature remains 
 intact and specific optical characteristics are un- 
 changed. 
 
 That substances so very different from H,O 
 chemically, occur as substituents in the space-lattice 
 
154 CRYSTALS AND MATTER 
 
 must, however, not be assumed. Here also such ideal 
 cases will be passed over as stratifications of the type 
 of macroscopic intergrowths as are found so often 
 in minerals; a similar arrangement may occur in 
 fine-structural dimensions. It appears to me that 
 the oxidation of graphite to graphitic acid is of this 
 type, the latter substance showing optical properties 
 (uniaxial) which graphite would show if it were 
 transparent. 
 
 RESISTANCE TO MECHANICAL DISRUPTION AND 
 CHEMICAL ATTACK 
 
 We are led from the foregoing to the view that, 
 in chemico-anatomical preparation and substitution 
 processes! (as have been mentioned above in a series 
 
 Fic. 182.—Undermining during bleaching (bauerite process) and chlorite 
 reconstruction of biotite: a, biotite, fresh; 4, in bleaching ; ¢, in chlorite 
 process. 
 
 of examples), there exists a correspondence with 
 reactions, especially those of undermining and recon- 
 struction of molecules, with which the chemist is con- 
 cerned, and which, particularly in organic chemistry, 
 he has so much under control. The resistances, too, 
 
 1These may be increased by turning to processes in nature. 
 The well-known bleaching (bauerite process) and very extensive 
 chloritisation of dark mica are examples (Fig. 182). 
 
CHEMICAL ACTIONS ON CRYSTALS 155 
 
 which occasionally oppose the transformation he 
 wishes to effect, have their counterparts in crys- 
 talline fine-structure. Leptonically considered, close- 
 built arrangements in which the neighbouring particles 
 to some extent screen one another, tend to oppose 
 chemical just as they oppose mechanical attack. 
 This is shown, for example, in the strong chemical 
 resistance of the leptonically close-built, hard dia- 
 mond, contrasting with the oxidation of the more 
 
 Graphite 
 
 0 
 
 Fic. 183. —Stereograms of the closely packed, hard, chemically resistive diamond, 
 and the loosely built, soft, chemically more easily attackable graphite. 
 
 loosely-built graphite. In the same way, zeolites 
 such as desmine and heulandite, with structures 
 much extended by the large water content, are 
 attacked by acids very much more readily than 
 are their anhydrous analogues, the felspars, which 
 are specifically heavier and harder. 
 
 We thus have a noteworthy connection between 
 the mechanical hardness, especially as resistance 
 to fracture, and the chemical reactivity of crystals. 
 It is certainly not merely by chance that gems 
 such as diamond, ruby, zirconia, tourmaline, topaz, 
 
156 CRYSTALS AND MATTER 
 
 rock-crystal, etc., are chemically and mechanic- 
 ally strong. For these close atom packing must be 
 assumed, but even then we shall not be surprised if 
 very close atom arrangements go parallel with great 
 softness of the material. This is the case for graphite 
 (Fig. 183). The structural form indicates at once 
 the origin of the very easy mechanical disrupture. 
 It is due to the weak connections between the 
 densely packed planes. Testing the hardness by © 
 scratching separates to some extent the rigidly built, 
 but loosely coupled, planes. The softness of many 
 organic compounds suggests a corresponding struc- 
 ture. In such compounds molecules more or less 
 rigidly constructed internally will be only loosely 
 linked up to one another. 
 
 With chemical series it appears quite under- 
 _standable from the fine-structural standpoint that 
 forms specially stable compared with their neighbours 
 should periodically arise. 
 
 This is, indeed, a striking feature of the natural 
 series of the atoms, in which the rare gases are singled 
 out as terms with very stable electron distributions 
 (p. 85), and which oppose, apparently with effect, 
 great resistance to chemical change. Their next 
 neighbours, the alkalies and halogens, on the other 
 hand, exhibit the greatest readiness to react chemic- 
 ally. We may add to these cases of periodically 
 recurring resistance the above-mentioned breaks in 
 the process of dehydration of zeolites, although here 
 the effect is much less marked. Ata point of simple 
 molecular ratio between silicate and water increased 
 resistance is offered to the separation of the com- 
 ponents. The “ lag points ’’ studied by G. Tammann 
 in the structural changes of mixed crystals (such 
 
CHEMICAL ACTIONS ON CRYSTALS 157 
 
 as gold-silver, gold-copper, silver chloride-sodium 
 chloride, etc.), in connection with the old metallur- 
 gical method of gold and silver separation by 
 “ quartation,’’ are to be judged similarly, assuming 
 enhanced fine-structural stability and consequent 
 increased chemical resistance. According to G. Tam- 
 mann, these points occur for especially simple 
 distributions of the atom varieties, as for molar 
 fractions 3, 4, 4, 3, etc., of the resistive component, 
 which acts as a protective substance for the second 
 component, which is, chemically, more easily attacked, 
 
XITI. AN ATTEMPT TO FORM SOME IDEA OF 
 THE COURSE OF CHEMICAL REACTIONS 
 FROM OBSERVATIONS ON CRYSTALS 
 
 molecules, as well as of their fine-structural 
 
 variations in material physical fields, is still 
 in its infancy, efforts to form some idea of the 
 mechanics of chemical reaction have, quite naturally, 
 a merely tentative character. In order to get at the 
 matter it will be advantageous to advance into this 
 unknown region from various sides. We are thus 
 justified, in the present undertaking, which treats 
 the question from a crystallographic standpoint, in 
 anticipating that the best-ordered materials are here, 
 as in other cases, likely to give us useful suggestions. 
 
 ‘\ S our knowledge of the structure of atoms and 
 
 CHEMICAL SYMMETRY ACTIONS 
 
 The space-lattice constitution of the crystal must 
 serve as the fundamental conception in this work. 
 In its particular fine-structural symmetry and special 
 tectonic nature are characterised the physico-chemical 
 connections of the particles. 
 
 As a result of this, for every particular case 
 we find at once certain indications as to the 
 mechanics of the chemical processes in the bodies 
 concerned, and a basis for generalisation is obtained. 
 Since, for example, CaCO, of calc-spar, which is 
 
 constructed from Ca:: and CO’,;’ ions in a ternary 
 158 
 
COURSE OF CHEMICAL REACTIONS 159 
 
 rhythm (Fig. 117, p. 95), undergoes, on heating, 
 the well-known reaction of splitting into CaO and 
 CO., it must be assumed for the fine mechanics of 
 this process that, on account of the increased heat 
 motions, first the geometrico-chemical radical CO , as 
 a ring of three O’s about a centre carbon, becomes 
 loosened in the fine-structure. With increasing tem- 
 perature these radicals, together with the calcium 
 atoms, which are free moving groups in the material 
 field, undergo a separation into CO, and O, which 
 links up with Ca to form CaO. 
 
 According to this, the loosening of the particles 
 in the fine-structure is always to be regarded from 
 the standpoint of symmetry action. Those particles 
 of the structure, coupled together by rhythm or 
 reflexion, participate simultaneously in the process, 
 and since such coupling thus occurs throughout the 
 entire crystal many million times, the process appears 
 to us in analytical chemistry as a discontinuous 
 change, possibly in a series of steps, if the new 
 arrangement contains the departing component again 
 in definite symmetry disposition, as is the case, for 
 example, in the ignition of gypsum to the subhydrate. 
 A practically steady variation can have its origin in 
 complicated re-groupings of the point system, closely 
 following on one another. 
 
 For non-crystalline substances such as gases and 
 liquids the relations in the molecule cannot, as 
 regards main principles, be thought of in any other 
 way. Crystal regularity is, indeed, only a special 
 case of fine-structural arrangement. The four H’s 
 of an individual CH, molecule are, in this sense, 
 coupled up in a symmetry arrangement just as the 
 three O’s in the CO, radical cale-spar. They must 
 
160 CRYSTALS AND MATTER 
 
 participate simultaneously in action as markers of 
 the equal valued corners of the tetrahedral molecule, 
 so long as this symmetry persists. . 
 
 PRE-CHEMICAL PROCESSES AND DISCONTINUOUS RE- 
 ACTIONS. MASS ACTION AND CATALYSTS. HEAT 
 AS AICATALYST, 
 
 In following up the above observations, a very 
 important point must be discussed. Since it actu- 
 ally happens that in the chemical field, i.e. in the 
 reciprocal action of several types of molecule, one 
 only of the four H’s of the CH, molecule may 
 participate in chemical action (say CH, + Cl, = 
 CH;Cl + HCl), then on the basis of the symmetry 
 action set forth above, it is necessary to assume that 
 the four H’s of CH,, before the completion of the 
 chemical reaction, that is, in a pre-chemical process, 
 will be differentiated by the fine-structural pro- 
 minence of one of their number. The tetrahedral 
 placing of the four H’s must have become changed 
 under the reciprocal anisotropic influence of neigh- 
 bouring molecules CH, and Cl, in such a way that 
 one of the four has obtained a singular position in 
 the fine-structure, the other three remaining equiva- 
 lent. The four H’s, instead of representing the cor- 
 ners of a tetrahedron, mark out those of a trigonal 
 pyramid (Fig. 184). The hydrogen at the apex is 
 in a certain sense connected to the remainder of the 
 molecule by very weak threads. It is these which 
 naturally give way first when the mutual change of 
 form of the deforming interacting molecules exceeds 
 a certain measure of tension. Substitution in this 
 stereochemical body occurs localised at the hydrogen 
 atom, which has become particularised in the fine- 
 
COURSE OF CHEMICAL REACTIONS 161 
 
 structure, and a new stable arrangement is set up. 
 When, in the case of more complicated substances, 
 splitting occurs, depending, of course, on the appro- 
 priate molecular structure, the process is directly com- 
 parable with the rupture of the internal connections 
 of crystals during cleavage which cuts through the 
 weaker bonds. In aliphatic compounds C — C coup- 
 lings are, according to Wollers, weak arrangements ; 
 for aromatic compounds separation occurs more 
 readily between C and H. 
 
 The law of mass action is, in the above sense, the: 
 expression of the fact that numerous deformation 
 forces keep the fine-structural displacement con- 
 stantly directed towards one side. 
 
 In addition, the analogous role of catalysts in the 
 fine-structure becomes evident. The tension neces- 
 sarily preceding chemical action may well be increased 
 by the presence of a third type of molecule. The 
 action may, in some cases, be initiated by such a 
 third party. In the actual chemical transformation 
 the auxiliary substance does not participate, and, 
 in consequence, suffers no loss: it can officiate in 
 innumerable cases, one after the other, in the mole- 
 cular swarm, and in so doing produces a great effect, 
 although present in very small quantity. Thus such 
 material catalytic factors function pre-chemically. 
 The substances concerned represent catalysts as they 
 deform the fine-structure. If it be desired to include 
 the preparatory tension process in the chemical 
 action, there is no formal objection. Physical and 
 chemical processes merge into one. I think the 
 observations of J. Stark and myself are pertinent in 
 this connection. 
 
 Raising the temperature, as the acceleration of the 
 IJ 
 
162 CRYSTALS AND MATTER 
 
 internal fine-structural motions, can be similarly 
 considered as catalytic. It is understood that in this 
 case, weak bonds between structural groups will give 
 way sooner than they would at lower temperatures ; 
 they are, we may say, pulled about at the higher 
 temperature. If, for example, in NaCl. 2H,O, the 
 water molecules, which do not concern the mono- 
 valency of Na and Cl, and are but loosely held by 
 co-ordination bonds, the thermal oscillation of the » 
 
 CMG 2, = GAG *H0 ee C72 
 a 
 
 My *Cl, 2 CHC? +HCL ” oe Ch; 
 Af 
 
 at 
 CH, CH, H Chg? 
 ‘ 44 
 KT 
 H 
 
 Vy 4 a 
 
 “4 cd 
 
 Fic. 184.—Fine-structural schemes for the action of chlorine on benzene and 
 methane respectively. 
 
 particles is considerably increased, the weak binding 
 forces will be overcome first. Water is suddenly 
 liberated in cases where the molecules are dissimilarly 
 attacked, as for BaCl,.2H,O, in two stages, one 
 after:the other {at 105° dnd 162°. see Digmergs 
 p. 150) ; sometimes in even more, as for CuSO, . 5H,O. 
 When the tension becomes sufficiently large under 
 the influence of the rise in temperature, “ valency 
 tensors’ also break apart such as those between the 
 ions Ca and CO; during lime-burning and in other 
 similar cases of chemical decomposition. 
 
COURSE OF CHEMICAL REACTIONS 163 
 
 To illustrate these points the schemes of Fig. 184 
 are shown, which refer to methane and benzene as 
 typical cases. The final structure there derived for 
 C,H;Cl appears to me to agree completely with the 
 diagram already published by J. Stark in his excellent 
 book, “ Die Elektrizitat im chemischen Atom,” a 
 happy case of the agreement of results derived from 
 different standpoints. 
 
 It was also of interest to me, on looking through 
 the literature, to learn from a hint by E. Farber 
 in “‘ Naturwissenschaften,’ that, in the delibera- 
 tions of the older chemical generation, representation 
 of a weakening of the bond in the molecule before 
 the occurrence of the chemical reaction occasionally 
 played a role. This is seen in the assertion of 
 A. Kekule, who says that “ during the approach of 
 the molecules the connections of the atoms in the 
 same are already weakened, for one part of the 
 chemical affinity is bound by the atoms of the other 
 molecule until finally the previously united atoms 
 entirely lose their interconnection and the newly 
 formed molecules separate.”’ 
 
 One ventures to extend the scheme in the above 
 to the assumption of a pre-chemical molecular 
 deformation. 
 
 CRYSTALLOGRAPHIC INDICATORS OF CHEMICAL 
 PROCESSES 
 
 Since the physical, chemical, and _ crystallo- 
 graphic considerations agree, as they do, we are 
 now in the position to corroborate, to some extent 
 at least, the assumption of preliminary structural 
 changes in crystallographic experiments. In par- 
 ticular, observation of the conditions for certain 
 
164 CRYSTALS AND MATTER 
 
 crystallisations lead once more to the postulation of 
 molecular fore-forms in solution from which crystals 
 are separating, an assumption which has already been 
 mentioned on page 41. While, for example, CaCO, 
 salt separates out from a pure calcium carbonate 
 solution in trigonal form of the 3m class as calc- 
 
 Fic. 185.—Calc-spar and aragonite. 
 
 spar, experiment shows that the addition of mag- 
 nesium sulphate to the solution causes the forma- 
 tion of a stereo-chemically different variety, digonal 
 aragonite of the 2m group. Thus one or other 
 modification of CaCO; must certainly be predeter- 
 mined by molecular pre-forms in the solution. A still 
 more varied example 
 of this has been in- 
 vestigated by O. Pauli 
 in my institute; his 
 experiment deals with 
 acid phenyl acridon- 
 Fic. 186,--Monodiinicand triclinic modifies- 14m sulphate as tis 
 tions of acid phenyl acridonium sulphate. appears in different 
 modifications according to the proportions of water, 
 sulphuric acid, and alcohol in the solution. Figs. 186 
 
 1 Probably as CaCO; or CaOCO,, possibly as a loose compound 
 with MgSQ,. 
 
COURSE OF CHEMICAL REACTIONS 165 
 
 and 187 give diagrammatically the appropriate} 
 conditions. 
 For the most part, then, we are supported in 
 the conclusion that chemical reactions do not occur 
 abruptly, but after preliminary actions, in cases 
 which permit a leisurely although indirect observation 
 of the changes of state by means of physical indi- 
 cators. Occasionally that is the case for changes of 
 crystallographic modifications, which are not, indeed, 
 
 Fic, 187,—Crystallization diagram for acid phenyl acridonium sulphate. 
 
 merely physical, but also chemical actions (p. 70). 
 For the investigation of the general course of the 
 fine-structural processes inside the substance, optical 
 methods may be used as in the elegant studies of 
 A. Hantzsch and his pupils, where absorption 
 phenomena in the ultra-violet were employed as 
 
 1 Different molecular pre-forms of crystallisation will arise if at 
 higher temperatures, or with certain other substances in the solution, 
 salts poor in water crystallise out. The same holds good if at low 
 temperatures, or in the presence of other substances, salts rich in 
 
 water are formed. The diagrams of van’t Hoff and D’Ans, in par- 
 ticular, furnish classical examples of this. 
 
166 CRYSTALS AND MATTER 
 
 indicators of chemical processes. For the stereo- 
 chemical changes to be determined here, investiga- 
 tion of the refractive index is helpful. In this 
 particular I have studied exactly, with R. Kolb, 
 
 15850 
 
 * 700° -s9° * 50° 100° 150° 200° o ° o * 450° ° * 650° a o oe 
 ST 8 oe i Oo” 250 300 540 400° 450° 500° 550) 600° 650° 700° 750° 800 
 
 Fic. 188.—Curves of the refractive indices w of 8 and a quartz for various kinds 
 of light. 
 
 S17 OF. 200° 400° 600° 800? 
 
 Fic. 189,—Curves of the refractive indices w and « of 8 and a quartz for sodium 
 light characterising the variation of the double refraction. 
 
 such a physico-chemical process for quartz, which, 
 on exceeding 575°, changes from the trigonal f into 
 the hexagonal a state (3827 6s) (Fig. 88, p. 71), as 
 
 the Laue diagrams show in very neat fashion (Fig. 89, 
 -p. 71). With respect to the refractive indices, 
 
COURSE OF CHEMICAL REACTIONS 167 
 
 Figs. 188 and 189 explain fully. It is clearly seen for 
 the case in question, 8 +a quartz, that on nearing 
 575. the gradient of the curves is much increased, 
 and at the temperature named exhibits a discon- 
 
 FIG. 190.—Curve of the angle variation of 8 and a quartz. 
 
 tinuous drop. This line must be regarded as a 
 definite indication that the process 3s ~ 6s quartz 
 is led up to by a gradually developing tension 
 in the structure; this increases as a pre-chemical 
 action until the sudden 
 rearrangement by a dis- 
 continuous change in the 
 fine-structure. 
 
 Similar conclusions to 
 those obtained above fol- ene ear 2 pieeners variation 
 low from the observations 
 which I carried out in collaboration with R. Kolb 
 on quartz with respect to its morphological variation 
 on transformation (Fig. 190). F.E. Wright obtained 
 similar results for the same mineral. 
 
 The thermo-goniometrical researches of R. Gross- 
 man made, under the direction of P. Niggliand myself, 
 
 Se esks, 
 
 400 600 
 
 S 
 
 200° 
 
168 CRYSTALS AND MATTER 
 
 on borazite and leucite (Fig. 191), show analogous 
 results. 
 
 Although the fine-structural relations of quartz 
 are not determined experimentally, still, by resorting 
 to the representations of J. Beckenkamp, and 
 especially by following the discussions of P. Niggli, 
 it is possible to make a provisional diagram for the 
 transition of the quartz modifications. In Figs. 192 
 and 193 such a scheme is depicted. The arrows in © 
 the diagram of the screw trigyric structure indicate 
 
 © 
 Hexagonal Quartz 
 
 Fics. 192 and 193.—Fine-structural diagram symbolising the transition 
 3s => 6s of the quartz modifications. 
 
 O 
 the tendency of the O particles in the SiC ii triangle 
 
 to set themselves in the screw hexagonal arrange- 
 ment, a tendency which steadily becomes more - 
 effective as the temperature rises; finally, the 
 sudden rupture of the tension which has increased 
 to the limit gives the 8 form. 
 
 Thus crystallographic considerations support the 
 assumption that one can, in ideal schematic fashion, | 
 represent a chemical transformation as the action of 
 a physical or chemical field of such a character that 
 the change in the chemical structure is led up to by 
 a state of fine-structural strain and deformation ; 
 
COURSE OF CHEMICAL REACTIONS 169 
 
 this becomes increasingly pronounced, and finally 
 leading, by a sudden adjustment, to the new stable 
 system. 
 
 Many gradations of the relation may, of course, 
 arise. In particular, owing to great resistance, the 
 period of strain may be more or less diminished so 
 that the discontinuous chemical action then occurs 
 almost or entirely without this intermediary state. 
 On the other hand, a pre-chemical deformation may 
 become very much extended, so that the discon- 
 tinuous change is correspondingly lessened or absent 
 altogether. The transition of modification, from 
 aragonite to calc-spar by heat, points to this. With 
 a view to learning more about the change, I suggested 
 to K. Wiinscher a thermo-goniometrical and thermo- 
 optical research. It was shown that angle and 
 refractive index variations, on heating the mineral 
 to 325°, are a function of the temperature, increasing 
 and decreasing with it. For higher temperatures, 
 however, the tension process in connection with the 
 transformation of aragonite into calc-spar occurs to 
 some extent in spontaneous glidings, for at constant 
 temperature the form and optical properties of the 
 mineral vary, the variations being more rapid the 
 higher the temperature taken. Finally, the sub- 
 stance completes the transition of one into the other 
 modification by a sudden adjustment. 
 
XIV. ANALOGY OF THE MORPHOLOGICAL 
 ACTION OF PHYSICAL AND CHEMICAL 
 FIELDS ON CRYSTALS 
 
 T is of considerable interest to compare the . 
 [ ecsis observable homogeneous deformations of 
 
 the crystal structure which occur under the in- 
 fluence of heat, with variations of crystallographic 
 form in the chemical field. 
 
 THERMAL INFLUENCES ON THE CRYSTAL FORM 
 
 The action of temperature change on the mor- 
 phology of the crystal becomes apparent, as is well 
 known, in explicit formal symmetry actions, the 
 general type of which is most easily studied for 
 spheres. Such forms remain, for uniform rise of 
 temperature, isoradial if they are composed of iso- 
 metric substances. The sphere remains for tem- 
 perature variations as such intact. The change is 
 restricted to an alteration of the radius. Crystallo- 
 graphic ternary, tetrad, or senary substances, on the 
 other hand, show transformation of the initial form 
 to a rotation ellipsoid, the axes of rotation coinciding 
 with the crystallographic main-axes. Spheres of 
 rhombic, monoclinic, and triclinic substances finally 
 give rise to triaxial ellipsoids, the principal direc- 
 tions of which are arranged in accordance with the 
 symmetry (Figs. 194-106). 
 
 Although the morphological reaction to change 
 
 in the heat motion of the particles appears usually 
 170 
 
MORPHOLOGICAL ACTION 171 
 
 simple, the complicated interlacing of fine-structural 
 force fields is shown here in the occasional contrac- 
 tion and not extension, with increase of temperature. 
 It may, in fact, happen that for anisotropic sub- 
 
 Fic. 196. FIG. 197. 
 
 Fics. 194-196.—Schemes for the homogereous deformation of isometric, 
 uniaxial and trimetric crystals on heating (initial sphere shaded). 
 
 Fic. 197..-Scheme for the homogeneous deformation of a calc-spar sphere 
 on heating. Coefficient of expansion in the direction a@ = 0:0,2621, 
 ¢ = — 0:0,0540 in the intermediate direction shown = 0 (initial sphere 
 shaded). 
 
 stances, in certain directions, dilatation occurs, and 
 in others contraction. For cuprite it happens that 
 with rise of temperature in the region below — 4:3°, 
 isoradial contraction takes place. On heating tri- 
 gonal calc-spar it expands along the rotation axis, at 
 
172 CRYSTALS AND MATTER 
 
 the same time contracting in all directions perpen- 
 dicular thereto. As a result, radii of the initial 
 sphere, inclined at 65° 49:5’ to the principal axis, 
 will not be altered in length by temperature change 
 (Fig. 197). 
 
 Hexagonal silver iodide has, on the other hand, 
 a negative expansion coefficient along the crystallo- 
 graphic vertical (a, = — 0:0,0397) ; in the horizontal 
 direction it expands on heating (a, = 0°0,065). The » 
 
 Fic. 198. Fic. 199. 
 
 Fics. 198 and 199.—Schemes for fine-structural variation in the physical field. 
 Preservation of the indices and zone relations. 
 
 cubic coefficient a, + 2a, 18 = — 00,0267. The 
 volume of the salt is therefore reduced by rise of 
 temperature. 
 
 From a fine-structural aspect such relations may 
 readily be understood in the symbolical representa- 
 tion, as an alteration of the distance between the 
 centres of heat motion corresponding to the visible 
 change of form. The Figs. 198 and 199 above show 
 this with the necessary diagrammatic exaggeration. 
 
 In these figures one recognises as the ruling con- 
 
MORPHOLOGICAL ACTION 173 
 
 ditions the preservation of the symmetry, of the 
 parallel edges (the zone relations), and of the indices 
 which, in the triangular surface shown retain their 
 unit values Ia: 1b: 1c. Angles and axial ratios alter 
 within the limits of the prevailing symmetry. 
 
 The extension coefficients, the order of which has 
 already been given in the special case of Fig. 197 
 give us an idea of the absolute value of the varia- 
 tions. They are usually very small. For example, 
 in the elementary cube of rock-salt (Fig. 24), p. 22), 
 
 200° -100° 0° ~~~-*400® ~~ *200° ~~ +300" +400" —+500° OOF *700° 
 FIG. 200,—Variation of the cleavage angle of the plagioclases, albite, labradorite, 
 and anorthite. 
 
 the side length increases merely from 5-63 x Io78 
 em, at 0° to 5:77 x 107* cm.at 500°. The angular 
 deformations of anisotropic substances known since 
 the time of Mitscherlich (1799-1863) are correspond- 
 ingly small. 
 
 To extend his studies on calc-spar I investigated 
 the angle variations of the rhombohedral cleavage 
 form of this mineral over the extensive tempera- 
 ture interval from — 165° C. to 596° C., i.e. for 
 761°. I found a change of angle of about 1° 9’ 20”, 
 i.e. about g:1’ to each 100° on an almost linear 
 
174 CRYSTALS AND MATTER 
 
 graph. Usually the thermo-morphological reaction 
 is even smaller. For quartz I measured a change 
 of the rhombohedron angle of only 14:0’ for 553° 
 (20° — 573°), 1e. not 3’ per 100° C. 
 
 Of course, in such variations of form sometimes 
 there arise very complicated fine-structural processes, 
 which may be inferred from appropriate diagrams. 
 Although the curve obtained in the example of calc- 
 spar rises almost linearly, in other cases well-marked ' 
 curvature is shown as for quartz, already referred to 
 (Fig. 190), and the plagioclases. In Fig. 200 the 
 second curve, which relates to an isomorphous mix- 
 ture of albite and anorthite, termed labradorite, also 
 deviates from the arithmetic mean of the other two, 
 showing that the angle and the accompanying fine- 
 structural variations in such cases are not. simply 
 additive. 
 
 CHEMICAL INFLUENCES ON THE CRYSTAL FORM 
 
 The analogous inquiry as to the morphological 
 action of a chemical field on the fine-structure fails in 
 general owing to the impossibility of magnifying the 
 effect sufficiently for observation. If, for example, 
 a crystal of calc-spar is suspended in water, then a 
 deformation of the crystal is certainly to be assumed, 
 but cannot be rendered visible experimentally. In 
 contrast to the thermal action, the influence of the 
 chemical field is restricted in the above case to the 
 surface. It can make itself felt in the peripheral 
 processes of growth and solution of the crystal, but 
 not markedly in transformation of the structure. 
 Moreover, a simultaneous action throughout the 
 whole body of the crystal, i.e. permeation of the 
 liquid to all parts, would be necessary to correspond 
 
MORPHOLOGICAL ACTION 175 
 
 to the effect of heat. For some crystals that is 
 actually brought about, as for those of albumen. 
 Indeed, the process exceeds in definiteness all expec- 
 tations. Albumen crystals take up water either from 
 the surrounding liquid or from an atmosphere con- 
 taining water vapour. In this case, then, the par- 
 ticles of the space-lattice are surrounded with water 
 molecules. In the reciprocal anisotropic action be- 
 tween the crystal structural groups and the water 
 particles regularly arranged about them, an unusually 
 large deformation of the crystal makes its appearance, 
 the crystallographic symmetry remaining unaltered. 
 
 Albumen crystal 
 
 Before imbibition ; Af ter t 
 imbibition 
 
 Fic. 201.—Homogeneous deformation of a crystal cf albumen by imbibition. 
 
 The isometric albumen crystals swell up, remain- 
 ing trigonal, with anisotropic variation of the angles. 
 According to A. F. W. Schimper, to whom chiefly we 
 are indebted for the appropriate observations, the 
 plane polar-angle of trigonal albumen (obtained from 
 Brazil nuts) changes from 604° to 39$°, and thus by a 
 very largeamount. Fig. 201 shows diagrammatically 
 a similar case of extensive deformation for the cubic 
 rhombohedra of albumen occurring in solanine. 
 These are drawn out to a very definitely acute- 
 angled form, the plane polar angle of which amounts 
 now to 68° instead of g0$°. It is especially interesting 
 that for albumen from Brazil nuts, perpendicular to 
 the ternary axis no observable swelling occurs, whilst 
 
176 CRYSTALS AND MATTER 
 
 the linear measure of the enlargement in the axial 
 direction is very considerable. 
 
 The optics of albumen crystals deformed by 
 swelling in water show a regular variation, as is also 
 the case for those expanded thermally. Isotropic 
 crystals remain, with variations of the refractive 
 index, isotropic; double refracting forms alter the 
 magnitude of this property. All return again on 
 evaporation of the water to their original states.1, 
 According to this, the morphological actions of 
 thermal and chemical fields in crystals are exactly 
 similar, probably an indication that the thermal 
 process also is to be regarded from a chemical stand- 
 point, and, as a bombardment of the structural 
 particles by electrons, is analogous to chemical 
 action. 
 
 COMPARISON OF THE THERMAL AND CHEMICAL 
 INFLUENCES ON THE CRYSTAL FORM 
 
 A comparison of the fine-structural effects follow- 
 ing temperature variation, on the one hand, and 
 under the influence of the chemical field, on the 
 other, may best be carried out with respect to iso- 
 morphous substances. In this connection the follow- 
 ing table (p. 177) will be of interest. It indicates the 
 relatively large effect of a chemical substitution of 
 Cl by Br or I in the potassium halides compared with 
 that of a temperature change of about 500°. The 
 molar cell and molecular domains (p. 107) relate to 
 cube forms. 
 
 For KCl, then, the effect on the external form, 
 measured by the axes of the molecular domain for a 
 rise of temperature of about 500°, is to that produced 
 
 1 Acids apparently break up the structure. 
 
MORPHOLOGICAL ACTION 177 
 
 Mol Domain. Cell Domain, Molecular Region. 
 KCl. 
 
 Volume Axes i en eer ela ae 
 
 tie nes crn Cm. Cc. Cm. om 
 20° ; eH) 39764 3°346 247°72 6°280 61°93 3°956 
 ROO | . | 40°19 3°425 205°72 6-429 66°43 4:050 
 2°55 0°079 18-00 O-149 4°50 0°094 
 20° KCl mh) 3704 3°346 247°72 6-280 61°93 3956 
 KBr .| 43°19 37508 285°52 6°585 71°38 4°148 
 5°55 o'162 37°80 0°305 9°45 0°192 
 4 Wl AGtTO 37508 285°52 6°585 71°38 4°148 
 Gee eh 5207 4) 3750 41) 350°24 1). FOO | 87-56. | agate 
 19°78 0°248 04°72 0°464 16-18 0:293 
 olen -| 37°64 3°346 247°72 6:280 61°93 3956 
 Bi -| 52°97 3°756 | 350°24 | 7:049 | 87°56 | 4-441 
 
 by a substitution of Br or I for Cl as 1: 2:04: 5:16. 
 Naturally, the greater the change of temperature the 
 larger will be the consequent variations, within the 
 limits of one modification. In Fig. 200 of the felspars 
 (p. 173), it is recognised how closely in such cases 
 thermal and chemical effects can resemble each other. 
 
 The phenomena in question possess further a 
 special interest, in that the action of a rise of tem- 
 perature in the crystal and the loosening effect of a 
 permeating substance are gradations in the process 
 of melting and solution, that is, in the process of “‘ren- 
 dering amorphous.’ Considered in connection with 
 
 the series of metamorphoses (p. 69), which substances 
 12 
 
178 CRYSTALS AND MATTER 
 
 pass through on raising the temperature, continuous 
 thermal homogeneous deformations figure as pre- 
 liminaries to the abrupt collapse of the space-lattice 
 arrangement, a process which generally runs parallel 
 with the external phenomenon of melting, i.e. of 
 “ flowing apart.’”’ Under the influence of increased 
 fine-structural agitation the crystal form is com- 
 pletely broken up, in accordance with Lindemann’s 
 views, if the vibrations of the particles about their, 
 positions of rest become commensurable with their 
 distance apart, and so lead to their collision. The 
 domains of the fine-structural groups merge together, 
 and the forces binding the lattice are overcome 
 by the disruptive action of the heat motions, the 
 crystal form is destroyed and scattered into irregu- 
 larly placed new kinetic units. In this sense the 
 fine-structural deformation which precedes the sudden 
 change appears analogous to the pre-chemical pro- 
 cesses referred to previously (p. 160). The albumen 
 crystals during imbibition strive in an exactly similar 
 way to become amorphous. It is of great interest, 
 and also characteristic, that the structure can be 
 linearly extended before dissolution to such a large 
 extent, often by several times its original length. 
 Even with extensive dilatation the interleptylic 
 fields of force give rise to some cohesive action. 
 That must be ascribed to a regular incorporation of 
 the water particles, which arrange themselves analo- 
 gously to the H,O in zeolites, and acting as a chemical 
 cement bring about the observed cohesion. The water 
 is present, however, in much greater quantity than 
 in zeolites. Finally, there occurs separation into 
 irregularly arranged particles, as may be observed 
 very neatly, according to A. F. W. Schimper, in 
 
MORPHOLOGICAL ACTION 179 
 
 small crystals of albumen from the seeds of the 
 castor oil plant. These, being cubic, swell isoradially 
 in dilute sulphuric acid to three or four times their 
 Original diameter, and then immediately go into 
 solution. To what extent, in structures of so many 
 atoms the space-lattice arrangement is lost on tran- 
 sition to the colloidal, and finally to the molecularly, 
 disperse state, further X-ray studies must show 
 (compare p. 65). 
 
XV.CRYSTAL PHYSIOLOGY: ANDi tis 
 CLASSIFICATION OF ATOMS 
 
 THE STANDARDS AND PHYSIOLOGY OF THE 
 CLASSIFICATION 
 
 CIENTIFIC classification, as a concise charac- 
 G essation of the peculiarities and relations of 
 
 the objects investigated, is of considerable 
 importance. Its development must move parallel 
 with the advance of knowledge, conforming to the 
 broader purpose of ensuring simple and natural 
 methods in our deliberations. In these times of 
 radical changes in our ideas of the nature of matter, 
 the systematic co-ordination of the results of investi- 
 gations merits careful attention. The importance of 
 the classification as the characterisation of the fine- 
 structural particles and their family relationships is 
 supported in a gratifying fashion by independent 
 lines of thought from many directions. Our ideas 
 of the constitution of atoms as neutral and ionised 
 forms, both of normal weight and as _ isotopes, 
 together with the conception of the element, play 
 the leading role here. It appears to me not inap- 
 propriate in these questions also to emphasise the 
 close connection of the various states which runs 
 through the fine-structural series, electrons, atoms, 
 ions, molecules, and crystals. The crystal, the 
 highest and especially regular member of the series, 
 
 easily observable in its external form and physical 
 180 
 
THE CLASSIFICATION OF ATOMS 181 
 
 conditions, enables the general idea of the principles 
 of classification to be readily grasped. In particular, 
 it is clearly seen that considerable physiological 
 breadth of property is to be ascribed to a leptonic 
 unit. The appearance of a rigid regular form and of 
 an inner homogeneity, say for ruby, is an illusion. 
 A change of temperature changes the volume of the 
 crystal, and in the case mentioned its form also. 
 Optical tests of the refraction, double refraction, and 
 absorption show that this crystalline form can 
 experience changes in its inner constitution which 
 are to be traced back, finally, to reversible rearrange- 
 ments of the fine-structure. 
 
 X-ray data testify that, in correspondence with 
 this general conclusion, the motion of the fine- 
 structural particles is highly variable. Even analy- 
 tical differences, as in the case of isomorphous 
 mixtures, with its powerful influence on the optical 
 absorption and the specific gravity, or the entrance 
 and exit of water which occurs for zeolites, can arise 
 without prejudicing the idea that we still are dealing 
 with the same kind of crystal whose physiology alone 
 changes within certain limits. In systematic classifi- 
 cation the type retains its place, despite these varia- 
 tions. It seems to me that the transfer of such views 
 to the classification of leptonic forms leads to a simple 
 and natural formulation. 
 
 ELECTRONS, ATOMS, AND MOLECULES 
 
 The fundamental constitution of all things lies 
 hidden in the electron as the elementary quantum of 
 electricity and the primary constituent of matter ;1 
 
 1If they have not to relinquish this rank to the archons as 
 vortex pairs as suggested by O. Wiener. 
 
182 CRYSTALS AND MATTER 
 
 electrons are therefore of the first importance. 
 Their division into e+ and e~ is, however, of so great 
 systematic simplicity that it has, up to the present, 
 sufficed. 
 
 The case is very different for the atoms, the 
 classification of which has developed into a special 
 study. Their general characteristic in the manifold 
 of forms lies in the presence of a nucleus within the 
 structural unit. | 
 
 The highest grade of individual leptonic structures 
 is represented by the molecule. Its special feature is 
 that of a combination of atoms to a new unit, thus 
 the presence of more than one nucleus in the kinetic 
 unit. 
 
 Everything else in fine-structural phenomena, as 
 they are presented in the gaseous, liquid, and crystal- 
 line states in quite inexhaustible abundance, comes 
 under the head of modes of aggregation of the elec- 
 tronic, atomic, or molecular fine-structural forms. 
 The force of the classification lies in the atomistic 
 structural gradations. 
 
 ATOM TYPES 
 
 Reviewing, therefore, the scientific facts relating 
 to the atomic units, there is now no further doubt 
 that these units must be arranged in the order of 
 their atomic numbers (corresponding to the X-ray 
 spectra, p. 20). This is done in the following table 
 (p. 183). In each case the atomic weight is sub- 
 joined thus, A.W., as it is not essential to the 
 classification. 
 
 The terms of this series will, in accordance with 
 scientific requirements, be designated atom types, 
 _and for the further systematic subdivision of these 
 
THE CLASSIFICATION OF ATOMS 
 
 183 
 
 the expression atom sub-types is introduced. The 
 table indicates the special property of atom types 
 —the atomic number which, in accordance with 
 the nuclear charge, fixes the position in the series 
 
 omer to Ur. 
 
 Atomic 
 Number. 
 
 O ON OAMUAWN 4 
 
 Name. 
 
 Hydrogen 
 Helium 
 Lithium 
 Beryllium 
 Boron 
 Carbon 
 Nitrogen 
 Oxygen 
 Fluorine 
 Neon 
 Sodium 
 Magnesium 
 Aluminium 
 Silicon . 
 
 Phosphorous 
 
 Sulphur 
 Chlorine 
 Argon 
 Potassium 
 Calcium 
 Scandium 
 Titanium 
 Vanadium 
 Chromium 
 Manganese 
 Iron 2 
 Cobalt . 
 Nickel . 
 Copper . 
 Zinc 4 
 Gallium 
 Germanium 
 Arsenic 
 Selenium 
 Bromine 
 Krypton 
 Rubidium 
 Strontium 
 Yttrium 
 Zirconium 
 Niobium 
 Molydenum 
 
 Ruthenium 
 Rhodium 
 Palladium 
 
 Atomic 
 Number. 
 
 Name. 
 
 Silver 
 Cadmium 
 Indium 
 Tin 
 Antimony 
 Tellurium 
 Iodine . 
 Xenon . 
 Caesium 
 Barium 
 Lanthanum 
 Cerium 
 
 Praesodymium 
 Neodymium . 
 
 Samarium 
 Europium 
 Gadolinium 
 Terbium 
 Dysprosium 
 Holmium 
 Erbium 
 Thullium I. 
 Ytterbium 
 Lutetium 
 Hafnium 
 Tantalum 
 Tungsten 
 
 Osmium 
 Tridium 
 Platinum 
 Gold 
 Mercury 
 Thallium 
 Lead 
 Bismuth 
 Polonium 
 
 Emanium 
 Radium 
 Actinium 
 Thorium 
 
 Protactinium . 
 
 Uranium 
 
 The characteristic of an atom is, 
 
184 CRYSTALS AND MATTER 
 
 therefore, its numbered place in the atomic series ; 
 
 briefly, its monotopy. 
 No term of the series can be dispensed with ; 
 
 ) ° i 
 70oF & A 
 <= t 1 
 s Mgt 
 sol = Rb Nias 
 S fon 
 2 t { 
 sok = Pe 
 : Be 
 Ba te 
 Sr : 
 30 ! 
 Nad 16n Br rs : ; Th 
 Pf Bi 
 Tl 
 Hy U 
 PL PAy 
 
 Atomic weights 
 200 220 
 
 180 2 
 
 £00 i420 0 T40 9460 
 Fic. 202.—Graph of atomic volumes. 
 
 each one fulfils the task of representing a necessary 
 
 number in a sequence. 
 The periodic table of L. Meyer and D. J. Mendeleeff 
 
 Logarithm of the diameter of 
 the crystallographic ion 
 domain 
 
 Logarithm of the atomic number 
 
 Fic. 203.—Graph showing series of atoms. After E. Schiebold. 
 
 accomplishes in the familiar way a grouping of the 
 atom types. The curve obtained for the atomic 
 volumes (Fig. 202) serves the same purpose. With 
 
THE CLASSIFICATION OF ATOMS 185 
 
 reference to our ideas on crystallographic atomic 
 domains, it appears of interest to indicate here a 
 graphical arrangement of important atomic groups 
 obtained by E. Schiebold. This is arrived at by 
 taking the logarithms of both atomic number and 
 the diameter of the atomic domain; the groups are 
 signified by the arrangement of the points in the 
 diagram which are often linear in series. 
 
 ATOM SUB-TYPES 
 
 The monotopy of the atom types is quite com- 
 patible with physiological differences which do not 
 interfere with the constancy of the nuclear charge 
 which is characteristic and equal to the atomic 
 number. In this way atomic sub-types may pos- 
 sibly arise in the series of monotopes, characterised 
 by fine-structural differences either in the region of 
 the satellite electrons or in the nucleus. 
 
 We may therefore assume within the atomic types 
 the following atomic sub-types. Differences in the 
 outer electron shell differentiate neutral atoms from 
 atom ions, which in turn separate into cations or 
 plus atoms and anions or minus atoms. Owing to 
 the extreme lightness of the electrons, the theoretical 
 changes of weight arising from omission of a few 
 negative particles from the outer shell, or by their 
 entrance, are inconsiderable ; neutral atoms are, with 
 their corresponding + or — ions, practically isobaric. 
 For other atomic sub-types a difference of the mass 
 content, actually in the central nucleus (without 
 change of its charge), of so extensive a nature occurs 
 that a difference of atomic weight is observable. 
 This is the case for isotopes, so termed in K. Fajan’s 
 fundamental papers on the subject. The members 
 of such a group are, therefore, heterobaric. 
 
186 CRYSTALS AND MATTER 
 
 ELEMENTS 
 
 Another criterion in classification is obtained from 
 our conception of the element. Its characteristic 
 is that its constitution consists entirely of monotopic 
 atoms ; thus elements are possible containing either 
 only one atom sub-type or several different varieties. 
 F. Paneth expresses this very neatly by differentiating 
 between pure elements and mixed elements. The 
 former contain only one atom sub-type, the latter — 
 more than one. 
 
 NORMAL MIXTURE AND SEPARATION OF ISOTOPES 
 
 Of primary importance in analytical chemistry 
 is the remarkable fact that for elements containing 
 heterobaric components a normal mixture is invari- 
 ably found. For chlorine, with isotopes of mass 
 35, 37, and 39, the ordinary atomic weight 35:46 is 
 always shown (whether the chlorine is extracted 
 from eruptive minerals or deposited sediment, 
 whether it is of.terrestrial or meteoric origin), a cir- 
 cumstance which reminds us, at least formally, of 
 the equilibrium phenomena of eutectics. Thus ordi- 
 nary. Brys-9,°) is. 0°46 Br,, + 0°54. Br, >) Cie 
 0:23 Cl,, + 0:77 Cl,;; (neglecting the small amount 
 Of .Cl,o}.3) Slpg-s ==’ 0230 [Oleg 4: 0:70.55, 5 ae 
 0:97 Ayo -+ 0°03 Age, etc. 
 
 The mixed isotopic constitution of many of the 
 atom types also enables us to understand why, in 
 the Mendeléeff system, there occur occasionally 
 deviations from the arrangement in order of atomic 
 weight. For atoms of nearly the same weight it 
 may easily happen that this admixture displaces an 
 atom type from the natural sequence to a false 
 
THE CLASSIFICATION OF ATOMS § 187 
 
 position, as is the case for argon. If the lighter 
 component (Ar;,) were present in somewhat greater 
 proportion than is the case in the normal argon 
 mixture (Afso.s3 = 0°97 Ar, + only 0:03 Ars,), the 
 rare gas would immediately assume its correct place 
 in the Mendeléeff system before potassium (A.W. 
 39°I0). 
 
 A partition of the isotopes by arrangement in 
 separate positions in the fine-structure occurs in every 
 crystallisation of a substance containing such atomic 
 mixtures (compare Fig. 119, p. 98). As is well 
 known, a separation of the components to a detectable 
 extent has been accomplished in the researches of 
 F. W. Aston who continued the work of Goldstein, 
 W. Wien, J. J. Thomson, and others. By the 
 different deviations of the ions in electrostatic and 
 magnetic fields, Aston separated and identified the 
 individual isotopes, one of the finest results of 
 general scientific endeavour in the direction of a 
 unified concept of matter as the aggregation of a 
 primary constituent. The integral atomic weights 
 of the iostopes point to this. Here the anomaly of 
 hydrogen, with its non-integral atomic weight 1-008 
 compared with oxygen = 16, is not yet explained, 
 but it now merely spurs us on completely to establish 
 this otherwise predominant concept by further experi- 
 mental work and study. 
 
 CONCLUSION 
 
 In reviewing all the various experiments and 
 arguments dealt with above, which in the present 
 early stages of fine-structural investigation naturally 
 more often pass in a mere mention than lead to 
 definite results, it is recognised that crystals are 
 
188 CRYSTALS AND MATTER 
 
 actually in many respects typical of the general 
 conception of the constitution of matter. In their 
 macroscopic form and their physico-chemical relations 
 are reflected, not only the fine-structure and the 
 physics and chemistry of their own particular micro- 
 cosm, but also of matter in general. With their 
 three-dimensional regularity and easy accessibility 
 to direct observation, they are specially suited to 
 serve as the starting point in the investigation of . 
 laws universally valid. In this way crystallography 
 stimulates the physicist, chemist, and natural phi- 
 losopher, as it itself, on the other hand, has gratefully 
 received so much help from these great sister sciences. 
 
 In such common endeavour the sowing of the 
 fine-structural soil cannot fail to germinate vigorously 
 and, on progressive cultivation, to develop into a 
 rich harvest. I hope that this present exposition 
 will serve as a small contribution to the great work. 
 
 Readers wishing to acquire a more detailed know- 
 ledge of crystal science, are referred, to the following 
 series of works on the subject :— 
 
 TEXT-BOOKS ON CRYSTALLOGRAPHY 
 
 J. BEcKENKAMP. Leitfaden der Kristallographie, 1919. 
 
 W. H. and W. L. Bracc. X-Rays and Crystal Structure, revised 
 edition, 1922. 
 
 R. Brauns. Mineralogie, 5th edition, 1918. 
 
 E.H.BoEKeE. Grundlagen der physikalisch-chemischen Petrographie, 
 1915. 2nd edition, by W. Eitel, 1923. 
 
 E. S. Dana. A Text-book of Mineralogy with an extended treatise 
 on Crystallography and Physical Mineralogy, 3rd edition, 1922. 
 
 C. DoEtTER. Physikalisch-chemische Mineralogie, 1905. 
 
 B. GossnER. Kristallberechnung und Kristallzeichnung, 1914. 
 
 P. GrotH. Physikalische Kristallographie, 4th edition, 1905. 
 
 P, GrotH. Elemente der physikalischen und chemischen Kristallo- 
 graphie, 1921. 
 
THE CLASSIFICATION OF ATOMS § 189 
 
 H. Hirton. Mathematical Crystallography, 1922. 
 
 F. M. JAEGER. A Treatise on the Principle of Symmetry, 1917. 
 
 F, KLrocKMANN, Lehrbuch der Mineralogie, 7th and 8th editions, 1922. 
 
 ST. KREUTz, Elemente der Theorie der Kristallstruktur, 1915. 
 
 Tu. LiEpiscu, Grundriss der physikalischen Kristallographie, 1896. 
 
 G. Linck. Grundriss der Kristallographie, 4th edition, 1920. 
 
 C, NAUMANN and F. ZrIrRKEL. Elemente der Mineralogie, 15th 
 edition, 1907. 
 
 P. NiaGt1. Geometrische Kristallographie des Diskontinuums, Ig19. 
 
 P. Nicei1. Lehrbuch der Mineralogie, 1920. 
 
 F, Rinne. Einfiihrung in die kristallographische Formenlehre und 
 Anleitung zu kristallographisch-optischen und réntgenograph- 
 ischen Untersuchungen, 4th and 5th editions, 1922. 
 
 A. SCHONFLIES. Kristallsysteme und Kristallstruktur, 1891. 
 
 G. TAMMANN. Kristallisieren und Schmelzen, 1913. 
 
 (GG. TSCHERMAK and F. BecKE. Lehrbuch der Mineralogie, 8th edition, 
 1g2t. 
 
 A. E. H. Turron. Crystallography and Practical Measurement 
 (2 vols.), 1922. 
 
 W. Voict. Die fundamentalen physikalischen Eigenschaften der 
 Kristalle in elementarer Darstellung, 1893. 
 
 W. Votet. Lehrbuch der Kristallphysik, 1910. 
 
 E. A. WULFING. Die 32 kristallographischen Symmetrieklassen und 
 
 ihre einfachen Formen, 2nd edition, 1914. 
 
 [Several of the older books given in the original are 
 omitted and some English works have been included 
 in the list—TRANSLATOR’S NOTE. ] 
 
INDEX 
 
 Numbers refer to pages 
 
 A Benzene, 58, 59, 91, 105, 163. 
 
 Benzophenone, 66. 
 
 Absolute zero, 68, 77. Beryl, 14. 
 
 Absorption, optical, 165. Biotite, 152. 
 
 Adsorption, 103, 127, 130. Borazite, 70, 168. 
 
 Adularia, 16. Brittleness, 55. 
 
 Affinity tensors, 82, 94. Bromine, 47, 176. 
 
 Albite, 173. Brucite, 173. 
 
 Albumen, 65, 173, 178. 
 Alcohol in crystals, 153. 
 
 Alkalies, 48. C 
 Alkali halides, 176. 
 Alkaline earths, 47. Cesium, 47. 
 Allomerism, 74. — salts, 109. 
 Allotropy, 69. Calcium, 47. 
 Aluminium, 22. Cale-spar, 7, 25, 31, 57, 90, 91, 99, 118, 
 Ammonium oleate, 133. 139, 158, 164, 169, I7I. 
 Amorphous bodies, 39. Canals in crystals, 146. 
 Anatase, 9o. Cane sugar, 76. 
 Andalusite, 132. Carbon, 44, 47, 75. 
 Anhydrite, 135, 149. — bisulphide in crystals, 153. 
 Anisotropy, 54, 56, 121, 134, 139, 140.|— tetrabromide, 75. 
 Anorthite, 173. Carborundum, 72, 117. 
 Aragonite, 164, 169. Catalysts, 161. 
 Argon, 84, 186. Cell axes, 106. 
 Arsenic type, 116. Cells, 106. 
 Atom, 40, I41, 181. Cellulose, 65. 
 — classification, 180. Centre symmetry, 29. 
 — domains, 47, 67, 82, 107, 126, 184. | Chabasite, 153. 
 — lattice, 90, 93. Chemical compounds, Ioo. 
 — number, 21, 183. —-- formule, 39, 68, 78. 
 — rings, 90. — reactions of crystals, 60, 139, 158, 
 — sub-types, 185. 170. 
 — theory, 5. — — — molecules, 140, 158. 
 — types, 182. Chlorine, 47, 186. 
 Augite or Pyroxene, 28, 132. Chlorite process, 154. 
 Axial sections, 8, 26, 50, 173. Chrysoberyl, 38. 
 Chrysolite, 114. 
 B Classification of atoms, I8o. 
 Cleavage, 7, 27, 55, 91, I6I. 
 
 Barium, 47. Close-packing of spheres, 46, 67. 
 — chloride dihydrate, 150, 162. Coarsening of the grain, 131. 
 
 _ — nitrate, 97. Cobaltite, 112. 
 Bauerite process, 154. Collective crystallization, 131. 
 Benitoite, 31. Colloidal, 45, 130. 
 
 Ig! 
 
192 CRYSTALS AND MATTER 
 
 Colloidal metals, 46. 
 Compounds, chemical, 98. 
 Contact, metamorphosis, 131. 
 Co-ordination, 80. 
 
 Copper, 22, 118. 
 
 — vitriol, 162. 
 
 Cordierite, 56. 
 
 Crystal classes, 31. 
 
 — nuclei, 211, 128. 
 
 — physiology, 180. 
 
 =~ Structure; §; 
 
 I SVStCiis, 631. 
 
 — undermining, 143. 
 Crystalline molecules, 63. 
 Crystallisation, 65, 121, 128. 
 Crystals, liquid, 63. 
 Cuprite, 171. 
 
 Cyanite, 56, 130. 
 
 D 
 
 Debye-Scherrer diagram, 19, 45, 46. 
 
 Decrescence, 7. 
 
 Deformation in physical and chemical 
 field, 170, 174, 176. 
 
 — morphotropic, 105, 117. 
 
 Desmine, 151. 
 
 Diamond, 22, 42, 43, 55, 66, 74, 96, 118, 
 
 1S6. 
 
 Dielectric constant, 119. 
 
 Diffraction, 11. 
 
 — equation, I2. 
 
 Digyric axis, 35. 
 
 = SYMIMEClEY,. 27. 
 
 Dolomite, 31. 
 
 Doma, 29. 
 
 E 
 
 Electron shells, 41, 84. 
 
 Electrons, 40, 41, 141, 181. 
 Elementary cell, 22, 34, 47, 78, 139. 
 Elements, 185. 
 
 Emanium, 86, 
 
 Enantiomorphy, 75. 
 
 Energy quanta, 77. 
 
 Etch figures, 134, 139. 
 
 F 
 
 Fayalite, 114. 
 
 Felspars, 99, 173. 
 
 Fibre diagrams, 17. 
 
 Fine-structural types, 34. 
 Fine-structure study, 5. 
 
 Flake diagrams, 17. 
 
 Fluorine, 47. 
 
 Fluor-spar, 22, 79, 118. 
 
 Fore-forms of crystallization, 41, 164. 
 
 Forsterite, 114. 
 Friction, internal, 62. 
 
 Fundamental law of crystallography, 
 8. 
 
 G 
 
 Garnet, 55, 132. 
 Gas, 59, 61. 
 
 Gel, 46. 
 
 Gems, 155. 
 Glancing angle, 12. 
 Gliadin, 65. 
 Globulites, 66. 
 Glutaminic acid, 65. 
 
 Gold, 22, 45. 
 
 Graphite, 20, 42, 74, 87, 92, 
 155, 
 
 —~) type, 210. 
 
 Graphitic acid, 154. 
 
 Growth, 121, 128, 131, 137; 
 — pyramids, 121. 
 
 Gypsum, 32, 57, 134, 149, 159. 
 Gyric symmetry, 30. 
 
 Gyroidal symmetry, 30. 
 
 H 
 
 Heemoglobin, 65. 
 Halogens, 47. 
 
 Hardness, 55, 62, 154. 
 Heat as a catalyst, 161. 
 — action, 103, 170. 
 
 — conduction, 56. 
 Helium, 84, 140. 
 Heterobars, 185. 
 Heulandite, 145, 151. 
 Hexagonal system, 31, 53. 
 Homoomeric, 71. 
 Hydrogen, 86, 89, 111, 187. 
 
 Ice, 70, 79, 120. 
 
 Imbibition of albumen, 175. 
 Indicators, crystallographic, 163. 
 Iodine, 47. 
 
 Ion, 40, 185. 
 
 — lattice, 93. 
 
 Tron, 73, 132: 
 
 — carbide, 78. 
 
 — glance, 122. 
 
 — pyrites, 19, 112, 118. 
 Isobars, 185. 
 Isodynamostasy, 58, 115. 
 Isometric system, 31. 
 Isomorphism, 105. 
 Isomorphotropy, 105. 
 
 132, 
 
INDEX 
 
 Isomorphous mixture, 97, 129. 
 — stratification, 129. 
 Isostasy, 54, 58, 115. 
 Isotopes, 97, 186. 
 
 Isotypy, I15. 
 
 Kolnenite, 152. 
 Krypton, 84. 
 
 L 
 
 Labradorite, 173. 
 
 Lag points, 77, 100, 150, 156. 
 
 Lattice types, 93, 116. 
 
 Laue, 13. 
 
 ri diagrams, II, 14, 15, 52, 71, 144, 
 146, 
 
 — effect, II. 
 
 Lead nitrate, 97. 
 
 Leptyles, 90. 
 
 Leptoblasts, 96. 
 
 Leptology, 5. 
 
 Leptonic axes, 106. 
 
 — volumes, 107. 
 
 Leptons, 5. 
 
 — shape of, 41. 
 
 Leptoscope, 24. 
 
 Leucite, 168. 
 
 Light figures, 134. 
 
 Liquid crystals, 63. 
 
 Liquids, 59, 62. 
 
 Lithium, 47, 184. 
 
 Longulites, 66. 
 
 Loschmidt number, 106. 
 
 M 
 
 Macrostereochemistry, 25. 
 Magnesium, 47, 116. 
 Magnetic pyrites, 71. 
 Margarites, 66. 
 Marmorization, 131. 
 Mass action, 100, I61. 
 Melting process, 178. 
 Metabiotite, 152. 
 Metabrucite, 149. 
 Metachabasite, 153. 
 Metaheulandite, 146. 
 Metals, 22, 46. 
 Metamorphoses, series of, 61. 
 Metascolecite, 147. 
 Methane, 43, 160. 
 
 Mica, 56, 152. 
 Microcline, 38. 
 
 Mimesy, 38. 
 
 Minimum symmetry, 41. 
 Mirror symmetry, 29, 33, 40, 140. 
 Mixed crystals, 97. 
 
 13 
 
 198 
 
 Mixed crystals, significance of, 98. 
 Mixture, physical, 99. 
 Modifications, 70, 76, 164. 
 Molar axes, 107. 
 
 — volume, 107. 
 
 — weight, 107. 
 
 Molecular lattices, 10, 91, 93. 
 
 — linkage, 94, 103, 130. 
 
 — magnitudes, 48. 
 
 Molecule, 40, 141, 181. 
 Molecules, additive, 7. 
 
 — asymmetrical, 27. 
 
 — crystalline, 65. 
 
 — in crystals, 87. 
 
 Monoclinic system, 31. 
 Monotopy, 183. 
 
 Morphology, 28. 
 
 Morphotropic constructions, 107. 
 Morphotropy, 105. 
 
 Multiple proportions, 79. 
 
 N 
 
 Nitrogen, 47, 142. 
 Nuclear charge, 183. 
 — sphere, I4I. 
 
 O 
 
 Occlusion, 130. 
 
 Olivine, 95, 113. 
 
 Opal, 44. 
 
 Organic compounds, 24, 90. 
 Outer shell, 142. 
 Outgrowths, 102. 
 
 Oxygen, 47. 
 
 Parisite, 144. 
 
 Pedion, 29. 
 
 Penecrystals, 63. 
 Phenylacridonium sulphate, 164. 
 Physical processes, 60, 165, 170, 176. 
 Pinacoid, 29. 
 
 Plagioclases, 99, 173. 
 
 Plasticity, 55. 
 
 Point system, Io. 
 
 Polanyi diagram, 18. 
 Polymorphism, 69. 
 
 Polytypy, 72. 
 
 Porosity, 126. 
 
 Potassium, 47. 
 
 — bromide, 100, 110, 176. 
 
 — chloride, 100, 102, I10, 176. 
 — cyanide, 97, IIo. 
 
 — iodide, 102, I10, 176. 
 Pre-chemical processes, 160. 
 Primitive bodies, 7. 
 
194 
 
 Primitive forms, 29. 
 
 Prisma, 29. 
 
 Projection diagrams, 32. 
 Pseudoisotropy, 59. 
 Pyroelectricity, 130, 147, 148. 
 Pyroxene or augite, 28, 132. 
 
 Q 
 
 Quartation, 157. 
 
 Quartz, 27, 31, 36, 44, 57, 70, 72, 73, 
 130, 166, 167. 
 
 — type, 116. 
 
 R 
 
 Radical lattices, 94. 
 
 Radicals, go. 
 
 Radium, 108. 
 
 — oxide, 108. 
 
 Rare gases, 86. 
 
 Rationality of axial sections, 8, 26, 50. 
 
 Reactions discontinuous, 160. 
 
 Reconstruction, 153. 
 
 Reflexion in ultra-red, 44, 92. 
 
 — of X-rays, I2, 152. 
 
 — with-translation, plane of, 33. 
 
 Regional metamorphosis, 131. 
 
 Regular polyhedra, 51. 
 
 Resistance to chemical attack, 154. 
 
 Rhombic system, 31, 53. 
 
 Rhythm in crystal structure, 28, 33, 50. 
 
 Rifts in structure, 126. 
 
 Ring structure, go. 
 
 Rock-salt, 22, 26, 47, 55, 82, 83, 97, 
 103, 127, 119, 125, 127,120. 
 
 Rotation axes, 28, 33, 50. 
 
 — methods, 16. 
 
 — with reflexion, 28, 30. 
 
 Rubidium, 47. 
 
 — chloride, 108. 
 
 Rutile, 90, 103. 
 
 S 
 
 Salting-out process, 103. 
 Sanidine, 147. 
 
 Scolecite, 147. 
 
 Screening action, 126, 155. 
 Screw axes, 33. 
 
 Secondary rays, II. 
 
 Silica gel, 46, 153. 
 
 Silver, 22, 46. 
 
 — iodide, 172. 
 
 Sodium, 46, 66. 
 
 — bromide, 97. 
 
 — chloride dihydrate, 162. 
 — hydrogen fluoride, 109. 
 —- periodate, 31. 
 
 Solution, 133, 137, 177. 
 
 ! 
 ‘ 
 
 CRYSTALS AND MATTER 
 
 Solution forms, 135, 137. 
 
 — process, 136. 
 
 Space lattice, 10, 94. 
 
 Sphenoid, 20. 
 
 Spherical crystals, 66. 
 
 Stability, 58, 60, 70, 84, 115, 129, 136, 
 I 
 
 57- 
 States of matter, 61. 
 Starch, 64. 
 Star figure, X-ray, 144, 149. 
 Staurolite, 130. 
 Step-rule, 72. 
 Stereochemical axes, 106. 
 Stereochemistry, 5. 
 Stereograms, 22. 
 Stereophysics, 5. 
 Succinic iodimide, 36. 
 Sulphur, 47. 
 Strontium, 47. 
 Structural chemistry, 81. 
 — groups, 90. 
 — rhythms, 28, 31, 50, 53. 
 — rigidity, 141. 
 Surfaces of crystals, 79, 125. 
 Symmetry actions, 158. 
 axes, 29. 
 planes, 29. 
 point, 29. 
 of crystals, 28. 
 — leptons, 40. 
 
 T 
 
 Tension processes, 160, 167. 
 Tetragonal system, 31, 53. 
 Tetragyral symmetry, 31. 
 Tetrahedral type, 118. 
 Thermal action, 170. 
 Topaz, 163. 
 
 Topic axes, 105, 107. 
 Topochemical reactions, 143. 
 Topotropy, 106. 
 Tourmaline, 31, 56, 163. 
 Transformation series, 69. 
 Trichites, 66. 
 
 Triclinic system, 27, 53. 
 Trigonal system, 31, 53. 
 Trigyric symmetry, 31. 
 Twin formation, 35, 80, 92. 
 — gliding, 92. 
 
 U 
 Uranium, 43, 85. 
 
 V 
 Valency, 80. 
 — changes, 87. 
 
 — distribution, 82. 
 — tensors, 82, 94. 
 
INDEX 
 
 WwW Z 
 
 Water in crystals, 144, 149, 162. Zeolites, 144, 155. 
 
 Wave length curves, 56. Zero absolute, 68, 77. 
 — valency, 84. 
 
 x Zinc blende, 22, 88, 118. 
 
 — oxide (zincite), 120. 
 Xenon, 85. Zone, 12, 173. 
 
 195 
 
‘ . ate. 
 
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