Transcriber’s Note:

The cover image was created by the transcriber and is placed in the public domain.

International Scientific Series

VOLUME XCVIII.

DIRECT REPRODUCTIONS OF AUTOCHROME PHOTOGRAPHS OF SCREEN PICTURES IN POLARISED LIGHT.

Fig. 90.—Screen Picture in Polarised Light, with Nicols crossed, of a thick Plate perpendicular to the Axis of a naturally twinned Crystal of Quartz, the left half being of right-handed Quartz and the right half of alternately left and right-handed Quartz, the Planes of Demarcation being oblique to the Plate.

Fig. 97.—Crystals of Benzoic Acid in the Act of Growth, as seen on the Screen in Polarised Light with crossed Nicols.

The International Scientific Series

CRYSTALS

BY

A. E. H. TUTTON

D.Sc., M.A. (New College, Oxon.), F.R.S.

VICE-PRESIDENT OF THE MINERALOGICAL SOCIETY MEMBER OF THE COUNCILS OF THE CHEMICAL SOCIETY AND THE BRITISH ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE

WITH 120 ILLUSTRATIONS

LONDON

KEGAN PAUL, TRENCH, TRÜBNER & CO. L^TD

DRYDEN HOUSE, GERRARD STREET, W.

1911

PREFACE

The idea underlying this book has been to present the phenomena of crystallography to the general reading public in a manner which can be comprehended by all. In the main the sequence is that of the author’s evening discourse to the British Association at their meeting at Winnipeg in the summer of 1909. It is hoped, however, that the book combines the advantages of sufficient amplification of the story there told to make it an adequately detailed account of the development of the subject, and of the immense progress which has been made in it during recent years, with a full description of the numerous experimental illustrations given in the lecture, involving some of the most beautiful phenomena displayed by crystals in polarised light. Such an account has not been otherwise published, the brief abstract appearing in the Report of the British Association for 1909 giving no account of the experiments, which were a feature of the lecture, owing to the employment of a fine projection polariscope of more or less novel construction, and including two magnificent large Nicol prisms, a pair of the original ones made by Ahrens. The author has been frequently requested to publish a fuller account of this discourse, and as the general plan of it so fully embodies the present aspect of this fascinating science, it was determined, when invited by the publishers to write a generally readable book on “Crystals,” to comply with these requests.

There is also included an account of the remarkable work of Lehmann and his fellow workers on “Liquid Crystals,” and the bearing of these discoveries on the nature of crystal structure is discussed in so far as the experimental evidence has gone. Similarly, the theory of Pope and Barlow, connecting crystalline structure with the chemical property of valency, is referred to and explained, as this theory has called forth deep and widespread interest. In both cases, however, the author has been careful to avoid any expression of opinion on purely theoretical questions for which there is as yet no definite experimental evidence, and has confined himself strictly to indicating how far such interesting theories are supported by actual experimental facts.

No forbidding mathematical formulæ and no unessential technical terms will be found in the book, the aim of the author being to make any ordinarily cultured reader feel at the conclusion that the story has been readily comprehensible, and that crystallography is not the abstruse and excessively difficult subject which it has so generally been supposed to be, but that, on the contrary, it is both simple and straightforward, and full of the most enthralling interest, as well for the exquisite phenomena with which it deals, as for the exceedingly important bearing which it has on the nature, both chemical and physical, of solid matter.

If any of its readers should be so impressed with the value of work in this domain of science as to be desirous of joining the very thin ranks of the few who are engaged in it, they will find a guide to practical goniometry and to the experimental investigation of crystals in all its branches and details, as well as the necessary theoretical help, in the author’s book on “Crystallography and Practical Crystal Measurement” (Macmillan & Co., 1911), and also an account of the author’s own contributions to the subject in a monograph entitled “Crystalline Structure and Chemical Constitution” (Macmillan & Co., 1910).

A. E. H. TUTTON.

January 1911.

CONTENTS

CRYSTALS

(INCLUDING LIQUID CRYSTALS)

CHAPTER I
INTRODUCTION.

It is a remarkable fact that no definition of life has yet been advanced which will not apply to a crystal with as much veracity as to those obviously animate objects of the animal and vegetable world which we are accustomed to regard in the ordinary sense as “living.” A crystal grows when surrounded by a suitable environment, capable of supporting it with its natural food, namely, its own chemical substance in the liquid or vaporous state or dissolved in a solvent. Moreover, when a crystal is broken, and then surrounded with this proper environment, it grows much more rapidly at the broken part than elsewhere, repairing the damage done in a very short space of time and soon presenting the appearance of a perfect crystal once more. In this respect it is quite comparable with animal tissue, the wonderful recuperative power of which after injury, exhibited by special growth at the injured spot, is often a source of such marvel to us. Indeed, a crystal may be broken in half, and yet each half in a relatively very brief interval will grow into a crystal as large as the original one again. The longevity and virility of the spores and seeds of the vegetable kingdom have been the themes of frequent amazement, although many of the stories told of them have been unable to stand the test of strict investigation. The virility of a crystal, however, is unchanged and permanent.

A crystal of quartz, rock-crystal, for instance—detached, during the course of the disintegration of the granitic rock of which it had originally formed an individual crystal, by the denuding influences at work in nature thousands of years ago, subsequently knocked about the world as a rounded sand grain, blown over deserts by the wind, its corners rounded off by rude contact with its fellows, and subjected to every variety of rough treatment—may eventually in our own day find itself in water containing in solution a small amount of the material of which quartz is composed, silicon dioxide SiO₂. No sooner is this favourable environment for continuing its crystallisation presented to it, than, however old it may be, it begins to sprout and grow again. It becomes surrounded in all probability by a beautiful coating of transparent quartz, with exterior faces inclined at the exact angles of quartz, although no sign of exterior faces had hitherto persisted through all the stages of its varied adventures. Or it may grow chiefly at two or three especially favourable places, and in the course of a few weeks, under suitable conditions, at each place a perfect little quartz crystal will radiate out from the sand grain, composed of a miniature hexagonal prism terminated by the well-known pyramid, really consisting of a pair of trigonal (rhombohedral) pyramids more or less equally developed, and together producing an apparently hexagonal one. Four such grains of sand, from which quartz crystals are growing, are shown in Fig. 1, as they appear under a microscope magnifying about fifty diameters. One of them shows a perfectly developed doubly terminated crystal of quartz growing from the tip of a singly terminated one, attached to and growing directly out of the grain.

Fig. 1.—Sand Grains with Quartz Crystals growing from them.

This marvellously everlasting power possessed by a crystal, of silent imperceptible growth, that is, of adding to its own regular structure further accretions of infinitesimal particles, the chemical molecules, of its own substance, is one of the strangest functions of solid matter, and one of the fundamental facts of science which is rarely realised, compared with many of the more obvious phenomena of nature.

A crystal in the ordinary sense of the word is solid matter in its most perfectly developed and organised form. It is composed of the chemical molecules of some definitely constituted substance, which have been laid down in orderly sequence, in accordance with a specific architectural plan peculiar to that particular chemical substance. The physical properties of the latter are such that it assumes the solid form at the ordinary temperature and pressure, leaving out of consideration for the present the remarkable viscous and liquid substances which will be specially dealt with in Chapter XVI. of this book, and which are currently known as “liquid crystals.” This term is not perhaps a very appropriate one. For the word “crystal” had much better be left to convey the idea of rigidity of polyhedral form and internal structure, which is the very basis of crystal measurement.

The solid crystal may have been produced during the simple act of congealment from the liquid state, on the cooling of the heated liquefied substance to the ordinary temperature. Sulphur, for instance, is well-known to crystallise in acicular crystals belonging to the monoclinic system under such conditions, a characteristic crop being shown in Fig. 2 (Plate I.); they were formed within an earthenware crucible in which the fusion had occurred, and became revealed on pouring out the remainder of the liquid sulphur when the crystallisation had proceeded through about one-half of the original amount of the “melt.”

PLATE I.
Fig. 2.—Monoclinic Acicular Crystals of Sulphur produced by Solidification of Liquid.

Fig. 3.—Octahedral Crystals of Arsenious Oxide produced by Condensation of Vapour.
Crystals formed by Different Processes.

PLATE II.
Fig. 4.—Cubic Octahedral Crystals of Potash Alum growing from Solution.

Fig. 10.—Micro-Chemical Crystals of Gypsum (Calcium Sulphate) produced by Slow Precipitation (see p. [14]).
Crystals formed by Different Processes.

Or the substance may be one which passes directly from the gaseous to the solid condition, on the cooling of the vapour from a temperature higher than the ordinary down to the latter, under atmospheric pressure. Oxide of arsenic, As₂O₃, is a substance exhibiting this property characteristically, and Fig. 3 (Plate I.) is a reproduction of a photograph of crystals of this substance thus produced. The white solid oxide was heated in a short test tube over a Bunsen flame, and the vapour produced was allowed to condense on a microscope glass slip, and the result examined under the microscope, using a 1½ inch objective. Fig. 3 represents a characteristic field of the transparent octahedral crystals.

Or again, the crystal may have been deposited from the state of solution in a solvent, in which case it is a question of the passage of the substance from the liquid to the solid condition, complicated by the presence of the molecules of the solvent, from which the molecules of the crystallising solid have to effect their escape. Fig. 4 (Plate II.) represents crystals of potash alum, for instance, growing from a drop of saturated solution on a glass slip placed on the stage of the microscope, the drop being spread within a hard ring of gold size and under a cover-glass, in order to prevent rapid evaporation and avoid apparent distortion by the curvature of an uncovered drop. The crystals are of octahedral habit like those of oxide of arsenic, but many of them also exhibit the faces of the cube.

In any case, however it may be erected, the crystal edifice is produced by the regular accretion of molecule on molecule, like the bricks or stone blocks of the builder, and in accordance with an architectural plan more elaborate and exact than that of any human architect. This plan is that of one of the thirty-two classes into which crystals can be naturally divided with respect to their symmetry. Which specific one is developed, and its angular dimensions, are traits characteristic of the substance. The thirty-two classes of crystals may be grouped in seven distinctive systems, the seven styles of crystal architecture, each distinguished by its own elements of symmetry.

A crystal possesses two further fundamental properties besides its style of architecture. The first is that it is bounded externally by plane faces, arranged on the definite geometrical plan just alluded to and mutually inclined at angles which are peculiar to the substance, and which are, therefore, absolutely constant for the same temperature and pressure. The second is that a crystal is essentially a homogeneous solid, its internal structure being similar throughout, in such wise that the arrangement about any one molecule is the same as about every other. This structure is, in fact, that of one of the 230 homogeneous structures ascertained by geometricians to be possible to crystals with plane faces. The first property, that of the planeness of the crystal faces, and their arrangement with geometrical symmetry, is actually determined by the second, that of specific homogeneity. For, as with human nature developed to its highest type, the external appearance is but the expression of the internal character.

When nature has been permitted to have fair play, and the crystal has been deposited under ideal conditions, the planeness of its faces is astonishingly absolute. It is fully equal to that attained by the most skilled opticians after weeks of patient labour, in the production of surfaces on glass or other materials suitable for such delicate optical experiments as interference-band production, in which a distortion equal to one wave-length of light would be fatal. In all such cases of ideal deposition, those interfacial angles on the crystal which the particular symmetry developed requires to be equal actually are so, to this same high degree of refinement. This fact renders possible exceedingly accurate crystal measurement, that is, the determination of the angles of inclination of the faces to each other, provided refined measuring instruments (goniometers), pure chemical substances, and the means of avoiding disturbance, either material or thermal, during the deposition of the crystal, are available.

The study of crystals naturally divides itself into two more or less distinct but mutually very helpful branches, and equally intimately connected with the internal structure of crystals, namely, one which concerns their exterior configuration and the structural morphology of which it is the eloquent visible expression, and another which relates to their optical characters. For the latter are so definitely different for the different systems of crystal symmetry that they afford the greatest possible help in determining the former, and give the casting vote in all cases of doubt left after the morphological investigation with the goniometer. It is, of course, their brilliant reflection and refraction of light, with production of numerous scintillations of reflected white light and of refracted coloured spectra, which endows the hard and transparent mineral crystals, known from time immemorial as gem-stones, with their attractive beauty. Indeed, their outer natural faces are frequently, and unfortunately usually, cut away most sacrilegiously by the lapidary, in order that by grinding and polishing on them still more numerous and evenly distributed facets he may increase to the maximum the magnificent play of coloured light with which they sparkle.

An interesting and very beautiful lecture experiment was performed by the author in a lecture a few years ago at the Royal Institution, which illustrated in a striking manner this fact that the light reaching the eye from a crystal is of two kinds, namely, white light reflected from the exterior faces and coloured light which has penetrated the crystal substance and emerges refracted and dispersed as spectra. Two powerful beams of light from a pair of widely separated electric lanterns were concentrated on a cluster of magnificent large diamonds, kindly lent for the purpose by Mr Edwin Streeter, and arranged in the shape of a crown, it being about the time of the Coronation of His late Majesty King Edward VII. The effect was not only to produce a blaze of colour about the diamonds themselves, but also to project upon the ceiling of the lecture theatre numerous images in white light of the poles of the electric arc, derived by reflection from the facets, interspersed with equally numerous coloured spectra derived from rays which had penetrated the substance of the diamonds, and had suffered both refraction and internal reflection.

CHAPTER II
THE MASKING OF SIMILARITY OF SYMMETRY AND CONSTANCY OF ANGLE BY DIFFERENCE OF HABIT, AND ITS INFLUENCE ON EARLY STUDIES OF CRYSTALS.

Fig. 5.—Natural Rhombohedron of Iceland Spar with Subsidiary Faces.

Nothing is more remarkable than the great variety of geometrical shapes which the crystals of the same substance, derived from different localities or produced under different conditions, are observed to display. One of the commonest of minerals, calcite, carbonate of lime, shows this feature admirably; the beautiful large rhombohedra from Iceland, illustrated in Fig. 5, or the hexagonal prisms capped by low rhombohedra from the Bigrigg mine at Egremont in Cumberland, shown in Fig. 6, appear totally different from the “dog-tooth spar” so plentifully found all over the world, a specimen of which from the same mine is illustrated in Fig. 7. No mineral specimens could well appear more dissimilar than these represented on Plate III. in Figs. 6 and 7, when seen side by side in the mineral gallery of the British Museum (Natural History) at South Kensington. But all are composed of similar chemical molecules of calcium carbonate, CaCO₃; and when the three kinds of crystals are investigated they are found to be identical in their crystalline system, the trigonal, and indeed further as to the subdivision or class of that system, which has come to be called the calcite class from the importance of this mineral.

PLATE III.
Fig. 6.—Hexagonal Prisms of Calcite terminated by Rhombohedra.

Fig. 7.—Scalenohedral Crystals of Calcite, “Dog-tooth Spar.”
Crystals of Calcite from the same Mine, illustrating Diversity of Habit.
(Photographed from Specimens in the Natural History Department of the British Museum, by kind permission.)

Moreover, many of the same faces, that is, faces having the same relation to the symmetry, are present on all three varieties, the “forms” to which they equally belong being the common heritage of calcite wherever found. A “form” is the technical term for a set of faces having an equal value with respect to the symmetry. Thus the prismatic form in Fig. 6 is the hexagonal prism, a form which is common to the hexagonal and trigonal systems of symmetry, and the form “indices” (numbers[^[1]] inversely proportional to the intercepts cut off from the crystal axes by the face typifying the form) of which are {2̄1̄1}; the large development of this form confers the elongated prismatic habit on the crystal. The terminations are faces of the flat rhombohedron {110}. The pyramidal form of the dog-tooth spar shown in Fig. 7 is the scalenohedron {20̄1}, and it is this form which confers the tooth-like habit, so different from the hexagonal prism, upon this variety of calcite. But many specimens of dog-tooth spar, notably those from Derbyshire, consist of scalenohedra the middle portion of which is replaced by faces of the hexagonal prism {2̄1̄1}, and the terminations of which are replaced by the characteristic rhombohedron {100} of Iceland spar; indeed, it is quite common to find crystals of calcite exhibiting on the same individual all the forms which have been mentioned, that is, those dominating the three very differently appearing types. The author has quite recently measured such a crystal, which, besides showing all these four forms well developed, also exhibited the faces of two others of the well-known forms of calcite, {3̄1̄1} and {310}, and a reproduction of a drawing of it to scale is given in Fig. 8. Instead of indices the faces of each form bear a distinctive letter; m = {2̄1̄1}, r = {100}, e = {110}, v = {20̄1} (the faces of the scalenohedron are of somewhat small dimensions on this crystal), n = {3̄1̄1}, and t = {310}.

Fig. 8.—Measured Crystal of Calcite.

It is obviously then the “habit” which is different in the three types of calcite—Iceland spar, prismatic calc-spar, and dog-tooth spar—doubtless owing to the different local circumstances of growth of the mineral. Habit is simply the expression of the fact that a specific “form,” or possibly two particular forms, is or are much more prominently developed in one variety than in another. Thus the principal rhombohedron r = {100}, parallel to the faces of which calcite cleaves so readily, is the predominating form in Iceland spar, while the scalenohedron v = {20̄1} is the habit-conferring form in dog-tooth spar. Yet on the latter the rhombohedral faces are frequently developed, blunting the sharp terminations of the scalenohedra, especially in dog-tooth spar from Derbyshire or the Hartz mountains; and on the former minute faces of the scalenohedron are often found, provided the rhombohedron consists of the natural exterior faces of the crystal and not of cleavage faces. In the same manner the prismatic crystals from Egremont are characterised by two forms, the hexagonal prism m = {2̄1̄1} and the secondary rhombohedron e = {110}, but both of these forms, as we have seen on the actual crystal represented in Fig. 8, are also found developed on other crystals of mixed habit.

This illustration from the naturally occurring minerals might readily be supplemented by almost any common artificial chemical preparation, sulphate of potash for instance, K₂SO₄, the orthorhombic crystals of which take the form of elongated prisms, even needles, on the one hand, or of tabular plate-like crystals on the other hand, according as the salt crystallises by the cooling of a supersaturated solution, or by the slow evaporation of a solution which at first is not quite saturated. In both cases, and in all such cases, whether of minerals or chemical preparations, the same planes are present on the crystals of the same substance, although all may not be developed on the same individual except in a few cases of crystals particularly rich in faces; and these same planes are inclined at the same angles. But their relative development may be so very unlike on different crystals as to confer habits so very dissimilar that the fact of the identity of the substance is entirely concealed.

Fig. 9.—Crystal of Gypsum.

A further example may perhaps be given, that of a substance, hydrated sulphate of lime, CaSO₄.2H₂O, which occurs in nature as the beautiful transparent mineral gypsum or selenite—illustrated in Fig. 9, and which is found in monoclinic crystals often of very large size—and which may also be chemically prepared by adding a dilute solution of sulphuric acid to a very dilute solution of calcium chloride. The radiating groups of needles shown in Fig. 10 (Plate II.) slowly crystallise out when a drop of the mixed solution is placed on a microscope slip and examined under the microscope, using the one-inch objective. These needles, so absolutely different in appearance from a crystal of selenite, are yet similar monoclinic prisms, but in which the prismatic form is enormously elongated compared with the other (terminating) form.

This difference of facial development, rendering the crystals of one and the same substance from different sources so very unlike each other, was apparently responsible for the very tardy discovery of the fundamental law of crystallography, the constancy of the crystal angles of the same substance. Gessner, sometime between the years 1560 and 1568, went so far as to assert that not only are different crystals of the same substance of different sizes, but that also the mutual inclinations of their faces and their whole external form are dissimilar.

What was much more obvious to the early students of crystals, and which is, in fact, the most striking thing about a crystal after its regular geometric exterior shape, was the obviously homogeneous character of its internal structure. So many crystals are transparent, and so clear and limpid, that it was evident to the earliest observers that they were at least as homogeneous throughout as glass, and yet that at the same time they must be endowed with an internal structure the nature of which is the cause of both the exterior geometric regularity of form, so different from the irregular shape of a lump of glass, and of the peculiar effect on the rays of light which are transmitted through them. From the earliest ages of former civilisations the behaviour of crystals with regard to light has been known to be different for the different varieties of gem-stones.

About the year 1600 Cæsalpinus observed that sugar, saltpetre, and alum, and also the sulphates of copper, zinc and iron, known then as blue, white and green vitriol respectively, separate from their solutions in characteristic forms. Had he not attributed this to the operation of an organic force, in conformity with the curious opinion of the times concerning crystals, he might have had the credit of being the pioneer of crystallographers. The first two real steps in crystallography, however, with which in our own historic times we are acquainted, were taken in the seventeenth century within four years of each other, one from the interior structural and the other from the exterior geometrical point of view. For in 1665 Robert Hooke in this country made a study of alum, which he appears to have obtained in good crystals, although he was unacquainted with its true chemical composition. He describes in his “Micrographia” how he was able to imitate the varying habits of the octahedral forms of alum crystals by building piles of spherical musket bullets, and states that all the various figures which he observed in the many crystals which he examined could be produced from two or three arrangements of globular particles. It is clear that the homogeneous partitioning of space in a crystal structure by similar particles building up the crystal substance was in Hooke’s mind, affording another testimony to the remarkably prescient insight of our great countryman.

Four years later, in 1669, Nicolaus Steno carried out in Florence some remarkable measurements, considering the absence of proper instruments, of the angles between the corresponding faces of different specimens of rock-crystal (quartz, the naturally occurring dioxide of silicon, concerning which there will be much to say later in this book), obtained from different localities, and published a dissertation announcing that he found these analogous angles all precisely the same.

In the year 1688 the subject was taken up systematically by Guglielmini, and in two memoirs of this date and 1705 he extended Steno’s conclusions as to the constancy of crystal angles in the case of rock-crystal into a general law of nature. Moreover, he began to speculate about the interior structure of crystals, and, like Hooke, he took alum as his text, and suggested that the ultimate particles possessed plane faces, and were, in short, miniature crystals. He further announced the constancy of the cleavage directions, so that to Guglielmini must be awarded the credit for having, at a time when experimental methods of crystallographic investigation were practically nil, discovered the fundamental principles of crystallography.

The fact that a perfect cleavage is exhibited by calcite had already been observed by Erasmus Bartolinus in 1670, and in his “Experimenta Crystalli Islandici” he gives a most interesting account of the great discovery of immense clear crystals of calcite which had just been made at Eskifjördhr in Iceland, minutely describing both their cleavage and their strong double refraction. Huyghens in 1690 followed this up by investigating some of these crystals of calcite still more closely, and elaborated his laws of double refraction as the result of his studies.

There now followed a century which was scarcely productive of any further advance at all in our real knowledge of crystals. It is true that Boyle in 1691 showed that the rapidity with which a solution cools influences the habit of the crystals which are deposited from it. But neither Boyle, with all his well-known ability, so strikingly displayed in his work on the connection between the volume of a gas and the pressure to which it is subjected, nor his lesser contemporaries Lemery and Homberg, who produced and studied the crystals of several series of salts of the same base with different acids, appreciated the truth of the great fact discovered by Guglielmini, that the same substance always possesses the same crystalline form the angles of which are constant. Even with the growth of chemistry in the eighteenth century, the opinion remained quite general that the crystals of the same substance differ in the magnitude of their angles as well as in the size of their faces.

We begin to perceive signs of progress again in the year 1767, when Westfeld made the interesting suggestion that calcite is built up of rhombohedral particles, the miniature faces of which correspond to the cleavage directions. This was followed in 1780 by a treatise “De formis crystallorum” by Bergmann and Gahn of Upsala, in which Guglielmini’s law of the constancy of the cleavage directions was reasserted as a general one, and intimately connected with the crystal structure. It was in this year 1780 that the contact goniometer was invented by Carangeot, assistant to Romé de l’Isle in Paris, and it at once placed at the disposal of his master a weapon of research far superior to any possessed by previous observers.

Fig. 11.—Contact Goniometer as used by Romé de l’Isle.

In his “Crystallographie,” published in Paris in 1783, Romé de l’Isle described a very large number of naturally occurring mineral crystals, and after measuring their angles with Carangeot’s goniometer he constructed models of no less than 500 different forms. Here we have work based upon sound measurement, and consequently of an altogether different and higher value than that which had gone before. It was the knowledge that his master desired to faithfully reproduce the small natural crystals which he was investigating, on the larger scale of a model, that led Carangeot to invent the contact goniometer, and thus to make the first start in the great subject of goniometry. The principle of the contact goniometer remains to-day practically as Carangeot left it, and although replaced for refined work by the reflecting goniometer, it is still useful when large mineral crystals have to be dealt with. An illustration of a duplicate of the original instrument is shown in Fig. 11, by the kindness of Dr H. A. Miers. This duplicate was presented to Prof. Buckland by the Duke of Buckingham in the year 1824, and is now in the Oxford Museum.

From the time that measurement of an accurate description was possible by means of the contact goniometer, progress in crystallography became rapid. Romé de l’Isle laid down the sound principle, as the result of the angular measurements and the comparison of his accurate models with one another, that the various crystal shapes developed by the same substance, artificial or natural, were all intimately related, and derivable from a primitive form, characteristic of the substance. He considered that the great variety of form was due to the development of secondary faces, other than those of the primitive form. He thus connected together the work of previous observers, consolidated the principles laid down by Guglielmini by measurements of real value, and threw out the additional suggestion of a fundamental or primitive form.

About the same time Werner was studying the principal forms of different crystals of the same substance. The idea of a fundamental form appears to have struck him also, and he showed how such a fundamental form may be modified by truncating, bevelling, and replacing its faces by other derived forms. His work, however, cannot possess the value of that of Romé de l’Isle, as it was not based on exact measurement, and most of all because Werner appears to have again admitted the fallacy that the same substance could, in the ordinary way, and not in the sense now termed polymorphism, exhibit several different fundamental forms.

But a master mind was at hand destined definitely to remove these doubts and to place the new science on a firm basis. An account of how this was achieved is well worthy of a separate chapter.

CHAPTER III
THE PRESCIENT WORK OF THE ABBÉ HAÜY.

The important work of Romé de l’Isle had paved the way for a further and still greater advance which we owe to the University of Paris, for its Professor of the Humanities, the Abbé Réné Just Haüy, a name ever to be regarded with veneration by crystallographers, took up the subject shortly after Romé de l’Isle, and in 1782 laid most important results before the French Academy, which were subsequently, in 1784, published in a book, under the auspices of the Academy, entitled “Essai d’une Théorie sur la Structure des Crystaux.” The author happens to possess, as the gift of a kind friend, a copy of the original issue of this highly interesting and now very rare work. It contains a brief preface, dated the 26th November 1783, signed by the Marquis de Condorcet, perpetual secretary to the Academy (who, in 1794, fell a victim to the French revolution), to the effect that the Academy had expressed its approval and authorised the publication “under its privilege.”

The volume contains six excellent plates of a large number of most careful drawings of crystals, illustrating the derivation from the simple forms, such as the cube, octahedron, dodecahedron, rhombohedron, and hexagonal prism, of the more complicated forms by the symmetrical replacement of edges and corners, together with the drawings of many structural lattices. In the text, Haüy shows clearly how all the varieties of crystal forms are constructed according to a few simple types of symmetry; for instance, that the cube, octahedron, and dodecahedron all have the same high degree of symmetry, and that the apparently very diverse forms shown by one and the same substance are all referable to one of these simple fundamental or systematic forms. Moreover, Haüy clearly states the laws which govern crystal symmetry, and practically gives us the main lines of symmetry of five of the seven systems as we now classify them, the finishing touch having been supplied in our own time by Victor von Lang.

Haüy further showed that difference of chemical composition was accompanied by real difference of crystalline form, and he entered deeply into chemistry, so far as it was then understood, in order to extend the scope of his observations. It must be remembered that it was only nine years before, in 1774, that Priestley had discovered oxygen, and that Lavoisier had only just (in the same year as Haüy’s paper was read to the Academy, 1782) published his celebrated “Elements de Chimie”; and further, that Lavoisier’s memoir “Reflexions sur le Phlogistique” was actually published by the Academy in the same year, 1783, as that in which this book was written by Haüy. Moreover, it was also in this same year, 1783, that Cavendish discovered the compound nature of water.

Considering, therefore, all these facts, it is truly surprising that Haüy should have been able to have laid so accurately the foundations of the science of crystallography. That he undoubtedly did so, thus securing to himself for all time the term which is currently applied to him of “father of crystallography,” is clearly apparent from a perusal of his book and of his subsequent memoirs.

The above only represents a small portion of Haüy’s achievements. For he discovered, besides, the law of rational indices, the generalisation which is at the root of crystallographic science, limiting, as it does, the otherwise infinite number of possible crystal forms to comparatively few, which alone are found to be capable of existence as actual crystals. The essence of this law, which will be fully explained in Chapter V., is that the relative lengths intercepted along the three principal axes of the crystal, by the various faces other than those of the fundamental form, the faces of which are parallel to the axes, are expressed by the simplest unit integers, 1, 2, 3, or 4, the latter being rarely exceeded and then only corresponding to very small and altogether secondary faces.

This discovery impressed Haüy with the immense influence which the structure of the crystal substance exerts on the external form, and how, in fact, it determines that form. For the observations were only to be explained on the supposition that the crystal was built up of structural units, which he imagined to be miniature crystals shaped like the fundamental form, and that the faces were dependent on the step-like arrangement possible to the exterior of such an assemblage. This brought him inevitably to the intimate relation which cleavage must bear to such a structure, that it really determined the shape of, and was the expression of the nature of, the structural units. Thus, before the conception of the atomic theory by Dalton, whose first paper (read 23rd October 1803), was published in the year 1803 in the Proceedings of the Manchester Literary and Philosophical Society, two years after the publication of Haüy’s last work (his “Traité de Minéralogie,” Paris, 1801), Haüy came to the conclusion that crystals were composed of units which he termed “Molécules Intégrantes,” each of which comprised the whole chemical compound, a sort of gross chemical molecule. Moreover, he went still further in his truly original insight, for he actually suggested that the molécules intégrantes were in turn composed of “Molécules Elémentaires,” representing the simple matter of the elementary substances composing the compound, and hinted further that these elementary portions had properly orientated positions within the molécules intégrantes.

He thus not only nearly forestalled Dalton’s atomic theory, but also our recent work on the stereometric orientation of the atoms in the molecule in a crystal structure. Dalton’s full theory was not published until the year 1811, in his epoch-making book entitled “A New System of Chemical Philosophy,” although his first table of atomic weights was given as an appendix to the memoir of 1803. Thus in the days when chemistry was in the making at the hands of Priestley, Lavoisier, Cavendish, and Dalton do we find that crystallography was so intimately connected with it that a crystallographer well-nigh forestalled a chemist in the first real epoch-making advance, a lesson that the two subjects should never be separated in their study, for if either the chemist or the crystallographer knows but little of what the other is doing, his work cannot possibly have the full value with which it would otherwise be endowed.

The basis of Haüy’s conceptions was undoubtedly cleavage. He describes most graphically on page 10 of his “Essai” of 1784 how he was led to make the striking observation that a hexagonal prism of calcite, terminated by a pair of hexagons normal to the prism axis, similar to the prisms shown in Fig. 6 (Plate III.) except that the ends were flat, showed oblique internal cleavage cracks, by enhancing which with the aid of a few judicious blows he was able to separate from the middle of the prism a kernel in the shape of a rhombohedron, the now well-known cleavage rhombohedron of calcite. He then tried what kinds of kernels he could get from dog-tooth spar (illustrated in Fig. 7) and other different forms of calcite, and he was surprised to find that they all yielded the same rhombohedral kernel. He subsequently investigated the cleavage kernels of other minerals, particularly of gypsum, fluorspar, topaz, and garnet, and found that each mineral yielded its own particular kernel. He next imagined the kernels to become smaller and smaller, until the particles thus obtained by cleaving the mineral along its cleavage directions ad infinitum were the smallest possible. These miniature kernels having the full composition of the mineral he terms “Molécules Constituantes” in the 1784 “Essai,” but in the 1801 “Traité” he calls them “Molécules Intégrantes” as above mentioned. He soon found that there were three distinct types of molécules intégrantes, tetrahedra, triangular prisms, and parallelepipeda, and these he considered to be the crystallographic structural units.

Fig. 12.

Having thus settled what were the units of the crystal structure, Haüy adopted Romé de l’Isle’s idea of a primitive form, not necessarily identical with the molécule intégrante, but in general a parallelepipedon formed by an association of a few molécules intégrantes, the parallelepipedal group being termed a “Molécule Soustractive.” The primary faces of the crystal he then supposed to be produced by the simple regular growth or piling on of molécules intégrantes or soustractives on the primitive form. The secondary faces not parallel to the cleavage planes next attracted his attention, and these, after prolonged study, he explained by supposing that the growth upon the primitive form eventually ceased to be complete at the edges of the primary faces, and that such cessation occurred in a regular step by step manner, by the suppression of either one, two, or sometimes three molécules intégrantes or soustractives along the edge of each layer, like a stepped pyramid, the inclination of which depends on how many bricks or stone blocks are intermitted in each layer of brickwork or masonry. Fig. 12 will render this quite clear, the face AB being formed by single block-steps, and the face CD by two blocks being intermitted to form each step. The plane AB or CD containing the outcropping edges of the steps would thus be the secondary plane face of the crystal, and the molécules intégrantes or soustractives (the steps can only be formed by parallelepipedal units) being infinitesimally small, the re-entrant angles of the steps would be invisible and the really furrowed surface appear as a plane one. Haüy is careful to point out, however, that the crystallising force which causes this stepped development (or lack of development) is operative from the first, for the minutest crystals show secondary faces, and often better than the larger crystals.

Fig. 13.

An instance of a mineral with tetrahedral molécules intégrantes Haüy gives in tourmaline, and the primitive form of tourmaline he considered to be a rhombohedron, conformably to the well-known rhombohedral cleavage of the mineral, made up of six tetrahedra. Again, hexagonal structures formed by three prismatic cleavage planes inclined at 60° are considered by him as being composed of molécules intégrantes of the form of 60° triangular prisms, or molécules soustractives of the shape of 120° rhombic prisms, each of the latter being formed by two molécules intégrantes situated base to base. This will be clear from Figs. 13 and 14, the former representing the structure as made up of equilateral prismatic structural units, and the latter portraying the same structure but composed of 120°-parallelepipeda by elimination of one cleavage direction; each unit in the latter case possesses double the volume of the triangular one, and being of parallelepipedal section is capable of producing secondary faces when arranged step-wise, whereas the triangular structure is not. The points at the intersections in these diagrams should for the present be disregarded; they will shortly be referred to for another purpose.

Fig. 14.

Probably, the most permanent and important of Haüy’s achievements was the discovery of the law of rational indices. At first this only took the form of the observation of the very limited number of rows of molécules intégrantes or soustractives suppressed. In introducing it on page 74 of his 1784 “Essai” he says: “Quoique je n’aie observé jusqu’ici que des décroissemens qui se sont par des soutractions d’une ou de deux rangées de molécules, et quelquefois de trois rangées, mais très rarement, il est possible qu’il se trouve des crystaux dans lesquels il y ait quatre ou cinq rangées de molécules supprimées à chaque décroissement, et même un plus grand nombre encore. Mais ces cas me semblent devoir être plus rares, à proportion que le nombre des rangées soutraites sera plus considérable. On conçoit donc comment le nombre des formes secondaires est néçessairement limité.”

The essential difference between Haüy’s views and our present ones, which will be explained in Chapter IX., is that Haüy takes cleavage absolutely as his guide, and considers the particles, into which the ultimate operation of cleavage divides a crystal, as the solid structural units of the crystal, the unit thus having the shape of at least the molécule intégrante. Now every crystalline substance does not develop cleavage, and others only develop it along a single plane, or along a couple of planes parallel to the same direction, that of their intersection and of the axis of the prism which two such cleavages would produce, and which prism would be of unlimited length, being unclosed.

Again, in other cases cleavage, such as the octahedral cleavage of fluorspar, yields octahedral or tetrahedral molécules intégrantes which are not congruent, that is to say, do not fit closely together to fill space, as is the essence of Haüy’s theory. Hence, speaking generally, partitioning by means of cleavage directions does not essentially and invariably yield identical plane-faced molecules which fit together in contact to completely fill space, although in the particular instances chosen from familiar substances by Haüy it often happens to do so. Haüy’s theory is thus not adequately general, and the advance of our knowledge of crystal forms has rendered it more and more apparent that Haüy’s theory was quite insufficient, and his molécules intégrantes and soustractives mere geometrical abstractions, having no actual basis in material fact; but that at the same time it gave us a most valuable indication of where to look for the true conception.

This will be developed further into our present theory of the homogeneous partitioning of space, in Chapter IX. But it may be stated here, in concluding our review of the pioneer work of Haüy, that in the modern theory all consideration of the shape of the ultimate structural units is abandoned as unnecessary and misleading, and that each chemical molecule is considered to be represented by a point, which may be either its centre of gravity, a particular atom in the molecule (for we are now able in certain cases to locate the orientation of the spheres of influence of the elementary atoms in the chemical molecules), or a purely representative point standing for the molecule. The only condition is that the points chosen within the molecules shall be strictly analogous, and similarly orientated. The dots at the intersections of the lines in Figs. 13 and 14 are the representative points in question. We then deal with the distances between the points, the latter being regarded as molecular centres, rather than with the dimensions of the cells themselves regarded as solid entities. We thus avoid the as yet unsolved question of how much is matter and how much is interspace in the room between the molecular centres. In this form the theory is in conformity with all the advances of modern physics, as well as of chemistry. And with this reservation, and after modifying his theory to this extent, one cannot but be struck with the wonderful perspicacity of Haüy, for he appears to have observed and considered almost every problem with which the crystallographer is confronted, and his laws of symmetry and of rational indices are perfectly applicable to the theory as thus modernised.

CHAPTER IV
THE SEVEN STYLES OF CRYSTAL ARCHITECTURE.

It is truly curious how frequently the perfect number, seven, is endowed with exceptional importance with regard to natural phenomena. The seven orders of spectra, the seven notes of the musical octave, and the seven chemical elements, together with the seven vertical groups to which by their periodic repetition they give rise, of the “period” of Mendeléeff’s classification of the elements, will at once come to mind as cases in point. This proverbial importance of the number seven is once again illustrated in regard to the systems of symmetry or styles of architecture displayed by crystals. For there are seven such systems of crystal symmetry, each distinguished by its own specific elements of symmetry.

It is only within recent years that we have come to appreciate what are the real elements of symmetry. For although there are but seven systems, there are no less than thirty-two classes of crystals, and these were formerly grouped under six systems, on lines which have since proved to be purely arbitrary and not founded on any truly scientific basis. It was supposed that those classes in any system which did not exhibit all the faces possible to the system owed this lack of development to the suppression of one-half or three-quarters of the possible number, and such classes were consequently called “hemihedral” and “tetartohedral” respectively. As in the higher systems of symmetry there were usually two or more ways in which a particular proportionate suppression of faces could occur, it happened that several classes, and not merely three—holohedral (possessing the full number of faces), hemihedral, and tetartohedral—constituted each of these systems.

Thanks largely to the genius of Victor von Lang, who was formerly with us in England at the Mineral Department of the British Museum, and to his successor there, Nevil Story Maskelyne, we have at last a much more scientific basis for our classification of crystals, and one which is in complete harmony with the now perfected theory of possible homogeneous structures. Victor von Lang showed that the true elements of symmetry are planes of symmetry and axes of symmetry. A crystal possessing a plane of symmetry is symmetrical on both sides of that plane, both as regards the number of the faces and their precise angular disposition with respect to one another.

It is quite possible, and even the usual case, that the relative development of the faces, that is their actual sizes, may prevent the symmetry from being at first apparent; but when we come to measure the angles between the faces, by use of the reflecting goniometer, and to plot their positions out on the surface of a sphere, or on a plane representation of the latter on paper, the exceedingly useful “stereographic projection,” we at once perceive the symmetry perfectly plainly.

Fig. 15.—Crystal of Potassium Nickel Sulphate.

Fig. 16.—Projection of Potassium Nickel Sulphate and its Isomorphous Analogues.

Thus in Fig. 15 is represented a crystal of the salt potassium nickel sulphate, K₂Ni(SO₄)₂.6H₂O, belonging to the monoclinic system of symmetry, and which, therefore, possesses only one plane of symmetry. In Fig. 16 its stereographic projection is shown, in which each face in one of the symmetrical halves is represented by a dot, the plane of symmetry, parallel to the face b, being the plane of the paper, so that each dot not on the circumference really represents two symmetrical faces, one above and one below the paper, while the circumferential dots represent faces perpendicular to the symmetry plane and paper. The mode of arriving at such a useful projection, or plan of the faces, will be discussed more fully later in Chapter VI. But for the present purpose it will be sufficient to note that the right and left halves of the crystal shown in Fig. 15 are obviously symmetrical to each other, and that the plan of either half, projected on the dividing plane of symmetry itself, may be taken as given in Fig. 16; that is, we may imagine the crystal shown in Fig. 15 to be equally divided by a section plane which is vertical and perpendicular to the paper when the latter is held up behind the crystal and in front of the eye, this section plane being the plane of symmetry and parallel to the face b = (010). It may thus be imagined as the plane of projection of Fig. 16.

An axis of symmetry is a direction in the crystal such that when the latter is rotated for an angle of 60°, 90°, 120°, or 180° around it, the crystal is brought to look exactly as it did before such rotation. When a rotation for 180° is necessary in order to reproduce the original appearance, the axis is called a “digonal” axis of symmetry, for two such rotations then complete the circle and bring the crystal back to identity, not merely to similarity. When the rotation into a position of similarity is for 120°, three such rotations are required to restore identity, and the axis is then termed a “trigonal” one. Similarly, four rotations to positions of similarity 90° apart are essential to complete the restoration to identity, and the axis is then a “tetragonal” one, each rotation of a right angle causing the crystal to appear as at first, assuming, as in all cases, the ideal equality of development of faces. Lastly, if 60° of rotation bring about similarity, six such rotations are required in order to effect identity of position, and the axis is known as a “hexagonal” one.

Now, there is one system of symmetry which is characterised by the presence of a single hexagonal axis of symmetry, and this is the hexagonal system. A crystal of this system, one of the naturally occurring mineral apatite, which has been actually measured by the author, is shown in Fig. 17. There is another system, the chief property of which is to possess a tetragonal axis of symmetry, and which is therefore termed the tetragonal system. A tetragonal crystal of anatase, titanium dioxide, TiO₂, which has likewise been measured on the goniometer by the author, is shown in Fig. 18. And there is yet another system, the trigonal, the chief attribute of which is the possession of a single trigonal axis of symmetry, and which is consequently named the trigonal system. In Fig. 19 is shown a crystal of calcite, within which the directions of the three rhombohedral crystallographic axes of the trigonal system, and that of the vertical trigonal axis of symmetry, are indicated in broken-and-dotted lines.

Fig. 17.—Measured Crystal of Apatite.

Fig. 18.—Measured Crystal of Anatase.

But there is one system of symmetry, the highest possible, and which has already been referred to as the cubic system, which combines in itself all but one (the hexagonal axis) of the elements of symmetry. Indeed, not only does it possess a tetragonal, a trigonal, and a digonal axis of symmetry, but also ten other symmetry axes; for these three automatically involve altogether the presence of no less than three tetragonal, four trigonal, and six digonal axes of symmetry, together with nine planes of symmetry, twenty-two elements of symmetry being thus present in all.

The perfections of the cube, the simple lines of which are illustrated in Fig. 20, as the expression of the highest kind of symmetry, with angles all right angles and sides and edges all equal, were so fully appreciated by the geometrical minds of the ancient Greek philosophers, imbued with the innate love of symmetry characteristic of their nation, that to them the cube became the emblem of perfection. We are reminded of this interesting fact in the Book of Revelation, which, in describing in its inimitable language the wonders of the Holy City, speaks of it as “lying foursquare,” and attributes to it the properties of the cube, that “The length and the breadth and the height of it are equal.”

Fig. 19.—Crystal of Calcite.

Fig. 20.—The Cube.

Fig. 21.—The Hexakis Octahedron.

The full symmetry of the cubic system is not realised, however, by a study of the cube alone; we only appreciate it when we come to examine the general form of the cubic system, that which is produced by starting with a face oblique to all three axes, and with different amounts of obliquity to each, and seeing how many repetitions of the face the symmetry demands. The presence of such a face involves as a matter of fact, when all the elements of symmetry are satisfied, the presence also of no less than forty-seven others, symmetrically situated, the forty-eight-sided figure produced being the hexakis octahedron shown in Fig. 21, and which is occasionally actually found developed in nature as the diamond. All diamonds do not by any means exhibit this form so wonderfully rich in faces, but diamonds are from time to time found which do show all the forty-eight faces well developed.

Fig. 22.—Measured Crystal of Topaz.

Besides these four more highly symmetrical systems or styles of crystal architecture, a fifth, the monoclinic system, characterised by a single plane of symmetry and one axis of digonal symmetry perpendicular thereto, has already been alluded to, and a typical crystal illustrated in Fig. 15. A sixth, the rhombic system, perhaps in some ways the most interesting of all, and certainly so optically, possesses three rectangular axes of symmetry, identical in direction with the crystallographic axes, and three mutually rectangular planes of symmetry, coincident with the axial planes and intersecting each other in the axes. The lengths of the three crystal axes are unequal, however, and herein lies the essential difference from the cube. A very typical rhombic substance is topaz, a crystal of which, about three millimetres in diameter, is shown very much enlarged in Fig. 22. Every face on this crystal has been actually investigated on the goniometer, and the interfacial angles measured.

Fig. 23.—Measured Crystal of Copper Sulphate.

Lastly, there is the seventh, the triclinic system, in which there are neither planes nor axes of symmetry, but, even in its holohedral class, only symmetry about the centre, each face having a parallel fellow. Sulphate of copper, blue vitriol, CuSO₄.5H₂O, shows this type of symmetry, or rather lack of it, very characteristically, and a crystal of this beautiful deep blue salt, measured by the author, is represented in Fig. 23.

Hence, we have arrived logically at seven systems of symmetry or styles of crystal architecture, distinguished by the nature of their essential axes of symmetry, and the planes of symmetry which may accompany them. Now the full degree of symmetry of each system may be reduced to a certain minimum without lowering the system, and in all the systems but the triclinic there are several definite stages of reduction before the minimum is reached, each stage corresponding to one of the thirty-two classes of crystals. Thus in the cubic system there are four classes besides the holohedral, in the tetragonal six, in the hexagonal four, in the trigonal six, in the rhombic and monoclinic two each, and in the triclinic one.

PLATE IV.
Fig. 24.—Octahedra of Potassium Cadmium Cyanide.

Fig. 25.—Octahedra of Cæsium Alum.
Cubic Crystals growing from Solution.

We have thus attained at length to a truly scientific classification of crystal forms, by using axes and planes of symmetry as criteria. There is no occasion whatever to imagine suppression of faces in the classes of lower than the holohedral or highest symmetry of any system. In these classes it is simply the fact that less than the full number of elements of symmetry possible to the system are present and characterise the class, which still conforms, however, to the minimum symmetry absolutely essential to the system.

The drawings of crystals of the seven systems in the foregoing illustrations will have given a correct idea of the nature of the symmetry in each case. But now it may be much more interesting to present a series of reproductions of photographs of some actual crystals of the different systems. Such a series is given in Figs. 24 to 33, Plates IV. to VIII. They were taken with the aid of the microscope, the substances being crystallised from a slightly supersaturated solution in each case, on a microscope slip. A ring of gold size was first laid on the slip, and allowed to dry for several days. The drop of solution, in the metastable supersaturated condition (corresponding to the region of solubility which lies between the solubility and supersolubility curves, Fig. 98, page [240]), was placed in the middle of the ring, and crystallisation just allowed to start, either owing to evaporation and consequent production of the labile condition for spontaneous crystallisation, or by access of a germ crystal from the air. It was then covered with a cover-glass, which had the desired effect of enclosing the solution in a parallelsided cell, a film of the thickness of thick paper, suitable for undistorted microscopic observation and photomicrography, and also the effect of arresting evaporation and therefore the rapidity of the growth of the crystals, so that a photomicrograph taken with the minimum necessary exposure was quite sharp.

The crystals shown in the accompanying photographic reproductions, Figs. 24 to 33 (Plates IV. to VIII.), as well as Fig. 4 (Plate II.), already described, were thus photographed in the very act of slow growth, employing a one-inch objective very much stopped down. Such photographs are infinitely sharper and more beautifully and delicately shaded than those taken of dry crystals.

Fig. 24, Plate IV., represents cubic octahedra of the double cyanide of potassium and cadmium, 2KCN.Cd(CN)₂, a salt which crystallises out in relatively large and wonderfully transparent and well-formed single octahedra on a micro-slip, and is particularly suitable for demonstrating the character of this highest system, the cubic, of crystal symmetry. Special development of the pair of faces of the octahedron parallel to the glass surfaces has occurred, owing to greater freedom of growth at the boundaries of these faces, as is usual in such circumstances of deposition, but the other pairs of faces are quite large enough to show their nature clearly.

Fig. 25, on the same Plate IV., shows a slide of cæsium alum, Cs₂SO₄.Al₂(SO₄)₃.24H₂O, in which the octahedra are smaller, and some of them, notably one in the centre of the field, are perfectly proportioned.

PLATE V.
Fig. 26.—Octahedra of Ammonium Iron Alum crystallising on a Hair.

Fig. 27.—Tetragonal Crystals of Potassium Ferrocyanide.
Crystals growing from Solution.

PLATE VI.
Fig. 28.--Rhombic Crystals of Potassium Hydrogen Tartrate.

Fig. 29.--Rhombic Crystals of Ammonium Magnesium Phosphate, showing Special Growth along Line of Scratch.
Rhombic Crystals growing by Slow Precipitation.

Fig. 26, Plate V., represents octahedra of ammonium iron alum (formula like that of cæsium alum, but with NH₄ replacing Cs and Fe replacing Al) crystallising on a hair. It illustrates the interesting manner in which crystallisation will sometimes occur, under conditions of quietude, when some object or other on which the crystals can readily deposit themselves is present or introduced, such as a silk or cotton thread, or a hair as in this case.

Fig. 27, on the same Plate V., represents tetragonal crystals of potassium ferrocyanide, K₄Fe(CN)₆, composed of tabular crystals parallel to the basal pinakoid, bounded by faces of one order, first or second, of tetragonal prism, the corners being modified at 45° by smaller faces of the other order of tetragonal prism.

Fig. 28, Plate VI., is a photograph of large rhombic crystals of hydrogen potassium tartrate, HKC₄H₄O₆, obtained by addition of tartaric acid to a dilute solution of potassium chloride. They are rectangular rhombic prisms capped by pyramidal forms, and also modified by other prismatic and domal forms.

Fig. 29, also on Plate VI., represents another rhombic substance, ammonium magnesium phosphate, NH₄MgPO₄.6H₂O, obtained by very slow precipitation of a dilute solution of magnesium sulphate containing ammonium chloride and ammonia with hydrogen disodium phosphate. It illustrates in an interesting manner how, when a saturated solution is kept quiet, and then the surface of the vessel containing it is scratched by a needle point, a line of small crystals at once starts forming along the line of scratch, even although the latter has made no actual impression on the glass itself. Such a line of crystals will be observed running across the middle of the slide.

Fig. 30, Plate VII., shows a monoclinic substance, ammonium magnesium sulphate (NH₄)₂Mg(SO₄)₂.6H₂O, which crystallises out splendidly on a micro-slip. The field includes several very well-formed typical crystals of the salt, which is one of the same exceedingly important isomorphous series to which potassium nickel sulphate, Fig. 15, belongs; it is obtained by mixing solutions containing molecularly equivalent quantities of ammonium and magnesium sulphates. The primary monoclinic prism is the chief form, terminated by clinodome faces and smaller strip-faces of the basal plane, the latter, however, being occasionally the chief end form. Small pyramid faces are also seen here and there modifying the solid angles.

Another beautifully crystallising monoclinic substance is shown in the next slide, Fig. 31, on the same Plate VII., namely, potassium sodium carbonate, KNaCO₃.6H₂O, obtained from a solution of molecular proportions of potassium and sodium carbonates. Numerous forms of the monoclinic system are developed, on relatively large and perfectly transparent and delicately shaded individuals.

A triclinic substance is represented in Fig. 32, Plate VIII., potassium ferricyanide, K₆F₂(CN)₁₂. The triply oblique nature of the symmetry is clearly exhibited by this salt, the absence of any right angles being very marked.

PLATE VII.
Fig. 30.—Monoclinic Crystals of Ammonium Magnesium Sulphate.

Fig. 31.—Monoclinic Crystals of Sodium Potassium Carbonate.
Monoclinic Crystals growing from Solution.

PLATE VIII.
Fig. 32.—Triclinic Crystals of Potassium Ferricyanide.

Fig. 33.—Tetrahedral Crystals of Sodium Sulphantimoniate, Cubic Class 28.
Crystals growing from Solution.

Fig. 33, also on Plate VIII., illustrates more particularly a class of one of the systems, the cubic, which is of lower than holohedral (full) systematic symmetry. This is the case also with hydrogen potassium tartrate and ammonium magnesium phosphate, but the forms shown of those salts on the slides represented in Figs. 28 and 29 are chiefly those which are also common to the holohedral classes of their respective systems, and the lower class symmetry is not emphasised. But here in Fig. 33, representing Schlippe’s salt, sodium sulphantimoniate, Na₃SbS₄.9H₂O, we have very clear development of the tetrahedron, belonging to the lowest of the five classes (class 28) of the cubic system. The crystals are almost all combinations of two complementary tetrahedra, one of which is developed so very much more than the other that the faces of the latter only appear as minute replacements at the corners of the predominating tetrahedron.

This is the last for the present of these fascinating growths of crystals under the microscope, but three more will be given subsequently, in Figs. 99 and 100, on Plate XXI., and Fig. 101, Plate XI., to illustrate crystallisation from metastable and labile solutions.

Fig. 34, Plate IX., represents another kind of phenomenon, equally instructive. It shows a field in a crystal of quartz, as seen under the same power of the microscope, a one-inch objective with small stop and an ordinary low power eyepiece. Just above and to the left of the centre of the field is a cavity, the shape of which is remarkable, for it is that of a quartz crystal, a hexagonal prism terminated by rhombohedral faces. The cavity is filled with a saturated solution of salt, except for a bubble of water vapour, and a beautiful little cube of sodium chloride which has crystallised out from the solution. This slide, therefore, gives us an example of a natural cubic crystal, and also an indication of the shape of quartz crystals, the cavity itself being a kind of negative quartz crystal. The crystal in which it occurs must have been formed very deep down in a reservoir of molten material beneath a volcano, under the great pressure of superincumbent rock masses. It was probably one of the quartz crystals of a granite rock which had crystallised under these conditions. Almost every crystal of quartz found in such granite rocks displays thousands of small cavities filled with liquid and a bubble, although it is very rare to find one with so good a cube of salt and having the configuration of a quartz crystal for the shape of the cavity. Many such cavities, however, contain as the liquid compressed carbonic acid, the very fact of the carbonic acid being in the liquefied state affording ample evidence of the pressure under which the crystal was formed. The proof that the liquid is carbonic acid in these cases is afforded by the fact that when the crystal is warmed to 32°C., the critical temperature of carbon dioxide, under which it can no longer remain liquid, but must become a gas, the bubble disappears and the cavity becomes filled with gas. Carbonic acid cavities are readily recognised, inasmuch as the bubble is extremely mobile, and is normally in a state of movement on the very slightest provocation.

PLATE IX.
Fig. 34.—Liquid Cavities in quartz Crystal (Trigonal) containing Saturated Solution and Cubic Crystals of Sodium Chloride.

Fig. 36.—Two characteristic Forms of Snow Crystals (Trigonal).

Fig. 35.—Negative Ice Crystals, or “Water Flowers,” in Ice.

The liquid cavity in the remarkable quartz crystal illustrated in Fig. 34, and the bubble of vapour formed on cooling, and consequent contraction of the liquid more than the solid quartz (the thermal dilatation of liquids being usually greater than that of solids) when it was no longer able to fill the cavity, remind one of the beautiful water flowers formed for the contrary reason in ice on passing a beam of light through a slab, owing to the warming effect of the accompanying heat rays. Water crystallises like quartz, in the trigonal system, its normal forms being the hexagonal prism and the rhombohedron. A slab of lake ice is generally a huge crystal plate perpendicular to the trigonal axis, or in the case of disturbed growth an interlacing mass of such crystals, all perpendicular to the optic axis, the axis of the hexagonal prism and of trigonal symmetry. When the heat rays from the lantern pass through such a slab of ice, the surface of which is focussed on the screen by a projecting lens, they cause the ice to begin to melt in numerous spots in the interior of the slab simultaneously; and the structure of the crystal is revealed by the operation occurring with production of cavities taking the shape of hexagonal stars, which when focussed appear on the screen as shown in Fig. 35. They are filled with water except for a bubble (vacuole), which contains only water vapour. For the liquid water occupies less room than did the ice from which it was produced, owing to the well-known fact that water expands on freezing. This abnormal expansion with cooling begins at the temperature of the maximum density of water, 4° C., and proceeds steadily until the freezing point 0° is reached, when, at the moment of crystallisation, the mass suddenly increases in volume by as much as 10 per cent. This expansive leap when the molecules of water marshal themselves into the organised order of the homogeneous structure, that of the space-lattice of the trigonal (rhombohedral) system, is one of the most remarkable phenomena in nature, and its exceptional character, so contrary to the usual contraction on solidification of a liquid, is of vital moment to aquatic life. For the layer of ice formed, being lighter than water, floats on the surface of the latter, and thus forms a protective layer and prevents to a large extent further freezing, except as a slow thickening of the layer, the total freezing of the water of a lake or river being rendered practically impossible, an obvious provision for the security of life of the piscatorial and other inhabitants of the waters.

Hence, as the molecules of the substance H₂O are one by one detached from their solid assemblage as ice, and become more loosely associated as the less voluminous liquid water, they cannot occupy the whole of the cavity formed in the solid ice, and a small vacuous space, occupied only by water vapour at its ordinary low tension corresponding to the low temperature, is formed and appears as the bubble. Moreover, the cavity itself takes the shape of a hexagonal star-shaped flower, the bubble showing at its centre, the cavity being thus a kind of negative ice crystal, like the negative quartz crystal shown in Fig. 34. Apparently in the production of these cavities, just as in the production of the well-known etched figures on crystal faces by the application of a minute quantity of a solvent for the crystal substance, the crystal edifice is taken down, molecule by molecule, in a regular manner, resulting in the formation of a cavity showing the symmetry of the space-lattice which is present in the crystal structure.

PLATE X.
Fig. 37.—Piz Palü and Snow-field of the Pers Glacier, from the Diavolezza Pass, Upper Engadine.
(From a Photograph by the author.)

The water flowers of Fig. 35 remind one very much of snow crystals, two of which, re-engraved from the wonderfully careful drawings of the late Mr Glaisher, are represented in Fig. 36, Plate IX. They all exhibit the symmetry of the hexagonal prism, which is equally a form of the trigonal system as it is of the hexagonal system. The snow crystals, being formed from water vapour condensed in the cold upper layers of the atmosphere, appear more or less as skeleton crystals, owing to the rarity of the semi-gaseous material condensed, compared with the extent of the space in which the crystallisation occurs. Indeed the exquisite tracery of these snow crystals appears to afford a visual proof of the existence of the trigonal-hexagonal space-lattice as the framework of the crystal structure of ice. When one considers the countless numbers of such beautiful gems of nature’s handiwork massed together on an extensive snow-field of the higher Alps—such as that of the Piz Palü in the Upper Engadine, shown in Fig. 37, Plate X., as seen from the Diavolezza Pass—produced in the pure air of the higher regions of the atmosphere, and frequently seen by the early morning climber lying uninjured in all their beauty on the surface of the snow-field, one is lost in amazement at the prodigality displayed in the broadcast distribution of such peerless gems.

CHAPTER V
HOW CRYSTALS ARE DESCRIBED. THE SIMPLE LAW LIMITING THE NUMBER OF POSSIBLE FORMS.

The most wonderful of all the laws relating to crystals is the one already briefly referred to which limits and regulates the possible positions of faces, within the lines of symmetry which have been indicated in the last chapter. Having laid down the rules of symmetry, it might be thought that any planes which obey these laws, as regards their mode of repetition about the planes and axes of symmetry, would be possible. But as a matter of fact this is not so, only a very few planes inclined at certain definite angles, repeated in accordance with the symmetry, being ever found actually developed. The reason for this is of far-reaching importance, for it reveals to us the certainty that a crystal is a homogeneous structure composed of definite structural units of tangible size, probably the chemical molecules, built up on the plan of one of the fourteen space-lattices made known to us by Bravais, and to be referred to more fully in Chapter VIII. In order to render this fundamental law comprehensible, it will be essential to explain in a few simple words how the crystallographer identifies and labels the numerous faces on a crystal, in short, how he describes a crystal, in a manner which shall be understood immediately by everybody who has studied the very simple rules of the convention.

It is a matter of common knowledge that the mathematical geometrician defines the position of any point in space with reference to three planes, which in the simplest case are all mutually at right angles to each other like the faces of a cube, and which intersect in three rectangular axes a, b, c, the third c being the vertical axis, b the lateral one, and a the front-and-back axis. The distances of the point from the three reference planes, as measured by the lengths of the three lines drawn from the point to the planes parallel to the three axes of intersection, at once gives him what he calls the “co-ordinates” of the point, which absolutely define its position. In the same way we can imagine three axes drawn within the crystal, by which not only the position of any point on any face of the crystal may be located, but which may be used more simply still to fix the position of the face itself. The directions chosen as those of the three axes are the edges of intersection of three of the best developed faces.

If there are three such faces inclined at right angles they would be chosen in preference to all others, as they would certainly prove to be faces of prime significance as regards the symmetry of the crystal. If there are no such rectangularly inclined faces developed on the crystal, then the three best developed faces nearest to 90° to each other are chosen, the two factors of nearness to rectangularity and excellence of development being simultaneously borne in mind in making the choice of axial planes, and discretion used.

Fig. 38.—The Cube and its Three Equal Rectangular Axes.

Fig. 39.—Tetragonal Prism and its three Rectangular Axes.

If the crystal belong to the cubic, tetragonal, or rhombic systems, for instance, three faces meeting each other rectangularly are possible planes on the crystal, and will very frequently be found actually developed; such would obviously be chosen as the axial planes. The edges of the cube, or of the tetragonal or rectangular rhombic prism, will be the directions of the crystallographic axes in this case, and we can imagine them moved parallel to themselves until the common centre of intersection, the “origin” of the analytical geometrician, will occupy the centre of the crystal, and the faces of the latter be built up symmetrically about it. When the crystal is cubic, the three axes will be of equal length as shown in Fig. 38; if tetragonal, the two horizontal axes will be equal, but will differ in length from the vertical axis, as represented in Fig. 39. If the crystal be rhombic, all three axes will be of different lengths, as indicated in Fig. 40, which represents the axes and axial planes of an actual rhombic substance, topaz, for which the lateral axis b and vertical axis c are nearly but not quite equal, while the front-and-back axis a is very different.

When the crystal is of monoclinic symmetry, as in Fig. 41, three axes will similarly be found as the intersection of three principal parallel pairs of faces, but two of them will be inclined at an angle other than 90° to each other, while the third, the lateral one in Fig. 41, will be at right angles to those first two and to the plane containing them; moreover, all three are unequal in length. In the case of a triclinic crystal, shown in Fig. 42, however, there can be no right angles, and the intersections of three important faces meeting each other at angles as near 90° as possible are chosen as the axes, regard being had to both factors of approximation to rectangularity and importance of development. These triclinic axes are the most general type of crystal axes, for not only are the angles not right angles, but the lengths of the axes are also unequal.

Fig. 40.—Axial Planes of a Rhombic Crystal.

Fig. 41.—Axial Planes of a Monoclinic Crystal.

Fig. 42.—Axial Planes of a Triclinic Crystal.

Fig. 43.—Hexagonal Prism of the First Order and its Four Axes.

Fig. 44.—Hexagonal Prism of the Second Order.

Fig. 45.—The Rhombohedron and its Three Equal Axes.

The cases of the hexagonal and trigonal systems are somewhat special. The hexagonal has four such axes, as represented in Fig. 43, the lines of intersection of the faces of the hexagonal prism closed by a pair of perpendicular terminal planes, namely, one vertical axis parallel to the vertical edges, and three horizontal axes inclined at 120° to each other, and parallel to the pair of basal plane faces, equal to each other in length, but different from the length of the vertical axis. The hexagonal axial-plane prism shown in Fig. 43 is known as one of the first order. The hexagonal prism corresponding to the tetragonal one of Fig. 39, in which the axes emerge in the centres of the faces, is said to be of the second order, and is shown in Fig. 44. The trigonal system of crystals is best described with reference to three equal but not rectangular axes, parallel to the faces of the rhombohedron, one of the principal forms of the system, so well seen in Iceland spar, and illustrated in Fig. 45. The rhombohedron may be regarded as a cube resting on one of its corners (solid angles), with the diagonal line joining this to the opposite corner vertical, and the cube then deformed by flattening or elongating it along the direction of this diagonal. The edges meeting at the ends of this vertical diagonal are then the directions of the three trigonal crystallographic axes.

In this last illustration the vertical direction of the altered diagonal is that of the trigonal axis of symmetry, for the rhombohedron is brought into apparent coincidence with itself again if rotated for 120° round this direction. But although a symmetry axis, this is not a crystallographic axis of reference. It is not shown in Fig. 45, therefore, but is given in Fig. 19. On the other hand, the singular vertical axis of reference of the tetragonal and hexagonal systems is identical with the tetragonal or hexagonal axis of symmetry of these systems, and the three crystallographic axes of reference of the cube are identical with the three tetragonal axes of symmetry of the cubic system. In the rhombic system also, the three rectangular axes of reference are identical with the three digonal axes of symmetry, and in the monoclinic system the one axis of reference which is normal to the plane of the two inclined axes is the unique digonal axis of symmetry of that system.

Having thus evolved a scientific scheme of reference axes for the faces of a crystal, it is necessary in all the systems other than the cubic and trigonal, in which the axes are of equal lengths, to devise a mode of arriving at the relative lengths of the axes; for on this depends the mode of determining the positions of the various faces, other than the three parallel pairs (or four in the case of the hexagonal system) chosen as the axial planes. This is very simply done by choosing a fourth important face inclined to all three axes, when one of this character is developed, as very frequently happens, as the determinative face or plane fixing the unit lengths of the axes. When no such face is present on the crystal, two others can usually be found, each of which is inclined to two different axes, so that between them all three axial lengths are determined. The faces of the octahedron, of the primary tetragonal pyramid and the primary rhombic pyramid, and of the corresponding forms of the other systems, are such determinative planes, fixing the lengths of the axes. This fact will be clear from the typical illustration of the most general of these primary or “parametral” forms, the triclinic equivalent of the octahedron, given in Fig. 46, the faces being obviously obtained by joining the points marking unit lengths of the three axes.

Fig. 46.—Triclinic Equivalent of the Octahedron.

Having thus settled the directions of the crystallographic axes and their lengths, it is the next step which reveals the remarkable law to which reference was made at the opening of this chapter. For we find that all other faces on the crystal, however complicated and rich in faces it may be, cut off lengths from the axes which are represented by low whole numbers, that is, either 2, 3, 4, or possibly 5, and very rarely more than 6 unit lengths. By far the greater number of faces do not cut off more than three unit lengths from any axis. Prof. Miller of Cambridge, in the year 1839, gave us a most valuable mode of labelling and distinguishing the various faces by a symbol involving these three values, employed, however, not directly but in an indirect yet very simple manner. If m, n, r be the three numbers expressing the intercepts cut off by a face on the three axes, a, b, c respectively, and if the Millerian index numbers be represented by h, k, l, then—

Each figure or “index” of the Millerian symbol is thus inversely proportional to the length of the intercept on the axis concerned. The intercepts themselves are used as symbols in another mode of labelling crystal faces, suggested by Weiss, but this method proves too cumbersome in practice.

The Millerian symbol of a face is always placed within ordinary curved brackets (  ), but if the symbol is to stand for the whole set of faces composing the form, the brackets are of the type {  }. Thus the Millerian symbol of the fourth face (that in the top-right front octant), determinative of the unit axial lengths, is (111), as shown in Fig. 46, the face in question being marked with this symbol; while the symbol {111} indicates the set of faces of the whole or such part of the double pyramid as composes the unit form. In the triclinic system this form only consists of the face (111) and the parallel one (̄1̄1̄1), but in the case of the regular octahedron of the cubic system it embraces all the eight faces. The triclinic octahedron, Fig. 46, is thus made up of four forms of two faces each. A negative sign over an index indicates interception on the axis a behind the centre, on the axis b to the left of the centre, or on the vertical axis c below the centre.

To take an actual example, suppose a face other than the primary one to make the intercepts on the axes 4, 2, 1; in this case h = a/4, k = b/2, and l = c/1, that is, when referred to the fundamental primary form for which a, b, c are each unity, h = ¼, k = ½, l = 1, or, bringing them to whole numbers by multiplying by 4, h = 1, k = 2, c = 4, and the symbol in Millerian notation is (124). Again, suppose we wish to find the intercepts on the three cubic axes made by the face (321) of the hexakis octahedron shown in Fig. 21. To get each intercept we multiply together the two other Millerian indices, and if necessary afterwards reduce the three figures obtained to their simplest relative values. For the face (321) we obtain 2, 3, 6. This means that the face (321) in the top-right-front octant of the hexakis octahedron cuts off two unit lengths of axis a, three unit lengths of axis b, and six unit lengths of axis c. No fractional parts thus ever enter into the relations of the axial lengths intercepted by any face on a crystal, and the whole numbers representing these relations are always small, the number 6 being the usual limit.

This important law is known as the “Law of Rational Indices,” and is the corner-stone of crystallography. A forecast of it was given in Chapter III., in describing how it was first discovered by Haüy, and it was shown how impressed Haüy was with its obvious significance as an indication of the brick-like nature of the crystal structure. What the “bricks” were, Haüy was not in a position to ascertain with certainty, as chemistry was in its infancy, and Dalton’s atomic theory had not then been proposed.

That Haüy had a shrewd idea, however, that the structural units were the chemical molecules, and that while the main lines of symmetry were determined by the arrangement of the molecules its details were settled by the arrangement of the atoms in the molecules, is clear to any one who reads his 1784 “Essai” and 1801 “Traité,” and interprets his molécules intégrantes and élémentaires in the light of our knowledge of to-day.

Before we pass on, however, to consider the modern development of the real meaning of the law of rational indices, as revealed by recent work on the internal structure of crystals, it will be well to consider first, in the next chapter, a few more essential facts as to crystal symmetry, and the current mode of constructing a comprehensive, yet simple, plan of the faces present on a crystal.

CHAPTER VI
THE DISTRIBUTION OF CRYSTAL FACES IN ZONES, AND THE MODE OF CONSTRUCTING A PLAN OF THE FACES.

It will have been clear from the facts related in the previous chapters that the salient property possessed by all crystals, when ideal development is permitted by the circumstances of their growth, and the substance is not one of unusual softness or liable to ready distortion, is that the exterior form consists of and is defined by truly plane faces inclined to each other at angles which are specific and characteristic for each definite chemical substance; and that these angles are in accordance with the symmetry of some particular one of the thirty-two classes of crystals, and are such as cause the indices of the faces concerned to be rational small numbers.

It will also be clear that, given the presence of any face other than the three axial planes, the symmetry of the class—supposing the crystal to exhibit some development of symmetry and not to belong to class 32, the general case possessing no symmetry—will require the repetition of this face a definite number of times on other parts of the crystal. Such a set of faces possessing the same symmetry value we have already learnt to call a “Form,” and the faces composing it will have the same Millerian index numbers in their symbols, but differently arranged and with negative signs over those which relate to the interception of the back part of the a axis, the left part of the b axis, or the lower part of the vertical c axis; that is, parts to the front and right, and above, the centre of intersection of the three crystal axes are considered as the positive parts of those axes.

A form, if of general character, that is, if composed of faces each of which is inclined to all three axes, will comprise more faces the higher the symmetry. Thus, in the cubic system, the form shown in Fig. 21, the hexakis octahedron, comprises as many as forty-eight faces, all covered by the form symbol {321}; while in the rhombic system the highest number of faces in a form is eight, in the monoclinic only four, and in the triclinic system two. It will also have become clear that the law of rational indices limits the number of forms possible of any one type. For instance, very few hexakis octahedra are known, the most frequently occurring ones besides {321} being {421}, {531}, and {543}. Forms, of any class, possessing higher indices than these are very rare, especially in the systems of lower symmetry.

Fig. 47.—The Spherical Projection.

We next come to a further very interesting fact about crystals. Let us imagine a crystal, on which the faces are fairly evenly developed, to be placed in the middle of a sphere of jelly, as indicated in Fig. 47 (reproduced from a Memoir by the late Prof. Penfield), so that the centre or origin of the axial system of the crystal and the centre of the sphere coincide. Let us now further imagine that long needles are stuck through the jelly and the crystal, one perpendicular to each crystal face, and so as to reach the centre. The crystal represented in Fig. 47 is a combination of the cube a, octahedron o, and rhombic dodecahedron d. If such a thing as we have imagined were possible, we should find that the needles would emerge at the surface of the sphere in points which would lie on great circles, that is, on circles which represent the intersection of the sphere by planes passing through the centre. Moreover, the points would be distributed along these circles at regularly recurring angular positions, corresponding to the symmetry of the crystal. If the crystal belonged to one of the higher systems of symmetry, it would happen that four of the points on at least one of these great circles, and possibly on three of them, would be 90° apart, that is, would be at the ends of rectangular diameters, which would most likely be the axes of reference. The other points would be distributed symmetrically on each side of these four points.

The great circles on which the points are thus symmetrically distributed—and they may legitimately be taken to represent the faces, for tangent planes to the sphere at these points would be parallel to the faces—are known as “zone circles,” and the faces represented by the points on any one of them form a “zone.” Now a zone of faces has this practical property, that when the crystal is supported so as to be rotatable about the zone axis—which is parallel to the edges of intersection of all the faces composing the zone, and is the normal to the plane of the great circle representing the zone—and a telescope is directed towards the crystal perpendicularly to the zone axis, while a bright object such as an illuminated slit is arranged conveniently so as to be reflected from any face of the crystal into the telescope, an image of it being thus visible in the latter, then it will be found that on rotating the crystal a similar image will be seen reflected in the telescope from every face of the zone in turn. Moreover, when the crystal is mounted on a graduated circle, the angle of rotation between the positions of adjustment to the cross-wires of the telescope of any two successive images, reflected from adjacent faces of the crystal, is actually the angle between the two points representing the faces concerned on the zone circle, and is the supplement of the internal dihedral angle between the two crystal faces themselves. It is, in fact, the angle between the normals (perpendiculars) to the two faces, the angle which is measured on the goniometer.

This is, indeed, the very simple principle of the reflecting goniometer, invented by Wollaston in the year 1809, and which in its modern improved form is the all-important principal instrument of the crystallographer’s laboratory. The work with it consists largely in the measurement of the angles between the faces in all the principal zones developed on the crystal. The very fact, however, that crystal faces occur so absolutely accurately in zones immeasurably lightens the labours of the crystallographer, and is one of prime importance.

Fig. 48.—The Reflecting Goniometer.

The most accurate and convenient modern form of reflecting goniometer, reading to half-minutes of arc, and provided with a delicate adjusting apparatus for the crystal, is shown in Fig. 48. It is constructed by Fuess of Berlin.

The graduated circle a is horizontal and is divided directly to 15′, the verniers enabling the readings to be carried further either to single minutes, which is all that is usually necessary, or to half-minutes in the cases of very perfect crystals. The divided circle is rotated by means of the ring b situated below, and the reading of the verniers is accomplished with the aid of the microscopes c. The circle which carries the verniers is not fixed, except when this is done deliberately by means of the clamping screw d, but rotates with the telescope e to which it is rigidly attached by means of an arm and a column f. A fine adjustment is provided with the clamping arrangement, so that the telescope can be adjusted delicately with respect to the divided circle. Both telescope and collimator are rigidly fixed at about 120° from each other during the actual measurements. The collimator g is carried on a column h definitely fixed to one of the legs (the back one in Fig. 48) of the main basal tripod of the instrument. The signal slit of the collimator is carried at the focus of the objective about the middle of the tube g, the outer half of the latter being an illumination tube carrying a condensing lens to concentrate the rays of light from the goniometer lamp on the slit. The latter is not of the usual rectilinear character, but composed of two circular-arc jaws, so that, while narrow in the middle part like an ordinary spectroscope slit, it is much broader at the two ends in order to be much more readily visible; the central part is narrow in order to enable fine adjustment to the vertical cross-wire of the telescope to be readily and accurately carried out. The shape of this signal-slit will be gathered from the images of the slit shown in Fig. 61 (page [126]) in Chapter X. The telescope carries an additional lens k at its inner, objective, end, in order that when this lens is rotated into position the telescope may be converted into a low power microscope for viewing the crystal and thus enabling its adjustment to be readily carried out.

The crystal l is mounted on a little cone of goniometer wax (a mixture of pitch and beeswax) carried by the crystal holder. The latter fits in the top of the adjusting movements, which consist of a pair of rectangularly arranged centring motions, and a pair of cylindrical adjusting movements; the milled-headed manipulating centring screws of the former are indicated by the letters m and n in Fig. 48, and those which move the adjusting segments are marked o and p. The top screw fixes the crystal holder.

The crystal on its adjusting apparatus can be raised or lowered to the proper height, level with the axes of the telescope and collimator, by means of a milled head at the base of the instrument, there being an inner crystal axis moving (vertically only) independently of the circle. Moreover, a second axis outside this enables the crystal to be rotated independently of the circle, the conical axis of which is outside this again. The two can be locked together when desired, however, by a clamping screw provided with a fine adjustment q. Freedom of movement of the crystal axis, unimpeded by the weight of the circle, is thus permitted for all adjusting purposes, the circle being only brought into play when measurement is actually to occur. With this instrument the most accurate work can be readily carried out, and for ease of manipulation and general convenience it is the best goniometer yet constructed.

The idea of regarding the centre of the crystal as the centre of a sphere, within which the crystal is placed (Fig. 47, page [62]), gives crystallographers a very convenient method of graphically representing a crystal on paper, by projecting the sphere on to the flat surface of the paper, the eye being supposed to be placed at either the north or south pole of the sphere, and the plane of projection to be that of the equatorial great circle. The faces in the upper hemisphere are represented by dots which are technically known as the “poles” of the faces, corresponding to the points where the needles normal to the faces emerge from the imaginary globe, and all these points or poles lie on a few arcs of great circles, which appear in the projection either also as circular arcs terminating at diametrically opposite points on the circumference of the equatorial circle, which forms the outer boundary of the figure and is termed the “primitive circle,” or else, when the planes of the great circles are at right angles to the equatorial primitive circle, they appear as diametral straight lines passing through the centre of the primitive circle.

Such a stereographic projection offers a comprehensive plan of the whole of the crystal faces, which at once informs us of the symmetry in all cases other than very complicated ones. A typical one, that of the rhombic crystal of topaz shown in Fig. 22 (page [40]), is given in Fig. 49.

It will happen in all cases of higher symmetry, as in that of topaz, for instance, that the poles in the lower hemisphere will project into the same points as those representing the faces in the upper hemisphere; but in cases of lower symmetry, where they are differently situated, they are usually represented by miniature rings instead of dots. From the interfacial angles measured on the goniometer the relative lengths and angular inclinations (if other than 90°) of the crystal axes can readily be calculated, by means of the simple formulæ of spherical trigonometry; and the stereographic projection constructed from the measurements as just described proves an inestimable aid to these calculations, by affording a comprehensive diagram of all the spherical triangles required in making the calculations.

Fig. 49.—Stereographic Projection of Topaz.

The relative axial lengths a : b : c (in which b is always arranged to be = 1), and the axial angles α (between b and c), β (between a and c), and γ (between a and b), form the “elements” of a crystal. These, together with a list of the “forms” observed, and a table of the interfacial angles, define the morphology of the crystal, and are included in every satisfactory description of a crystallographic investigation. They are preceded by a statement of the name and chemical composition and formula of the substance, the system and the class of symmetry, and the habit or various habits developed by crystals from a considerable number of crops. An example of the mode of setting out such a description will be found on pages [157] to 160.

Having thus made ourselves acquainted with the real nature of the distribution of faces on a crystal, and learnt how the crystallographer measures the angles between the faces by means of the reflecting goniometer, plots them out graphically on a stereographic projection, and calculates therefrom the “elements” of the crystal, it will be convenient again to take up the historical development of the subject so far as it relates to crystal forms and angles, and their bearing on the chemical composition of the substance composing the crystal, by introducing the reader to the great work of Mitscherlich, whose influence in the domain of chemical crystallography was as profound as that of Haüy proved to be as regards structural crystallography.

CHAPTER VII
THE WORK OF EILHARDT MITSCHERLICH AND HIS DISCOVERY OF ISOMORPHISM.

During the height of the French Revolution, which caused the work of the Abbe Haüy to be suspended for a time, although he was fortunately not one of the many scientific victims of that terrible period, there was born, on the 7th of January 1794, in the village of Neuende, near Jever, in Oldenburg, the man who was destined to continue that work on its chemical side. Eilhardt Mitscherlich was the son of the village pastor, and nephew of the celebrated philologer, Prof. Mitscherlich of Göttingen. His uncle’s influence appears to have given young Mitscherlich a leaning towards philological studies, for during his later terms at the Gymnasium at Jever, where he received his early education, he devoted himself with great energy to the study of history and languages, for which he had a marked talent, under the able direction and kind solicitude of the head of the Gymnasium at that time, the historian Schlosser. He eventually specialised on the Persian language, and when Schlosser was promoted to Frankfort young Mitscherlich accompanied him, and there prosecuted these favourite studies until the year 1811, when he went to the university of Heidelberg.

For some time now he had cherished the hope of proceeding to Persia and conducting philological investigations on the spot, and in 1813, an opportunity presenting itself in the prospect of an embassy being despatched to Persia by Napoleon, he transferred himself to the university of Paris, with the object of obtaining permission from Napoleon to accompany the embassy. This visit to Paris must have been one of Mitscherlich’s most exciting and interesting experiences. For Napoleon had just returned from the disastrous Russian campaign of 1812, and was feverishly engaged in raising a new army wherewith to stem the great rise of the people which was now re-awakening patriotic spirit throughout the whole of Germany, and which threatened to sweep away, as it eventually did, the huge fabric of his central European Empire.

Indeed Mitscherlich appears to have been detained in Paris during the exciting years 1813 and 1814, and with the abdication of Napoleon on April 4th of that year, he was obliged to give up all idea of proceeding to Persia. He decided that the only way of accomplishing his purpose was to attempt to travel thither as a doctor of medicine. He therefore returned to his native Germany during the summer of 1814, and proceeded to Göttingen, which was then famous for its medical school. Here he worked hard at the preliminary science subjects necessary for the medical degree, while still continuing his philology to such serious purpose as to enable him to publish, in 1815, the first volume of a history of the Ghurides and Kara-Chitayens, entitled “Mirchondi historia Thaheridarum.” It is obvious from the sequel, however, that he very soon began to take much more than a merely passing interest in his scientific studies, and he eventually became so fascinated by them, and particularly chemistry, as to abandon altogether his cherished idea of a visit to Persia. Europe was now settling down after the stormy period of the hundred days which succeeded Napoleon’s escape from Elba, terminating in his final overthrow on June 18th, 1815, at Waterloo, and Mitscherlich was able to devote himself to the uninterrupted prosecution of the scientific work now opening before him. He had the inestimable advantage of bringing to it a culture and a literary mind of quite an unusually broad and original character; and if the fall of Napoleon brought with it the loss to the world of an accomplished philologist, it brought also an ample compensation in conferring upon it one of the most erudite and broad-minded of scientists.

In 1818 Mitscherlich went to Berlin, and worked hard at chemistry in the university laboratory under Link. It was about the close of this year or the beginning of 1819 that he commenced his first research, and it proved to be one which will ever be memorable in the annals both of chemistry and of crystallography. He had undertaken the investigation of the phosphates and arsenates, and his results confirmed the conclusions which had just been published by Berzelius, then the greatest chemist of the day, namely, that the anhydrides of phosphoric and arsenic acids each contain five equivalents of oxygen, while those of the lower phosphorous and arsenious acids contain only three. But while making preparations of the salts of these acids, which they form when combined with potash and ammonia, he observed a fact which had escaped Berzelius, namely, that the phosphates and arsenates of potassium and ammonium crystallise in similar forms, the crystals being so like each other, in fact, as to be indistinguishable on a merely cursory inspection.

Being unacquainted with crystallography, and perceiving the importance of the subject to the chemist, he acted in a very practical and sensible manner, which it is more than singular has not been universally imitated by all chemists since his time. He at once commenced the study of crystallography, seeing the impossibility of further real progress without a working knowledge of that subject. He was fortunate in finding in Gustav Rose, the Professor of Geology and Mineralogy at Berlin, not merely a teacher close at hand, but also eventually a life-long intimate friend. Mitscherlich worked so hard under Rose that he was very soon able to carry out the necessary crystal measurements with his newly prepared phosphates and arsenates. He first established the complete morphological similarity of the acid phosphates and arsenates of ammonium, those which have the composition NH₄H₂PO₄ and NH₄H₂AsO₄ and crystallise in primary tetragonal prisms terminated by the primary pyramid faces; and then he endeavoured to produce other salts of ammonia with other acids which should likewise give crystals of similar form. But he found this to be impossible, and that only the phosphates and arsenates of ammonia exhibited the same crystalline forms, composed of faces inclined at similar angles, which to Mitscherlich at this time appeared to be identical. He next tried the effect of combining phosphoric and arsenic acids with other bases, and he found that potassium gave salts which crystallised apparently exactly like the ammonium salts.

He then discovered that not only do the acid phosphates and arsenates of potassium and ammonium, H₂KPO₄, H₂(NH₄)PO₄, H₂KAsO₄, and H₂(NH₄)AsO₄ crystallise in similar tetragonal forms, but also that the four neutral di-metallic salts of the type HK₂PO₄ crystallise similarly to each other.

He came, therefore, to the conclusion that there do exist bodies of dissimilar chemical composition having the same crystalline form, but that these substances are of similar constitution, in which one element, or group of elements, may be exchanged for another which appears to act analogously, such as arsenic for phosphorus and the ammonium group (although its true nature was not then determined) for potassium. He observed that certain minerals also appeared to conform to this rule, such as the rhombohedral carbonates of the alkaline earths, calcite CaCO₃, dolomite CaMg(CO₃)₂, chalybite FeCO₃, and dialogite MnCO₃; and the orthorhombic sulphates of barium (barytes, BaSO₄), strontium (celestite, SrSO₄), and lead (anglesite, PbSO₄). Wollaston, who, in the year 1809, had invented the reflecting goniometer, and thereby placed a much more powerful weapon of research in the hands of crystallographers, had already, in 1812, shown this to be a fact as regards the orthorhombic carbonates (witherite, strontianite, and cerussite) and sulphates (barytes, celestite, and anglesite) of barium, strontium, and lead, as the result of the first exact angular measurements made with his new instrument; but his observations had been almost ignored until Mitscherlich reinstated them by his confirmatory results.

While working under the direction of Rose, Mitscherlich had become acquainted with the work of Haüy, whose ideas were being very much discussed about this time, Haüy himself taking a very strong part in the discussion, being particularly firm on the principle that every substance of definite chemical composition is characterised by its own specific crystalline form. Such a principle appeared to be flatly contradicted by these first surprising results of Mitscherlich, and it naturally appeared desirable to the latter largely to extend his observations to other salts of different groups. It was for this reason that he had examined the orthorhombic sulphates of barium, strontium, and lead, and the rhombohedral carbonates of calcium, magnesium, iron, and manganese, with the result already stated that the members of each of these groups of salts were found to exhibit the same crystalline form, a fact as regards the former group of sulphates which had already been pointed out not only by Wollaston but by von Fuchs (who appears to have ignored the work of Wollaston) in 1815, but had been explained by him in a totally unsatisfactory manner. Moreover, about the same time the vitriols, the sulphates of zinc, iron, and copper, had been investigated by Beudant, who had shown that under certain conditions mixed crystals of these salts could be obtained; but Beudant omitted to analyse his salts, and thus missed discovering the all-important fact that the vitriols contain water of crystallisation, and in different amounts under normal conditions. Green vitriol, the sulphate of ferrous iron, crystallises usually with seven molecules of water of crystallisation, as does also white vitriol, zinc sulphate; but blue vitriol, copper sulphate, crystallises with only five molecules of water under ordinary atmospheric conditions of temperature and pressure. Moreover, copper sulphate forms crystals which belong to the triclinic system, while the sulphates of zinc and iron are dimorphous, the common form of zinc sulphate, ZnSO₄.7H₂O, being rhombic, like Epsom salts, the sulphate of magnesia which also crystallises with seven molecules of water, MgSO₄.7H₂O, while that of ferrous sulphate, FeSO₄.7H₂O, is monoclinic, facts which still further complicate the crystallography of this group and which were quite unknown to Beudant and were unobserved by him. But Beudant showed that the addition of fifteen per cent. of ferrous sulphate to zinc sulphate, or nine per cent. to copper sulphate, caused either zinc or copper sulphate to crystallise in the same monoclinic form as ferrous sulphate. He also showed that all three vitriols will crystallise in mixed crystals with magnesium or nickel sulphates, the ordinary form of the latter salt, NiSO₄.7H₂O, being rhombic like that of Epsom salts.

The idea that two chemically distinct substances not crystallising in the cubic system, where the high symmetry determines identity of form, can occur in crystals of the same form, was most determinedly combated by Haüy, and the lack of chemical analyses in Beudant’s work, and the altogether incorrect “vicarious” explanation given by von Fuchs, gave Haüy very grave cause for suspicion of the new ideas. The previous observations of Rome de l’Isle in 1772, Le Blanc in 1784, Vauquelin in 1797, and of Gay-Lussac in 1816, that the various alums, potash alum, ammonia alum, and iron alum, will grow together in mixed crystals or in overgrowths of one crystal on another, when a crystal of any one of them is hung up in the solution of any other, does not affect the question, as the alums crystallise in the cubic system, the angles of the highly symmetric forms of which are absolutely identical by virtue of the symmetry itself.

It was while this interesting discussion was proceeding that Mitscherlich was at work in Berlin, extending his first researches on the phosphates and arsenates to the mineral sulphates and carbonates. But he recognised, even thus early, what has since become very clear, namely, that owing to the possibility of the enclosure of impurities and of admixture with isomorphous analogues, minerals are not so suitable for investigation in this regard as the crystals of artificially prepared chemical salts. For the latter can be prepared in the laboratory in a state of definitely ascertained purity, and there is no chance of that happening which Haüy was inclined to think was the explanation of Mitscherlich’s results, namely, that certain salts have such an immense power of crystallisation that a small proportion of them in a solution of another salt may coerce the latter into crystallisation in the form of that more powerfully crystallising salt. Mitscherlich made a special study, therefore, of the work of Beudant, and repeated the latter observer’s experiments, bringing to the research both his crystallographic experience and that of a skilful analyst. He prepared the pure sulphates of ferrous iron, copper, zinc, magnesium, nickel and cobalt, all of which form excellent crystals. He soon cleared up the mystery in which Beudant’s work had left the subject, by showing that the crystals contained water of crystallisation, and in different amounts. He found what has since been abundantly verified, that the sulphates of copper and manganese crystallise in the triclinic system with five molecules of water, CuSO₄.5H₂O and MnSO₄.5H₂O; in the case of manganese sulphate, however, this is only true when the temperature is between 7° and 20°, for if lower than 7° rhombic crystals of MnSO₄.7H₂O similar to those of the magnesium sulphate group are deposited, and if higher than 20° the crystals are tetragonal and possess the composition MnSO₄.4H₂O. The Epsom salts group crystallising in the rhombic system with seven molecules of water consists of magnesium sulphate itself, MgSO₄.7H₂O, zinc sulphate ZnSO₄.7H₂O, and nickel sulphate NiSO₄.7H₂O. The third group of Mitscherlich consists of sulphate of ferrous iron FeSO₄.7H₂O and cobalt sulphate CoSO₄.7H₂O, and both crystallise at ordinary temperatures with seven molecules of water as indicated by the formulæ, but in the monoclinic system. Thus two of the groups contain the same number of molecules of water, yet crystallise differently. But Mitscherlich next noticed a very singular fact, namely, that if a crystal of a member of either of these two groups be dropped into a saturated solution of a salt of the other group, this latter salt will crystallise out in the form of the group to which the stranger crystal belongs. Hence he concluded that both groups are capable of crystallising in two different systems, rhombic and monoclinic, and that under the ordinary circumstances of temperature and pressure three of the salts form most readily the rhombic crystals, while the other two take up most easily the monoclinic form. Mitscherlich then mixed the solutions of the different salts, and found that the mixed crystals obtained presented the form of some one of the salts employed. Thus even so early in his work Mitscherlich indicated the possibility of dimorphism. Moreover, before the close of the year 1819 he had satisfied himself that aragonite is a second distinct form of carbonate of lime, crystallising in the rhombic system and quite different from the ordinary rhombohedral form calcite. Hence this was another undoubted case of dimorphism.

During this same investigation in 1819, Mitscherlich studied the effect produced by mixing the solution of each one of the above-mentioned seven sulphates of dyad-acting metals with the solution of sulphate of potash, and made the very important discovery that a double salt of definite composition was produced, containing one equivalent of potassium sulphate, one equivalent of the dyad sulphate (that of magnesium, zinc, iron, manganese, nickel, cobalt, or copper), and six equivalents of water of crystallisation, and that they all crystallised well in similar forms belonging to the monoclinic system. Some typical crystals of one of these salts, ammonium magnesium sulphate, are illustrated in Fig. 30 (Plate VII., facing page [44]). This is probably the most important series of double salts known to us, and is the series which has formed the subject of prolonged investigation on the part of the author, no less than thirty-four different members of the series having been studied crystallographically and physically since the year 1893, and many other members still remain to be studied. An account of this work is given in a Monograph published in the year 1910 by Messrs Macmillan & Co., and entitled, “Crystalline Structure and Chemical Constitution.”

This remarkable record for a first research was presented by Mitscherlich to the Berlin Academy on the 9th December 1819. During the summer of the same year Berzelius visited Berlin, and was so struck with the abilities of Mitscherlich, then twenty-five years old, that he persuaded him to accompany him on his return to Stockholm, and Mitscherlich continued his investigations there under the eye of the great chemist. His first work at Stockholm consisted of a more complete study of the acid and neutral phosphates and arsenates of potash, soda, ammonia, and lead. He showed that in every case an arsenate crystallises in the same form as the corresponding phosphate. Moreover, in 1821 he demonstrated that sodium dihydrogen phosphate, NaH₂PO₄, crystallises with a molecule of water of crystallisation in two different forms, both belonging to the rhombic system but with quite different axial ratios; this was consequently a similar occurrence to that which he had observed with the sulphates of the iron and zinc groups.

It was while Mitscherlich was in Stockholm that Berzelius suggested to him that a name should be given to the new discovery that analogous elements can replace each other in their crystallised compounds without any apparent change of crystalline form. Mitscherlich, therefore, termed the phenomenon “isomorphism,” from ἰσός, equal to, and μορφή, shape. The term “isomorphous” thus strictly means “equal shaped,” implying not only similarity in the faces displayed, but also absolute equality of the crystal angles. The fact that the crystals of isomorphous substances are not absolutely identical in form, but only very similar, was not likely to be appreciated by Mitscherlich at this time. For the reflecting goniometer had only been invented by Wollaston in 1809, and accurate instruments reading to minutes of arc were mechanical rarities. It will be shown in the sequel, as the result of the author’s investigations, that there are angular differences, none the less real because relatively very small, between the members of such series. But Mitscherlich was not in the position to observe them. It must be remembered, moreover, that he was primarily a chemist, and that he had only acquired sufficient crystallographic knowledge to enable him to detect the system of symmetry, and the principal forms (groups of faces having equal value as regards the symmetry) developed on the crystals which he prepared. His doctrine of isomorphism, accepted in this broad sense, proved of immediate and important use in chemistry. For there were uncertainties as to the equivalents of some of the chemical elements, as tabulated by Berzelius, then the greatest authority on the subject, and these were at once cleared up by the application of the principle of isomorphism.

The essence of Mitscherlich’s discovery was, that the chemical nature of the elements present in a compound influences the crystalline form by determining the number and the arrangement of the atoms in the molecule of the compound; so that elements having similar properties, such for instance as barium, strontium, and calcium, or phosphorus and arsenic, combine with other elements to form similarly constituted compounds, both as regards number of atoms and their arrangement in the molecule. Number of atoms alone, however, is no criterion, for the five atoms of the ammonium group NH₄ replace the one atom of potassium without change of form.

This case of the base ammonia had been one of Mitscherlich’s greatest difficulties during the earlier part of his work, and remained a complete puzzle until about this time, when its true chemical character was revealed. For until the year 1820 Berzelius believed that it contained oxygen. Seebeck and Berzelius had independently discovered ammonium amalgam in 1808, and Davy found, on repeating the experiment, that a piece of sal-ammoniac moistened with water produced the amalgam with mercury just as well as strong aqueous ammonia. Both Berzelius and Davy came to the conclusion that ammonia contains oxygen, like potash and soda, and that a metallic kind of substance resembling the alkali metals, potassium and sodium, was isolated from this oxide or hydrate by the action of the electric current, which Seebeck had shown facilitated the formation of the so-called ammonium amalgam. Davy, however, accepted in part the views of Gay-Lussac and Thénard, who, in 1809, concluded from their experiments that ammonium consisted of ammonia gas NH₃ with an additional atom of hydrogen, the group NH₄ then acting like an alkali metal, views which time has substantiated. But their further erroneous conclusion that sodium and potassium also contained hydrogen was rejected by him. Berzelius, however, set his face both against this latter fallacy and the really correct NH₄ theory, and it was not until four years after Ampère, in 1816, had shown that sal-ammoniac was, in fact, the compound of the group NH₄ with chlorine, that Berzelius, about the year 1820, after thoroughly sifting the work of Ampère, accepted the view of the latter that in the ammonium salts it is the group NH₄, acting as a radicle capable of replacing the alkali metals, which is present.

The fact that this occurred at this precise moment, four years after the publication of Ampère’s results, leads to the conclusion that the observation of Mitscherlich, that the ammonium compounds are isomorphous with the potassium compounds, was the compelling argument which caused Berzelius finally to admit what has since proved to be the truth.

While still at Stockholm Mitscherlich showed that the chromates and manganates are isomorphous with the sulphates, and also that the perchlorates and permanganates are isomorphous with each other. Although these facts could not be properly explained at the time, owing to the inadequate progress of the chemistry of manganese, it was seen that potassium chromate, K₂CrO₄, contained the same number of atoms as potassium sulphate, K₂SO₄, and that potassium permanganate KMnO₄ and perchlorate KClO₄ likewise resembled each other in regard to the number of atoms contained in the molecule.

As a good instance of the use of the principle of isomorphism, we may recall that when Marignac, in 1864, found himself in great difficulty about the atomic weights of the little known metals tantalum and niobium which he was investigating, he discovered that their compounds are isomorphous; the pentoxides of the two metals occur together in isomorphous mixture in several minerals, and the double fluorides with potassium fluoride, K₂TaF₇ and K₂NbF₇ are readily obtained in crystals of the same form. The specific heat of tantalum was then unknown, so that the law of Dulong and Petit connecting specific heat with atomic weight could not be applied, and the vapour density of tantalum chloride, as first determined by Deville and Troost with impure material, did not indicate an atomic weight for tantalum which would give it the position among the elements that the chemical reactions of the metal indicated. Yet Marignac was able definitely to decide, some time before the final vapour density determinations of Deville and Troost with pure salts, from the fact of the isomorphism of their compounds, that the only possible positions for tantalum and niobium were such as corresponded with the atomic weights 180 and 93 respectively. Time has only confirmed this decision, and we now know that niobium and tantalum belong to the same family group of elements as that to which vanadium belongs, and the only difference which modern research has introduced has been to correct the decimal places of the atomic weights, that of niobium (now also called columbium, the name given to it by its discoverer, Hatchett, in 1801) being now accepted as 92.8 and that of tantalum 179.6, when that of hydrogen = 1.

Applying the law of isomorphism in a similar manner, Berzelius was enabled to fix the atomic weights of copper, cadmium, zinc, nickel, cobalt, iron, manganese, chromium, sulphur, selenium, and chlorine, the numbers accepted to-day differing only in the decimal places, in accordance with the more accurate results acquired by the advance of experimental and quantitative analytical methods. But with regard to several other elements, owing to inadequate data, Berzelius made serious mistakes, showing how very great is the necessity for care and for ample experimental data and accurate measurements, before the principle of isomorphism can be applied with safety. Given these, and we have one of the most valuable of all the aids known to us in choosing the correct atomic weight of an element from among two or three possible alternatives. We are only on absolutely sure ground when we are dealing not only with a series of compounds consisting of the same number of atoms, but when also the interchangeable elements are the intimately related members of a family group, such as we have since become familiar with in the vertical groups of elements in the periodic table of Mendeléeff.

Before leaving Stockholm Mitscherlich showed, from experiments on the crystallisation of mixtures of the different sulphates with which he had been working, that isomorphous substances intermix in crystals in all proportions, and that they also replace one another in minerals in indefinite proportions, a fact which has of recent years been wonderfully exemplified in the cases of the hornblende (amphibole) and augite (pyroxene) groups.

In November 1821 Mitscherlich closed these memorable labours at Stockholm and returned to Berlin, where he acted as extraordinary professor of the university until 1825, when he was elected professor in ordinary. His investigations for a time were largely connected with minerals, but on July 6th, 1826, he presented a further most important crystallographic paper to the Berlin Academy, in which he announced his discovery of the fact that one of the best known chemical elements, sulphur, is capable of crystallising in two distinct forms. The ordinary crystals found about Etna and Vesuvius and in other volcanic regions agree with those deposited from solution in carbon bisulphide in exhibiting rhombic symmetry. But Mitscherlich found that when sulphur is fused and allowed to cool until partially solidified, and the still liquid portion is then poured out of the crucible, the walls of the latter are found to be lined with long monoclinic prisms. These have already been illustrated in Fig. 2, Plate I., in Chapter I.

Here was a perfectly clear case of an element—not liable to any charge of difference of chemical composition such as might have applied to the cases of sodium dihydrogen phosphate, carbonate of lime, and iron vitriol and its analogues, which he had previously described as cases of the same substance crystallising in two different forms—which could be made to crystallise in two different systems of symmetry at will, by merely changing the circumstances under which the crystallisation occurred. His explanation being thus proved absolutely, he no longer hesitated, but at once applied the term “dimorphous” to these substances exhibiting two different forms, and referred to the phenomenon itself as “dimorphism.” The case of carbonate of lime had given rise to prolonged discussion, for the second variety, the rhombic aragonite, had been erroneously explained by Stromeyer, after Mitscherlich’s first announcement in 1819, as being due to its containing strontia as well as lime, and the controversy raged until Buchholz discovered a specimen of aragonite which was absolutely pure calcium carbonate, so that Mitscherlich’s dimorphous explanation was fully substantiated.

Dimorphism is very beautifully illustrated by the case of the trioxide of antimony, Sb₂O₃, a slide of which, obtained by sublimation of the oxide from a heated tube on to the cool surface of a glass microscope slip, is seen reproduced in Fig. 50, Plate XI. The two forms are respectively rhombic and cubic. The rhombic variety usually takes the form of long needle-shaped crystals, which are shown in Fig. 50 radiating across the field and interlacing with one another; the cubic variety crystallises in octahedra, of which several are shown in the illustration, perched on the needles, one interesting individual being poised on the end of one of the needles. The two forms occur also in nature as the rhombic mineral valentinite and the cubic mineral senarmontite, which latter crystallises in excellent regular octahedra. Antimonious oxide, moreover, is not only isomorphous, but isodimorphous with arsenious oxide, a slide of octahedra of which has already been reproduced in Fig. 3, Plate I., in Chapter I. For besides this common octahedral form of As₂O₃ artificial crystals of arsenious oxide have been prepared of rhombic symmetry, resembling valentinite. Hence the two lower oxides of arsenic and antimony afford us a striking case of the simultaneous display of Mitscherlich’s two principles of isomorphism and dimorphism.

Thus the position in 1826 was that Mitscherlich had discovered the principle of isomorphism, and had also shown the occurrence of dimorphism in several well-proved specific cases, and that he regarded at this time isomorphism as being a literal reality, absolute identity of form.

PLATE XI.
Fig. 50.—Rhombic Needles and Cubic Octahedra of Antimony Trioxide obtained by Sublimation. An interesting Example of Dimorphism.

Fig. 101.—Ammonium Chloride crystallising· from a Labile Supersaturated Solution (see p. [248]).
Reproductions of Photomicrographs.

These striking results appeared at once to demolish the theory that any one substance of definite chemical composition is characterised by a specific crystalline form, which was Haüy’s most important generalisation. Mitscherlich, however, soon expressed doubts as to the absolute identity of form of his isomorphous crystals, and saw that it was quite possible that in the systems other than the cubic (in which latter system the highly perfect symmetry itself determines the form, and that the angles shall be identically constant), there might be slight distinctive differences in the crystal angles. For he caused to be constructed, by the celebrated optician and mechanician, Pistor, the most accurate goniometer which had up till then been seen, provided with four verniers, each reading to ten seconds of arc, and with a telescope magnifying twenty times, for viewing the reflections of a signal, carried by a collimator, from the crystal faces. Unfortunately in one respect, he was almost at once diverted, by the very excess of refinement of this instrument, to the question of the alteration of the crystal angles by change of temperature, and lost the opportunity, never to recur, of doing that which would at once have reconciled his views with those of Haüy in regard to this important matter, namely, the determination of these small but real differences in the crystal angles of the different members of isomorphous series, and the discovery of the interesting law which governs them, a task which in these later days has fallen to the lot of the author.

Another remarkable piece of crystallographic work, this time in the optical domain, which has rendered the name of Mitscherlich familiar, was his discovery of the phenomenon of crossed-axial-plane dispersion of the optic axes in gypsum. (The nature and meaning of “optic axes” will be explained in Chapter XIII., page [185].) During the course of a lecture to the Berlin Academy in the year 1826 Mitscherlich, always a brilliant lecturer and experimenter at the lecture table, exhibited an experiment with a crystal of gypsum (selenite) which has ever since been referred to as the “Mitscherlich experiment.” He had been investigating the double refraction of a number of crystalline substances at different temperatures, and had observed that gypsum, hydrated calcium sulphate, CaSO₄.2H₂O, was highly sensitive in this respect, especially as regards the position of its optic axes. At the ordinary temperature it is biaxial, with an optic axial angle of about 60°, but on heating the crystal the angle diminishes, until just above the temperature of boiling water the axes become identical, as if the crystal were uniaxial, and then they again separate as the temperature rises further, but in the plane at right angles to that which formerly contained them; hence the phenomenon is spoken of as “crossed-axial-plane dispersion.” Mitscherlich employed a plate of the crystal cut perpendicularly to the bisectrix of the optic axial angle, and showed to the Academy the interference figures (see Plate XII.) which it afforded in convergent polarised light with rising temperature. At first, for the ordinary temperature, the usual rings and lemniscates surrounding the two optic axes were apparent at the right and left margins of the field; as the crystal was gently heated (its supporting metallic frame being heated with a spirit lamp) the axes approached each other, with ever changing play of colour and alteration of shape of the rings and lemniscates, until eventually the dark hyperbolic brushes, marking by their well defined vertices the positions of the two optic axes within the innermost rings, united in the centre of the field to produce the uniaxial dark rectangular cross; the rings around the centre had now become circles, the lemniscates having first become ellipses which more and more approximated, as the temperature rose, to circles. Then the dark cross opened out again, and the axial brushes separated once more, but in the vertical direction, and the circles became again first ellipses and then lemniscates, eventually developing inner rings around the optic axes; if the source of heat were not removed at this stage the crystal would suddenly decompose, becoming dehydrated, and the field on the screen would become dark. If, however, the spirit lamp were removed before this occurred, the phenomena were repeated in the reverse order as the crystal cooled.

This beautiful experiment is now frequently performed, as gypsum is perhaps the best example yet known which exhibits the phenomenon of crossed-axial-plane dispersion by change of temperature alone. A considerable number of other cases are known, such as brookite, the rhombic form of titanium dioxide TiO₂, and the triple tartrate of potassium, sodium, and ammonium, but these are more sensitive to change of wave-length in the illuminating light than to change of temperature.

Fig. 51.—The Mitscherlich Experiment with Gypsum.

PLATE XII.
Fig. 52.—Appearance of the Interference Figure half a Minute after commencing the Experiment. Temperature of Crystal about 40° C.

Fig. 53.—Appearance a Minute or so later, the Axes approaching the Centre. Temperature of Crystal about 85° C.

Fig. 54.—The Two Optic Axes coincident in the Centre of the Figure, two or three Minutes from the commencement. Temperature of Crystal 106° C.

Fig. 55.—The Axes re-separated in the Vertical Plane a Minute or two later. Temperature of Crystal about 125° C.
The Mitscherlich Experiment with Gypsum.
Four Stages in the Transformation of the Interference Figure in Convergent Polarised Light, from Horizontally Biaxial through Uniaxial to Vertically Biaxial, on Raising the Temperature To 125° C.
(From Photographs by the author.)

The author has recently exhibited the “Mitscherlich experiment” to the Royal Society,[^[2]] and also in his Evening Discourse to the British Association at their 1909 meeting in Winnipeg, in a new and more elegant manner, employing the large Nicolprism projection polariscope shown in Fig. 51, and a special arrangement of lenses for the convergence of the light, which is so effective that no extraneous heating of the crystal is required. The convergence of the rays is so true on a single spot in the centre of the crystal plate about two millimetres diameter, that a crystal plate not exceeding 6 mm. is of adequate size, mounted in a miniature holder-frame of platinum or brass with an aperture not more than 3 mm; the thickness of the crystal should remain about 2 mm., in order that the rings round the axes may not be too large and diffuse, the crystal being endowed with very feeble double refraction, which is one of the causes of the phenomenon. Such a small crystal heats up so rapidly in the heat rays accompanying the converging light rays—even with the essential cold water cell two inches thick between the lantern condenser and the polarising Nicol, for the protection of the balsam of the latter—that any extraneous heating by a spirit or other lamp is entirely unnecessary. The moment the electric arc of the lantern is switched on, the optic axial rings appear at the right and left margins of the screen, when the crystal is properly adjusted and the arc correctly centred, and they march rapidly to the crossing point in the centre, where the dark hyperbolæ unite to produce the rectangular St Andrew’s cross, the rings, figure-eight curves, and other lemniscates passing through the most exquisite evolutions and colour changes all the time until they form the circular Newton’s rings, around the centre of the cross; after this the cross and circles again open out, but along the vertical diameter of the screen, into hyperbolæ and rings and loop-like lemniscates surrounding two axes once more. It is wise as soon as the separation in this plane is complete and the first or second separate rings have appeared round the axes, to arrest the heating by merely interposing intermittently a hand screen between the lantern and polariser, or by blowing a current of cool air past the crystal, which will cause the axes to recede again, and the phenomena to be reversed, the crossing point being repassed, and the axes brought into the original horizontal plane again. By manipulation of the screen, or air-current, the axes can thus be caused to approach or to recede from the centre at will, along either the horizontal or vertical diameter. Four characteristic stages of the experiment are shown in Figs. 52 to 55, Plate XII. Fig. 52 exhibits the appearance just after commencing the experiment, the optic axes being well in the field of view. Fig. 53 shows the axes horizontally approaching the centre. Fig. 54 shows the actual crossing, which occurs for different crystals at temperatures varying from 105°.5 to 111°.5 C.; and Fig. 55 represents the axes again separated, but vertically.

The experiment as thus performed is one of the most beautiful imaginable, and it can readily be understood how delighted were Mitscherlich’s audience on the occasion of its first performance by him. The author has since discovered no less than six other cases of substances which exhibit crossed-axial-plane dispersion of the optic axes, in the course of his investigations, one of which is illustrated in Plate XIII., facing page [108]; and, moreover, has arrived at a general explanation of the whole phenomenon, the main points of which are that such substances, besides showing very feeble double refraction (the two extreme of the three refractive indices being very close together), also exhibit very close approximation of the intermediate refractive index β to either the minimum index α or the maximum index γ. Also, change of temperature, or of wave-length, or most usually both, must so operate as to bring the two indices closest together into actual identity and then to pass beyond each other, these two indices thus exchanging positions, the extreme one becoming the intermediate index. In other words, the uniaxial cross and circular rings are produced owing to two of the three refractive indices (corresponding to the directions of the three rectangular axes of the ellipsoid which, in general, expresses the optical properties of a crystal) becoming equal at the particular temperature at which the phenomenon is observed to occur, and for light of the specific wave-length in question. The ellipsoid of general form which represents the optical properties of a biaxial crystal thus becomes converted into a rotation ellipsoid corresponding to a uniaxial crystal. Brookite and the triple tartrate are excellent examples of the predominance of the effect of change of wave-length, for the optic axes are separated in both cases widely in one plane for red light and almost equally widely in the perpendicular plane for blue light. The new cases observed by the author are sensitive both to change of wave-length and to change of temperature, and so fall midway between the cases just quoted and the case of gypsum. The cause of it, in four of these new instances, is a very interesting one, connected with the regular change of the refractive indices in accordance with the law of progression in an isomorphous series according to the atomic weight of the alkali metal present, which will be discussed in Chapter X.

A further most important discovery was made by Mitscherlich in the year 1827, which also profoundly concerns the work of the author, namely, that of selenic acid, H₂SeO₄, analogous to sulphuric acid, and of the large group of salts derived from it, the selenates, analogous to the sulphates. He showed first that potassium selenate, K₂SeO₄, is isomorphous with potassium sulphate, K₂SO₄, and subsequently that the selenates in general are isomorphous with the corresponding sulphates; consequently it followed that selenium is a member of the sulphur family of elements. This element selenium had only been discovered ten years previously by his friend Berzelius, and doubtless Mitscherlich had seen a great deal of the work in connection with it during the two years which he spent in the laboratory of Berzelius at Stockholm, and was deeply interested in it.

The discovery has proved a most fruitful one, for the selenates are beautifully crystalline salts, particularly suitable for crystallographic researches, and their detailed investigation has afforded a most valuable independent confirmation of the important results obtained for the sulphates.

Again in 1830 Mitscherlich, following up the preliminary work already referred to, definitely established another fact bearing on the same series, namely, the isomorphism of potassium manganate K₂MnO₄ with the sulphate and selenate of potash; moreover, on continuing his study of the manganese salts he further substantiated the isomorphism of the permanganates with the perchlorates, and isolated permanganic acid. This also proved a most important step forward, as these salts likewise afford admirable material for crystallographic investigation, and such an examination, carried out by Muthmann and Barker, has yielded most valuable results.

Much later in his career Mitscherlich also described the dimorphous iodide of mercury, HgI₂, one of the most remarkable and interesting salts known to us, the unstable yellow rhombic modification being converted into the more stable red tetragonal form by merely touching with a hard substance. Also we are indebted to him at the same later period for our knowledge of the crystalline forms of the elements phosphorus, iodine, and selenium, when crystallised from solution in bisulphide of carbon.

From the record of achievements which has now been given in this chapter it will be obvious how much chemical crystallography owes to Mitscherlich. The description of his work has taken us into almost every branch of the subject, morphological, optical, and thermal, and although it has consequently been necessary to refer to phenomena which have not yet been explained in this book, it has doubtless proved on the whole most advantageous thus to present the life work of this great master as a complete connected story.

CHAPTER VIII
MORPHOTROPY AS DISTINCT FROM ISOMORPHISM.

It has been shown in the last chapter how Mitscherlich discovered the principle of isomorphism, as applying to the cases of substances so closely related that their interchangeable chemical elements are members of the same family group; and also how the principle enabled him to determine the chemical constitution of two hitherto unknown acids which he isolated, selenic H₂SeO₄ and permanganic HMnO₄. For he observed that the selenates were isomorphous with the sulphates, and the permanganates with the perchlorates. It was further made clear that the principle as bequeathed to us by Mitscherlich was only defined in very general terms, and its details have only recently been precisely decided.

Before proceeding further (in Chapter X.) with the elucidation of the true nature of isomorphism, however, some important crystallographic relationships between substances less closely related than family analogues must be referred to, as the outcome of a series of investigations by von Groth, chiefly between the derivatives of the hydrocarbon benzene. Also, some suggestive results obtained by the author from an investigation of an organic homologous series, that is, one the members of which differ by the regular addition of a CH₃ group, may be briefly referred to.

The interval between the work of Mitscherlich and that of von Groth was one of doubt, discouragement, and somewhat of discredit for chemical crystallography. The chemists Laurent[^[3]] and Nicklès[^[4]] carried out during the years from 1842 to 1849 measurements of numerous organic substances and of some inorganic compounds, the former chiefly halogen or other derivatives of particular hydrocarbons or salts of homologous fatty acids. Laurent, for instance, found that naphthalene tetrachloride, C₁₀H₈.Cl₄, and chloronaphthalene tetrachloride, C₁₀H₇Cl.Cl₄, crystallise in different systems, the former in the monoclinic and the latter in the rhombic system. Yet the primary prism angles of the two are less than a degree different, namely, 109° 0′ and 109° 45′. Laurent named this kind of similarity “hemimorphism,” a most unfortunate term as it was already employed in crystallography in its other well-known geometrical significance, that is, to denote a crystal differently terminated at the two ends of an axis. Many other like similarities were discovered by Laurent, and he again coined an objectionable term, now discarded, to represent the cases of similarity extending over more than the same system, namely, “isomeromorphism.”

Nicklès observed similar facts in connection with the barium salts of the fatty acids, which crystallise in different systems with different amounts of water of crystallisation. But their prism angles are all within a couple of degrees of each other, varying from 98° to 100°. Thus the phenomenon of “isogonism,” a term much less objectionable than those invented by Laurent, appears to be a common observance not only for different kinds of derivatives of the same original hydrocarbon or other organic nucleus, but also for the case of homologous series. But Nicklès missed the real point by including salts with different amounts of water, which, it will be shown later, entirely upset the crystalline structure. When this is eliminated the resemblance between true similarly constituted homologues, differing by regular increments of CH₃, is very much closer than would appear from Nicklès’ results.

Unfortunately, some of the work of Laurent and Nicklès was not carried out with the care and accuracy which is indispensable for researches which are to retain permanent value, and critics were not slow to arise. Kopp,[^[5]] in 1849, unmercifully exposed these failings, so that the real kernel of the work, which was of considerable value, came into discredit.

Pasteur,[^[6]] however, in 1848, besides the important observations regarding enantiomorphism, to be described in Chapter XI., had noticed similar zonal likenesses between related tartrates, amounting only therefore to isogonism and not to isomorphism; for here again the system often differed, particularly when the members of a series compared differed in their water of crystallisation. Thus there was ample evidence of a really significant series of facts in the work of these authors, but they were not properly arranged and explained.

So high was the feeling against the whole subject carried, however, after Kopp’s memoir, that had it not been for the steadying influence of Rammelsberg and Marignac, who themselves carried out many crystallographic measurements as new substances continued to be discovered with great rapidity, the science would have suffered a serious set-back. Moreover, even Rammelsberg was led astray in the direction of the views of the chemists of the time, that isomorphism could be extended over the crystal system. Frankenheim, whose discovery of the space-lattice, to be referred to in the next chapter, will ever render his name famous, strongly opposed this view. Delafosse, on the other hand, recognised some truth in both views, and assumed that there were two kinds of isomorphism, that of Mitscherlich on the one hand, and the broader one of Laurent on the other hand, and that in the case of the latter kind the overstepping of the system is no bar.

Hjortdahl,[^[7]] in the year 1865, supported the views of Delafosse more or less, at any rate so far as to assume the possibility of the existence of partial isomorphism, that is, of isogonism. He was very definite, however, against accepting the proposition that any general law could be applied. He himself discovered a partial similarity of angles in several homologous series of organic compounds.

About this time Sella[^[8]] uttered a warning which is one worthy of being prominently posted in every research laboratory, namely, that It is unwise to make hasty generalisations from the results of a small number of observations. Were this principle more generally followed, much greater progress would in the end be achieved, and without the discouragement and discredit which inevitably follows the detection of errors due to lack of broad experimental foundation. It is certainly an incontrovertible fact that only such generalisations as find themselves in accordance with all new but well-verified experimental facts as they are revealed can stand the test of time and become accepted universally as true laws of nature. And it is unreasonable to expect any generalisation to be of such a character unless it is already based on so large a number of facts that there is little fear of other new ones upsetting them.

Some order was, however, introduced into this chaotic state of chemical crystallography in the year 1870 by P. von Groth.[^[9]] He investigated systematically the derivatives of the hydrocarbon benzene, C₆H₆, many of which are excellently crystallising solids suitable for goniometrical measurement. He showed that although the crystal system may be and often is altered, yet there is a striking similarity in the angles between the faces of certain zones, which for the purposes of comparison he arranged to be parallel to each other in his descriptions of the crystals, so that the relationship would then consist in an elongation or a shortening of this particular zone axis, which was usually a crystallographic axis. He recognised that this was a totally different phenomenon from isomorphism, and called it “morphotropy.” Although it may possibly be permissible from one point of view to regard isomorphism as a particular case of complete morphotropy along all zones, such a course is not advisable, as morphotropic similarities are frequently of a comparatively loose and often indeed of a somewhat vague character, while isomorphous relationships are governed by very precise laws.