Nothing in connection with the subject of crystallography is more surprising than the neglect and apathy with which it has been for long treated by the chemical world. That crystalline structure is intimately related to chemical constitution will have been made abundantly plain during the course of this book. Yet in spite of the great work of Mitscherlich, essentially a chemist, and of a large amount of striking work which has been steadily accumulating during the last thirty years, with results of vital importance to chemistry, it is only at the eleventh hour that chemists are really awakening to the vast significance which crystal structure has for them.

The explanation undoubtedly is, that the long interregnum of conflicting investigations, doubt, and controversy, which followed the work of Haüy and Mitscherlich, and preceded the beginning of really accurate and painstaking investigation of an organised and systematic character, had caused chemists to regard with more or less of indifference the work of the crystallographers. Added to this we must remember that the subject of crystallography has hitherto been taught, when taught at all, merely as an appanage of mineralogy, although the pure chemical substances which crystallise well infinitely outnumber the naturally occurring minerals, and the results afforded by them frequently possess a much greater value by reason of the purity of the substances and their more definite chemical constitution. Also the mathematical and geometrical side has usually been unduly emphasised, and carried on in lectures without any practical goniometrical work at all. Moreover, the current text-books have often proved forbiddingly full of calculations and formulæ, and of the obsolete and unenticing symbols of Naumann.

At last we have come to see that the subject is one of fascinating interest when its study is commenced in a practical manner from the beginning, armed from the very first lesson with the goniometer. The crystal itself is then our main and highly interesting study; its exterior form unravels itself in a most delightfully simple manner when we follow the arrangement of its faces in zones on the goniometer itself; and its symmetry becomes immediately patent to our eyes in all ordinary simple cases, when we construct for ourselves its plan in a stereographic projection, drawn at first in freehand while still at the goniometer. The calculations also become perfectly simple when we have learnt that only the simplest of the very easy formulæ of spherical trigonometry are required, and which a knowledge of only elementary plane trigonometry enables us to apply. Aided by a few very helpful rules, such as those of Napier for calculating right-angled spherical triangles, and the rule of the anharmonic ratio of four poles in a zone—which, when the positions of three crystal faces of the zone are known, at once enables us to calculate the situation of any fourth face of the zone—we have at once a stock-in-trade which carries us over all difficulties in the way of calculation, and relegates this side of the work to an altogether subordinate position, although accuracy in carrying it out is, of course, absolutely essential and even vital.

Especial interest has recently been attracted on the part of chemists to the bearing of crystallography on their science by a remarkable theory which has been advanced by Pope and Barlow, connecting the internal structure of crystals with chemical valency, the power which the atom of a chemical element possesses of combining with the atoms of other elements, and which is generally expressed by the number of atoms of a monad element, such as hydrogen or chlorine, with which it is capable of combining chemically. Thus the electro-positive metal potassium is said to have monad valency because it is capable of combining with one atom of the electro-negative element chlorine to form the salt potassium chloride, KCl; calcium possesses dyad valency because it can unite with two atoms of chlorine to form calcium chloride, CaCl₂; aluminium is triadic as it can combine with three atoms of chlorine producing aluminium chloride, AlCl₃; carbon is a tetrad because it can take up four chlorine atoms, forming carbon tetrachloride, CCl₄, and phosphorus a pentad as it can fix five with production of phosphorus pentachloride, PCl₅; while sulphur is a hexad because it can take up as many as six atoms of chlorine, forming sulphur hexachloride, SCl₆.

Occasion has already been taken in Chapter XI. to refer to the able work of Prof. Pope with regard to optically active carbon compounds, and in Chapter IX. the important contribution of Mr Barlow to the completion of the theory of the homogeneous partitioning of space has likewise been discussed. In collaboration these two investigators have now propounded a theory which connects the chemical and geometrical sides of crystallography in a somewhat startling manner, which has naturally aroused very considerable discussion, and which, whether right or wrong, cannot fail to have the best results in attracting investigators to the subject.

Starting from the facts which have now been laid down in this book as having been firmly established by the most careful measurement and experimental investigation—notably the constancy of the interfacial angles of the crystals of the same substance, the fixed positions of the atoms or their spheres of influence in the molecule and in the crystal, and the arrangement of the molecules in space-lattices and the atoms in point-systems—Pope and Barlow assume that valency, the expression of the relative combining power of the chemical elements, is a question of the size of the sphere of influence of the atom of an element, and that the relative sizes of such spheres of influence determine the modes in which they can be packed, that is, the nature of the homogeneous crystal structure which they can build up. The theory consequently renders the chemical phenomenon of valency and the physical phenomenon of crystalline form mutually interdependent.

It will thus be apparent that the essence of their conception is that the chemical molecule may be considered as made up of a number of spheres corresponding to, and representing, the spheres of influence of the atoms composing it, and that the volume of each sphere is roughly proportional to the valency of the atom which it represents. They then assume that the sum of the valencies of the atoms present in the molecule may be substituted for the molecular volume, and the quantity thus arrived at is termed by them the “valency volume.” By using the valency volume instead of the molecular volume in the author’s formulæ for calculating the molecular distance ratios, which have been shown in Chapter X. to afford the relative dimensions of the unit cell of the space-lattice, and the distances of separation of the centres of gravity of the molecules from each other along the directions of the crystal axes, Pope and Barlow arrive at new ratios, which they term equivalence parameters. By the use of these latter they have attempted to account for the crystalline structure of a number of substances—chiefly organic compounds, in the investigation of which Prof. Pope has proved so adept—which are connected morphotropically in the manner described in Chapter VIII., and of others which are still less intimately connected.

Unfortunately, the equivalence parameters do not make clear the relationships in an isomorphous series, as do the molecular distance ratios; for they are, from their very nature and mode of derivation, almost identical for all the members of an isomorphous series, the valencies of the interchangeable elements being the same. The molecular distance ratios have also the great advantage of being derived from the three measurements which have now been brought to the highest pitch of experimental accuracy, namely, atomic weight determinations, density determinations by the Retgers immersion method, and goniometrical and physical measurements (optical and thermal) with instruments now available of the utmost refinement. Structural constants thus derived are obviously of especial value. It would thus appear that the theory requires modification so as to take account of the experimentally proved regular increase in volume and in the directional dimensions of the structural unit cell of the crystal space-lattice, when one element of the same family group and of the same valency is interchanged for another. Indeed, as the theory stands at present it entirely ignores and fails to offer any explanation of the highly important physical property of density, specific gravity. That this physical constant, and the equally important constant molecular volume, derived by dividing the molecular weight by the density, possess a real and significant meaning in isomorphous series, formed by the interchange of elements of the same family group, has been clearly proved in Chapter X. This fact is, indeed, so obvious that any further development of the theory must of necessity take account of it.

A precise statement of their conception has recently been made by Prof. Pope in an excellent Report on the Progress of Crystallography, issued by the Chemical Society early in the year 1909. He states that they (Pope and Barlow) “regard the whole of the volume occupied by a crystalline structure as partitioned out into polyhedra, which lie packed together in such a manner as to fill the whole of that volume without interstices. The polyhedra can be so selected that each represents the habitat of one component atom of the material, and are termed the spheres of atomic influence of the constituent atoms. Up to this point no assumption is made other than that clearly indicated by the result of crystallographic measurements, namely, that each atom present in a crystalline structure exerts a distinct morphological effect—or, what is the same thing, appropriates a certain definite volume. The assumption is next made that the crystalline structure, which is resolvable into individual molecules and ultimately into individual atoms, exists as such by reason of equilibrium set up between opposing attractive and repulsive forces operating between the component atoms, and that this equilibrium results in the polyhedra representing the spheres of atomic influence assuming shapes which are as nearly as possible spherical.... The polyhedra thus arrived at may be regarded as derived by compression of a close-packed assemblage of deformable, incompressible elastic spheres,^[29] the compression sufficing for the practical extinction of the interstitial space. When such an assemblage is released from pressure it is evident that in place of polyhedra, the shapes of which approximate as closely as possible to the spherical, closely packed spheres are presented; the distances between the sphere centres can be substantially in the same ratios as the distances between the centres of the corresponding polyhedra in the unexpanded mass, and the equilibrium condition of maximum sphericity of the polyhedra will be presented in the expanded mass of spheres by the existence of the maximum number of contacts between spheres. The whole method of treating the primary assumption thus resolves itself into finding close-packed assemblages of spheres of various sizes representing by their relative volumes the spheres of influence of the component atoms of any particular crystalline structure.”

Some very interesting evidence of the validity of their fundamental assumption of spheres of influence of the component atoms as the ultimate structural units is brought forward. They show that there are two modes of closely packing equal spheres, which give rise respectively to a cubic and a hexagonal crystal structure, the latter having a specific axial ratio of the vertical to the three equal equatorial horizontal axes; and that the chemical elements which are solids and crystallise, and the structural units of which can naturally be assumed to be equal spheres, being those of the similar atoms of the same chemical element, do practically all crystallise either in the cubic system or in the hexagonal system with the specific axial ratio indicated by them. The theory as it concerns chemical valency is obviously not affected by these interesting facts, as the spheres of influence present are those of the identically similar atoms of the same element. But the theory has received considerable support from the results of the investigation of a number of carbon compounds, chiefly derivatives of benzene.

Thus, in spite of the very wise decision of Barlow at the time of developing his theory of the homogeneous partitioning of space, to keep quite clear of attributing shape to the structural unit atoms or molecules, and to consider them as points, he, in common with the other contributors to that splendid geometrical work, appears driven to consider the question of shape when, in collaboration with his chemical colleague, he endeavours to apply his geometrical results to the practical problems of chemistry. It may be inevitable that we cannot get away from the idea of shape of the fundamental structural units. Yet the moment we do admit the idea, and begin to talk of polyhedra, or even of spheres, in close or any other packing, we enter the debatable land, concerning which the experimental evidence is as yet but shadowy and liable to many interpretations. Hence it is that we have the parallelohedra of Von Fedorow, having volumes proportional to the molecular volumes, the more general plane-faced cell of fourteen faces, the tetrakaidecahedron of Lord Kelvin and its deformed derivatives, and now the polyhedra of Pope and Barlow. The more indefinite “Fundamentalbereich” of Schönflies appears to be left behind, and we have embarked on a definite course of attributing shape to the component atoms or their regions of influence in the crystal structure. Von Fedorow has developed his particular theory in a very masterly manner, and with the aid of it professes, and with very considerable success in many cases, to determine the correct mode of setting up a crystal for truly comparative descriptive purposes, and has derived therefrom a remarkable method of crystallochemical analysis.

Moreover, there is yet another view, that of Sollas,^[30] that the packing of the molecules is a more open one altogether, a view to which he has been guided by consideration of the molecular volume. Sollas has offered some remarkable explanations of crystal structure, notably in the case of the dimorphous forms of silver iodide. The abnormal contraction which occurs on heating this interesting substance, and its sudden transformation at 146° from the ordinary hexagonal into a cubic modification as discovered by Lehmann, appear capable of very clear explanation on the basis of his theory. According to this theory of Sollas the volumes of the spheres of influence of the atoms of the different elements of the same family group, such as those of the group of alkali metals or those of the halogens, chlorine, bromine, and iodine, vary progressively in a manner which is dependent on the atomic volumes of the elements, which have a real comparative significance when the elements belong to the same family group.

It is probable that there is a considerable substratum of truth behind these various apparently conflicting views, and what is now required is that the germ of real fact shall be winnowed from the husk of fallacious speculation, just as occurred in the happy recent settlement of the old issue between Haüy and Mitscherlich. As in that case, moreover, it will be experimental work of superlative accuracy which can alone offer the desirable evidence on which a satisfactory arbitration can be founded.

It is thus obvious that we have now arrived at a stage in the history of crystallography when more experimental data, and many more measurements of the most carefully conducted character, on pure materials and excellently developed crystals, are most urgently needed, in order to decide these important, indeed fundamental, questions, the present state of which the author has endeavoured to present with judicial impartiality. When one looks around, and sees the almost complete lack of opportunities for the training of investigators in this rapidly growing branch of science, the importance of which to chemistry and physics is increasing every day, while the field is ripe for the harvesters, one is inclined to feel depressed with the thought of the opportunities which are being lost. Our country has, in this science at any rate, a fine record, having with few breaks led the van of progress from the time of Wollaston, the inventor in the year 1809 of the reflecting goniometer, and of Miller, the originator of our method of describing crystals and the pioneer of accurate experimental work, down to the present day. It may be, also, that our country’s reputation is safe at this moment. But it is in the hands of a band of investigators so small, and often of the private and not professional nature, carrying on the work for sheer love of it and deep interest in it, that the wonder is that so much has been done, and it is the provision for carrying on our national tradition of leadership in crystallography in the future that is a matter for the deepest concern.

If the perusal of this book should have awakened sufficient interest in the minds of some of its readers to prompt them to offer themselves as recruits to this small band of investigators, and especially if it should have inspired the zeal and enthusiasm of a few young students looking around for a promising and fascinating field of work, and, finally, if it should prove to be of assistance in obtaining the means of training such recruits with the help of the best and most accurate experimental apparatus which can be obtained, the author’s main objects in writing it will have been attained.