Bibliography

N. V. Belov. The Structure of Ionic Crystals and Metallic Phases. Publishing House of the Academy of Sciences of the USSR, 1947, 226 pp., 174 figs. Price 19 rubles, bound.

For about 30 years structural analysis of crystals has existed; on the basis of experimental data on the diffraction of X-rays by a crystal, it is possible to determine the positions of the centers of atoms within the elementary cell. This work has already been done for a great many crystals; among simple compounds of the type \(AB\) (anion \(A\), cation \(B\)) it is almost impossible to find a compound that has not been subjected to X-ray investigation. It is quite natural that, with the passage of time, the center of gravity of X-ray structural investigations has shifted to ever more complex compounds: the structure of molecules consisting of 10–20 or more atoms is what interests the investigator of the present day. How, then, is structural investigation carried out in this case? If we leave out of consideration a certain number of fortunate cases (for example, those in which the molecule contains one heavy atom occupying a center of “symmetry” in the lattice), then it may be said that, in principle, no other method than trial and error is possible in solving a structural problem. This means that the solution of a structural problem must begin with a model for which the investigator can obtain, by calculation, figures comparable with experiment. If experiment does not confirm the first model, then the second is tested, and so on. This is the method of trial and error. It is perfectly clear that, without any guiding thread in selecting possible models, we make the work of determining the structure of a more or less complex molecule for the most part impossible and, at best, practicable only at the cost of titanic labor.

As our knowledge of the structure of the crystal has accumulated, a certain confidence has gradually begun to appear that there are guiding principles in the construction of a crystal from atoms and ions. The very chief of these may boldly be declared to be the principle of “fear of voids,” i.e., the principle of closest packing.

In N. V. Belov’s book it is shown how, using the general principles of crystal chemistry, one can find a limited number of possible solutions—structural models—from among which a further choice may be made by X-ray experiment. This book shows how, from a single point of view, the whole world of crystals can be considered. It shows the presence of analogies where they could not have been noticed without a crystallochemical approach, and makes comprehensible a number of features in the behavior and properties of crystals which until now had been riddles. For the first time a book has appeared in which what is given is not a description, but an explanation of the structure of crystals. At the same time, it represents chiefly the author’s own research.

N. V. Belov’s work may be regarded almost as a direct continuation of the work of the great Fedorov. The author himself notes this connection in the preface, pointing out that one of the works—“The work on ditopes, in which space is filled by two sorts of polyhedra”—for the reader”】【

will have to be encountered more than once in the text. It may be noted with satisfaction that Soviet science has not lost its leading place in the field of structural crystallography: the honor of developing the ideas of close packing as the basic idea of crystal chemistry belongs to our “structuralists” and, above all, to N. V. Belov.

Proceeding from what has been said, the reviewer regards N. V. Belov’s book as an outstanding event in the world crystallographic literature.

The book is divided into six chapters: 1) Symmetry of the closest sphere packings. 2) On the voids between spheres in the closest packing. 3) Structures of the AX₁, AX₂, etc. types. 4) Structures with coordination numbers 4 and 6, 8 and 12. 5) Pseudosymmetry and twinning. 6) New structures solved on the basis of the principles of closest packing. The main theme of the book is, in essence, set forth in four chapters; the last two chapters, to which only 8 pages are allotted, are small, though important, remarks.

In the first chapter all possible kinds of close packings of spheres are considered. The space groups are derived in which the formation of the closest sphere packing is possible. Rules are given by which it is easy to investigate all possible cases of closest packings with repetition of the structure after a given number of layers. A simple and clear symbolism is proposed, reflecting the way in which the closest spherical layers are arranged relative to one another. A complete list of all closest packings is presented. A detailed analysis of the symmetry features of closest packings and the presence of good illustrations fully introduce the reader to the course of the basic ideas of the theory of sphere packings.

In the second chapter the necessity of passing from modeling structures with spheres to representing them by means of polyhedra is convincingly demonstrated. Two types of voids in closest sphere packings make it possible to confine oneself to two polyhedra—an octahedron and a tetrahedron, the centers of which are located at the centers of the voids, and the vertices at the centers of the spheres. “A large spherical-anion body, its flesh, is distributed among the surrounding holes,” the author writes figuratively. The arrangement of the polyhedra is illustrated by exceptionally clear, beautifully executed drawings.

The first two chapters are a theoretical introduction. In the following two chapters N. V. Belov applies his method to the consideration of concrete structures. This survey is carried out systematically and leaves a sense of completeness in the crystallochemical edifice. In chapter three compounds are considered in which the cations have coordination numbers 4 and 6. In the first case they fall into the centers of tetrahedra, in the second into the centers of octahedra. The whole “game,” in the author’s expression, consists in how, through how many periods and with what symmetry the layers of the spherical anion packing are repeated, and in what pattern the voids of the packing are filled by cations. As follows from the title, along with ionic crystals, metallic phases are also considered. The structures are combined according to the geometrical principle. And this is as it should be, since close packing, along with anions, can also be formed by metallic atoms of larger size, while in the voids in this case there may be silicon, carbon, perhaps boron, nitrogen, and the same oxygen, which in oxides plays the role of the “anion of the sea” (this is also one of the author’s figurative expressions). The consideration of crystals with the above-mentioned coordination numbers presents, from our point of view, an exceptionally orderly picture. The enormous number of structures is examined with the aid of only two types of polyhedra—octahedron and tetrahedron. At the end of chapter 3 the structures in which the coordination number is achieved not in the form of an octahedron, but in the form of a prism, are considered. This is already a departure from closest packing.

Chapter 4 is constructed in the same way as chapter 3, but in it structures are considered in which the cations have coordination 12 and 8. The consideration

coordination. 12 adds to the polyhedra considered above one more—a cuboctahedron. Structures with coordination number 8 lead to a cube (eight-vertexed).

A number of structures are examined in detail in which coordination 8 is realized at the vertices of a twisted cube, i.e., one whose “lid” is turned relative to the “bottom.” For a very considerable majority of structures these polyhedra are sufficient to fill crystalline space. The author also analyzes more complex atomic arrangements, which can be described by means of more complex polyhedra.

Thus the author applies his method of explaining the structure of crystals not only to cases of densest packing (structures of octahedra and tetrahedra), clearly showing that where the dimensions of the particles composing the crystal do not permit the formation of a densest packing, a dense arrangement of particles or of polyhedra replacing them is nevertheless produced. It is impossible to convey fully the elegance of the explanations of particular structures. For this one must read the book and look at its excellent illustrations.

The book contains a large number of examples—explanations, from the standpoint of the author’s theory, of physical facts that had previously remained incomprehensible. Such are, for example, the explanation of the laws of rutile intergrowth with hematite (p. 83), the peculiarities of metallic β-phases, etc.

Chapter 6 shows the significance of the author’s method for structural analysis. Using as examples several new structures determined under the author’s direction, the enormous importance of the method is made perfectly clear: it makes it possible to avoid the tormenting, and in these examples perhaps even simply unfeasible, blind method of trial and error almost completely.

Let us now dwell on certain points of the work that seem to us unclear or unjustified. It seems to us, first of all, that the use of a coordination polyhedron is advisable only in cases where the number of its vertices corresponds to the coordination number. If this is not so, then filling space with such polyhedra hardly gives a representation of the distribution of matter in the crystal. From this point of view, the representation of the pyrite structure by a deformed cube at coordination 6 leaves a feeling of dissatisfaction, and especially the structure of carbon monoxide at coordination 2 (if one can speak of it at all in this case).

It also seems to us that attempts to represent molecular compounds by means of polyhedra are unsatisfactory. Fig. 140 for benzene only obscures the true character of the dense packing in this compound, which is expressed in an entirely different form. Just as forced seems the inclusion in the general scheme of such a compound as potassium azide, in which one would have to “conditionally” deprive one of the three nitrogen atoms of volume. It is scarcely advisable to extend the ideas of the sphere packing theory to such a structure as diamond, treating it as a zinc-blende-type structure with two ions—one tetravalent positive and one tetravalent negative. Calculation of the interatomic distance in diamond as the mean value of the ionic radii does not lead to the required value and can hardly serve as a justification for such an artificial treatment.

It also seems to us that the distinction between the basic structures, so naturally explained by densest packing, and the structures, say, of the cementite type, composed of a complex pattern of arranged prisms, might have been emphasized somewhat more sharply. If in the first, fundamental case the representation of a structure by means of polyhedra is an explanation of the structure, i.e., its subsumption under a general principle, then in the second case this representation can be regarded only as—

Bibliography

an interesting description showing the genetic kinship of this structure with the simplest ones.

In Chapter 5 the author gives an extraordinarily witty explanation of the tendency of crystals toward twinning and of Fedorov’s law of limits. Here one would like to note that the law of crystallographic limits must also have more general foundations, since it is observed in the crystals of very complex organic compounds, where it does not seem possible to speak of octaves as elements of structure.

I should like to say a few more words about the exceptional impression of significance and integrity that remains after reading the book. This is also aided by the author’s splendid figurative language, examples of which we have already given above. Beyond all praise is the graphical work, so essential in a work on geometrical crystallography. In general, the book is well produced, although a few minor omissions in the placement of the figures could be wished corrected. It seems that one could have avoided 6–7 pages of distance between a figure and the reference to it. Fig. 41 has been pasted in out of place. There are also misprints: on p. 61, instead of Fig. 38 there should be Fig. 39; on p. 169 the word “motif” has been omitted, etc.

We note these small defects in the external appearance of the book and certain inconveniences of use, and the small number of particular questions raised, only as the duty of a reviewer. Of course, these remarks cannot diminish the feeling of joy for Russian crystallography, enriched by a work whose significance can hardly be overestimated.

A. I. Kitaigorodskii