THE CRYSTAL AND ITS CONSTANTS

B. F. Ormont, Moscow

1. Statement of the Question

Crystals have long attracted general interest, since in the varied forms of crystals matter acquires a visible orderliness corresponding to regular geometrical figures. Modern physical methods for studying the structure of crystals—above all X-ray methods—have confirmed the a priori assumptions that the molecules (atoms, ions) of which a crystal consists are arranged in space near one another at strictly definite distances and at strictly definite angles, thereby determining the strictly definite form of the elementary cell and of the crystal as a whole.

It has long been established that crystals of some definite substance—for example, common salt—which arise in different places and very often even under dissimilar conditions, in general possess not only the same form, but also the same internal properties: optical, electrical, mechanical, and others; that is, they are physically analogous. Hence arose the conception of the physical constants of the crystal of a given substance.*

As a result, the investigator—and especially the production worker dealing with crystals—becomes accustomed to this picture, which is correct only as a first approximation, and as a rule assumes that a crystal which has undergone a definite cycle of external influences (for example, a change in pressure or temperature), upon return to the initial external conditions, must again acquire the totality of its initial properties. Indeed, reversibility of the process is very often observed: a return to the initial state, at any rate of certain properties of the crystal and of its constants. Nevertheless, facts of this kind in no way give grounds for judgments that ascribe to the crystal the property of a physical pendulum—of necessarily returning, after the end of a disturbing action, to the state

* Of course, it is assumed here that with a change in the external conditions in which the crystal is found (for example, temperature and pressure), the properties of the latter change: continuously—between transition points, for example from one modification to another—and discontinuously—at transition points (for example, sulfur from monoclinic to rhombic), which also finds confirmation in numerous known facts.

equilibrium*. Unfortunately, however, judgments of this kind—if not stated explicitly, then tacitly admitted—are constantly encountered both in scientific works and in the production design of many processes, which in many cases causes considerable harm.

Such an incorrect approach to evaluating the significance of lattice constants has in very recent times become, as it seems to us, a stimulus to the failure of the principal lines of classical crystal physics (see below).

The point is that the basic regularities of the latter have been established primarily on crystals obtained in systems in which the crystal, with respect to the coexisting phase (solution, melt), was in a state differing very little from equilibrium. The probability of coincidence of the constants of crystals obtained under such conditions is maximal. Meanwhile, an enormous number of crystals obtained for technical purposes arise under conditions that are far from so ideal, but, for example, very often from highly supersaturated media, under conditions of temperature inequality not only within the entire reaction space but even within the limits of the growing crystal.

Only in very recent times has the investigation of such systems attracted various authors. The main attention in this has been devoted to the question of the nucleation of the solid phase, i.e. of the crystal (which takes place at large supersaturations; this question was recently critically considered by H. Fuchs 43 and will not be touched upon in this article). However, not only nucleation but also the growth of crystals can occur under nonequilibrium conditions, with all the consequences that follow from this. Moreover, in practice there are frequent cases when the substance of which the crystal consists is itself the product of a simultaneously proceeding chemical reaction, so that crystallization is the result not only of a physicochemical process of transition of a substance from one phase to another, but also of a complex chemical process of formation of the substance of which the crystal consists. The crystal obtained as a result must possess specific properties. Meanwhile, such processes have hardly been touched upon by crystal physics; as a result, these crystals are usually approached with the same incautious evaluation of their constants of which we spoke above and which is impermissible even in the case of equilibrium crystals**. Despite

* If a given crystal cannot in principle always be returned to its initial state, then, obviously, two crystals obtained under approximately analogous conditions also need not necessarily, and not in all properties, show identical behavior.

** The concept of an “equilibrium crystal” may have two different meanings: 1) a crystal formed under conditions in which it was in practical equilibrium with the medium, forming a system differing very little from an equilibrium one, and 2) a crystal possessing a minimum of energy. To avoid confusion we shall call crystals satisfying the first requirement equilibrium crystals, and those satisfying the second—stable crystals.

that although the necessity of such an assessment of crystal constants is almost obvious, the enormous number of scientific works by various authors, especially in the field of nonequilibrium crystals, testifies at best to an insufficient understanding of what has been set forth above; as a result, a number of figures published as constants have very doubtful value.

Before proceeding further, it is necessary once again to emphasize that the enormous number of crystal constants is determined by the conditions under which the crystals are found. Meanwhile, for the correct evaluation of any properties of a crystal one must establish the dependence of these latter both on the conditions of its formation and growth, and on its subsequent treatment, and on the peculiarities of the process in which particular properties of the crystal are used and manifested. It must not be forgotten that a crystal is not an immutable monument to itself, but an object subject to change, passing during the time of its formation and existence through various phases of development and degradation, and that the seemingly observed subordination of matter to form in crystals is apparent, and this not only from a fundamental point of view but also practically.

Consequently, it is of practical importance not only to strive to obtain a crystal of good quality, but also to know by what influences it can be improved or spoiled. Therefore, the study of the laws governing the formation of “full-value” crystals of a given substance and their alteration must be regarded not only as a most interesting theoretical problem, but also as an urgent need of branches of industry that use crystals (the production of hard alloys, abrasive materials, processes of mechanical treatment of metals and rocks—drilling, cutting, boring, etc.). Here we shall dwell only on certain fundamental premises that underlie our views on the problem indicated.

II. Principal Directions of Crystal Physics and the Study of the Question of the Formation and Properties of the Crystal

The thermodynamic investigation of chemical and physicochemical systems, which in its time yielded such significant results in theoretical chemistry, also left a vivid imprint on the development of classical crystal physics. Beginning in the 1880s, the formation and growth of the crystal were considered as a thermodynamic problem; moreover, this direction of crystal physics, especially clearly formulated by Curie ¹ and Wulff ², remained predominant almost until the very recent past, being reflected not only in a number of research works—Ritzel ³ (1911), Liebmann ⁴ (1914), Ehrenfest ⁵ (1915), Yamada ⁶ (1923–1924), Bimendler ⁷ (1926), and others—but also in the content of textbooks.

From the standpoint of this school, a crystal is regarded as an aggregate of atoms, ions, or molecules correctly arranged in space relative to one another; a direct consequence of such a correct arrangement, depending on the properties of the substance, is also the correct form of the crystal.

Irrespective of whether the crystal is isotropic or not, it was assumed that in a good specimen of a crystal the matter uniformly fills space, in accordance with the structure of the elementary cell, and that the development of particular faces is ultimately determined by the Gibbs–Curie principle: “The equilibrium form of a crystal, as the most stable form, must be considered to be that form whose surface energy, for a given volume of the crystal, is the least,” i.e.

\[ \sum \sigma S = \min \]

with

\[ V = \text{const}. \]

This principle, proposed for a crystal in a saturated solution (which is often forgotten), according to Wulff’s opinion may also be generalized to the case of a crystal growing at any supersaturation, provided only that this supersaturation is not too large.

The desired minimum of surface energy for a given volume is attained, according to Wulff’s theorem (the Curie–Wulff principle), with that mutual arrangement of its faces in which they are removed from one and the same point by distances proportional to their capillary constants.

\[ h_1 : h_2 : h_3 = \sigma_1 : \sigma_2 : \sigma_3. \]

This theorem was proved (1895) by Wulff and by another method (1924) by Frenkel.^10

That the Gibbs–Curie principle is valid in general admits of no doubt, and connected with it is the well-known fact that in a saturated solution small crystals pass into large ones, while an isolated crystal of irregular form acquires a regular form corresponding to a decrease in surface energy. Calculations by Born and Stern^11 and by Yamada^12 (on the basis of the theory of the ionic crystal, taking into account the forces of “Born repulsion” of the ions) of the values of the surface energy of crystals have shown that, indeed, for example, for common salt the faces of the cube possess the minimum surface energy. According to these calculations,

\[ \frac{\sigma_{(110)}}{\sigma_{(100)}} = 2.7, \]

so that, for example, attempts to split an NaCl crystal along the face of the rhombic dodecahedron should have led to the formation of a stepped surface (Fig. 1a). Approximately the same picture is obtained with respect to splitting along the face of the octahedron.

(Fig. 1b). From the experimental side this question was studied by Kuznetsov and Kudryavtseva[^1] (1917), and the order of magnitude found by calculation was confirmed. However, the Gibbs–Curie principle, as was already noted in Gibbs’s older works, has practical significance only for the smallest crystals. In the case of larger ones, the equilibrium form often does not correspond to the Gibbs–Curie principle; in such crystals the influence of the term depending on surface energy in the expression for the thermodynamic potential is negligible.*

Thus, for example, in grinding a gram-molecule of NaCl down to cubes with an edge length of \(1\mu\), i.e., increasing the surface by \(1.7 \cdot 10^{6}\ \mathrm{cm}\), it is necessary to expend energy equal to \(0.006\ \mathrm{kg\ cal}\), whereas

Fig. 1. Surface formed in attempts to split an NaCl crystal along the (100) and (111) faces.

the lattice energy is of the order of \(180\ \mathrm{kg\ cal}\). In confirmation of this, a considerable body of experimental material has been assembled in crystal physics, showing that, as a general rule, during crystallization growth forms arise (for example, needles are formed instead of cubes) that correspond very little, in the ratio of the distances of the faces from the common center, to the values of the capillary constants of these faces, etc.[^15]

The formation of growth forms was also well known to Wulff. Proceeding from the thermodynamic point of view on the formation of a crystal, Wulff sought an explanation of these facts also in an argument that characterizes the system in a purely statistical way: “the equilibrium form depends on the degree of supersaturation of the solution”[^16]—an argument that is quite insufficient for any acceptable clarification of the question.

* The sharp criticism of this principle by some authors, for example by Follmer[^14] and others, who declared it unsuitable, is unfair because the edge of the criticism should be directed not against this principle but against applying it to cases where it is knowingly unsuitable (and to which Gibbs himself pointed), i.e., against a phenomenon that often occurs in physics with respect to many laws, where the fault lies not so much with the laws, which have their own limits of application, as with the authors who go beyond these limits. Curiously, critics who call this principle the Curie–Wulff principle cite the works of Gibbs himself as evidence.

Thus, the doctrine of the form of a crystal, developed in this direction, was based on the Gibbs–Curie principle, whose limited applicability was provided for by Gibbs himself, and still more so on a purely thermodynamic approach to the crystal and to its formation from a coexisting phase.

The question of the structure and physical properties of the crystal was treated in the same plane: under given physical conditions the modification stable under those conditions must form. To explain numerous deviations, especially frequent in substances of a certain kind, for example S, Se, As₂O₃, etc., Ostwald’s rule of stages (Stufenregel) was developed in the classical theory of the crystal; as applied to the case under consideration it is usually stated as follows1: “from a supersaturated phase there must first form predominantly unstable modifications, which then pass into more stable ones.” However, the applicability of this rule has been disputed by many, in particular by Elinek1. We shall return to the question of the applicability of this rule in Section V.

Often the statistical approach to the study of a crystal made possible a more or less accurate measurement of those constants and properties of the latter which are determined as a statistical mean of the properties of many atoms (molecules), and whose reproducibility is not affected by “microdefects” or, in general, “microfeatures” of the crystal; the conception of isotropic or anisotropic crystals is also based on this approach. It is interesting to note that, for example, the X-ray method of investigation, which has such great importance in crystal physics, gives, in essence, a statistical picture of the structure of the crystal . The ideal filling of space with matter, as if following from X-ray photographs, is an appearance through and through. To an equal degree the theory of the ionic crystal (Born2, Goldschmidt3, and others), i.e. the most elaborated branch of the theory of crystal structure, also characterizes the crystal as a whole, i.e. as an aggregate of + and − charges. All these theoretical and experimental methods are unable to establish the separate elements of the microstructure of the crystal, for example the presence of discontinuities of molecular size (voids, cracks, etc.); thus the thermodynamic and statistical approach to the crystal*—the identification of properties and constants governed by the interaction of the billions of atoms, ions, or molecules composing the crystal—constitutes the main direction of classical crystal physics, in which what is essentially described is a certain ideal crystal; the variety of properties of real crystals is considered rather as a deviation from the rule, and possible regularities of such deviations are almost not established.

* The same also applies to the vast majority of other physical methods of crystal investigation (refraction, etc.).

** Both from the standpoint of theory and from that of experiment.

The second direction, relying on Gibbs’ conclusion that, for example, the dependence of the solubility of a sufficiently large crystal on surface energy may be neglected, developed views according to which growth and, consequently, the form of a crystal are an exclusively kinetic problem. Representatives of this direction are Berto ^20 (1912), Friedel ^21 (1913), Masing ^22 (1920), Nigli ^23, Valeton ^24 (1920–24), Spangenberg ^25 (1923), and others.

This, again, purely statistical approach to the problem of crystal formation (and, consequently, of properties) was in essence an attempt to transform the theory of crystal form in accordance with the experimentally established variety of growth forms; moreover, beginning in the 1920s, the conclusions of the concurrently developing doctrine of the applicability of electrostatic laws to the calculation of the strength of bonds in salt crystals (Born ^18, Lennard-Jones ^26, and others) formed the basis of this direction. As is known, Born’s theory showed that it is possible not only to calculate the energy of the crystal lattice of this type, but also that the results of the calculation are close to those found from experimental data (by the Born–Haber cycle method). Let us emphasize that, if the energy calculations gave excellent agreement with the results of experiment, then calculations of the mechanical properties of a crystal, for example its strength, repeatedly led to figures of an entirely different order of magnitude (we shall return to this below).

The doctrine of the structure of the ionic crystal, relying on electrostatic calculations, made the first steps toward differentiating the properties of the crystal surface both in the static aspect (the finished crystal) and in the kinetic aspect (the forming crystal).

Already in 1919 Madelung ^27 calculated the energy of detachment of surface ions, proceeding from the fact that in the finished crystal the distance between ions of the first and second layers is decreased (as a result of one-sided attraction), while that between the second and third is increased. In addition, it is possible that these distances for cations and anions of one and the same layer are different, i.e., a double electric layer arises. Later this question was discussed by Davisson and Germer ^27a, Bethe ^27b, and Zwicky ^30. The magnitude of the contraction of the distance was calculated (1928) for NaCl by Braunbek ^28a. In connection with the development of the doctrine of polarization (Fajans, Debye, and others), Bimöller ^28 (1926), at M. Born’s suggestion, assumed in these same calculations that only a one-sided deformation of the surface ions takes place, while their position in the lattice remains unchanged.

Finally, Stranski ^29 (1928) considered the same problem of the structure of the surface layer of a crystal, allowing for the simultaneous action of both factors. According to Stranski, on the one hand, there must occur an approach of the second layer to the first and a removal of the second layer from the third, caused by the one-sided polarization of the first layer of ions; on the other hand, taking into account that the lattice parameter

For NaCl \(a = 5.6\,\text{Å}\), i.e. the distance between the centers of the Na\(^+\)- and Cl\(^-\)-ions is \(r = 2.8\,\text{Å}\), whereas in a free NaCl molecule, according to the calculations of Born and Heisenberg, the distance between the ion centers is \(r = 2.29\,\text{Å}\). Stranski assumed that, in the surface layer, a considerable approach of Na\(^+\)- and Cl\(^-\)-ions is not excluded, with the formation of a kind of crack of atomic dimensions (Fig. 2). Similar views, however, had been set forth considerably earlier by Tswikki \({}^{30}\) (see also Lennard-Jones \({}^{30}\) and Dent \({}^{31}\)).

Fig. 2. I. Kossel (1920), II. Madelung (1919), Braunbek (1928), III. Tswikki (1923), Stranski (1928)

Thus, by 1928 the idea had become established that the structure even of an isotropic equilibrium crystal in its surface layers differs substantially from the structure inside the crystal, and that, as a result of various kinds of deformations, discontinuities of the type of cracks and cavities appear at the surface.

At this same time a discussion was in full swing concerning the so-called Ioffe effect, discovered in 1867 by Engelhardt and carefully studied by Ioffe and Levitskaya \({}^{32}\), which consists in an enormous increase in the tensile strength of NaCl crystals when they are stretched under water. We shall not go into the details of the dispute here, but note that this effect was attributed in particular* to surface cracks (Ioffe), or to loosened places in the body of the whole crystal (Smekal) into which water entered, leading to the elimination of cracks (Ioffe) or to an increase in the plasticity of the crystal (Smekal).

Thus, even equilibrium NaCl crystals by this time had begun to be regarded as formations not devoid of discontinuities (pores, cavities, cracks) of atomic dimensions.

On the other hand, by the beginning of the 1920s it becomes obvious that the problem of the formation and properties of a crystal cannot be solved within the framework of the old kinetic theory, that the problem comes down to the need to clarify the mechanism of growth of individual faces and of the whole crystal as a whole, with a study, as far as possible, of elementary processes. By this time, in considering the question of the kinetics of growth of individual faces

* We do not cite other points of view for lack of space.

crystal, using sodium chloride as an example, Valeton4 showed, on the basis of the distribution of electrostatic forces on the faces (100) and (111), that the former holds ions more weakly, as a result of which the faces (111) should grow faster and, consequently, disappear. In the development of these views a new direction arose, which is to a certain extent a synthesis of the two preceding ones, formulated by H. Fölmer and his collaborators; Kossel7 (1927), Brandes6 (1927), Stranski8,10–12 (1927–35), Balarev9, and others.

As is well known, the new trend proceeds from the fact that the energy of attachment of particles to different points of the crystal surface is not the same; and, for example, the attachment of a particle to the vertex of a cube, as is easy to calculate for the case of ionic crystals on the basis of the laws of electrostatics, releases more energy than attachment to a face. In other words, the probability of retention by the crystal surface of molecules (ions) from outside that fall upon it is different for different points of the surface. Thus, following upon the teaching on anisotropy created by classical crystal physics, there arises first the idea of the different strength of bonding in the case of the surface and internal layers of a crystal, and then—in the case of different points of the surface itself. At the same time, Fölmer, in contrast to Kossel, advanced the proposition5 that ions arriving at the crystal surface are not laid down directly into the lattice of the latter, but are adsorbed on the surface, forming a mobile two-dimensional nucleus, the particles (ions) belonging to which may become fixed in the crystal lattice not at that point of the face, and even not on that face, where they were adsorbed. Stranski took an intermediate position on this question, admitting the Fölmer mechanism of crystal growth for certain cases.

Since this doctrine has been presented in review form in the Soviet literature by N. Fuks13, A. Shubnikov14, and V. Kuznetsov15, we shall not dwell on it, but shall point out that when a crystal is in equilibrium with a solution, this means, according to the theory being expounded, equilibrium only with respect to definite points of the crystal surface.

Proceeding from these views, Stranski and Kaishev11,12 considered the mechanism of equilibrium of small crystals, and Stranski and Tomanov10 the Ostwald rule of stages, which, according to their conclusions, fits within the framework of the new theory. It is extremely important that this direction is based on definite models of the crystallization process (although, admittedly, the opinions of different authors concerning some details do not always coincide).

From the point of view of this theory, for the beginning of the formation of a new layer of crystal a certain supersaturation is necessary in comparison with that required for the continuation of growth of an already begun layer. In this case, for example, from Kossel’s point of view, the crystallization process is most strongly retarded at the initial moment of the stage of formation of a new layer. From Fölmer’s point of view, the process is most strongly retarded at the stage of formation of a two-dimensional nucleus, etc. But

with an increase in supersaturation, the possible number of embryos beginning a new layer on a face increases, and their size decreases.

As a result, the rate of formation of a new face may become sufficiently large even when the preceding face has not yet had time to “finish building itself,” i.e., on unfinished planes a new layer may begin to grow (Brandes). But if the formation of the latter does take place, then a greater rate of its growth is very likely in comparison with the lower layer, access to which is more difficult for the substance of the crystal.

Thus crystals with voids, pores, etc., must arise, and the occurrence of the latter is connected with the fact that, for the growth of a new face to begin, a certain supersaturation of the solution is necessary; in other words, the very mechanism of crystal growth contains the causes for the appearance of defects in its structure. Another possibility for the formation of pores, as some authors note,^46 manifests itself in the case when a face grows at the expense of molecular aggregates. Since the latter will always differ in size and shape, their dense packing must prove impossible.

As a result, instead of ideal crystals, real crystals with a definite sum of individual properties must be obtained. In connection with such ideas, the concept of the “real” crystal is being used more and more often.

However, unfortunately, the question that interests us—concerning the constancy of the constants and properties of the crystal and its parts—cannot, up to the present time, be considered to have been posed by crystal theory clearly and sharply enough. In essence, in recent years the latter has only “permitted” the crystal to be “real,” has only sanctioned the existence of facts discovered in experimental investigations. Therefore, before touching upon certain fundamental questions, let us see how the problem considered here is interpreted from the experimental side. Unfortunately, for lack of space we can cite only a few examples.

III. Various cases of variation in the properties of crystals discovered experimentally

If, from the theoretical side, as we have said, the question of the properties and constants of the crystal had until recently remained not very clearly and sharply posed, then from the experimental side many facts have long been known whose significance was, perhaps, not always sufficiently appreciated.

1. Variations in the form of crystals

This question has been studied most fully and in detail, and for a long time.

If we leave aside the fact of the existence, under different conditions, of several stable modifications of a whole series of substances (for example, S₂ and S₃)—a phenomenon excellently illuminated by thermodynamics and the phase rule—then, under conditions in which only one modification is stable, other forms have repeatedly been observed in many substances in connection with Ostwald’s rule of stages (for example, at room temperature, alongside graphite there exists diamond; alongside rhombic sulfur, a monoclinic form can be obtained). It also happens that a modification metastable under the given conditions is observed more often than the stable one. This applies, for example, to white tin, whose transition into the stable modification (gray tin) was regarded in inorganic chemistry as an extraordinary phenomenon (“tin pest”), and so on. We shall return to this question in Section V.

On the other hand, the variety of growth forms of crystals of a given modification has likewise long been known, in accordance with Steno’s law.

Among the external factors causing the variety of growth forms, the following have been established experimentally: changes in the concentrations at individual faces (concentration flows, etc.) and the presence of impurities. It is known, for example, that in the presence of traces of urea, NaCl forms octahedra rather than cubes. Recently, a change in the form of KClO₃ crystals in the presence of dyes was observed by Buckley and Cocker⁴⁷. A series of observations was described by Royer⁴⁸.

It is very important to emphasize that, since classical crystal physics was based on the Gibbs–Curie principle, it attributed essentially the entire diversity of growth forms to external factors, since internal factors, from this point of view, should have led the crystal to the single stable form satisfying this principle. Although on this question the new theories have taken a clearer position (see above on calculations explaining the growth tendencies of the (100), rather than the (111) or (110), faces in NaCl crystals), there are, in general, no systematic experiments establishing the dependence of growth forms on internal factors, or at any rate they in no way compare with the enormous number of works devoted to investigating the influence of external factors (see Section IV).

2. Variation of Color

It is known that crystals of many substances noticeably change their color with increasing degree of dispersion and usually become lighter. The composition (according to chemical analysis) and the structure of the unit cell (according to X-ray diagrams) remain unchanged. This phenomenon is caused by well-known physical reasons, so from the theoretical side we shall not dwell on it.

Among recent works in which a very considerable change in coloration was observed, we shall refer to one recently completed in our ...

laboratory* investigation of certain properties of chromium oxide obtained by thermal decomposition of chromic chloride. In this process an anhydrous black chromium oxide is formed, which on grinding readily turns into a rather light green powder; a similar powder is also obtained when $Cr_2O_3$ is captured and precipitated from the gases leaving the reaction zone.

An X-ray investigation of the samples we obtained, carried out at our suggestion by V. I. Kasatochkin at TsNIIMASh, showed that the structure of the unit cell of the black and green chromium oxide is the same. The size of the particles of the first is measured in fractions of a millimeter, of the second—usually in fractions of a micron, but it may also be considerably larger (the color, with increasing layer thickness, gradually changes to black).

3. Variations of the apparent specific gravity

It has long been known that the apparent specific gravity of crystals of one and the same substance may vary. Accordingly, in any handbook, even in Landolt’s tables, such an important constant as specific gravity is sometimes represented not by a few figures, but by a good dozen of them (we have in mind, of course, figures referring only to a definite modification of a substance).

TABLE 1

Specific gravities of certain substances according to Landolt

Crystal	Characteristic	Specific gravity	Mean specific gravity
$Al_2O_3$	Amorphous, calcined	3.73–3.99	3.85
$Al_2O_3$	Corundum, ruby, sapphire	3.95–4.02	4.00
$Al_2O_3$	X-ray data	—	3.96
$Sb_2O_3$	Regular	5.11–5.30	5.20
$Sb_2O_3$	Rhombic	5.56–5.78	5.67
$As_2O_3$	Rhombic	3.85–4.15	4.10
$PbS$	—	6.77–7.51	7.13
$PbS$	Artificial galena	7.51–7.76	7.65
$PbS$	Additional data	7.48–7.59–7.766	—

* Being prepared for publication (M. A. Khavankaya et al.).

The range of fluctuations is sometimes quite considerable—on the order of 2–8%, far exceeding the limits of the high precision allowed by the weighing method. We shall cite some figures as an example (see Table 1).

The figures given in Landolt’s tables for gold are very indicative; they range from 18.8840 for gold distilled in vacuum to 19.2685 for gold pressed at 10,000 atm, and 19.431 for gold obtained by crystallization through slow reduction with formaldehyde [Averkiev^49 (1903)]. The value corresponding to X-ray data is 19.37 ± 0.03 [Davey^50 (1925)]. These discrepancies are especially noteworthy against the background of the fact that the specific gravity of gold has been accurately determined since the most ancient times, and that over the last millennium progress has essentially consisted in establishing the fact of significant fluctuations in the first digit after the decimal point.* Moreover, gold is a substance that can easily be obtained in pure form.

The causes of such considerable fluctuations in the values of specific gravity (in the case of one and the same modification) have not given rise to disagreement—they were ascribed to pores and to occluded gas bubbles, mainly on the surface of the crystal, and partly also within it. In this connection there was a tendency to overestimate the significance of external pores. During recent decades crystals and, in general, powders have been used for purposes of catalysis and adsorption by utilizing their surface. It has often been assumed that the difference \(\Delta d = d_2 - d_3\) between the true \((d_r)\), i.e. calculated from the parameters of the unit cell, and the apparent \((d_s)\) (found pycnometrically) specific gravity characterizes the preparation from the point of view of its surface (determining the volume of external pores). Although in a number of cases this is indeed true (or although sometimes there is a certain proportionality between the difference \(\Delta d\) and the magnitude of the surface), in the case of many preparations this difference must be attributed almost entirely to internal pores and in no way characterizes the surface of the preparation. Moreover, naturally, a change in the volume of the external pores still says nothing about the surface of these pores.

Indeed, if we denote by: \(V_r\)—the volume occupied by the substance of the crystal, \(V_i\)—by internal pores, \(V_a\)—by external ones, and \(V_0\)—the volume of air (gas) bubbles occluded by the crystal; if \(d_r\)—the specific gravity of the substance of the given crystal corresponding to the parameters of the unit cell, \(d_i\)—the mean specific gravity of the substance located in the internal pores, and \(d_a\)—the specific gravity of the gas in the external pores, then

\[ V_r d_r + V_i d_i + (V_a + V_0)d_a = (V_r + V_i + V_a + V_0)d_s = V_s d_s, \]

* Vitruvius’s reference to Archimedes’ determination of the specific gravity of gold is well known, i.e. 2000 years ago.

* In the book of Al-Khazini (1137), The Balance of Wisdom*, the specific gravity of gold is given as 19.05—a figure that could quite successfully have appeared even in Landolt.

where \(d_s\) is the apparent specific gravity of the substance. Hence

\[ d_s=\frac{V_r d_r+V_i d_i+(V_a+V_0)d_a}{V_s}. \]

Thus \(d_s\) is a very complex function of the origin of the crystal, of the size of the pores and the substance filling them (provided that \(d_r=\mathrm{const}\)), and also of the method of determination.

Let us note that the volume \(V_a\) is often greatly overestimated, taking instead of it the sum \(V_a+V_0\) (it is known that \(V_i\) may be very large, reaching the value \(V_r\)). If the removal of bubbles is ensured, then \(d_s\) comes very close to \(d_r\). There is no doubt that the considerable fluctuations in old determinations of the specific gravities of various preparations (made by different authors) are connected with unsatisfactory dehydration.

On the other hand, external pores should enter into the body of the crystal only in certain cases (see Sec. IV). Therefore \(V_i\), in comparison with \(V_a\), must be a rather significant quantity (at least for many preparations).

It is very important that even a pressure of the order of \(10\,000\) atm, applied to a fairly plastic substance (gold), is insufficient to destroy voids in a preparation where they arose as a consequence of the method of preparation. Regardless of whether the slow crystallization of gold from solution really gives crystals of such exceptional specific gravity as that found by Averkiev, there is no doubt that it is more expedient, for example, to obtain a continuous crystal (of substances of this type; see below) by growth under equilibrium and similar conditions than to remove pores by means of superhigh pressure.

Let us emphasize that whereas the specific gravities of crystals of hundreds of substances are characterized by many figures pertaining to one and the same modification, the crystals of a whole series of substances have a specific gravity fluctuating within considerably narrower limits.

For example (according to data of various authors compiled by Landolt), the specific gravity of \(\mathrm{K}_2\mathrm{SO}_4\) is

\[ d_1=2{,}666,\quad d_2=2{,}670,\quad d_3=2{,}6633\ \text{etc.} \]

This circumstance cannot be overlooked in discussing the question (see below).

4. Variations of the specific gravity of crystals under mechanical treatment

During grinding and, in general, pressure on a powder, the modification, or allotropic form, may of course change. Thus, for example, amorphous substances, when rubbed, often pass into crystalline ones; some even do so violently. It has long been known, for example, that explosive antimony, i.e. amorphous antimony containing in its pores

SbCl$_3$ and, upon grinding, passes explosively into another modification [Kohen and Ringer$^{51}$ (1904)]. Under a pressure of 12,000 atm at 200° Bridgman$^{52}$ obtained (1922) graphite-like, hard-to-grind phosphorus with a specific gravity 1.5 times greater than usual. Facts of this kind present nothing especially difficult—it is enough to take X-ray photographs of the preparation before and after treatment in order to establish the appearance of a different form. However, having obtained completely similar X-ray photographs of two preparations with apparently different specific gravities $d_s$, one can, of course, in no case conclude that they differ only in the external surface of the crystals and in their pores, enlarged during abrasion. Internal microcavities and microcracks (of atomic dimensions) may arise in them, with an increase in the volume of internal pores $V_i$; on the other hand, during abrasion part of the internal pores, through which fracture surfaces pass, will become external, which will act in the direction of decreasing $V_i$ (the latter factor may predominate, obviously, only if the crystal contains rather large pores along which splitting will occur). Particularly considerable changes during abrasion must be expected in the case of brittle crystals (incidentally, these include a large part of the substances used in the technique of abrasive processes).

One is compelled to be surprised that this phenomenon and its possible limits have, up to the present time, not only not been properly studied, but have also been very scantily described.

A typical example of such work (which did not penetrate to the essence of the matter) is the study by Ray$^{53}$, who found that, upon grinding, the specific gravity of quartz decreases from $2.638 \pm 0.002$ to $2.528 \pm 0.002$, and its heat of solution in hydrofluoric acid increases from 30.20 to 32.46 cal. Such a considerable increase in the heat of solution testifies to the formation of a very large surface, i.e. to the formation of micropores.

To the same kind of work belongs, for example, the study (1932) by Tammann$^{72}$, “Change in the Properties of Nonmetallic Substances during Their Treatment.” A decrease in the specific gravity of AgCl (as well as a decrease in electrical conductivity and an increase in hardness) upon rolling is described. On heating to 125–200° the preparation undergoes changes in the reverse direction.

As we see, in these (typical) works it has not been clarified at all owing to the change of which of the factors the change in specific gravity occurred.

Against this background, of special interest is the communication by Tammann and Moryts$^{73}$ (1934), who showed that, when quartz is ground in an agate mortar under a pressure of up to 20 kg, the specific gravity of the latter decreases by 20%, while the structure of the quartz, judging from the X-ray photograph, does not change, just as the volume of external pores $V_a$ does not change (i.e. the true specific gravity $d_r$ and $V_a$ are constant). Thus, during grinding, internal voids seem in fact to appear in quartz crystals.

As we have said, mechanical treatment may lead also

to an increase in \(d_s'\); too large an increase*, however, is not to be expected, unless \(V_i\) is sufficiently large and, moreover, is due to large pores. Their formation should above all be expected in very nonequilibrium crystals subjected to severe heat treatment, in viscous glassy masses, etc., containing air inclusions to which access of liquid is facilitated by attrition. Incidentally, in the cited work of Tammann and Moritz it is indicated that, upon grinding quartz glass and crystobalite, an increase in specific gravity by 2% is observed.

In any case, such phenomena cannot at present be regarded as either at all seriously studied or even put forward as elements worthy of study in the problem, despite their considerable theoretical and applied interest**. To draw attention to them is the task of this section.

5. Change in the structure of a crystal as a function of the structure of the adjoining solid phase

It has long been known that, in crystallization in the presence of a seed whose structure corresponds to a metastable modification, under favorable conditions it is precisely the latter that is readily formed, in accordance with Ostwald’s rule of stages. Thus, for example, in the presence of PbS, from aqueous solutions of NaBr there crystallizes, as Stranski and Totomanow have shown\(^{40}\), below \(50.7^\circ\) not NaBr \(2\mathrm{H_2O}\), but NaBr metastable under these conditions (isomorphous with PbS).

There are, however, apparently also facts in which, under certain conditions, a stable form may disappear with the formation of a form less stable under ordinary conditions. Thus, for example, Finch, Quarrell, and Rebek (1934)\(^{54}\) recently observed that Zn vapors, condensing on the surface of polished copper, at first give a clear electron diffraction pattern (i.e., have a crystalline structure), but after a few seconds the rings of the electron diffraction pattern disappear. Phenomena of this kind deserve attention from many points of view. On the one hand, their reality would testify to those dangers to which the existence of two-dimensional Vollmer nuclei on the surface of certain types of substrates is subject. Secondly, processes of this kind may precede, as it seems to us, certain reactions between solid bodies, disturbing the structure and altering the reactivity of the components. This disturbance of structure may, thirdly, occur in some cases when a solid is worked by the cutting surface of another solid, for which degeneration of the structure of the cutting and similar surface may mean unsuitability or a reduction in suitability for further work. It is not superfluous to recall here the old work of Beilby\(^{55}\) (1904), who supposed that the thin surface layer of a polished metal has an amorphous structure.

* With constancy of the modification.
** This direction has been included by us in the plan of our laboratory.

Beilby’s views were subsequently subjected to severe criticism; indeed, they had been extended without sufficient grounds to a number of other, more complex phenomena (in particular to the sliding of two layers of a crystal relative to one another). However, in the light of the experiments of Finch and others, the question of the formation of amorphous forms on polished planes again arises. If at the time Beilby’s work appeared the experimental technique was insufficient to elucidate the problem, modern physics possesses an excellent tool—the electron-diffraction method. The question of experimental investigations of changes in the structure of the surface layer is, from our point of view, a very urgent question for solid-state theory and surface-treatment technology.

6. The influence of annealing on the mechanical properties of crystals

A rather considerable group of works* is devoted to the question of the influence of annealing on the mechanical properties of crystals. These works concern almost exclusively either rock-salt crystals or single crystals of metals.

As is known, annealing NaCl crystals has an extremely strong effect on their mechanical properties. Thus, for example, as a result of annealing at \(600^\circ\), the elastic limit of NaCl crystals proved to be so low that the crystals bent easily under careless handling,\(^{57}\) while the strength of annealed NaCl specimens proved to be two times lower than that of natural ones.\(^{58,52}\) It is worth noting that, both in experiments with rock salt and with metal crystals, a considerable influence of negligible impurities of extraneous substances on the strength of the crystal was established. Thus, for example, the presence in an NaCl crystal of \(10^{-5}\) moles of \(\mathrm{PbCl}_2\) (Blan and Smekal\(^{60}\)) produced a significant increase in strength.**

The list of individual experiments showing the variability of many properties of crystals depending on their origin and treatment could be greatly enlarged. However, our task is not so much to accumulate experimental material as to draw certain general conclusions.

First of all, it is striking that all the experimental material (in the aspect that interests us) relates chiefly to metal crystals and, because of the convenience of the experiment and especially because of the discussion concerning the Joffe effect, to experiments with rock salt. As for crystals of other substances and other structures, here we have isolated investigations, not united, moreover, by any general theoretical framework.

Secondly, as we have already noted, despite the excellent sim—

* Polanyi and Sachs\(^{56}\), Obreimov and Shubnikov\(^{57}\), Schmid and Faupel\(^{58}\), Blan\(^{59}\), and others.

** The study of changes of this kind as a function of annealing temperature and of impurities must be recognized as a most important task for the technology of production of hard alloys.

The comparison of experimental thermochemical data with theoretical data obtained on the basis of the theory of ionic crystals (Born), and of mechanical and certain other properties of NaCl crystals, proves to be very far from what corresponds to the theory. Smekal^61 (1929) called properties that do not depend on impurities in the crystal, on the degree of their treatment, etc., structurally insensitive, while properties that depend on the origin of the crystal—structurally sensitive; to the first he assigned the thermal and chemical energy of the crystal, the lattice structure, thermal expansion, etc.; to the second—strength, thermal conductivity, electrical conductivity, etc. In recent years a number of properties of the first group have had to be transferred to the second, for example the modulus of elasticity (Sementsov and Kuznetsov^62, etc.); moreover, as V. Kuznetsov quite rightly notes, it is highly probable that other insensitive properties also do not remain unchanged. Smekal and other authors see the reason for such a division of properties into two groups in a certain degree of loosening of the crystal lattice (an explanation that must be admitted to be rather primitive, although not contrary to the truth). As a result, as many authors point out, the existing electrical theory of the solid state is not in a position to explain the variability of a number of properties of crystals and therefore cannot be regarded as satisfactory.

It seems to us that even of the most excellent theory one cannot demand more than is permitted by the basic initial premises of that theory and by the paths along which it develops.

The modern theory of the crystal is (by a tradition inherited from classical crystallophysics) still a theory of the ideal crystal, despite the fact that the direction initiated by Vollmer, Kossel, Stranski, and others essentially sets itself the task of establishing the regularities determining the properties of the real crystal. However, since the systematics of crystals and of crystallization methods remains the same, since a number of problems connected with the behavior of the real crystal have not been posed or advanced for solution, since the theory, as a whole, still treats the real crystal as a kind of exception (a product of crystallization at excessively high supersaturations, etc.), and not as the rule; since the theory must be acknowledged to stand, to a fairly large extent, on the positions of classical crystallophysics—necessary, but not sufficient for solving the problems of the present day.

Intending to speak on a number of questions in special articles, here, in the form of a general statement of the problem, we intend to touch upon several points which seem to us not without interest.

IV. On the Systematics of Crystals

The generally accepted systematics of crystals at the present time, according to the criterion of symmetry elements, lies at the basis of geometrical crystallography, which on this basis constitutes a coherent whole.

Unfortunately, the classical school, which constructed the theory of the crystal as applied to the ideal crystal, tried, in accordance with this trend, to use the systematics of the elements of syngony for constructing physical crystallography, studying the symmetry of a phenomenon in a crystal as a function of the symmetry of the medium (Wulff).

This trend was, in its time, absolutely necessary and quite expedient, having yielded quite definite results. On the basis of the geometrical characterization of the crystal, physical crystallography constructed:

a) the doctrine of the isotropy or anisotropy of the crystal as a whole,

b) the conception of the equilibrium and non-equilibrium forms of the real crystal (in this connection attempts were also made to characterize the relation of the capillary constants of individual faces).

In connection with this specific character of the systematics, crystal physics adapted it for its own purposes by means of physical criteria, namely the properties inherent in definite syngonies of the crystal, and so forth (double refraction, rotation of the plane of polarization, etc.). As an example of such a search for the dependence of the symmetry of a phenomenon on the symmetry of the medium, let us quote lines from the work of G. V. Wulff^63.

“Double refraction occurs in media not of every symmetry: among crystals there are five classes that do not possess double refraction. In the remaining classes, in nineteen of them, this phenomenon cannot occur along one direction, called the optical axis: a light wave traveling in the direction of such an axis is not divided into two,” etc.

The dependence of the symmetry of a phenomenon on the symmetry of the medium—there, in one phrase, is the essence of the systematics of the classical trend in theoretical crystal physics.

Besides this “subordinate” systematics, there were attempts to classify the factual material on the basis of methods for obtaining the crystal.

In general, the results of this trend are excellently reflected by V. D. Kuznetsov in his interesting book^64. The methods for obtaining a crystal are as follows:

1) crystallization from solution,
2) crystallization from a melt,
3) crystallization from a vapor-like state,
4) recrystallization.

This classification of methods, embracing all the numerous cases of crystal formation—from gaseous, liquid, and solid substances and solutions—at first glance seems extremely complete.

However, if one poses the problem of the variation of the constants and properties of the crystal, or, what is the same thing, the problem of the theory and, consequently, the systematics of real crystals, this classification is, in our opinion, insufficient.

In particular, the classification method described above is the method by following which attempts have hitherto been made to obtain crystals as close as possible to ideal ones, but it is not a general classification of methods for obtaining real crystals (see below).

To clarify our starting point, let us return to Smekal’s division of the properties of a crystal into structurally sensitive and structurally insensitive ones. This division is too schematic. Indeed, it is not difficult to show, on the basis of quite general considerations, that one and the same property in some crystals will belong to the first group, and in others—to the second. Let us imagine a crystal having the form of a cube with side length, say, \(1\) mm. If through the center of the cube and the centers of two of its opposite faces one draws a plane and assumes that in the crystal, along this plane, a crack has formed with a thickness of \(10\) atomic distances (\(10\) Å) (Fig. 3), and that the atoms (ions) which occupied this space have been removed (so that the crystal is split in two), but the two halves are not separated, then a whole series of properties of the crystal will not change—thermochemical, chemical, many optical properties, etc. However, other properties will change within very wide limits. Thus, the thermal and electrical conductivity in the direction parallel to the cleaving plane of the crystal decreases by \(10^{-6}\) of the original value (the ratio of the cross section of the crack to the area of the face); in the direction perpendicular to the plane of cleavage, it will fall to zero; the same will be found in studying the tensile strength of the crystal. The resistance to compression will be equal to zero within \(10^{-6}\) of the original volume (the volume of the crack), and then will increase sharply to the original value.

Fig. 3. Crack, \(\parallel\) to the face (100) of a crystal

It is not difficult to imagine that if these microcracks are uniformly distributed throughout the entire crystal, then the specificity of direction, so exaggerated in the extreme case considered, will disappear. Such a real crystal will behave more isotropically when its statistical constants and properties are investigated.

In the general case, a real crystal differs from an ideal one in that, upon the anisotropy caused by the properties of the substance of the crystal, there is superposed the anisotropy of its defects, the resultant of which determines the properties of the crystal. But the anisotropy of the substance of the crystal is a structural phenomenon (depending on the structure of the atoms, etc.—we shall call it structural), whereas the anisotropy of the defects is genetic (depending on the origin of the crystal—we shall call it genetic). It may be anticipated in advance that, with respect to very few properties, both components will always act in the same direction or retain constant values. In the majority of cases there will be observed variations either in magnitude or in direction of one or both components, with all the consequences following from this.

Let us consider, as an example, the case when the structural anisotropy of the crystal

\[ A_s=\mathrm{const}, \]

and let the genetic anisotropy be caused by the simplest reason: the presence of pores in the crystal.

Let the cross-section of the crystal be equal to \(1\ \mathrm{cm}^2\), and its length \(4\ \mathrm{cm}\); then the volume

\[ V_s=4\ \mathrm{cm}^3=4\cdot 10^3\ \mathrm{mm}^3. \]

Let the pores have the form of cubes whose sides are equal to \(2\cdot 10^{-6}\ \mathrm{mm}\); if their total volume

\[ V_i=1\cdot 10^2\cdot 2\cdot 10^{-6}\ \mathrm{mm}^3 =2\cdot 10^{-4}\ \mathrm{mm}^3, \]

then the ratio

\[ \frac{V_s}{V_i}=\frac{4\cdot 10^3}{2\cdot 10^{-4}}=2\cdot 10^7. \]

The total number of pores

\[ \sum=\frac{2\cdot 10^{-4}}{8\cdot 10^{-18}}=0.25\cdot 10^{14}. \]

It is not difficult to see that the conditions have been chosen by us so that the area of the projection of all the pores placed in one plane \(\alpha\), perpendicular to the longitudinal axis of the crystal, is exactly equal to the cross-section of the crystal. Suppose now that these pores diffuse (Fig. 4) through the crystal parallel to its longitudinal axis, i.e. in such a way that the sum of the projections of their bases onto the bases of the crystal \(\beta\) and \(\gamma\) is still equal to the area of these bases.

In case \(b\), by applying electrodes to the bases \(\beta\) and \(\gamma\), we obtain an electrical conductivity equal to zero, for the pores collected in the layer \(\alpha\) will serve as an insulator. In case \(a\), the electrical conductivity will have a finite value.

On diffusing, the pores may also be arranged in chains or steps (\(c\)), i.e. in such a way that the sum of the projections of their sections onto the bases \(\beta\) and \(\gamma\) will be smaller than the areas of the latter. In this case the electrical conductivity again increases appreciably.

If we assume enlargement of the pores, then, for example in the limiting case, the pores will gather into one large cubic (\(e\)) or parallelepipedal (\(d\)) cavity; then, with their negligible total volume, their existence will not at all affect the magnitude of the electrical conductivity in comparison with a crystal devoid of pores.

Thus, the electrical conductivity of the crystal in case \(b\) will be very sensitive to the existence of pores; in cases \(d\) and \(e\), insensitive; cases \(a\) and \(c\) are intermediate. Many other properties and constants of a real crystal will behave analogously.

From the above picture there clearly follows the following conclusion: extremely uniformly distributed defects (\(a\)) and especially extremely concentrated defects (\(e\)) are least of all reflected in the properties of a crystal that depend on genetic anisotropy, while planar defects (\(b\)) are reflected extremely strongly.

In that case, accepting together with Folmer, Kossel, Stranski, and others that crystals grow or tend to grow layer by layer (by lattice nets), we at once establish that the mechanism of growth

Fig. 4. Various cases of the distribution of micropores in crystals

of crystals greatly contributes to increasing the role of defects. It is obvious that if a crystal grows under not very favorable, but not strongly fluctuating, conditions, under which a uniform distribution of small defects takes place, then genetic anisotropy should manifest itself to a lesser degree than if disturbances suddenly intrude into the conditions ensuring ideal growth of the crystal,

…factors, which in the shortest time disrupt the regularity of filling a series of networks.

Let us now return to the question of the systematics of crystals. It is quite obvious that, in addition to the well-known dependence of the symmetry of phenomena on the symmetry of the medium, i.e. in addition to the systematics which we shall below call morphological, it is necessary to establish the dependence both of the symmetry of phenomena and of the symmetry of the medium on the character and distribution of interatomic bonds in the crystal, i.e. to establish a structural systematics of crystals.

Finally, it is also necessary to have, if possible, a more complete genetic systematics, which would facilitate the creation of a general theory of the genetic anisotropy of the crystal. Up to the present time, unfortunately, the situation both in the question of structural systematics and in that of genetic systematics leaves much to be desired. Therefore we shall allow ourselves to set forth here briefly our point of view, which will be discussed by us in more detail elsewhere.

Structural systematics of crystals (S type)

Various types of crystal lattices have long been known*:

1. ionic lattices, for example \([NaCl]\),
2. atomic lattices, ” \([C]_{\text{diamond}}\),
3. metal lattices, ” \([Au]\),
4. molecular lattices ” \([CCl_4]\)

and a series of intermediate ones, for example layered \([C]_{\text{graphite}}\), chain (paraffins), and so on. The general characterization of both the basic and the intermediate types has been set forth many times by various authors (considerable space is devoted to it also in one of our works^65) and has been used by crystal physics.

However, this use had a somewhat one-sided character. From the properties of lattices there follow, first of all, ideas about structural anisotropy, which determines the properties of an ideal crystal. In this direction, corresponding, incidentally, to the classical conceptions of crystal physics, research thought has also worked.

Among recent works in this direction we shall point to the interesting investigation by Stranski^66, who studied the question of the equilibrium form of an ideal crystal with a homeopolar structure, in contrast to crystals of the \([NaCl]\) type, and came to the conclusion that if in \([NaCl]\) the corners and edges are occupied by ions beginning a new layer, then in homeopolar* crystals the corners and edges must be rounded. Other works are also devoted to this same question (for example, Stranski and Kaischew^67).

* We shall use the following designations: \([\,]\) — crystal; \(\|\) — amorphous substance; \(\{\}\) — liquid; \(()\) — gas, vapor.
* Stranski’s terminology.
** An obsolete term (reproduced by us after Stranski). It should not be used.

Since here we are interested in the theory of a real crystal, we intend to approach the problem from an entirely different side: to consider briefly (in Section V) the phenomenon of genetic anisotropy as a function of the structure of the crystal. In establishing dependencies of this kind, it seems to us, the decisive factor is first of all the character and distribution of the bonding forces between the components of the crystal, and only secondarily the structure of its unit cell and the morphological features (syngony, etc.) determined by the latter. With this understanding of the term “structural systematics,” we consider it expedient to construct it in the following manner.

I. Crystals formed exclusively by interatomic forces having a relatively uniform spatial distribution

In crystals of this type all atoms form, as it were, one large molecule.

1. Crystals with ionic bonds:

\[ [\mathrm{NaCl}]. \]

These crystals have a coordination lattice.

1. Crystals with atomic bonds and localized valence electrons.

Such crystals include:

\[ [\mathrm{C}]_{\text{diamond}};\ [\mathrm{CSi}]. \]

1. Crystals with atomic bonds and free valence electrons.

These include metal lattices

\[ [\mathrm{Au}]. \]

II. Crystals formed by interatomic forces noticeably concentrated in certain directions

1. Crystals with atomic bonds and a predominantly planar distribution of interatomic forces. They form layer lattices:

\[ [\mathrm{C}]_{\text{graphite}}. \]

1. Crystals with a predominantly linear (chain) distribution of atomic forces,

III. Crystals formed by a) atomic (or ionic), and b) molecular forces

Such crystals consist of molecules: a) in whose composition the atoms have not completely realized the maximum possible valences and are inclined to realize them when the molecules approach one another,

or b) the external force field of which manifests itself when molecules approach one another in their considerable interaction (rather high sublimation energies and temperatures). The latter type is transitional to IV.

IV. Crystals formed by intermolecular forces

(conditioned by permanent or induced dipole moments, in general by polarization effects; in the limiting case—by van der Waals forces):

1. Molecules with a predominantly three-dimensional distribution of the internal force field (\([{\rm CCl}_4]\); \([{\rm SF}_6]\)).
2. Molecules with a predominantly planar distribution of the internal force field (\([{\rm C}_6{\rm H}_6]\)).
3. Molecules with a predominantly linear (chain) distribution of the force field (\([{\rm C}_n{\rm H}_{2n+2}]\)).

In addition to types 1—IV, certain less important combinations are possible. Let us note that crystals of type S II 1 and S IV (respectively S II 2 and S IV 3) differ from one another in that in the former the planes (or chains) may consist of any number of atoms joined by identical (only interatomic) forces, whereas in the latter both the planes and the chains consist of separate molecules and, consequently, interatomic and intermolecular forces alternate.

It is obvious that two crystals of one and the same syngony but of different types of bonding must have different possibilities for exhibiting genetic syngony.

Since the latter, in turn, also depends on the type of reaction of crystal formation, it remains for us to dwell on the systematics of the processes of crystal formation which seems expedient to us, in order in section V to touch upon certain conclusions.

*

Genetic systematics of crystals (\(G\)-type)

It is necessary to distinguish the following processes of crystal formation:

1. Processes of crystallization of a solid phase in a one-component monovariant (or invariant) system, the crystal being formed from a previously formed substance, the amount of which in the system remains unchanged during the formation of the crystal, i.e.

\[ \frac{dq}{dt}=0. \]

Here should be assigned the following types of reactions of crystal formation:

1. Condensation of vapors of sublimed substances

\[ ({\rm AB}) \rightleftharpoons [{\rm AB}], \quad \text{for example } ({\rm I}_2) \rightleftharpoons [2{\rm I}]. \]

1. Crystallization of liquid substances, in particular melts of the same composition as the crystal

\[ \{AB\} \rightleftarrows [AB], \]

for example,

\[ \{KCl\} \rightleftarrows [KCl]. \]

1. Recrystallization of solid or amorphous substances

\[ [AB]_{\alpha} \rightleftarrows [AB]_{\beta}; \quad |AB| \to [AB], \]

for example

\[ [S]_{\alpha} \rightleftarrows [S]_{\beta}. \]

II. Processes of crystallization of a solid phase in a two- or multicomponent system, where the crystal is formed from a previously formed substance present in a solvated state, the quantity of which in the system during formation of the crystal remains unchanged, i.e.

\[ \frac{dq}{dt}=0. \]

The following two types of reactions should be assigned here:

1. The solvent molecules do not enter into the composition of the elementary cell of the crystal (and may be present in the crystal only by virtue of defects), i.e.

\[ AB_{\text{solution}} \rightleftarrows [AB], \]

for example

\[ KCl_{\mathrm{aq}} \rightleftarrows [KCl]. \]

1. The solvent molecules enter into the composition of the crystal

\[ AB_{\text{solution}} \rightleftarrows [AB_{(\mathrm{solv})n}], \]

for example

\[ LiCl_{\mathrm{aq}} \rightleftarrows [LiCl2H_2O]. \]

III. Processes of crystallization of a solid phase in a multicomponent system, where the crystal nucleates and grows from a substance formed as the result of a parallel chemical reaction proceeding outside the crystal or on its surface, i.e.

\[ \frac{dq}{dt}\ne 0. \]

The following types of crystal-formation reactions should be assigned here:

1. Thermal decomposition of some liquid or vaporous substance

\[ [AB] \to [A]+B, \]

for example

\[ 2(CrO_2Cl_2) \to [Cr_2O_3]+\frac{1}{2}O_2+2Cl_2. \]

1. Interaction of vapors or solutions of two substances

\[ A_m+B_n=C_0+D_p, \]

for example

\[ (NH_3)+(HCl)=[NH_4Cl], \]

\[ 2(CrCl_3)+3(H_2O)=[Cr_2O_3]+6(HCl), \]

\[ Ba^{++}_{\mathrm{aq}}+SO_4^{\prime\prime}{}_{\mathrm{aq}}=[Ba^{++}SO_4^{\prime\prime}]+\mathrm{aq}. \]

IV. Processes of crystallization of the solid phase in a multicomponent system, where the crystal is formed from a crystal (or amorphous substance) of another chemical composition, i.e., as a result of a chemical reaction proceeding within the space occupied by the particles of the new crystal; in this case \(\dfrac{dq}{dp} \ne 0\). The following types of reactions of crystal formation should be included here:

1. Dissociation of the initial crystals or amorphous substances with separation of part of the substance

\[ [AB] \longrightarrow [A] + B, \]

\[ |AB| \longrightarrow [A] + B, \]

for example

\[ 2|Cr(OH)_3| \longrightarrow [Cr_2O_3] + 3H_2O. \]

1. Absorption by the initial solid substance of other substances

\[ [A] + B = [AB], \]

for example

\[ [CaCl_2] + 8(NH_3) = [CaCl_2 8(NH_3)]. \]

1. Interaction of the initial crystal with another substance, with the formation, as a result of the reaction, among other products, of a crystal of new composition

\[ [AB] + C = [A] + BC, \]

for example

\[ [CSi] + 2(Cl_2) = [C] + (SiCl_4). \]

We have already noted that crystal physics, until very recently, had created a theory of the ideal crystal.*

This could not but be reflected also in the approach to the methodology of crystal formation. It is enough to compare V. D. Kuznetsov’s classification with the one given above in order to be convinced that the former encompasses only the reaction types \(G I\) and \(G II\), which were attributed by theoretical chemistry to physicochemical reactions, which the formation of a crystal formally is. However, even between these groups of processes there is already a substantial difference, consisting in the fact that in reactions of group \(G II\) the dissolved substance must free itself from the solvent either completely (\(G II\ 1\)) or partially (\(G II\ 2\)) before entering the crystal lattice. And since desolvation proceeds on the surface of the crystal, it is obvious that in the surface layer a very complex

* As a result, a considerable gap arose between the theory, which strove to obtain a crystal under ideal conditions, to grow separate nuclei, and so forth, and technology, which obtained crystals under directly opposite conditions of enormous supersaturations and growth rates, in the presence of millions of nuclei, nonuniform temperature and concentration regimes, etc. The absence of a special investigation of such crystals could be tolerated from the point of view of those branches of production which are interested in the output of a particular substance and where the specific properties inherent in the crystalline form play no special role (the production of granulated sand, distillation and sublimation, etc.). However, for many industries (for example, abrasive materials and hard alloys) the structural elements of the crystal are not secondary. This must be especially emphasized when we speak of the need to create a theory of the real crystal.

an environment that cannot be compared with the case, for example, of the formation of a crystal from vapor ($G$ I).

A wholly exceptional neglect has befallen the crystallization processes of groups $G$ III and $G$ IV, whose “misfortune” is that they lie on the boundary between crystal physics and chemistry, as a result of which both disciplines have passed by these processes, as though yielding to one another the pleasure of dealing with them. Indeed, any type of reaction of groups $G$ III and $G$ IV has been perfectly well known to chemistry for a long time. For example, reactions such as

\[ 2(\mathrm{CrO_2Cl_2})=[\mathrm{Cr_2O_3}]+\frac{1}{2}(\mathrm{O_2})+2(\mathrm{Cl_2}) \]

have been known by the dozens. Chemistry, however, was interested in all their details except one—the structure of the solid phase being formed and the process of formation of the crystal. Crystal physics, on the other hand, at the mere sight of equations like the one written above, classified them as chemical.

It is extremely important to emphasize that in all processes of group $G$ III there are two reactions superposed upon one another:

1) the formation of the substance of the crystal,
2) the formation of the crystal,

so that, for example, the reaction

\[ 2\mathrm{HJ}=\mathrm{H_2}+[2\mathrm{J}] \]

in essence consists of two

\[ 2\mathrm{HJ}=\widetilde{\mathrm{J}}_2+\mathrm{H_2},\qquad \widetilde{\mathrm{J}}_2+\mathrm{H_2}=[2\mathrm{J}]+\mathrm{H_2}, \]

where $\widetilde{\mathrm{J}}_2$ denotes the emergence of the substance iodine, which forms iodine crystals. It is quite obvious that ordinary chemical equations obscure the essence of the matter, as it were identifying the process of formation of iodine with the process of formation of the iodine crystal. Incidentally, we write $[2\mathrm{J}]$, and not $[\mathrm{J}_2]$, since iodine forms an atomic-molecular lattice. It seems expedient to us to distinguish the notation in the following way:

\[ \begin{array}{ccc} 2[\mathrm{A}] & [2\mathrm{A}] & [\mathrm{A}_2] \\ \text{atomic lattice} & \text{atomic-molecular} & \text{molecular lattice,} \\ & \text{lattice} & \text{consisting of } \mathrm{A}_2\text{-molecules} \end{array} \]

i.e., for iodine one would have to write in the corresponding cases:

\[ 2[\mathrm{J}] \qquad [2\mathrm{J}] \qquad [\mathrm{J}_2] \]

In processes of group $G$ III one cannot confine oneself to the study of the crystal-formation reaction while trying to separate it from the reaction of formation of the substance of the crystal—this yields only negative results. To an equal extent, the regularities of crystal formation established for processes of group $G$ I and partly $G$ II cannot be applied in this case without reservations.

The processes of group \(G\) IV are very complex; they differ from cases of recrystallization, or of mutual transition of modifications \(G\) I 3, in that within the crystal a chemical reaction simultaneously takes place, and the new crystal must be formed from a substance distributed in the space occupied by the old crystal, which consisted of another substance, i.e., in a space in which there occurs continuous diffusion of foreign products or components of the reaction.

Processes of this group have repeatedly been investigated in chemistry, but only from a purely chemical standpoint. At best, certain properties of the crystals formed were taken into account, such as, for example, the apparent specific gravity and the structure of the elementary cell according to X-ray diagrams taken at some indefinite moment, most often after the completion of the process, i.e., at the least interesting stage.

Among such experimental works that have clarified at least the specific gravities of preparations, we shall note, for example, the interesting investigations of Le Blanc and Richter \(^{68}\), who determined the specific gravity of \(MgO\) obtained in different ways. It turned out that the specific gravity of \(MgO\) obtained from calcined \(Mg(OH)_2\) reaches 3.593; from \(MgCO_3 \cdot 3H_2O\), 3.538; from magnesite \(MgCO_3\), 3.525; from \(Mg(NO_3)_2 \cdot 6H_2O\), 3.332 (whereas from X-ray diagrams, as Runge found, it follows to be 3.58), while the specific gravity of periclase (natural \(MgO\)) reaches 3.75.

In the same work it is noted that small hexagonal needles of \(MgCO_3 \cdot 6H_2O\), upon calcination, transform into \(MgO\) without losing their former appearance; with preservation of the crystal volume, the specific gravity of the latter falls to 1.80, i.e., the crystal (as should be expected) has pores.

Other authors characterize, with analogous data, the specific gravities of various substances obtained in the reaction \(G\) IV 1. Deeper investigations, unfortunately, are not known to us. The situation is even worse with reactions \(G\) IV 2 and \(G\) IV 3*.

Meanwhile, the substances obtained as a result of reactions of group \(G\) IV have great industrial significance**. This is yet another weighty argument in favor of the mechanism of formation of these crystals and their properties becoming the object of serious investigations.

We shall now return to the question of the properties of a real crystal as dependent on its structural and genetic characteristics.

V. On Certain Regularities in the Formation and Properties of Real Crystals

Here we shall touch upon only some of our views on this complex problem.

* Some investigations have been begun in our laboratory.

** These preparations have an enormous surface; in a number of cases they must form extremely dispersed grains (polishing magnesia); in bulk they must conduct electric current and heat extremely poorly (insulators), etc.

1. Structural Inertia. Quasi-stable State

As is known, the valence angles of a molecule consisting of more than two atoms, owing to the vibrational motion of the latter, change slightly, as do the distances between the centers of the atoms. If the crystal belongs to type S IV, these changes are very little hindered by the interaction of the molecules, owing to the insignificance of the intermolecular forces.

On the contrary, in crystals of type S I, in which all the atoms are connected into one continuous whole, a spontaneous decrease in the valence angles or in the distances between some atoms is accompanied by an increase in them between neighboring atoms, which impedes this change. On the other hand, if in some part of the crystal a certain energy barrier has been overcome and the system has passed at that point into a metastable state, the return to the former (stable) configuration of this group of atoms is impeded for the same reasons. As a result, unlike gases, fluctuations in crystals may be metastable. Here we are dealing with the structural inertia of matter carried to its limit.

If a given substance can form several modifications, then, according to the step rule, the less stable forms tend to arise first, passing into more stable ones (Ostwald). Whereas recently some authors have considered this rule applicable also to crystals (Stranskii and Totomanow^40 and others), others, for example Ellinek^17, regard it as theoretically unfounded, since sometimes several modifications arise simultaneously, sometimes only a stable one, and sometimes only an unstable one. It seems to us that, in the general formulation of the question, the discussion will be entirely fruitless.

In the familiar game, balls, rolling down an inclined plane, fall into intermediate hollows. Depending on its supply of energy, a ball may either skip over the upper hollow without falling into it, or be detained in it for a moment in order to continue its fall, and, finally, become firmly stuck in it, requiring activation energy (a push) in order to leave this metastable state (Fig. 5a).

In gases, in liquids, and even in many amorphous substances, the components are such more or less independent balls with stores of energy that differ among different molecules (at a given temperature and pressure) within fairly wide limits in accordance with the Maxwell distribution curve. As a result, if several intermediate states are possible, there will always be found many molecules passing into the stable state by each of the various paths.

Naturally, each molecule is by no means obliged (moreover, by its energetic possibilities it cannot) to be detained in all the metastable states. Therefore the step rule would better be formulated as follows: “If, in addition to the final stable-

... state, possibly several metastable ones; different components or parts of the system may pass through some of them, or through all of them, depending on their stores of energy and on the height of the potential barriers of each of these states.”

In contrast to other aggregate states in crystals, especially of types $S$ I and $S$ II, thousands of spheres, falling into a potential well or overcoming a potential barrier, are bound together. Depending on the character of these bonds, the structural inertia of the substance must manifest itself in different ways; however, in most cases it leads to the metastable state in crystals acquiring a specifically stable character (Fig. 5b). Therefore, if one compares, in the sense of reaction conditions, a medium in which there are no strong binding forces

Fig. 5. Metastable and quasi-stable states relative to the stable state

between atoms (molecules) with an inclined board having shallow hollows, then the crystalline state of types $S$ I and $S$ II corresponds to the relief of a mountain range gradually descending toward the sea, with the highest peaks and deep abysses into which, once fallen, it is difficult to climb out. The most generally known example of the difficulty of such a transition from a metastable form to a stable one is the coexistence of diamond and graphite; their heats of combustion are very close and are respectively equal to 7869 and 7856 cal. Thus, for such crystals the metastable state has an entirely different physical meaning than for other aggregate states: with sufficient depth of the potential well, the probability that the system will get out of it is so small that it becomes immaterial whether there is nearby an even deeper well of the stable state into which the system could pass. Such metastable states we shall call quasi-stable. In contrast to crystals of type $S$ IV,* crystals of types $S$ I and $S$ II need not necessarily occur in stable modifications, but in those modifications that follow from the method of obtaining the crystal. It is precisely to this that quasi-stable white tin owes its everyday prevalence.

* In which the height of the potential barrier is considerably lower.

This special character of the metastable states of definite types of crystals has been noted by many prominent investigators*. Let us emphasize that from what has been said there follows a series of important practical consequences concerning the dependence of the modification and structure of a crystal on the method of obtaining it, a dependence reflected in different ways in the properties of crystals of different types of chemical bonds. Another aspect of structural inertia, manifested in real crystals, consists in the following: by analogy with the transition of the solid state into the liquid (and vice versa), it is customary to think that the transition of one form into another occurs at a strictly defined temperature. States similar to supercooling (supersaturation) are regarded in classical chemistry rather as exceptions.

In crystals, however, the process of transition of one form into another must proceed considerably more complexly, with a “smearing out,” in certain cases, of the transition points of quasi-stable forms into stable ones, and of a form stable under former conditions into a stable one under new conditions.

For example, structural inertia must manifest itself in the fact that a quasi-stable form passes into the stable one over a very broad interval, for instance of temperatures, so that instead of transition points (or smeared-out points), transition regions must appear.

The appearance of transition regions has often been observed before in cases where the modification undergoing transition contained a dissolved impurity (for example \(SO_2\) in sulfur), and in this the effect of the presence of so-called mineralizers during crystallization of substances was seen above all. However, the action of mineralizers may be much subtler. Thus, in a recently completed work in our laboratory** it was shown that amorphous chromium oxide passes into the crystalline state, depending on the composition of the gas present, within the temperature interval from 380 to \(630^\circ\). The transition itself is also extended in time. The process proceeds in a very complex manner.

An extremely interesting report on another kind of retarded transitions, using \(NaNO_3\) as an example, was published by Krachak and Poznyak \(^{69}\) (1931); references to other authors are also given there.

In one way or another, structural inertia, characteristic of the crystalline state, may manifest itself in rather varied forms depending on the \(S\)- and \(G\)-types of the crystal.

2. On certain regularities of crystal growth

The picture of crystal growth adopted by Kossel-Stranski and Volmer seems to us somewhat one-sided; it was developed chiefly as applied to \(NaCl\) crystals; moreover, despite taking into account a number of details of the process, it is not devoid of schematic character.

* This will be considered in a special article.
** Responsible executors: M. A. Khachvankyan and A. M. Yakubovich; the work is being prepared for publication.

Let us try to consider the phenomenon more generally.

First, it is by no means necessary that the particles impinging on a growing face be monatomic or monoionic. Under conditions of \(t^\circ\) and \(p\), when growth of the crystal is possible, interatomic forces are so great that interaction of atoms (ions) with one another before they reach the crystal face becomes unavoidable, with the formation of microaggregates (for example, in crystals \(S\) I, \(S\) II, and \(S\) III).

Such formation of microaggregates can be avoided either in those cases where the forces of interaction between the particles of the vapor or liquid from which the crystal is being built are very small (\(S\) IV), or if some cause can keep the particles from interacting. Such causes may include:

a) the presence of a solvating shell (solvent molecules), from which the component can free itself only at the crystal face (\(G\) II),

b) the occurrence of the reaction forming the substance (\(G\) III), of which the crystal consists, on the wall of this crystal, when, consequently, the atoms being formed come into contact with the wall earlier than with one another,

c) in systems \(G\) I, by cooling not the entire mass, but the growing plane of the crystal (nucleus).

Of course, microaggregates, once they reach the surface of the crystal, may in some cases be sources of micropores, i.e., of the creation of a certain volume \(V_i\).

Second, however, such microaggregates cannot be regarded purely mechanically as a heap of stones, once and for all piled upon the surface of the crystal, of unchanging form irrespective of the size and structure of these microaggregates. If, as Folmer has shown, on the surface of some crystals there exists a two-dimensional mobile layer in which migration of particles is possible, then above all one must remember that the conditions of migration, depending on the type of crystal, must vary within the broadest limits in accordance with the character of the force field and the binding energy of the atom with the surface [i.e., on the height and character of the potential barrier which the atom (molecule, etc.) must overcome in passing from one point of the surface to another]. It is quite obvious that in the case of very strong bonds of atoms with the surface (for example, diamond) the conditions of migration are very unfavorable, which cannot be said of crystals with weak bonds (for example, benzene).

Likewise, the correction of growth defects as a result of detachment (evaporation) of the most weakly bound molecules (atoms), with condensation of other molecules in places with a more significant force field (cracks, etc.), is also facilitated in the case of low bond strength.

Thus crystals of the type \(S\) I, \(S\) II, \(S\) III have an entirely different migrational characteristic of the surface, and therefore different possibilities for filling surface pits and, consequently, eliminating defects (voids). (Of course, for one and the same crystal this characteristic changes depending on...

dependence on temperature. At a sufficiently high temperature, or with a low bond strength of the given \(S\)-type, the probability of migration increases greatly not only on the surface and within cavities inside the crystal, but also in layers filled with atoms, which have “holes” of atomic dimensions.

In connection with this, for example, in the crystallization not only of substances of one and the same \(S\)-type, but even of one and the same substance from solution with or without the formation of a crystal solvate, the conditions for correcting defects change sharply. Since the energy of detachment of an unsolvated ion is considerably higher than that of a solvated one, the production of good crystal solvates should in general be more successful than the production of crystals of anhydrous salts. Equally, the formation of good fluoride crystals, generally speaking, should proceed with more difficulty than that of chlorides, bromides, or nitrates. (Of course, one must not disregard also the \(G\)-type of the crystal, which may introduce corrections caused by the method of obtaining the crystal.) Crystals of type \(S\) IV have a greater possibility, when formed from vapor, of turning out better than crystals of type \(S\) I (compare sublimed gold \(S\) I and benzoic acid \(S\) IV).

With sufficiently easy migration, the formation of a crystal from microaggregates whose magnitude does not exceed the dimensions of the elementary cell may not be accompanied by so noticeable a number of voids as in the case where migration is hindered, owing to the possibility that these voids are filled at the moment the microaggregate reaches the surface, or even after that. On the contrary, with considerable potential barriers, the formation of microaggregates proves to be an irreparable evil; if one takes into account that, for example, the solid compounds used in industry must all have very high potential barriers, it is not difficult to conclude that in obtaining them, for example, according to type \(G\) III 2, the growth of crystals from microaggregates means a high probability of lowering the mechanical quality of the product. In such cases the \(G\)-type of the reaction of crystal formation from gaseous components (\(G\)-characteristic) may play a decisive role.

Let us consider, as an example, two reactions:

\[ 1)\ 2(\mathrm{CrO_2Cl_2}) = [\mathrm{Cr_2O_3}] + (\mathrm{Cl_2}) + \frac{1}{2}(\mathrm{O_2}) \]

and

\[ 2)\ 2(\mathrm{CrCl_3}) + 3(\mathrm{H_2O}) = [\mathrm{Cr_2O_3}] + 6(\mathrm{HCl}). \]

In both cases gaseous components decompose with the formation of \([\mathrm{Cr_2O_3}]\).

However, in the first case the reaction proceeds almost exclusively on the crystal surface, so that the decomposition product \([\mathrm{Cr_2O_3}]\) can immediately join the surface of the face. In the second case the reaction also readily proceeds in the volume, where microaggregates of \(\mathrm{Cr_2O_3}\) are formed, from which the crystal must grow. It is obvious that in the sec—

case, the conditions for the formation of large crystals are worse; on the contrary, the formation of the smallest disaggregated crystals is stimulated.

Thirdly, the structure of microaggregates may prove to be different from the structure of the unit cell. In this case, even in very small aggregates, a restructuring of the microaggregates will be required, etc.

What has been said may, in general, be summarized as follows. Minimal manifestations of genetic anisotropy should be observed in the growth of crystals from particles: a) whose size is, if possible, smaller than the unit cell of the crystal and whose structure corresponds to the structure of the latter; b) whose components (atoms, molecules, ions) are characterized by small values of the potential energy barriers during migration on the surface, or, in a first approximation, by small energies of detachment or excitation of valence bonds. In all cases, the formation of a crystal from smoke or mist must lead to exceptionally pronounced genetic anisotropy. Therefore, for example, the formation of crystalline structures of substances of the type \(M(OH)_n\) (where \(M\) is a heavy metal), which usually form colloidal systems, should be possible under conditions such that the substance of the crystal is obtained by a reaction proceeding on the wall of the embryo. And indeed, as is known, for example, in the cathodic reduction of ammonium chromate (and in other cathodic processes), crystalline \(Cr(OH)_3\) is obtained (as are crystalline hydroxides of other elements).

3. On certain defects of real crystals of various \(S\)-types

Of course, the defects of a crystal may vary to a greater extent depending on the \(G\)-type of the crystal: on the method of preparation, on the type and order of the reaction, etc. In the present article, unfortunately, for lack of space, we are not able to deal with this question to the proper extent. However, we would like to dwell briefly on the question of the extent to which the \(S\)-type entails a tendency toward the occurrence of defects.

§ 11. Crystals with ionic bonding

Bonds can be calculated according to the laws of electrostatics. An ion undergoes uniform polarization and creates a field of high symmetry. Thus, with mutual displacement of the ions

* Let us note that, by reaction (1), in our laboratory there were obtained not only single crystals of chromium oxide up to 11 mm in diameter with an apparent specific gravity of 5.19 (the true specific gravity is 5.23 according to the X-ray pattern of V. I. Kasatochkina), but also a finely dispersed powder with a specific gravity of 5.20.

under the action of an external force, ions automatically tend to form configurations possessing a minimum of energy, and the number of ions surrounding a given ion is not limited by the numbers of valence electrons or by definite valence angles.

Ions do not form separate molecules inside the crystal, so that the displacement of one of them does not require overcoming additional intramolecular forces. Owing to these properties of the force field, an ionic crystal is fairly plastic, despite the large values of the lattice energy. Schematically this property of ionic configurations is shown in Fig. 6a.

Fig. 6. Different behavior of structures with ionic (a) and atomic (b) bonding upon the incorporation of a new ion (respectively, atom)

The specific character of the ionic force field excludes the capture of a number of impurities, the formation of chemical compounds with oxygen and other substances that contaminate the crystal or its surface. As a result of considerable magnitudes of the solvation energy with polar and readily polarizable molecules (H₂O, NH₃, etc.), in the presence of the latter the energies of detachment of ions from the crystal must sharply decrease, migration must be facilitated, and plasticity must increase.*

The tendency of the electrostatic force field automatically to create energetically more favorable configurations leads to the fact that the phenomenon of allotropy in typical salts is weakly expressed, but appears at once as soon as the bond acquires a more atomic character (compare KJ and AgJ).

This circumstance and the facilitated migration of ions in the presence of a solvent, etc., strongly soften the range of fluctuations of the specific gravities (see above) of crystals with ionic bonding, caused by large values of \(V_i\). Incidentally, in fluorides the tendency toward fluctuations should be greater than in chlorides, and in anhydrous salts greater than in crystal hydrates; the growth of large crystals of good quality should proceed more easily in the case

* Incidentally, in addition to studying the behavior of NaCl in water (when testing mechanical properties), it would be useful to study in detail the change in the strength of crystals of different S- and G-types in the presence of various solvents or their vapors.

crystal solvates and be hindered in anhydrous salts. Let us note that crystallization of the latter in the presence of molecules of substances capable of solvating the ion and, consequently, of aiding its detachment and migration, can improve the properties of the crystal of an anhydrous salt. The presence of solvents can therefore, in general, and should, have a sharp effect on the genetic anisotropy of the crystal.

§ 12. Crystals with Atomic Bonding

(localized electrons)

Between the atoms of such substances there appear forces that cannot be calculated according to the laws of electrostatics. Atoms tend to distribute bonds at definite angles (valence angles) and to realize a strictly definite number of bonds in accordance with the quantum characteristics of the atoms and the conditions in which the system is found.

As a consequence of this character of the bonds, all kinds of displacements of atoms, including those involving an external force both in the lattice and in the surface layer, are extremely difficult, i.e. the potential barrier is very high. In contrast to the electrostatic field, which automatically arranges charges, while the number of ions of the opposite sign around a given ion may increase without limitation imposed by the valence of the element (for example, reaching 6 or 8 for “monovalent” ions), in the case of an atomic lattice with localized electrons each atom can retain around itself no more than a strictly definite number of atoms; and an introduced new atom will be pushed out by this system: until some one of the bonds existing at this site is broken, it will be unable to form any bond (Fig. 6).

To this it must be added that any distortions of the valence angles cause additional repulsive forces. This is why, if in crystals of this type some part of the bonds proves to be broken, the free atoms will adapt themselves at other points only with great difficulty, i.e. there will simply be a rupture of some bonds without the immediate establishment of new ones, since for this it is necessary to break still another group of bonds in order to create free places.

The character of the surface of crystals of type S I 2 is also quite specific in comparison with S I 1.

Whereas in an ionic crystal the ions create an unsaturated force field, into which ions of the opposite sign, once they enter it, are readily retained; in a crystal of type S I 2, the surface atoms strive mutually to saturate the free valences (as in the benzene ring). Therefore even a free atom that reaches the surface must overcome a peculiar energy barrier; having overcome it and entered the potential well, it will with still greater difficulty move to a new position, since it will have to break both old bonds and new ones.

These rather high barriers hinder the formation of surface layers through interaction with the oxygen of the air.

and even adsorption, which on diamond or carborundum is probably not great.

Since migration, which is a means of self-perfection of the crystal, is hindered, good crystals can be obtained only under conditions of very slow crystallization.

§ I 3. Metals

In metals the bonding energy varies within very wide limits. However, even in cases of very strong lattices, the specific properties of the metallic bond ensure that a shearing atomic residue can relatively easily assume a new configuration, and, consequently, also considerable plasticity.

In one of our works it was shown 70, 71 that the melting temperatures of simple substances with an atomic lattice, including metals, must correspond and do correspond to the values of the maximum valence of these elements in their compounds. There is no doubt that the strength of interatomic bonds in metals is closely connected with the number of valence electrons effecting the bond. Within a period, with transition to elements with high values of the number of $w$-electrons, not only does the melting temperature of the metal increase, but also its brittleness; the most plastic are the elements with the smaller number of $w$-electrons (compare Ta—W—Os—Ir—Pt—Au).

For elements with large values of $w$ within a subgroup, brittleness also increases with atomic number (compare Cr—Mo—W; Fe—Ru—Os) in parallel with the growth of the value of $w$. Conversely, for elements with a small value of $w$, which in such cases is the same within a subgroup, there is observed, as was to be expected, rather the reverse course of brittleness (compare Cu, Ag, Au) *.

The presence of free electrons on the surface of metallic crystals facilitates the ready formation of oxide and other films.

In the molten state metals dissolve gases. Accordingly, metals easily form good crystals even with comparatively rapid crystallization, provided extraneous impurities, which may be occluded and chemically incorporated, are excluded; in the case of metals with a great bond strength it is nevertheless desirable that the release of atoms should occur on the growing surface, and not in the volume above it, in order to avoid the formation of microaggregates. In the presence of conditions permitting the occlusion of gases, owing to the high viscosity of metals, the specific gravities of the crystals vary within very wide limits.

§ II 1st type. Crystals with an atomic-layer lattice

In this case, in general, what was said concerning § I 2-crystals is applicable, with the correction that characteristic for type § II 1 is the presence of weaker bonds perpendicular

* The question will be considered in greater detail elsewhere.

planes where force fields are concentrated (cleavage planes); on the one hand, this weakening of certain bonds may facilitate the migration of atoms parallel to the cleavage planes; on the other hand, the rupture of the lattice by external actions, both mechanical (splitting) and chemical (for example, oxidation), is also facilitated. Thus there is a rather complex interaction of opposite tendencies, whose influence must be taken into account in each particular case.

However, in cases where lattices of this kind were formed according to type \(G\) III and especially \(G\) IV, the components and products of the reaction have the possibility of being retained between the layers. As a result, lattices with a layered distribution of impurities and with considerable fluctuations in specific gravities and other properties are formed. A good example is graphite: its specific gravity, according to Landolt, ranges from 2.10 to 2.32 (10%). However, after purification and pressing, a whole group of graphite varieties acquires a specific gravity of \(2.25 \pm 0.01\) (0.5%).

§ III-d. Crystals with an atomomolecular lattice

Molecules of a number of substances (sulfur, selenium, etc.) possess the clearly expressed property, with increasing temperature, of dissociating many times into parts and, conversely, as the vapor cools, of assembling into polyatomic microaggregates with a highly fluctuating number of atoms in the molecule. Thus the force field of such molecules can carry out interaction not only by means of secondary, but also by means of principal valence. This exceptional ability of molecules to combine, forming from small molecules ever larger and larger ones (for example, in sulfur \(S^1 \to S^2 \to S^4 \to S^8\), etc.), where the elementary cell already consists of tens of atoms and is very complex, leads to the fact that, when a crystal is formed from vapor, the optimum conditions of crystallization are very rarely observed: large microaggregates of the most diverse forms, grown with exceptional speed and, consequently, not free of voids, reach the surface of the crystal. The structure of the elementary cell of such microaggregates, owing to the sharply changing composition of the molecules, has every chance of differing from the stable structure of the crystal; and the rather considerable intermolecular forces, causing an impressive structural inertia, facilitate the formation of metastable and even quasi-stable forms. Hence it is understandable that precisely among such substances the phenomenon of allotropy is widely widespread (sulfur, selenium, phosphorus, arsenic, \(As_2O_3\), \(SO_3\), etc.); in these same substances one should expect considerable fluctuations in the specific gravities of crystals of one and the same modification depending on the \(G\)-type (reaction of formation) of the crystal. In fact (according to Landolt)

Substance	Form	Specific gravity
\(As_2O_3\)	rhombic	3.85—4.15
\(As_2O_3\)		3.985—4.250
\(Sb_2O_3\)	regular	5.11—5.30
\(Sb_2O_3\)	rhombic	5.56—5.78

Thus even the first digit after the decimal point already fluctuates.

Compare also the properties of sulfur obtained in the reaction

\[ S_{\text{carbon disulfide}} \longrightarrow [S] \]

\[ (S) \longrightarrow [S] \]

\[ 2(H_2S) + (SO_2) \longrightarrow 2(H_2O) + [3S]. \]

Closely related to these substances are \(Al_2O_3\), \(Fe_2O_3\), etc., which are important for the technology of abrasive processes; their specific gravities fluctuate almost within the same limits.

It is interesting that substances transitional from typical salts \((KCl)\) to substances of type S III also reveal the appearance of noticeable fluctuations in specific gravities. If the specific gravity for \(KCl\) agrees quite well: \((25/4)\ d = 1.987;\ (30/4)\ d = 1.934\), then \(AgCl\) (fused) already shows specific gravities \(5.517—5.594\) (mean \(5.56\)), i.e., strong fluctuations in the second digit.

S IV, etc. Crystals with a molecular lattice

Interatomic bonds are saturated within the molecules forming the lattice. In the case of dissociation, not more than one stage is possible \((2NO_2 = N_2O_4)\). There are no free valence electrons readily entering into interaction in the lattice, so that the formation of layers of foreign atoms chemically bound to the atoms of the lattice (with the formation of oxide, hydride, etc. films) does not occur. The weakness of intermolecular forces facilitates the migration of molecules and in general their detachment from the surface, and consequently the correction of certain genetic defects of the crystal, especially in the case of molecules of spherical symmetry. On the contrary, long, linear molecules (chain molecules, etc.) must be less capable of displacement; with rapid crystallization their disorderly piling up on the growing surface is quite possible, like a heap of logs thrown down haphazardly.

In the case of considerable molecular length, various complications are possible depending on the \(G\)-type of the crystal. The presence in chain molecules of groups with a considerable dipole moment, when crystallization is not too rapid, should promote the orientation of the molecules and, consequently, in a number of cases the formation of more regular crystals than in the case of chain nonpolar molecules*.

In a similar manner, a number of other properties of real crystals must depend sharply on the \(S\)- and \(G\)-types of the latter. Being unable, for lack of space, to expand the scope of the material considered here, we shall return to this subject in detail in special articles.

A few more words about the handling of crystals in branches of industry interested in properties inherent specifically in the crystalline state.

* The dependencies, however, must be considerably more complex (this will be considered elsewhere).

A crystal is not something immutable. By unskillful processing one can ruin even the best crystal. For example, by subjecting crystals of hard compounds to a sintering operation (in the production of hard alloys), one can sometimes spoil a good crystal or (rarely) improve a bad one, and it is difficult to preserve it unchanged.

We consider it necessary to emphasize this once again, for from a failure to understand this, or from inattention to it, defects and the irreproducibility of the results of the technological process of manufacturing hard alloys may arise.

Finally, one example relating to abrasive substances. That the abrasive properties of crystals change noticeably when they are ground is known to everyone. Not everyone, perhaps, realizes how significant these changes may be. A striking example* is black chromium oxide. If it is ground in a mortar, it is hard to believe that the resulting green powder, quite reminiscent of a lubricating powder, has the same unit-cell structure as the ground black, lustrous crystal that scratched glass and quartz. Incidentally, such metamorphoses should be expected especially in crystals, for example, of hexagonal structure, i.e., with bonds considerably weakened in some definite direction, which, as is known, contributes to the emergence of lubricating properties. However, similar changes in the properties of crystals as a result of various actions are to a large extent inherent in crystals of all syngonies.

In one way or another, a whole series of branches of industry that deal with crystals must not forget the characteristic peculiarities of real crystals as compared with ideal ones; the task of science now is to create a theory of real crystals of various $S$- and $G$-types, and this means a theory of the dependence of the defects of a crystal on the character of the bonds in its lattice, on the methods of obtaining it, and on its subsequent processing.

LITERATURE

1. P. Curie, Z. Kryst., 12, 651, 1887.
2. G. Wulff, Z. Kryst., 34, 512, 1901.
3. A. Rietzel, Z. Kryst., 49, 172, 1911.
4. H. Liebmann, Z. Kryst., 53, 171, 1914.
5. P. Ehrenfest, Ann. Phys., 48, 360, 1915.
6. M. Jamada, Physik. Z., 24, 364, 1923; 25, 52, 1924.
7. Biemüller, Z. Physik, 38, 573, 1926.
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* Observed in the work mentioned above with chromium-oxide preparations studied in our laboratory.

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