On the Question of Melting Eutectic Alloys

D. D. Saratovskii and P. A. Savintsev

Introduction

Many works have been devoted to investigations of eutectic alloys; however, the question of the causes of their low melting point has not yet been clarified. In most cases attempts are made to explain this low melting point by the properties of the eutectic structure.

We approached the question of the formation and melting of eutectics from the standpoint of the modern theory of the structure of solids, set forth in recent times chiefly in the works of the Soviet scientist Ya. I. Frenkel. Therefore, before describing our experiments, we preface them with a brief exposition of the question of the structure of real crystalline bodies and liquids and of the mechanism of melting.

The Structure of Real Crystalline Bodies

Crystalline bodies are usually ascribed a regular structure in the form of a spatial lattice, at the nodes of which atoms are situated. Thermal motion is explained by oscillations of atoms about positions of equilibrium. The theory of such a regular structure of crystals cannot explain a whole series of processes occurring in solids, such as, for example, evaporation, dissolution, and diffusion. To explain these processes one must abandon the notion according to which the atoms of a crystal merely perform small oscillations about positions of equilibrium. It is necessary to suppose that considerable displacements of an irreversible kind are possible, associated with the departure of atoms from their initial positions of equilibrium.

An atom that has broken away from its position of equilibrium passes into a new position of equilibrium in a previously free node, or interstice. Such transitions are associated with the appearance of defects in the regular crystal lattice in the form of “holes,” i.e., free internal lattice nodes, or else in the form of dislocated atoms, i.e., atoms that have penetrated into the spaces between nodes. This may occur both inside the crystal and on its surface. The former was considered by A. F. Ioffe [7] in 1923 and was called by him the dissociation of crystals. Dissociation cannot be considered complete as long as the dislocated atom is in the immediate vicinity of the hole it has left; therefore such a state is called predissociation [1]. It may be followed either by complete dissociation, when the dislocated atom moves to a position more distant from the original node, or by recombination of the atom with the hole, i.e., the return of the atom to its original position. Complete dissociation may be accomplished not only by displacement of the dislocated atom, but also by the transition of the hole left by it to a more distant lattice node, as a result of the transition of an atom located in that node into the node left by the dislocated atom [8]. When the dislocated atom and its hole are not in immedi-

owing to their substantial closeness to one another, they may be regarded as completely dissociated, and they can move independently of one another.

If, inside a crystal, dissociated atoms and holes arise jointly, then on the surface of the crystal they can form independently of one another. Their displacement into the interior of the crystal, through interstices or sites, leads to the formation of two independent types of disturbances of the crystal structure.

Holes may also arise by means of the process of “dissolution in the crystal of the surrounding void” [1]. This process begins with the outward displacement of a surface atom, without separation from the surface of the crystal, and with its attachment to the surface of the original layer, which initiates the formation of a more external layer. As a result, a “hole” appears, as it were adsorbed on the surface of the crystal, which is then replaced by the atom that had previously been beneath it. Ultimately the hole is, as it were, absorbed by the crystal from the surrounding space.

The process by which dissociated atoms arise may proceed independently of the formation of holes. It is analogous to the penetration of holes from the surface into the depth of the crystal and begins with the transition of one of the surface atoms into the nearest interstice. After this, the given atom can move into a neighboring interstice lying farther from the surface of the crystal. The binding energy, upon displacement through the interstices, practically does not change.

At first glance it seems that an atom cannot be accommodated in the interstices of the lattice, especially in the case of a close-packed lattice. This would be true if the nondissociated atoms were firmly bound to the lattice sites. In fact, they can be displaced from their normal positions under the action of the dissociated atom so as to give it the necessary space. As a consequence, the dissociated atom is surrounded by an elastically strained region.

The number of holes \(N'\) and of dissociated atoms \(N''\) in a crystal with total number of atoms \(N\) is determined by its temperature and volume or pressure. The values of \(N'\) and \(N''\) need not be identical, since both holes and dissociated atoms that have arisen inside the crystal can diffuse independently of one another to its surface and disappear there if their number exceeds the statistically equilibrium value; in exactly the same way, additional holes or dissociated atoms can diffuse inward from the surface of the crystal, arising by “dissolution of the void” or by “self-dissolution,” i.e., by the transition of atoms of the surface layer into interstices. Consequently, at a given temperature and pressure, a real crystal in a state of statistical equilibrium may be regarded as a “saturated solution” of \(N'\) holes and \(N''\) dissociated atoms in a crystalline lattice consisting of \(N - N''\) correctly arranged atoms and containing \(N - N'' + N'\) sites, among which these atoms and holes may be distributed.

Holes form in the crystalline lattice a solid solution of the substitutional type, in the sense that each hole replaces a missing lattice atom. Dissociated atoms form a solid solution of the interstitial type, being situated in the interstices, i.e., being inserted into the lattice. The formation of holes is associated with an increase in the macroscopic volume of the crystal, while the formation of dissociated atoms is associated with a decrease.

Upon a sudden lowering of the temperature of the crystal, holes and dissociated atoms form a kind of supersaturated solution, which must approach the new saturation point by recombination and diffusion. In this process, excess holes can coagulate with one another, forming microscopic cavities inside the crystal, with a considerable internal surface.

Thermal Motion of the Atoms of a Crystal

Disturbers of order in a crystal (holes and dislocated atoms) retain their position only for some average time \(\tau\). After this they pass into one of the equivalent positions at a distance \(\delta\). The number of equivalent positions is equal to the coordination number of the lattice, i.e., to the number of nearest neighbors. The ratio \(\dfrac{\delta}{\tau}\) may be regarded as the average velocity of motion of a “disturber” in the crystal lattice. This motion is of a disordered character, since the successive displacements of which it is composed have directions independent of one another. The displacement of “disturbers” may therefore be considered as the motion of gas molecules and likened to the mean free path \(\lambda\). The difference is that in a gas the path lengths are not identical, whereas “disturbers” have elementary displacements always equal to \(\delta\). In addition, the direction of flight of a gas molecule is arbitrary, whereas the elementary displacements have directions limited by the coordination number of the lattice.

Thus, thermal motion in crystals, at least at high temperatures, is not merely oscillations about fixed equilibrium positions, but is complicated by a series of elementary displacements of equilibrium positions along lattice sites or interstices. In addition, displacements from a site into a neighboring interstice (dissociation), or from an interstice into the nearest free site (recombination), are possible. As a result, atoms wander throughout the entire crystal, continuously mixing, like particles of a gas, only much more slowly. This makes it possible to regard a solid as a synthesis of an ideal crystal and an ideal gas.

Self-Diffusion and Diffusion of Impurities in Crystals

Mixing, or self-diffusion, of atoms in crystals, according to Frenkel’s exposition [1], may occur by three routes:

1) by displacement of holes;
2) by displacement of dislocated atoms in connection with their appearance at one place in the lattice, as a result of dissociation, and disappearance at another, as a result of recombination with holes;
3) by direct transposition of two or more neighboring atoms.

Self-diffusion of atoms situated at lattice sites may proceed through the appearance and disappearance of dislocated atoms on the surface of the crystal, without any participation of holes, or else through their appearance by dissociation and disappearance by recombination with a hole. The participation of holes in this recombination process is passive and is not essentially connected with their mobility. Hevesy assumes that mixing of atoms in a crystal may take place by the simultaneous transposition of two neighboring atoms, which, as it were, exchange places with one another. In this case one may suppose that the transposition of two neighboring atoms occurs through an intermediate state in which both atoms prove to be dislocated, and simultaneously two holes arise, which immediately thereafter close up.

The mechanism by which self-diffusion proceeds may, with slight changes, be transferred to the diffusion of impurity atoms forming a solid solution in the crystal under consideration, by the introduction of foreign atoms into the interstices of the basic crystal lattice (for example, a solution

carbon in iron). In this case diffusion proceeds by displacement through interstices.

Diffusion of impurities by substitution may proceed by the same three paths indicated above as self-diffusion, i.e., by displacement through interstices, by transition into a mobile hole that happens to be nearby in the lattice of the base substance, and by direct exchange of position of the foreign atom with a neighboring atom of the given substance.

Self-diffusion in crystalline bodies must account for their fluidity. Frenkel [2] considers it possible to explain the sintering of crystalline powder at high temperatures, not too far below the melting temperature, by viscous flow of crystals. The process amounts to the following: small crystalline grains, under the influence of molecular (atomic) cohesion, coalesce with one another into a compact crystalline body. The voids between them, which initially communicate with one another, in the end find themselves surrounded on all sides by compact crystalline substance and gradually decrease in size, tending with time to disappear completely.

Surface Creep of Atoms on Crystals

Frenkel [3] advances the supposition that, in a state of statistical equilibrium, the surface of a crystal should be formed not only by the principal faces, but also by faces close to them. The appearance of the latter may be regarded as the result of thermal fluctuations. Therefore, at temperatures different from absolute zero, the natural faces of a crystal must possess a certain “roughness.” On them there form, as it were, steps made of atomic layers. The length of a step changes with time by the addition or removal of individual atoms capable of moving along the steps independently of one another. In the absence of statistical equilibrium this leads to a change in the shape of the crystal. A similar change in the shape of a crystal takes place in the experiments of Łukirskii [4] with the heating of single crystals of rock salt. Spheres, from one to several centimeters in diameter, were turned from the crystal; these were polished and heated in a furnace for several hours at temperatures considerably lower than the melting temperature of rock salt. The temperature ranged from 720 to 760°. After the spheres had been heated, faces appeared on them corresponding to the crystallographic symmetry of its axes.

The formation of an equilibrium figure from a nonequilibrium one may occur either by means of repeated evaporation and condensation, or at the expense of the creep of surface atoms. The heat of activation of creep is considerably less than the heat of evaporation. Therefore, a change of shape due to the creep of atoms is more probable.

Order in the Arrangement of Lattice Atoms

Crystals formed by atoms of one kind are called unary. These include atoms of metals. Lattices consisting of atoms of two different kinds are called binary.

In binary ionic crystals, of the rock-salt type, four kinds of violations of the regularity of the crystal lattice are possible: dislocated positive and negative ions and the corresponding holes [1]. In binary nonionic crystals, for example intermetallic alloys, more substantial disturbances of order are observed, which are expressed not in partial dissociation of the lattice at high

temperatures, but in the disturbance of the regular alternation of atoms at its sites [1]. As an example one may cite the alloy $CuAu$ (50% $Au$, 50% $Cu$), where atoms of both kinds form a simple cubic lattice, at the sites of which, at low temperatures, they alternate in the same way as the ions $Na$ and $Cl$ in the rock-salt lattice. Beta brass $CuZn$ has an analogous structure, except that here each atom of one kind is surrounded not by six but by eight atoms of the other kind. With increasing temperature the indicated regular alternation of atoms is disturbed more and more and, beginning at a certain temperature, becomes disordered.

The decrease of order in the alternation of atoms of the lattice may be likened to melting and is called alternational melting [1].

If, with increasing temperature, the degree of order in the alternation of unlike atoms continuously decreases and, at a certain temperature, abruptly becomes zero (which is associated with the absorption of internal heat), then such a transition is called a transition of the first kind (observed, for example, in the alloy $Cu_3Au$).

In other alloys, for example $CuAu$, the degree of order decreases continuously, up to its complete disappearance at the temperature $T_0$. This is accompanied by an anomalous increase in heat capacity, which again becomes normal above the temperature $T_0$, called the Curie temperature. Such a transition is called a transition of the second kind [1].

In ionic crystals the regular alternation of atoms is determined by the forces of electrical attraction and repulsion. In nonionic crystals this may be explained by the fact that atoms of different kinds attract one another more strongly than atoms of the same kind. In this case the potential energy of the whole system has a minimum value.

In intermetallic alloys, the interchange of two atoms of different kinds is accompanied by an increase in energy by an amount equal to the mean energy of thermal motion at comparatively low temperatures, and may occur at temperatures far from the melting temperature. Such an interchange may proceed by direct exchange of places or by diffusional mixing as atoms move through interstices and holes.

Order in which the positions of atoms $A$ and $B$ are taken into account not with respect to one another, but with respect to lattice sites, divided into their own and foreign sites, is called order at large distances, or long-range order [1]. If, however, regular arrangement of atoms is understood as the predominance of atoms $A$ around an atom $B$, or conversely, irrespective of how the central atom is situated, i.e., whether it is on its own or on a foreign site, then such order is called short-range order.

Melting

At high temperatures and low pressures liquids resemble gases; however, at temperatures close to the crystallization temperature, liquids may be compared with solids. Being a phase intermediate between the solid and gaseous phases, liquids exhibit transitional properties and, in the region of high temperatures and large specific volumes, approach gases, while in the region of low temperatures and small specific volumes they approach solids. Until recently, however, physics predominantly emphasized the similarity of liquids to gases and ignored the similarity of liquids to solids. The reason for this may be explained by the success of the van der Waals theory.

Let us consider the properties of a solid that bring it close to a liquid. On melting, the volume of most substances increases only slightly (about 10%). The latent heat of fusion is very small in comparison ...

in comparison with the latent heat of vaporization. This may be regarded as confirmation that the forces of cohesion are weakened only insignificantly upon melting; when the volume increases by 10%, the distance between particles increases by 3.3%. Experiments show that upon melting the heat capacity either remains almost entirely unchanged or increases somewhat. This argues that the thermal motion in a liquid at a temperature close to the melting temperature will be practically the same as in solids, i.e., it will be expressed in small oscillations of atoms about positions of equilibrium, and, for diatomic or more complex molecules, also in rotational librations about equilibrium orientations.

Liquids, like solids, can withstand very large tensile stresses without rupture when these are reduced to an all-round negative pressure that excludes the possibility of flow. This can be accomplished by cooling mercury placed in a sealed glass vessel, which it completely fills at a high temperature. Meyer [1] showed that in this way mercury can be stretched by 1.4%, which corresponds to a negative pressure of about 100 atmospheres.

By applying large pressures to a liquid, it can be brought into the solid state even at a temperature above the usual crystallization temperature.

Finally, the visible crystalline structure of solids cannot be the cause of a sharp difference between the solid and liquid states. It has been found by X-ray methods that the arrangement of particles at temperatures close to the crystallization temperature is not, as in gases, disordered, but proves to be similar to that regular arrangement into which the liquid passes upon crystallization. It is as if it consists of a very large number of randomly oriented submicroscopic crystallites.

Summarizing the comparison of the liquid and solid states, we must say that with respect to density, the magnitude of the forces of cohesion, and the character of thermal motion, the difference between them is insignificant. There is a greater difference in the arrangement and orientation of the particles, but even it is not sharp. Proceeding from this, one may regard the transition from the solid state to the liquid as a continuous change in the degree of order at the melting temperature from unity to zero.

In 1932 Frenkel advanced a theory according to which disturbances of long-range order in a crystal reduce to “holes” (empty sites of the crystal lattice), and melting begins when the number of these holes exceeds a certain value.

In 1939 Lennard-Jones and Devonshire proposed a scheme of melting based on holes and dislocated atoms. They reduce the process of weakening of the crystal to the gradual dissociation of the lattice as atoms pass from sites into interstices. In this case the interstices of a given crystal lattice were regarded as sites of another crystal lattice inserted into the first. At low temperatures all sites of the second lattice are free, i.e., they represent holes, while the sites of the main lattice are occupied. Therefore the crystal may be regarded as a binary alloy of \(A\) atoms and \(B\) holes, arranged in strict alternation. With increasing temperature the degree of long-range order in the alternation of atoms and holes gradually decreases, and when it becomes zero, the crystal turns into a liquid.

Upon heating, the volume of a body, as a rule, increases. When the volume of the body has increased, in comparison with the minimum volume \(V_0\) (corresponding to \(p = 0\) and \(T = 0\)), by approximately 50%, the limit of strength of the body is reached; it becomes absolutely “soft.” A considerable softening of the body must take place long before the volume \(V^*\), at which this body

exists in the form of an amorphous mass. The preservation of a regular crystalline structure is possible in this case when the temperature is lowered in comparison with the ordinary melting temperature. In the limit \((V = V^*)\) the melting temperature should tend to absolute zero.

At \(T > 0\), thermal vibrations of the atoms may be the cause of a local rupture of the body at volumes far from the limiting \(V^*\). Such local ruptures must be temporary in character and must quickly “heal” again. However, as a consequence of the indicated local and short-lived ruptures, the regular arrangement of atoms at lattice sites, and their connection with the sites as positions of equilibrium, must be disturbed—and the more so, the higher the temperature and the larger the volume.

Properties of Liquids

A liquid, at a temperature close to the temperature of equilibrium with a crystal, differs from the crystal at not very high pressures by a relatively large specific volume. The latter increases upon melting by approximately 10%, i.e., by an amount approximately corresponding to the theoretical limiting strength of solids under all-round tension. In practice a solid ruptures at smaller extensions, which can be explained by the presence of microvoids (cavitation) in the form of cracks, holes, etc.

In a liquid, as a result of fluctuations caused by thermal motion, cavitations, or “atoms of emptiness,” likewise arise. In a macroscopically homogeneous body the fluctuations may be homophase and heterophase. As a result of homophase fluctuations, the homogeneity of the body in small volumes is not altered in any essential way. Heterophase fluctuations lead to the formation of embryos of a new phase (solid or gaseous, depending on the temperature).

The appearance of holes in a liquid may be regarded as a particular case of heterophase fluctuations, and the holes may be regarded as vapor bubbles, if they were large enough to contain a sufficient number of vapor particles. This can occur in a liquid only at a temperature close to the boiling temperature. At lower temperatures the holes are too small, and it is meaningless to speak of their being filled with vapor. In these cases the holes should be considered “empty.” It is necessary to note that the holes in liquids differ from the holes in solids. In liquids the concept of sites and interstices loses its meaning. Holes must now be understood as enlarged spaces between molecules, which have neither definite dimensions nor a definite shape, and which can arise spontaneously, expand, contract, and disappear again, and also move by closing up in one place and appearing in another.

The “hole” scheme of the liquid state set forth above is applicable in the region of not very high temperatures and pressures. Near the critical temperature the density of the liquid becomes small and the concept of holes loses all meaning.

Liquid Phase at the Contact Point of Two Crystals That Do Not Form Solid Solutions

In the case of contact between two crystals whose atoms can form solid solutions, we obtain diffusion of atoms \(A\) into crystal \(B\) and of atoms \(B\) into crystal \(A\). The mechanism of this diffusion was described above. The matter is otherwise in the case when a solid solution cannot form. Developing the positions described above, we proceed from the fact that every impurity atom distorts the lattice of the crystal not only when it is located

within it, but also being on its surface. In the case of contact between two crystals that do not dissolve in one another, the surface of each crystal is in contact with foreign atoms and, consequently, the surface layer to some depth must prove to be distorted. The degree of long-range order in the alternation of atoms and holes in this layer must be less than in the interior of the crystal, and thus the melting temperature of this layer is lowered. In the case of melting of the surface layer of one of the contacting crystals, this melting cannot propagate inward, since its own atoms are in contact with the deeper layers and do not cause a lowering of the melting temperature there. It should be remembered, however, that both contacting crystals are under equal conditions; the surface layers of both melt at a lower temperature than their melting temperatures. When this lowered melting temperature of the surface layer of the more refractory crystal is reached, at the contact point we shall have two thin layers of two melted crystals—two liquids. The presence of vacancies in these liquids makes it possible for them to diffuse into one another, i.e. to mix, which could not have taken place for crystals. As a result of such mixing, foreign atoms penetrate into the deeper layers of each crystal, which had not previously melted, come into contact with them, loosen them, and cause further melting. Thus, both crystals melt at a temperature lower than the melting temperature of either of them. This temperature is the melting temperature of the eutectic.

It should be noted that only up to the moment when melting of the more refractory component begins is the surface of its crystals in continuous contact with foreign atoms. When a mixture of melts of both components appears, the surface of each crystal will be in contact with a liquid in which the foreign atoms constitute only a certain percentage. We then have a liquid of a quite definite percentage composition. It is not difficult to see that a decrease in the number of foreign atoms in contact with the crystal must lead to a decrease in the rate of melting of this crystal, i.e. of the mass of it melted per unit time; from this follows the constancy of the eutectic composition at the melting temperature of the eutectic.

Experiments Confirming the Effect of Contact Melting for Two Components

Our assumptions about the existence of the above-described effect were first tested by us on organic compounds that give eutectic pairs [5]. The work was carried out as follows. Into a test tube lowered into a jar of water, crystals of both substances were poured, and the water was heated slowly. At a temperature somewhat above the eutectic one, but far below the melting temperature of the individual substances, a liquid appeared. In all, we investigated seven eutectic pairs (Table 1).

These experiments showed that, for organic compounds, contact (eutectic) melting exists. Further investigation was carried out on metallic pairs. In the first experiments, small pieces of both metals were taken in the eutectic proportion and placed in a crucible with rosin; the crucible was heated to a temperature 10–20° above the eutectic temperature and held for about half an hour, after which cooling followed. As a result, a piece of alloy fell out of the crucible. Rosin was used in order to protect the metals from oxidation, thereby making possible contact between the pure metals.

Table 1

In the following experiments the same was done with an arbitrary quantity of metals. The results were the same, except that, together with the melt, pieces that had not entered into the eutectic fell out. After this, the experiments were carried out with massive pieces of metals placed one on top of another. Melting always began at the place of contact and, depending on the duration of exposure, the metals either melted completely or ate into one another.

We investigated four pairs, given in Table 2.

Table 2

As a result of these experiments, it may be said that the metals constituting the eutectic pair also melt in contact when they are heated to a tem-

temperatures somewhat higher than the eutectic temperature. It is only necessary to ensure the removal of oxides from both metals at the point of contact. Special experiments with a lead–bismuth pair showed that melting begins as soon as the temperature at the point of contact, on rising, passes the melting point of the eutectic.

The effect of contact melting for three and four components

Passing to the ternary eutectic [6], it should be noted that it has a melting temperature lower than the melting temperature of any pair of the three components entering into its composition. Where, then, should the liquid phase appear at the moment when melting begins? If the ternary eutectic is a mixture of crystals of three components, then each of them, by different parts of its surface, is in contact with the surfaces of crystals of the other components. In these places, consequently, a liquid phase can appear at the melting temperatures of the binary eutectics. However, it is not difficult to see that in a conglomerate of crystals of three kinds there exist lines along which the crystals of all these kinds meet; here there is, as it were, contact of three crystals. It is precisely here that the liquid phase of ternary composition should appear, upon reaching the melting temperature of the ternary eutectic.

Since bringing three crystals into contact along a line is considerably more difficult than contact of the surfaces of two crystals, and the contact-melting effect itself is more difficult to observe, we carried out the experiments in the following manner. A piece of the eutectic alloy of tin and lead was polished, and the polished section was pressed against a piece of cadmium. When such a contact was heated to a temperature five degrees above the melting temperature of the ternary eutectic, but lower than the melting temperatures of the binary eutectics tin–cadmium, lead–tin, and lead–cadmium, it was possible, after a 30-minute holding period, to obtain sintering of both pieces. Since the piece of eutectic consisted of a conglomerate of tin and lead crystals, and neither of these metals sinters with cadmium at such a temperature, there is no doubt that the liquid phase was formed where the network of boundaries between the lead and tin grains was in contact with the cadmium.

Passing to a eutectic of quaternary composition, we obtain contact of four crystals in the form of points; it is precisely at these points that the liquid phase should appear. We took a piece of the ternary eutectic lead–cadmium–tin; on the surface of the polished section of such a specimen there must undoubtedly be points at which the crystals of all three components meet. A polished section of bismuth, applied to this surface, was in contact with all three components at the indicated points. After holding for one hour at a temperature of 80°, the two pieces sintered. This temperature is 10° above the melting temperature of the quaternary eutectic lead–tin–cadmium–bismuth, but considerably below that of any ternary eutectic composed of the four available components.

Conclusions

Having considered in general terms the modern theory of the structure of crystals and liquids, we have attempted, from the standpoint of this theory, to prove the necessity for the appearance of a liquid phase at the point of contact of two such crystals whose atoms cannot form solid solutions. The appearance of liquid here must occur at a temperature considerably lower than the melting temperature of both crystals.

The assumptions set forth have been confirmed by experiments, from which it follows that the melting temperature of eutectics is not a property of eutectic structures. This melting occurs whenever crystals of different substances capable of forming a eutectic are in contact along a surface, line, or point. Eutectic structures merely facilitate the existence of such contacts. Thus it would be more correct to speak not of eutectic melting, but of the effect of contact melting, which is the cause of the melting of eutectic structures.

Cases in which a liquid phase appears at the point of contact of three or four crystals are, of course, different from the case we have considered of the contact of two crystals. However, there is no reason to doubt that here, too, the explanation must be sought with the aid of the theory set forth by Frenkel in his works.

LITERATURE

1. Frenkel, Ya. I.—Kinetic Theory of Liquids, 1945.
2. Frenkel, Ya. I.—Viscous flow in crystalline bodies, ZhETF, 16, 20, 1946.
3. Frenkel, Ya. I.—On the surface creep of particles in crystals and the natural roughness of crystal faces, ZhETF, 16, 39, 1936.
4. Lukirskii, P. I.—Experiments with single crystals of rock salt, DAN, XVI, 300, 1945.
5. Saratovkin, D. D. and Savintsev, P. A.—Formation of a liquid phase at the point of contact of two crystals constituting a eutectic pair, DAN, XXXIII, 4, 1941.
6. Saratovkin, D. D. and Savintsev, P. A.—The effect of contact melting as the cause of the low melting point of eutectics, DAN, LVIII, 9, 1947.
7. Joffe, A. Ann. d. Phys. 72, 461, 1923.
8. Frenkel, I., ZS. f. Phys. 35, 652, 1926.