V. D. Kuznetsov.

Some Problems of Solid-State Physics.

§ 1. Introduction.

Several years ago, the physics of the solid state was in complete neglect. It was not considered interesting because there was no general idea in it that would unite the entire group of phenomena in this branch. Very few physicists were engaged in the study of individual phenomena of solid-state physics. They obtained certain regularities, which were so complex that they could not serve as a basis for a theory. This was especially characteristically emphasized by Bouasse¹, who, as a result of his many years of research, came to the conclusion that in solid-state physics there are no general laws and that each individual case must be investigated and described separately. It is not without reason that Prof. B. P. Weinberg², in his report at the Mendeleev Congress in 1910, said that “with regard to solid-state physics one may state without exaggeration that this is a neglected and very slowly developing field.” In those years, one paper relating to solid-state physics accounted for almost 200 papers in other branches.

An enormous number of specimens of various materials had been stretched, compressed, twisted, destroyed, and it would seem that a great body of experimental material had been accumulated for the construction of a theory; yet no general theory appeared; the more experimental material there was, the less hope there was for the creation of a theory. The nature of elastic forces was still incomprehensible, and solid-state physics was completely cut off from other branches.

Theory appeared from the very quarter from which no one expected it. It was born out of a whole series of unsuccessful attempts to obtain the phenomena of interference and diffraction of X-rays. Laue supposed that the wavelength of X-rays was too small for interference phenomena to be obtained by the usual method. All the finest optical instruments, for example diffraction gratings, were too coarse for these rays. To obtain the diffraction spectrum of X-rays, he used that natural grating in which the molecules of a crystal are arranged. This idea was justified experimentally and opened to physicists entirely new paths for studying the nature of X-rays and the structure of crystals.

Attractive and repulsive forces, whose nature had been completely obscure, were reduced to electrical interactions between the separate elements of the crystalline lattice. Solid-state physics became a chapter of the theory of electrical phenomena; an electrical theory of solids appeared. In the design of experiments there appeared a definite guiding idea, and their results became considerably more fruitful. At the present time every experimenter in solid-state physics must adopt the point of view of the electrical theory in order to participate in the general development of this branch.

Solid-state physics, which several years ago had been completely isolated, has found itself in close connection with other branches. It is now possible to connect into one whole such remote questions as elasticity, plasticity, strength, relaxation, internal friction, and the surface energy of crystals with one

on the one hand, electrification upon contact, dissociation, ionization, electrical conductivity, the photoelectric effect, etc.—on the other hand.

Temperature plays a major role in all these phenomena; theoretically its influence is still not entirely clear and comprehensible.

In the present article I should like to consider several problems in solid-state physics which, in my opinion, are of interest from both the theoretical and practical sides and which should be studied experimentally.

§ 2. Internal Friction of Crystals.

The study of the internal friction of crystals—bodies with a definite, known structure—may shed light on the internal friction of solids in general. For the investigation it is best to take such crystals whose structure is sufficiently well known. The experiment must be arranged so that it is possible to study the shear velocity in different directions. For example, for rock salt or for metals crystallizing in the cubic system, one should determine the angular velocity when layers are displaced parallel to the planes \((1, 0, 0)\), \((1, 1, 0)\), and \((1, 1, 1)\). It may be expected that in the plane \((1,1,0)\) the internal friction will be the smallest, since sliding occurs most easily parallel to this plane.

It is necessary to determine what happens to crystals when such stresses are applied to them under the action of which they begin to flow and a constant shear velocity is established. Here three assumptions may be made.

1. When a small external stress is applied to a crystal, the elements of the lattice—atoms or ions—are displaced by small distances relative to one another, and when the stress is removed they return to their original positions. This is the elastic part, obeying approximately Hooke’s law. If the displacement of the lattice elements exceeds a certain limit, then sliding of one part of the crystal relative to another is possible. During sliding, atoms or ions may jump by a whole number of normal distances between them. The crystalline lattice may not change at all as a result of such displacement or jumping, and its X-ray photograph will have the ordinary appearance.

2. It may happen, as A. F. Ioffe and M. V. Kirpicheva discovered for NaCl, that under sufficiently large deformation the crystal breaks up into a number of small crystallites. Internal friction is nothing other than the displacement of separate crystallites relative to one another.

3. Under sufficient stress there occurs both displacement of crystallites and sliding within them.

To study the internal friction of crystals, various methods should be used as far as possible. It seems to me that the most suitable method is the following. Two identical crystalline plates are clamped between three metal plates with rough surfaces. The two outer plates remain immobile, and a load is suspended from the middle one; by pulling out the middle part, it produces shear in the two crystalline plates. At the same time it is of interest to X-ray the crystals in order to observe the moment when destruction and displacement of the separate crystallites begin in them.

It is necessary to study the dependence of the angular velocity of sliding at various loads and at various temperatures, up to temperatures close to the melting point.

In particular, it is interesting to study the internal friction of ammonium and potassium alums, because the electrical conductivity of these crystals has been sufficiently studied.

The electrical conductivity of crystals depends on dissociation and on internal friction. Dissociation, in all probability, is a scalar quantity; as for internal friction, it is natural to assume that it differs in different directions. The experiments of A. F. Ioffe and M. V. Kirpicheva¹ showed that the electrical conductivity of ammonium alum (of the cubic system), within the accuracy of the experiments, is the same in directions perpendicular to the faces of the cube and of the octahedron. If this is so, then the internal friction of ammonium alum crystals should not depend on direction. This question requires experimental investigation.

If one takes mixed crystals of ammonium and potassium alum, it turns out that their electrical conductivity is somewhat less than the electrical conductivity of pure crystals; consequently, one may suppose that the internal friction in mixed crystals is somewhat greater than in crystals with an unimpaired crystalline structure.

§ 3. Internal friction of metals.

If a metal specimen with a polished surface is subjected to some deformation and the change in its microstructure is followed under a microscope, two kinds of changes can be detected. Either, during deformation of the specimen, individual crystallites move relative to one another without a change in the form and structure of the individual grains, or else residual deformations occur in the crystalline grains, consisting of simple or twinning slips. If slips occur, the surface of the crystallite becomes stepped.

Under the action of external forces every crystal is deformed and, when the stress exceeds the elastic limit, slips occur in it along slip planes, if it is plastic, or the crystal breaks down if it is brittle. As the temperature is raised, a crystal changes from brittle to plastic, provided its chemical composition does not change.

In crystals of copper and gold, deformation is accompanied by slips, and the crystals separate into individual layers which slide relative to one another. For copper, which crystallizes in the cubic system, the slip planes are the planes of the rhombic dodecahedron.

In twinning slip in a crystal, parallel layers are rotated about axes lying in a certain plane, this rotation being through such an angle that all these layers are inclined to this plane at the same angle as before, but only turned to the other side.

At a certain stress, on the polished surfaces of crystallites of copper and gold there first appear slip lines, and then a displacement of individual crystallites is observed. In zinc, tin, lead, cadmium, magnesium, iron, and nickel, at a certain stress the displacement of crystallites begins without residual deformations in them. The crystallites, as they move, protrude onto the surface and make it rough. This displacement of crystallites is accompanied by a characteristic crackling in tin, zinc, and other metals. In a third group of metals, to which aluminum and silver belong, at a sufficient stress both slip lines and the displacement of crystallites appear simultaneously.

To clarify the nature of the internal friction of metals, it seems to me that one should study the influence of the size of crystallites on internal friction. Consequently, in parallel with measuring viscosity, it is necessary to study thin sections of specimens under the microscope. A change in the sizes of crystallites may be achieved by annealing specimens obtained in an almost amorphous, glassy—

in this way by forcing metals through an opening. One may expect that, for those metals in which deformation first involves displacements of crystallites, the internal friction increases with the increase in the sizes of the individual grains.

§ 4. Influence of the Size of Crystallites on the Strength of Metals.

If one stretches two wooden rods, one cut parallel and the other perpendicular to the fibers, the second rod breaks under a considerably smaller stress than the first. The tensile strength of a crystalline specimen cut parallel to the cleavage planes is greater than the strength of the same specimen cut perpendicular to these planes. If the cleavage planes are situated at some angle to the axis of the specimen, then the stress \(P\) required for rupture will lie between \(P_{\max}\) and \(P_{\min}\).

Let us take a specimen consisting of small crystallites, which are oriented with respect to the cleavage planes in all possible directions. Examining such a specimen along its length, we shall always find in it such a cross section in which the greatest number of crystallites are arranged so that the cleavage planes lie in this section. When such a specimen is stretched, it ruptures at some stress \(p_1\). If we take a specimen of the same material, but with larger crystallites, it may happen that in some place a large crystallite will be situated with its cleavage planes perpendicular to the length. This will be the weakest section, and the specimen will rupture along it at a stress \(p_2 < p_1\). Consequently, one may expect that the tensile strength of specimens will be the greater, the smaller the crystallites.

During forging, rolling, extrusion of metals through openings, drawing through a drawplate, and similar mechanical actions, the crystallites are crushed; the specimen acquires a fine-crystalline structure and its strength increases. If the specimen is annealed, the crystallites grow, with the large crystallites growing at the expense of the small ones, just as large drops of liquid grow at the expense of small ones if these drops are located near the large ones. This is explained by the fact that the surface energy of small drops, in relation to their total internal energy, is considerably greater than that of large drops. The strength of an annealed specimen must be less than that of an unannealed one.

When a specimen is stretched, a neck is formed in some place. At this place there occurs a flow of the substance, consisting in the displacement and crushing of crystallites. If, without bringing the specimen to rupture, one turns it on a lathe so that it has the same cross section along its entire length, and then stretches it again, the neck will form in some new place, while the place of the original necking is guaranteed against narrowing. This can be explained by the fact that, at the place of narrowing, a flow of the substance occurs, consisting in the displacement and crushing of crystallites; at this place the crystallites become finer, and therefore the strength of the specimen increases.

It seems to me interesting to study the dependence of the strength of metals on the size of the crystallites, testing specimens to rupture and measuring under a microscope the size of the crystallites on polished surfaces of the fracture sections.

§ 5. Surface Energy of Crystals.

Ya. I. Frenkel, in his book Electrical Theory of Solids, pp. 195–210 (1924), theoretically considers the question of the surface energy of various crystalline faces. This energy is equal to the work that must be done in order to separate one face from another.

the experimental determination of the surface energy of crystals is of great interest. Theory gives an unexpectedly small value for the magnitude of the energy. Thus, to separate \(1 \text{ cm}^2\) of the surface \((1,0,0)\) of rock salt (NaCl), work of about \(0.3\ \text{g}\times\text{cm}\) is required. For the face \((1,1,0)\) this work is \(2.50\) times greater, and for the face \((1,1,1)\) it is \(5.81\) times greater.

For determining the work, simple tearing off, splitting, shearing, etc., are suitable. It is not sufficient to measure the force that is necessary for separating one face from another, since this does not give the magnitude of the work.

If it were possible to make a knife with an edge whose thickness were of the order of \(10^{-8}\) cm, i.e., somewhat less than the distance between the elements of the crystal lattice, and if it were possible to set the knife edge exactly along the line between the elements of the lattice, then the work could easily be determined. But an ordinary knife edge has a thickness of the order of several thousand distances between lattice elements, and therefore the determination of the energy is attended by difficulties.

§ 6. The phenomenon of relaxation.

If, by some force \(F\), a deformation is produced in a solid body, then, in order to preserve the constancy of the magnitude of this deformation over time, it is necessary gradually to diminish the force. The phenomenon of relaxation, i.e., the decrease with time of the ability to resist an external action, is closely connected with the phenomenon of internal friction.

Despite a number of investigations, it may be considered that relaxation has scarcely been studied and that its law has not been established. As yet we have no theoretical approach to this phenomenon. Maxwell’s and Shvedov’s formulas are not valid in most cases. Thus, the relation established by Maxwell:

\[ \eta = NT,\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (1) \]

where \(\eta\) is internal friction, \(N\) is the shear modulus, and \(T\) is the relaxation time, i.e., the interval of time during which the force decreases by a factor \(e\), where \(e\) is the base of natural logarithms, cannot be justified, since the magnitude of the internal friction depends on the angular velocity of shear.

The establishment of the law of relaxation, the clarification of the nature of this phenomenon, the influence of the size of crystallites on relaxation, etc.—all these questions are so urgent both for theory and for practice that they are worth taking up.

It seems to me that relaxation must obey the following differential equation:

\[ \frac{dF}{dt}\, t = \mathrm{Const.} \qquad (2) \]

i.e., the rate of decrease of the force, at constant deformation, is inversely proportional to the interval of time \(t\) from the beginning of the application of the force.

§ 7. The influence of a magnetic field on the strength of metals.

The following phenomenon is known. If an iron specimen is stretched, it becomes magnetized; when it is broken, both its halves prove to be magnets.

If one assumes that elementary magnets are certain particles not symmetrical with respect to the electrons moving in them, then it may be supposed that, when an iron rod is stretched, the elementary magnets rotate and arrange themselves with their magnetic axes along the axis of the rod. Instead of a chaotic distribution of particles, a definite orientation is obtained, and, as a consequence of it, magnetization appears; at the same time the strength of the metal may change.

It may be expected that the influence of a magnetic field on para- and diamagnetic metals will be opposite, i.e., if the strength of paramagnetic metals—

...in a longitudinal magnetic field increases, then for diamagnetic ones it must decrease; in a transverse magnetic field the opposite phenomenon should occur.

Probably belonging to this group of phenomena is Dufour’s old observation³, who wrote that the tensile strength of copper wires, when an electric current is passed through them for a long time, decreases, whereas for iron wires it increases. Here there may be two causes: either under the influence of the electric current even a small quantity of heat is produced, which promotes the growth of crystals, or the strength of the metals changes under the influence of the magnetic field created around the wire by the electric current.

In any case, the influence of a magnetic field on the strength of metals has apparently scarcely been studied, whereas this question is of interest from the theoretical side.

In addition to the influence of constant longitudinal and transverse magnetic fields, one should also study the influence of an alternating field, which causes elemental magnets to turn now in one direction, now in the other.

§ 8. Conclusion.

In the present article I have by no means exhausted those questions which should be studied experimentally. Even I have more of these questions than I have indicated. For example, I have not at all touched upon the question of so-called single crystals, whereas the study of the properties of metallic single crystals is of great interest from the theoretical side. Further, I have not touched upon the question of investigating solids by means of X-rays. The application of these rays to the investigation of the structure of bodies is so extensive and so fruitful that in a brief article it is impossible even to give a cursory survey.

For me one thing is clear: it is necessary to establish the closest possible connection between pure physics and the technical sciences. It is necessary that physicists and engineers work on the same questions, those having significance for industry and technology. Thanks to such a connection, technology will be able to find the best methods for one or another production process. The necessity of such a connection is clearly recognized abroad, where large industrial enterprises, for the improvement of production, establish enormous scientific-research institutes in which scholars carry out purely scientific work, sometimes not directly connected with production. The expenses of maintaining such institutes are quickly and amply repaid.

I have noted here only those questions which may be of equal interest both to physicists and to engineers.

Tomsk.

December, 1924.

1. Zh. R. F. O. 48, 261 (1916). ↩↩

2. B. P. Weinberg. Zh. R. F. O., Issue 2 (1913). ↩

3. Dufour. Pogg. Ann. 99, 611 (1856). ↩