TO THE THIRTIETH ANNIVERSARY OF SOVIET PHYSICS

DEVELOPMENT IN THE SOVIET UNION OF THE THEORY OF SUPERFLUIDITY AND SUPERCONDUCTIVITY

E. L. Andronikashvili and K. A. Tumanov*)

I. SUPERFLUIDITY.

1. Some properties of helium-II

In 1908 the last of the gases—helium—was liquefied ^(1a**) (boiling point \(4.2^\circ\mathrm{K}\) at atmospheric pressure). Soon after this it was found that liquid helium possesses very interesting anomalies, thanks to which it occupies an altogether exceptional place among all other liquids.

Thus, for example, liquid helium, when under the pressure of its saturated vapor, does not pass into the solid state down to absolute zero and can be crystallized only at elevated pressures (not less than 23 atm).

In addition, at a temperature of \(2.19^\circ\mathrm{K}\) this liquid undergoes a phase transition of the second kind***)—the so-called \(\lambda\)-transition. Correspondingly, at \(2.19^\circ\mathrm{K}\), or, as one says, at the \(\lambda\)-point, a number of physical properties of helium, such as heat capacity, speed of sound propagation, coefficient of thermal expansion, thermal coefficient of elasticity, and other thermoelastic coefficients, suffer a discontinuity, changing in a jump-like fashion. Conversely, other properties, such as density and vapor elasticity, change smoothly with temperature (Figs. 1 and 2).

*) The chapter on superfluidity was written by E. Andronikashvili; the chapter on superconductivity by K. Tumanov.
**) In view of the fact that the present article is devoted to the achievements of Soviet science, the list of Soviet works and of works by foreign authors are given separately, and the works of foreign authors are marked with the corresponding digit with the sign \(a\).
***) A phase transformation of the second kind is a phase transition that occurs without the release of latent heat.

Helium found at temperatures above the λ-point has been called helium I, in contrast to helium II, which occupies the temperature range from 2.19° K to 0° K.

Despite their perfect transparency, the two modifications differ sharply from one another in appearance: while He-I boils vigorously throughout its entire volume, He-II, even under intensified pumping, is a completely calm liquid with a sharply defined meniscus. The reason underlying this difference

Fig. 1. Temperature dependence of the heat capacity of liquid helium:
1 — at saturated-vapor pressure, 2 — at \(p = 19\) atm.,
3 — at \(p = 25\) atm.

in behavior lies in the unusually high thermal conductivity, reaching \(10^5\) watt/deg·cm, which occurs in He-II and which exceeds by \(10^8\) times the thermal conductivity existing in He-I, and by tens of times the thermal conductivity of copper at low temperatures—one of the best conductors of heat. This phenomenon was discovered by Keesom and Miss Keesom in Holland \(^{2a}\) in 1936 and was subsequently studied repeatedly by many scientists.

Such a high thermal conductivity served as the basis for calling He-II “super-thermally-conductive,” and this term for some time remained firmly established in the scientific literature.

Subsequent experiments \(^{3a}\) showed that this thermal conductivity apparently is not genuine, since, contrary to the usual case, it depends on the power supplied to the heater, increasing sharply as the latter decreases.

But these unusual phenomena are not the only ones that can be observed in He-II. No less striking is the ability of He-I to form on solid surfaces thin films \(^{4a}\), rapidly moving against the gradient

temperature, owing to which a test tube sealed at the bottom, half-immersed in He-II, is quickly filled with liquid. (What is remarkable in this case is not the effect of wetting a solid surface by the liquid, but the rate at which this liquid can flow along such a film.)

In He-II one can also observe the so-called fountain effect \(^{5a}\), the essence of which is as follows. An open U-shaped tube, at both ends, terminates on one side in a capillary. In the wide part of the tube a porous plug is placed (finely powdered carborundum, tightly compressed), to which heat is supplied, for example by means of intense illumination. As a result of the heat input, a fountain of liquid helium issues from the capillary, the height of which in some cases reaches 30 cm above the level of the liquid in the bath.

Fig. 2. Temperature dependence of the density of liquid helium.

Despite numerous attempts, both experimental and theoretical, to unite within at least a qualitative theory these seemingly disparate facts, the mysterious behavior of helium-II nevertheless remained incomprehensible. It was clear that new fundamental facts were still lacking.

The rapid development of the doctrine of helium-II began with the publication in print of the results of the first Soviet works devoted to this question.

2. Discovery of Superfluidity

In 1938 P. L. Kapitza^35 published a brief communication containing the results of measurements of the viscosity of liquid helium. The very first experiments led to extremely important conclusions: the viscosity of He-I was found to be equal to \(10^{-5}\) poise, whereas the upper limit of the viscosity of He-II, estimated on the assumption that the flow is laminar, did not exceed \(10^{-9}\) poise. (Let us recall that the viscosity of water is \(10^{-2}\) poise.) Thus at the \(\lambda\)-point there occurs a jump by a factor of \(10^4\).

Fig. 3. Kapitza’s capillary viscometer.

The viscometer used by Kapitza consisted of two optically polished quartz disks, the gap between which could be reduced to \(0.5\mu\). The upper disk had a central opening into which a quartz tube was inserted (Fig. 3), serving as a reservoir for the liquid flowing out through the slit-shaped gap. By means of a thread the apparatus was suspended in liquid helium and could be lowered and raised in it, thereby making it possible to regulate the head under which the outflow took place. The flow rate of the liquid and the pressure difference were measured with a cathetometer.

It is important to emphasize that in this case the viscosity of He-II was measured precisely from the rate of its flow through a very narrow slit.

Kapitza’s paper was the first of a whole series of works^6,7a devoted to the investigation of the viscosity of He-II.

It is of interest to compare the results obtained by Kapitza with the results of the work of Keesom and MacWood^7a. In the experiments of the latter, the viscosity was investigated from the damping of axial oscillations of a disk immersed in He-II (a method fundamentally different from the method of forcing a liquid through a slit). Their results, though perhaps not entirely reliable if one speaks of the exact temperature dependence of the viscosity, nevertheless left no doubt that the value measured by them exceeds the viscosity measured by Kapitza by a factor of \(10^3\), and near the \(\lambda\)-point by a factor of \(10^4\). More precisely, in the experiments of Keesom and MacWood the viscosity of He-II decreases smoothly from

\[ \eta = 2.10^{-5}\ \text{poise} \]

at the \(\lambda\)-point to

\[ \eta = 1.5 \cdot 10^{-6}\ \text{poise} \]

at \(T = 1.3^\circ\mathrm{K}\). The sudden jump of this quantity, observed by Kapitza at the \(\lambda\)-point, is absent in their experiments. The meaning of this paradox will be explained later.

3. Mechanism of Heat Transfer in He-II

The discovery by Kapitza of superfluidity was a fundamental fact not only for understanding the phenomena connected with the flow of He-II, but also for understanding other anomalies observed—

... present in it. It gave all subsequent work, both in our country and in other countries, the direction thanks to which the He-II problem, in its main features, proved to be solved in an extraordinarily short time.

In particular, it became possible to approach the phenomena of propagation in He-II bodies from an entirely new point of view.

Having established that a nonviscous flow can exist in He-II, Kapitsa suggested that the phenomena of heat transfer are connected with the existence of certain convective currents.

His subsequent work was devoted to testing this hypothesis. Measuring the heat transfer of helium-II filling sufficiently narrow capillaries, Kapitsa\(^ {36}\) found a number of external factors which can strongly change its heat-transfer coefficient. Thus, for example, it increases noticeably when the cross-section of the capillary is reduced, into which at a certain moment of the experiment a glass rod is inserted; heat transfer is reduced if, at the moment when the thermal conductivity is measured, the helium flows through the capillary; it decreases when the helium in the capillary is stirred by means of a rapidly rotating glass rod inserted into it; it is noticeably reduced if pressure pulsations, always present during the operation of fore-vacuum pumps, are transmitted to the He-II.

Fig. 4. Study of the topography of a jet by means of a torsion balance, after Kapitsa.

All the facts listed above undoubtedly convince us that the thermal conductivity of He-II in capillaries is not in fact genuine. After this it became important to trace the mechanism of convective transfer. For this purpose Kapitsa constructed an apparatus whose principal parts were: a sensitive torsion balance, provided with light vanes and immersed in helium, and a small vessel made in the form of a Dewar, communicating with the helium bath by means of a thin capillary surrounded by a vacuum jacket (Fig. 4). Inside the vessel were placed a constantan heater and a sensitive resistance thermometer made of phosphor bronze. The experiment showed that, when power is liberated inside the vessel, a jet issues from its nozzle, exerting a noticeable pressure on the vane of the torsion balance. The jet issues from the vessel for the entire time during which current flows through the heater. The vessel, however, does not become empty; in other words, no expenditure of liquid is observed. The topography of the jet was

was also investigated by Kapitsa. It turned out that at considerable distances, of the order of 10 mm (which is twenty times the diameter of the nozzle), the jet preserves its direction well, and its cross-section, approximately equal to the cross-section of the capillary nozzle, is blurred only very slightly.

It should be pointed out that the vane gives noticeable deflections even in those cases when a bronze thermometer (the sensitivity of the measuring arrangement corresponds to a temperature difference \(\Delta T = 6 \cdot 10^{-6}\) degrees) proves incapable of detecting a temperature drop between the vessel and the helium bath. This experiment undoubtedly proved that the release of heat in He-II produces powerful convective flows.

In view of the fact that, despite the presence of a flow, no consumption of liquid in the vessel is observed, it would seem that we should, in one way or another, detect a counterflow. However, no matter how close the vane is brought to the capillary nozzle, it indicates an unchanged one-sided pressure. Thus the counterflow entering the vessel, in its motion, exerts no pressure on the bodies around which it flows. This fact finds its confirmation in the following very demonstrative experiment of Kapitsa. The annular space of a small double-walled glass vessel is connected to the helium bath by means of several bent capillaries. The inner bottom of the vessel rests on a needle. The heating of the helium located in the annular space is produced by focusing a light beam on the wall blackened from inside. The device was named a “spider.” It operates on the principle of a Segner wheel: the reaction of the jets issuing from the capillaries causes the spider to rotate (Fig. 5). However, if opposite each capillary nozzle a vane is placed, rigidly connected with the rotating spider, then the latter stops. Undoubtedly, if the counterflow exerted any pressure at all on the apparatus, then in the last experiment the spider would change its speed of rotation, but would not stop completely. This experiment may be interpreted in two ways: either the counterflow “creeps” along the wall toward the temperature gradient, or else the counterflow occupies the entire cross-section of the capillary,

Fig. 5. “Spider”—an apparatus for demonstrating the reaction of a jet according to Kapitsa.

but, possessing the properties of an ideal liquid moving without friction, it exerts no pressure on the bodies it flows around—its motion is potential.

As will be shown below, theory and subsequent experiments unequivocally decided this question in favor of the second point of view.

These experiments establish with complete certainty the fact that the thermal conductivity of He-II not only is not true thermal conductivity, as Keesom supposed, but that heat transfer occurs thanks to the existence of peculiar convective flows that have nothing in common with gravitational convection.

This is the situation in experiments on the study of heat transfer in helium-II through thin capillaries. The experiments of P. G. Strelkov^78 showed that an analogous picture of two flows can also be observed when heat transfer takes place in a volume. The radiometer he used to detect convective flows consisted of a torsion balance from whose beam two hollow glass plano-convex small lenses were suspended. The plane sides of the lenses were ground and blackened. A beam of light fell on these blackened surfaces. The experiment showed that in He-I the radiometer was “attracted” to the beam, whereas in He-II it was “repelled” by it. The behavior of the radiometer in He-I appears quite natural and is explained by the interaction of the blackened surface with ordinary convective flows. The presence of a considerable effect in He-II indicates that this liquid is not “super-heat-conducting” in Keesom’s sense and that strong convective flows exist in it. However, as follows from the sign of the effect, the nature of these convective flows in the case of free helium-II is not ordinary gravitational convection, but something quite distinctive. Apparently, just as in Kapitza’s experiments, the flow directed away from the heated surface exerts a reactive action on it.

Fig. 6. Strelkov’s “optical Dewar.”

The study of heat transfer in bulk He-II was also carried out by an optical method (Strelkov^79). In a specially constructed optical Dewar, consisting of a single four-walled vessel with flanges sealed with optically polished plane-parallel glass plates (Fig. 6), heat sources of various forms were placed: a plane, a wire stretched in free helium, and a wire clamped between two small glass plates. The author sought

detect optical inhomogeneities caused by flows moving inside an unevenly heated liquid. However, no such flows were detected, which indicates that the ordinary convective mechanism of heat transfer has no place here. A sharp optical inhomogeneity, recorded with the aid of a motion-picture apparatus, was observed only when a gas bubble arose around the heater. An approximate estimate of the conditions under which the gas bubble is formed enabled Strelkov to determine the order of magnitude of the apparent thermal conductivity. In the case of a freely suspended wire it proved to be approximately \(1.5 \cdot 10^{1}\ \text{watt}\cdot\text{cm}^{-1}\cdot\text{deg}^{-1}\) (for temperatures lying not far from the \(\lambda\)-point). For the same wire, clamped in a slit between two glass plates, and at the same temperatures, it was at least \(1.5 \cdot 10^{3}\ \text{watt}\cdot\text{cm}^{-1}\cdot\text{deg}^{-1}\). This result, too, clearly contradicts Keesom’s conception of the “superthermal conductivity” of He-II, since the true thermal conductivity, however approximate the estimates may be, cannot change by two orders of magnitude depending on the experimental conditions.

It should be noted that in some cases Strelkov was able to observe the formation of a gas bubble from the superheated liquid. Kapitza also points to an analogous phenomenon—the appearance of a certain temperature gradient near solid surfaces freely washed by He-II, to which heat is supplied.^36

4. Reversibility of thermohydrodynamic processes in He-II

Thus, the experiments described in the preceding section unambiguously establish the character of the two counterflows by means of which heat is transferred in He-II. The next step must be taken in the direction of clarifying their nature. Let us repeat that one of these flows has the properties of a viscous liquid, whereas the other flow corresponds to the potential motion of an ideal liquid. It is quite natural that, if one attempts to explain heat transfer in He-II by means of such a picture, one must assume that the two flows are in fundamentally different energetic states. The merit of such a formulation of the question and the experimental solution of the problem belong to Kapitza.^38 The experiment is based on a phenomenon observed by a number of authors, the essence of which is as follows: a vessel, separated from a helium bath by a narrow capillary, fills with He-II and is cooled. In this process a certain temperature difference arises, which prevents further inflow of liquid. If heat is generated inside the vessel, then it will fill with helium much more rapidly. If, consequently, such a power is generated in this vessel that no temperature difference arises, then, knowing the volume of liquid that has flowed through the capillary, one can determine the difference

between its specific heat content and the heat content of the liquid in the volume.

The result of one of the experiments is shown in Fig. 7. Along the abscissa is plotted the power supplied to the vessel (in ten-thousandths of a calorie per gram); along the ordinate—the volume rate of flow in \(\mathrm{cm^3\cdot sec^{-1}}\) (on the left) and the resulting temperature difference in hundredths of a degree (on the right). As is seen from the curves, at small powers the rate of flow is proportional to the power. As for the temperature difference, in the initial stage of the experiment it is absent. However, upon reaching a certain power, the temperature difference begins to increase rapidly, while the rate of flow slows down. Obviously, this regime corresponds to a certain critical flow velocity, at which viscous forces appear and the phenomenon of superfluidity disappears.

Fig. 7. Typical behavior of the volume rate of flow (solid curve) and of the temperature difference arising when He-II flows through a capillary (dashed curve).

The difference in heat contents \(Q\), discussed above, was measured by Kapitza over a wide temperature interval and under different experimental conditions (mainly the width of the slit was varied). Its numerical value is obtained by processing curves analogous to Fig. 7. The form of the curve \(Q=f(T)\) is shown in Fig. 8.

Fig. 8. Temperature dependence of the heat content of He-II, according to Kapitza.

E. L. Andronikashvili and K. A. Tumanov

If He-II is forcibly pressed through a slit, then the interior of the vessel into which the flow takes place will be cooled. This phenomenon was studied in detail by Kapitza, who established a linear dependence of the resulting temperature difference on the pressure difference. From these experiments one can also obtain the difference in heat contents. The results obtained fit excellently on the curve in Fig. 8, which is incontrovertible proof of the complete reversibility of hydrothermal processes. From general thermodynamic considerations it follows that such reversibility can occur only if the helium flowing through the slit carries no entropy with it; consequently the quantity \(Q - ST\), plotted in Fig. 8, will represent the absolute value of the specific heat content of He-II contained in the volume (\(S\) is the entropy of He-II).

Thus, if helium under hydrostatic pressure flows through a slit along which there is a certain temperature gradient, then the total head \(\Pi\) will be composed not only of the ordinary hydrodynamic head \(gh\rho\) (\(g\) is the acceleration of gravity, \(h\) the height of the liquid column, \(\rho\) the density of helium), but also of the temperature head, equal, as Kapitza showed, to \(A\rho Q \dfrac{\Delta T}{T}\), where \(A\) is the mechanical equivalent of heat and \(Q\) the heat content. Consequently,

\[ \Pi = gh\rho + A\rho Q\,\frac{\Delta T}{T}. \]

It follows directly from this that an effect must exist in which the presence of a temperature difference in two vessels connected by a capillary causes the appearance of a pressure difference, while the presence of a pressure difference, conversely, causes the appearance of a temperature difference. This phenomenon, experimentally discovered before Kapitza’s work, was known as the thermomechanical effect. This effect is especially striking during the fountain effect of He-II.

It should be particularly emphasized that failure to take account of the temperature head in all earlier investigations of the viscosity of He-II carried out with capillary viscometers led to an incorrect value of this quantity. The determination of the viscosity made by Kapitza with allowance for the thermomechanical effect led to a new value,

\[ \eta \leq 10^{-11}\ \text{poise}. \]

On the basis of these experiments of his, Kapitza proposed and partially realized a new method for obtaining low temperatures, which apparently will not have the fundamental limitations inherent in all other methods now in use. His idea consists in “filtering off” helium under high pressure, by means of a system of very narrow capillaries, from the heat contained in it. In this case the helium entering the receiver will have a lower temperature than that at the filter pores. Once there, it will be heated the less, the lower the receiver has been cooled and the

less its heat capacity. Conversely, the helium remaining in the original reservoir will gradually raise its temperature, in accordance with the fact that an unchanged quantity of heat will be distributed in a smaller volume of liquid.

5. Quantum theory of superfluidity

The first quantitative theory of superfluidity arose on the basis of Kapitsa’s experiments already described here. Its creator, L. D. Landau\(^ {56,59}\), succeeded in explaining practically all the experimental facts known at the time of its appearance, and in predicting a number of new and important phenomena.

The theory postulates a definite form of the energy spectrum of the quantum liquid—helium II. The lowest excited level of the short-wavelength part of the spectrum (the elementary excitations belonging to it were called rotons) is separated from the unexcited level of the liquid, realized at \(0^\circ\) K, by a certain energy gap \(\Delta\), whose magnitude is expressed in several degrees. The excited levels of the long-wavelength part of the spectrum (the elementary excitations belonging to it are called, as in a solid, phonons), on the contrary, adjoin directly the ground level. In essence, the long-wavelength part of the spectrum consists of ordinary longitudinal sound waves.

Fig. 9. Spectrum of a quantum liquid according to Landau.

It should be understood with particular clarity that, when speaking of energy levels, in the present case we mean not the energy levels of individual atoms, but the energy levels of the entire liquid as a whole, in other words, of the aggregate of all atoms participating in the collective motion. The relation between energy and momentum for both parts of the spectrum (both the long-wavelength and the short-wavelength parts) is shown in Fig. 9. The shaded parts of the drawing correspond to levels occupied by phonons and rotons. For the long-wavelength part there is a proportionality between energy and momentum, which then gives way to a more complex dependence. The short-wavelength part of the spectrum is characterized by a minimum. The ordinate corresponding to this minimum is the energy gap, while its abscissa has the meaning of a certain momentum \(p_0\), possessed by a stationary roton. In addition to these two constants \(\Delta\) and \(p_0\), in the theory

superfluidity there is also another constant, namely the effective mass of the roton \(\mu\), which plays a significant role. On the basis of more recent experimental data, the following values should be assigned to these three constants:

\[ \frac{\Delta}{k}=9.6^\circ;\qquad \mu=0.77m_{\mathrm{He}};\qquad \frac{p_0}{\hbar}=1.95\cdot 10^8\ \mathrm{cm}^{-1}. \]

Let us again stipulate that at \(T=0^\circ\mathrm{K}\) both types of excitations are completely absent.

Considering the ensemble of elementary thermal excitations existing at temperatures different from \(0^\circ\), as a mixture of two ideal gases: a roton gas and a phonon gas, obeying Bose–Einstein statistics, Landau succeeded in determining the temperature dependence of the heat capacity of He-II. It is composed of a phonon heat capacity proportional to \(T^3\), and a roton heat capacity exponentially dependent on temperature. Since, correspondingly, the number of phonons grows as \(T^4\), whereas the number of rotons increases exponentially with \(T\), already at rather low temperatures the roton gas begins noticeably to prevail over the phonon gas. The calculated values of the roton heat capacity and entropy agree brilliantly with the values obtained experimentally (see Fig. 8).

The presence of a gap in the energy spectrum of thermal excitations is an extremely fundamental fact. As Landau showed, it is precisely this that explains the superfluidity of He-II, i.e. the absence of energy exchange between the wall of a vessel and the liquid moving relative to it. However, at velocities of motion of the order of tens of meters per second, a quantum of excitation may arise in the liquid, and thus the phenomenon of superfluidity must disappear (critical velocity).

In practice, for reasons as yet unclear, the critical velocities lie considerably lower. Consideration of the roton-phonon gas leads to the conclusion that, at a constant temperature different from zero, motion of a solid wall inside liquid helium, occurring with sufficiently small velocity, does not cause the generation of new thermal excitations. However, the existing excitations are carried along by the wall, dissipating its energy. Hence it becomes clear that the flow of a thermal gas through a thin capillary or slit, accompanied by friction, occurs the more slowly the thinner the capillary. In the limiting case of an infinitely thin capillary, the thermal quanta, owing to their “viscosity,” have no possibility at all of penetrating through it. At the same time the “liquid,” which possesses no friction, freely seeps through a slit of any thickness, provided only that this seepage occurs at a velocity less than the critical one. It is precisely these properties of the теп-

...of thermal quanta is explained by the presence in helium-II of two viscosities, differing from one another by at least a factor of a million (compare the experiments of Kapitsa, on the one hand, and of Keesom and MacWood, on the other). Naturally, a liquid flowing through a capillary is depleted of thermal excitations, and its temperature differs from the temperature of the reservoir from which it flows the more, the thinner the capillary.

Thus the experimental fact according to which the inflow into a vessel of helium-II through a narrow capillary is accompanied by cooling, while outflow along the same path is accompanied by heating of the contents of the vessel, becomes completely understandable.

The thermomechanical effect, of which mention was made above, from this point of view is a simple analogue of osmotic pressure; the role of the semipermeable partition is played by the capillaries, and the role of the dissolved molecules by the thermal excitations.

The theory also explains the mechanism of heat transfer in He-II. In particular, Kapitsa’s observations, already described by us, of the outflow of He-II streams from capillaries occurring under the influence of heat (measurement of the pressure on a vane, topography of the jet, reaction of the jet) receive an especially elegant interpretation.

Let us again imagine a small vessel with a heater inside it, filled with He-II and connected with a helium bath by means of a thin capillary. Naturally, thermal excitations will form near the heater, the concentration and, consequently, the “pressure” of which inside the vessel will be somewhat greater than their concentration and “pressure” in the remaining volume. The pressure difference will cause a gas of thermal excitations to flow through the capillary. And, despite the presence of friction against the walls, this gas will burst out of the nozzle with considerable velocity. In any case this velocity is sufficiently great for the stream of quanta to preserve over a distance of centimeters the form of a jet and to exert a noticeable pressure both on bodies that it encounters (a vane) and on the vessel from which it has burst out (reactive rotation of the spider).

Thus Kapitsa’s experiments, in essence, reveal in a new way the character of thermal processes at low temperatures: heat, existing in the form of an aggregate of elementary excitations, possesses inertia.

N. N. Bogolyubov²⁰, with the aid of an ingenious method, showed that a rarefied Bose–Einstein gas with repelling particles does indeed possess an energy spectrum similar to that shown in Fig. 9. Bogolyubov’s results, although they are not directly applicable to real liquid helium, can nevertheless be regarded as a convincing theoretical substantiation of Landau’s conception.

6. Direct Proof of Two Kinds of Motion in Helium-II. Hydrodynamics of Helium-II

We have considered a number of phenomena taking place in He-II from the point of view of the behavior of “elementary particles”—thermal quanta. Let us now turn to a description of the properties of He-II from the macroscopic point of view.

Let us imagine an axially symmetric vessel filled with helium-I and rotating about its axis. If the moment of inertia of such a system is measured, it will obviously be equal to the sum of the moments of inertia of the vessel itself and of the liquid filling it; an analogous measurement carried out at absolute zero would show that the moment of inertia of the system is exactly equal to the moment of inertia of the vessel: under these conditions helium-II could certainly not be set into rotation because of its superfluidity. What, then, will be observed at temperatures different from zero but lying below the \(\lambda\)-point? As follows from Landau’s theory, the rotation of the vessel will entrain the thermal excitations. It turns out that the momentum of the thermal quanta, and consequently also their moment of inertia, will be different from zero, and its magnitude will be a function of temperature. At the \(\lambda\)-point the moment of inertia of the gas of thermal excitations—mainly rotons—reaches its maximum value, the moment of inertia of the whole liquid, which participates entirely in the rotation. This prediction of the theory was checked experimentally by E. L. Andronikashvili\({}^{16}\), who observed the temperature dependence of the period of axial-torsional oscillations of a vessel filled with He-II (Fig. 10). The vessel consisted of a stack of light parallel disks (thickness \(0.01\) mm), separated from one another by a distance of \(0.2\) mm. The stack was surrounded by a very light little bucket rigidly connected with it.

Fig. 10. Temperature dependence of the period of oscillation of a stack of parallel disks with helium-II, according to Andronikashvili.

With the aid of this experiment the possibility of the existence of two kinds of motion in He-II is directly proved: one normal-

of one (the liquid participating in it is capable of performing both potential and vortical motion—$\operatorname{rot}\mathbf{V}\ne0$—and of being carried along by the walls of the rotating vessel) and of the other—the superfluid one (the liquid participating in it is not capable of twisting—$\operatorname{rot}\mathbf{V}=0$—and can perform only potential motions).

Associated with the normal type of motion is a certain effective mass or density $\rho_n$, whose temperature dependence was measured in the experiments of Andronikashvili$^{16,17}$ and is shown in Fig. 11.

Fig. 11. Temperature dependence of the normal density of helium-II, according to Andronikashvili (circles) and according to Peshkov (dashed line).

On the same curve, for comparison, are plotted the data obtained by V. P. Peshkov$^{68}$ by calculating $\dfrac{\rho_n}{\rho}$ from the velocity of propagation of second sound in He-II, using formulas following from Landau’s theory.

From the curve in Fig. 11 one can calculate the width of the energy gap $\Delta$, as well as one of the quantities $\mu$ or $p_0$ (if the other is assumed known).

To the other type of motion—the potential, or superfluid, one—it is natural likewise to assign its own effective mass, or density—

density \(\rho_s\), while the sum \(\rho_n+\rho_s=\rho\) is the total true density of the liquid.

These two kinds of motion interpenetrate each other without exchanging momentum.

Assigning to each of the kinds of motion its own velocity and its own density, we may speak of the flow of two liquids interpenetrating each other: a normal one and a superfluid one. It should be firmly remembered, however, that this division is entirely formal and is nothing more than a convenient way of describing the phenomena.

The most important equation for He-II is the equality

\[ \rho \mathbf{v}=\rho_s \mathbf{v}_s+\rho_n \mathbf{v}_n, \]

where \(\mathbf{v}\), \(\mathbf{v}_s\), and \(\mathbf{v}_n\) are the velocities of motion of helium as a whole, of its superfluid part, and of its normal part, respectively. If \(\rho \mathbf{v}=0\), then the motion of He-II can take place without mass transfer.

The most important parameter of He-II, determining its properties at any given temperature, is the ratio of the normal and superfluid densities \(\dfrac{\rho_n}{\rho_s}\), with \(\rho_s=\rho\) and \(\rho_n=0\) at \(T=0\), and at the \(\lambda\)-point \(\rho_s=0\), \(\rho_n=\rho\). At a temperature of \(2.19^\circ\mathrm{K}\) there occurs a smooth disappearance of the superfluid liquid, with which are associated all the phenomena accompanying phase transformations of the second kind, for example, the absence of latent heat of transition, the presence of a jump in the heat capacity, and so on.

Since superfluid motion, regardless of the specific conditions in which it occurs (a narrow slit, a wall film, a volume of free helium), carries no entropy with it, this method of describing the properties of He-II can also be used successfully when considering the mechanism of heat transfer in He-II. According to the new method of description, heat transfer takes place by means of the normal liquid, toward which and “through which” the superfluid liquid moves in the direction of the heat source. Near the heat source it is transformed into normal liquid, and all the heat released is expended on this. So long as the critical regime has not been reached, a temperature gradient should not arise near the heat source, since the rate of supply of the superfluid part and the rate of removal of the normal part are automatically regulated by the amount of heat released. This, however, is true only as a first approximation.

The Navier–Stokes equation for He-II, written in general form, as Landau has shown\(^{58}\), splits into two independent equations, one for the superfluid and the other for the normal part. The latter differs in no way from the ordinary equation of a viscous liquid.

Therefore, considering the motion only of the normal part, occurring under the influence of heat, one may use the ordinary Poiseuille equation. Owing to viscous forces, a “pressure” difference (for the normal mass) must arise at the ends of the capillary. But an increase in pressure, i.e. an increase in the concentration of thermal excitations, obviously entails an increase in temperature, which, as we already know, is given by the ratio \(\rho_n/\rho\), or, what is the same, by the density of thermal excitations. Thus, in reality, at the ends of the capillary there always arises a certain temperature difference, determined from the equation \(\Delta T=\dfrac{\Delta p}{\rho S}\). However, this temperature difference is usually so small that in many experiments it is not taken into account at all.

7. Normal viscosity of He-II

As we have already indicated, some investigators attempted to measure the viscosity of He-II by methods different from the method of capillary viscometers developed by Kapitza.

Fig. 12. Temperature dependence of the viscosity of helium-II, according to Keesom and Macwood.

The most reliable were considered to be the experiments of Keesom and Macwood, who obtained for the viscosity the curve shown in Fig. 12. However, these authors did not take into account the circumstance that in their experiments only the normal part takes part in the transfer of momentum,

mass. Therefore, the results of their measurements were erroneously referred by them to the whole mass of helium-II.

Fig. 13. Temperature dependence of the viscosity of helium-II, according to Andronikashvili (circles on the solid curve) and according to Landau and Khalatnikov (crosses on the dashed curve).

Recently Andronikashvili\(^{18}\), also using the oscillating-disk method, carried out measurements of the viscosity of the normal

DEVELOPMENT OF THE DOCTRINE OF SUPERFLUIDITY AND SUPERCONDUCTIVITY

mass. The results of his experiments are shown in Fig. 13 (solid curve). The curve falls smoothly from the λ-point, approximately as \(\rho_n\). Around \(1.9^\circ\) K it turns into a straight line parallel to the abscissa axis. This is the region in which the gas of thermal excitations may be regarded as an ideal gas: its viscosity does not depend on temperature.

In the present case temperature plays the same role as pressure for ideal gases, since it is temperature that determines the concentration of thermal excitations. Apparently, at these temperatures the viscosity of the phonon gas plays no role. From the value of the viscosity coefficient one can calculate the effective cross-section of a roton, which proves to be close to the cross-section of a helium atom.

At lower temperatures, of the order of \(1.5^\circ\) K, the viscosity coefficient again increases sharply.

On the basis of Andronikashvili’s experiments, L. D. Landau and I. M. Khalatnikov\({}^{60}\) succeeded in calculating the temperature dependence of the mean free path of phonons (phonon scattering by rotons) and, on this basis, in predicting the behavior of the purely phonon viscosity. It turns out that already at temperatures of \(1.5^\circ\) K the mean free path of phonons becomes a very appreciable quantity and continues to grow rapidly as the temperature is lowered. This explains the exponential behavior of the viscosity in the low-temperature region. The theoretical data are shown by the dashed curve in Fig. 13. Thus, at present it is possible to speak of the viscosity of the superfluid and of the normal component. But in the normal viscosity as well, one must distinguish roton viscosity and phonon viscosity, although in all experiments they appear together.

It should be noted that when the paths of the phonons become comparable with the dimensions of the vessel, the normal viscosity will again begin to decrease. The position and magnitude of this maximum on the curve of the temperature dependence of viscosity will, naturally, be determined by the specific experimental conditions. However, in principle, in a vessel of sufficiently large dimensions and at sufficiently low temperatures the viscosity of the phonons can reach any prescribed values.

Apparently, the anomalous behavior of the viscosity should also explain the following very interesting fact observed by P. P. Savich and A. I. Shalnikov\({}^{77}\). Into a Dewar with liquid helium under pumping, gaseous helium containing an admixture of air was admitted. The impurities condensed, as a result of which a colloid was formed from the finest crystals of solid air. The colloidal air behaved very stably in He-I, which at the same time became opalescent under side illumination. As soon as the temperature fell below the λ-point, the colloid rapidly coagulated, precipitating in the form of large and loose flakes. The true causes of this phenomenon have so far remained unexplained.

8. Second sound in He-II

Landau’s theory was able to predict yet another new and extremely interesting phenomenon. In He-II two kinds of waves can propagate, with two different velocities differing from one another by one order of magnitude. The first kind of wave is ordinary sound, with a propagation velocity equal to \(240 \text{ cm/sec}\) \(^{58a}\). The other kind of wave has been given the name “second sound,” or thermal waves. Its propagation velocity depends sharply on the temperature and, in order of magnitude, is equal to \(20 \text{ m/sec}\).

The conditions for exciting second sound were considered by E. M. Lifshitz \(^{61}\), who, on the basis of Landau’s theory, showed that the most favorable case is its radiation either by a surface with a periodically varying temperature, or by a plate oscillating in its own plane.

In addition, he succeeded in calculating in advance the temperature dependence of its propagation velocity (Fig. 14).

Fig. 14. Temperature dependence of the propagation velocity of second sound, according to Lifshitz.

Part of the experimental discovery of this unusual effect belongs to V. P. Peshkov \(^{67,68}\). As a generator of second sound he used an electric heater made in the form of a flat spiral of thin wire, through which an alternating current was passed. As a result, a periodic oscillation of the heat flux occurred at the surface of the wire, and this propagated through space in the form of a thermal wave. Whereas in ordinary viscous media a thermal wave rapidly dies out, in He-II, owing to the inertia possessed under these conditions by heat (see § 5), thermal waves turn out to be practically undamped.

In a certain sense an analogy can be drawn between an ordinary sound wave in a gas and a thermal wave. A sound wave propagates by means of gas molecules moving under the influence of a periodically varying pressure (concentration). The temperature oscillations that occur owing to the adiabatic nature of the process are usually small. A thermal wave propagates by means of rotons and phonons moving under the influence of a periodically varying temperature (concentration). The small pressure oscillations here are a consequence of the thermal expansion and compression, which for He-II is very small. If in ordinary sound

takes place, then in second sound it gives way to the heat flux.

As we already know, the flux of heat quanta moving in some direction can formally be replaced by two mutually opposite motions of the normal and superfluid components. Since a periodic distribution of temperature means the same periodic distribution of the concentration of rotons

Fig. 15. Temperature dependence of the propagation velocity of second sound, according to Peshkov.

and phonons, i.e. of the normal density \(\rho_n\), then, owing to the equality \(\rho_n+\rho_s=\rho=\mathrm{const}\), an outflow of the superfluid component must be observed from places with increased normal density, while at the same time the superfluid mass will flow into places with a decreased value of \(\rho_n\).

Thus, the heat wave can again be represented as the superposition of two motions of the normal and superfluid masses, oscillating in mutually opposite directions.

Peshkov was able to detect the heat wave with the aid of a receiver—a resistance thermometer made of phosphor bronze, which, as

like the heater, was made in the form of a flat spiral and was placed parallel to the radiator. Observations were carried out by means of a cathode oscilloscope, both on standing and on traveling waves.

In addition to the method already described, Peshkov[^69] used another, no less ingenious method for generating second sound. A metallic

Fig. 16. Isotherms of the velocity of second sound, according to Peshkov and Zinov’eva.

cylinder, closed on one side by a membrane and on the other by a fine-pored filter, was immersed in a helium bath. With the aid of a loudspeaker, intense oscillations of the membrane were excited, transmitted in the form of periodically varying pressure to the helium enclosed inside the cylinder; this caused only one superfluid component to be forced out and drawn in through the filter. Thus, on the outer side of the filter facing the helium bath, a periodic flow of cold was excited, i.e., conditions were created for the emission of second sound. The form of the curve of the temperature dependence of the propagation velocity of second sound, obtained by Peshko-

ing, is shown in Fig. 15. It reproduces almost exactly the form of the curve calculated in advance by Lifshitz. However, the numerical discrepancy amounts to about 20% and cannot be explained by experimental errors.

This discrepancy served as the reason for revising the form of the energy spectrum originally proposed by Landau^56.

Fig. 17. Isobars of the velocity of second sound, according to Peshkov and Zinov’eva.

As was indicated in § 6, from Peshkov’s data one can also obtain values of $\frac{\rho_n}{\rho}$, which, throughout the entire temperature interval in which the measurements were made, are in good agreement with the results of Andronikashvili^17.

The question of the velocity of propagation of a heat pulse in He-II had long attracted the attention of scientists. Interesting attempts were made, for example, by Gantz^9a. However, his experiments did not give unambiguous results.

The investigations we have discussed on the nature of second sound received further development both in the Soviet Union and in the work of foreign scientists^10a.

The next step in this direction was the work of Peshkov and Zinov’eva,^70 who studied the dependence of the velocity of second sound on pressure; they covered the entire region of existence of He-II at temperatures from 1.3°K to the λ-point. Numerous curves are given in Figs. 16 and 17. In addition, by means of this method they succeeded in determining the transition curves of He-II into He-I and into solid helium (Fig. 18). Their results confirm the results of Keesom,^11a obtained by an entirely different method.

Determination of the dependence of the propagation velocity of second sound on pressure is an important stage for further quantitative verification of the theory of superfluidity and for obtaining refined values of the fundamental constants of the theory, \(\mu\) and \(\rho_0\).

Fig. 18. Transition curves of He-II into He-I and into solid helium according to Peshkov and Zinov’eva.

9. Motion of He-II Films

As was indicated in § 1, the enormous mobility of thin wall-adjacent layers of He-II long ago attracted the attention of scientists.

One of the first works devoted to the study of the physical properties of films was the work of A. K. Kikoin and B. G. Lazarev,^41 who succeeded in establishing that intense heat transfer can occur along a thin wall-adjacent layer. In their experiments, wetting films rose along glass tubes to a height of 15 cm. The upper end of this tube was connected to a copper tube, on which were placed a heater and a resistance thermometer made of phosphor bronze. Despite the current passing through the heater, the thermometer showed an unchanged temperature, differing little from the temperature of the bath \((\Delta T \simeq 10^{-4}\ \text{degree})\). Only at a certain critical current strength did the bronze thermometer begin to show an increase in temperature. In the authors’ opinion, these experiments indicate that the film possesses a high true thermal conductivity, which increases smoothly from zero at the λ-point and reaches a constant value at a temperature of 1.5°K and below. The phenomenon described is not observed in He-I.

In a subsequent experiment Kikoin and Lazarev^42 studied the thickness of the film. By immersing the tip of a tube connected with a strongly developed copper surface into a helium bath, they were able to register a noticeable lowering of the liquid level in the Dewar. Obviously, a considerable quantity of He-II passed in the form of a film onto a large

surface of the metal. The thickness of the film, measured in this way, turned out to be \(2\)—\(3\cdot 10^{-6}\) cm. The data of Kikoin and Lazarev are in good agreement with the results of other authors\(^{12a}\), obtained subsequently by an entirely different method.

Nevertheless, it still cannot be considered that the thickness of the film has been determined with sufficient accuracy, although it is a very important quantity for the theory of He-II. Thus, for example, Kapitsa, by direct weighing of the film, established that its thickness is not less than \(10^{-5}\) cm.

The conclusions which Kikoin and Lazarev\(^{41}\) draw from their experiments on determining the thermal conductivity of the He-II wetting film are refuted by Strelkov’s experiment\(^{80}\). An inverted U-shaped vessel, both arms of which were soldered shut and did not communicate with the helium bath, was filled with some amount of He that was in thermal equilibrium with the liquid surrounding the vessel. The liquid in one arm of the tube could communicate with the contents of the other arm only through the He-II film covering the walls of the connecting tube. As Strelkov found, a slight heating of the liquid in one arm, even by means of a weak beam of light, is sufficient to cause a rise of the liquid level in the heated part of the vessel at the expense of a lowering of the level in its other part. It is clear that in this case we are dealing with a thermomechanical effect occurring through the film, which plays the role of a thin capillary. Along this “capillary” there occurs a non-viscous flow of He directed against the temperature gradient.

Fig. 19. Temperature dependence of the quantity \(vd\), according to Strelkov.

A detailed study of the siphon effect, not complicated by a temperature gradient and occurring only under the influence of the difference of levels in vessels communicating by means of a film, was undertaken by Strelkov\(^{23}\). Having no independent method for determining the film thickness, he measured the quantity \(vd\)—the product of the creep velocity along the film by its thickness. The experiments were carried out both in open vessels and in closed ones. In the latter case the vessel was filled with He by condensation. The temperature dependence of \(vd\) is shown in Fig. 19. As is seen from the drawing, the creep velocity tends to zero as the temperature approaches the \(\lambda\)-point. At lower temperatures the creep velocity tends toward saturation. Taking the film thickness to be \(5\cdot 10^{-6}\) cm, one may suppose that the velocity of motion of the film does not exceed \(20\ \text{cm}\cdot\text{sec}^{-1}\).

In addition, he succeeded in discovering a dependence of the quantity \(vd\) on the length of the siphon (the distance from the upper liquid level to the edge of the vessel). For small lengths this dependence is approximately linear. A weak dependence of the overflow velocity on the head was also observed: when the head was reduced by a factor of 80, the overflow velocity decreased only by a factor of 1.5. However, when the difference in levels was reduced within the range from \(0.5\) mm to zero, the overflow velocity also rapidly fell to zero.

Naturally, the rate of equalization of the levels is determined by the perimeter of the vessel or, more precisely, by the perimeter of the narrowest part of the vessel, if the constriction is located above both liquid levels.

All the results obtained by Strelkov in studying the siphon effect were confirmed in the work of Daunt and Mendelssohn\(^{12a}\), who independently arrived at the same conclusions.

Strelkov also undertook an interesting attempt to measure the momentum of the film. For this purpose he constructed sensitive spring balances, from which there was suspended either a strip of thin foil heated by light, or a wire heated by current.

Creeping, under the influence of the temperature gradient, onto the strip or the wire, the film should have, in the author’s opinion (in accordance with the principle of conservation of momentum), stretched the sensitive spring of the balance. Having obtained an effect of the opposite sign, Strelkov was forced to abandon this experiment.

The ability of He-II to form rapidly creeping films is a substantial obstacle to the attainment of low temperatures by the pumping method. Indeed, even with the aid of very powerful pumps and in sufficiently narrow Dewars, it is rarely possible to reach temperatures below \(1^\circ\) K. The point is that the He-II film covering the walls of the Dewar very greatly increases the evaporation surface. In the upper part of the Dewar this evaporation proceeds especially rapidly. Owing to its great mobility, however, the film continuously maintains its thickness. There is therefore nothing surprising in the fact that Keesom\(^{13a}\) succeeded in reaching a temperature of \(0.726^\circ\) K in a volume of \(3\ \text{cm}^3\) only with the aid of a powerful battery of pumps with a total capacity of 675 liters per second.

Some investigators succeeded in increasing the effectiveness of pumping by means of a narrow constriction of the inner vessel of the Dewar. The perimeter of the constriction determined the amount of film able to penetrate into the upper part of the Dewar.

Analogous ideas underlie the work of Lazarev and Esselson\(^{49}\), who divided the Dewar into two parts by means of a flat diaphragm with an aperture diameter down to \(0.05\) mm. The authors of this work succeeded in obtaining a temperature of \(0.71^\circ\) K (registered by measuring the elasticity of saturated vapors) with the aid of a single unit consisting of a fore-vacuum pump and a mercury diffusion pump of small

productivity. This temperature should be regarded as the lowest achieved by pumping.

All the experimental results set forth in this paragraph are readily explained by Landau’s theory, if the film is thought of as an analogue of an extremely thin capillary along which a superfluid mass flows. The normal part of He-II is practically immobile relative to the solid surface. The large thermal conductivity of films, observed by Kikoin and Lazarev[^41], is explained by the unimpeded transport along the film of the superfluid part, flowing with zero entropy. The fact that the temperature in a reservoir rises when the film flows out of it, and falls when the film flows into it, receives a quite straightforward explanation, as does the fact of the temperature creep (the creeping of the film against the temperature gradient).

If one takes into account that only the superfluid part flows, whose quantity depends on temperature, it turns out that the curve of Fig. 19 should be recalculated[^69] in terms of the superfluid density, and for the quantity \(vd\) under the given experimental conditions one obtains an approximately straight line, i.e. the velocity of overflow proves to be independent of temperature.

The dependence of \(vd\) on the height of the siphon, in all likelihood, must be explained, as Strelkov[^80] notes, by the nonconstancy of the film thickness along the solid wall: near the liquid level the film thickness is greater.

The theory also makes it possible to interpret Strelkov’s experiment on measuring the momentum of the film. As has already been noted, the superfluid part of He-II exerts no pressure on bodies around which it flows. Therefore it is hopeless to detect the momentum of a superfluid film. But at the moment of heating the ribbon of the spring balance, the superfluid mass passes into the normal state, imparting to it its momentum. The normal mass, however, owing to frictional forces, is rapidly brought to rest, in turn giving up its momentum to the wall along which the motion takes place. This effect, apparently, is what Strelkov observed. Such an interpretation of his experiments, so far as we know, is given here for the first time.

Of course, the thickness of the film cannot be calculated from Landau’s theory, since determining it requires postulating the law of interaction that exists between the helium atoms and the solid wall. A calculation of the film thickness was carried out by Ya. I. Frenkel[^86], who assumed that the film, being under the action of gravity, is acted upon by van der Waals–type forces from the solid wall. For the case of complete wetting of the wall by the liquid he obtained an expression for the maximum height of the film. Its thickness depends on the height, varying from approximately \(10^3\) atomic layers near the surface to the thickness of a monomolecular layer at the maximum height.

10. Critical Velocities

As was indicated in § 5, Landau’s theory necessarily implies the existence of a critical velocity.

A direct determination of critical velocities has apparently been carried out only in the experiments of Kapitsa^38, who observed the disappearance of the reversibility of the motion of helium through a capillary and the slowing down of the rate at which it flowed into a vessel.

The values obtained by him, other conditions being equal, depend little on the temperature and quite strongly on the width of the slit. Unfortunately, as the author himself notes, the slit was somewhat wedge-shaped, and it is difficult to indicate the true value of the linear velocity of flow at which viscous forces appeared. The possibility is also not excluded of a breakdown of superfluidity when the edges of the slit are flowed around. With these reservations, on the basis of Kapitsa’s experiments the following numerical values of the critical velocity are obtained:

\[ \begin{aligned} &\text{at } T=1.557^\circ \mathrm{K} \text{ and slit width } 0.14\,\mu,\quad v_k=110\ \mathrm{cm/sec},\\ &\text{at } T=1.935^\circ \mathrm{K} \text{ and the same slit width } v_k=80\ \mathrm{cm/sec},\\ &\text{at } T=1.64^\circ \mathrm{K} \text{ and slit width } 3\,\mu,\quad v_k=14\ \mathrm{cm/sec},\\ &\text{at } T=1.64^\circ \mathrm{K} \text{ and slit width } 0.3\,\mu,\quad v_k=40\ \mathrm{cm/sec}. \end{aligned} \]

It follows from this that the critical velocity probably decreases with increasing size of the vessel as

\[ \frac{1}{\sqrt{d}}, \]

where \(d\) is the characteristic dimension.

The existence of critical velocities explains many experimental facts. For example, the independence of the velocity of outflow from very narrow capillaries on the head, observed^6a for large pressure differences, may occur owing to the presence of critical regimes that automatically regulate the speed of the flow process. It is precisely by the fact that critical velocities exist that one can explain the independence of the rate of flow on the head observed by Strelkov^80 and others. At the same time, the sharp dependence of the velocity of motion of the film on the head, occurring for level differences from 0 to 0.5 mm of helium column, becomes comprehensible.

The presence of critical velocities makes qualitatively understandable such a seemingly strange phenomenon as the dependence of the apparent thermal conductivity on the power supplied to the heater. Undoubtedly the thermal conductivity must begin to decrease rapidly as soon as the critical velocity is exceeded.

Connected with the theory of superfluidity is the conception, essentially, of an ideal mechanism of heat transfer, automatically ensuring the complete removal of heat from the heater. The phenomenon already noted in this review—the heating of heat-dissipating surfaces—was therefore also explained by the fact that critical regimes exist.

However, in many cases the counterflow velocities of the normal and superfluid components are so small, and the temperature of the surface so …

of the heat source so greatly exceeds the mean temperature of the bath that it is now becoming obvious that critical velocities are of no consequence here.

Despite the abundance of work carried out under deliberately “closed” conditions, not only has the question of the physical essence of the breakdown of superfluidity not been clarified, not only have the true causes that, under one set of conditions or another, give rise to the irreversibility of helium motion not been formulated, but even the dependence of \(v_k\) on such factors as temperature, the characteristic dimensions of the problem, etc., has remained essentially undetermined.

The study of questions connected with this phenomenon undoubtedly constitutes a rich field of activity.

II. SUPERCONDUCTIVITY

1. Superconducting State

The discovery of the phenomenon of superconductivity, which in some respects remains mysterious even in our day, was made by Kamerlingh Onnes\(^{15a}\) in 1911. The first metal to be brought into the superconducting state was mercury: at a temperature of \(4.2^\circ\text{K}\) its resistance suddenly drops to a value not measurable with the aid of even the most sensitive modern instruments. An estimate of the upper limit of the resistance was obtained from Kamerlingh Onnes’\(^{6a}\) classical experiment with a superconducting ring, in which a once-induced current did not die away to any noticeable extent over several hours. From this it may be concluded that the resistance of a metal in the superconducting state is, at the very least, \(10^{12}\) times smaller than the resistance of the metal at room temperatures. Therefore, in all theoretical considerations the resistance of a superconductor is taken to be exactly equal to zero.

At present most metals have been investigated with respect to superconductivity, many of them down to temperatures of several hundredths of a degree of the absolute scale. Of all metals, 19 have proved to be superconductors. Their list is given in Table 1. In addition to pure metals, some alloys and chemical compounds possess superconductivity.

The latest of the superconductors—uranium—was discovered only a year ago by Alekseevskii and Migunov\(^{14}\).

Each of the metals listed in Table 1 passes into the superconducting state at one or another temperature, which is called the critical temperature—\(T_k\).

At the transition point, besides the disappearance of resistance, a change is observed in a number of physical properties; for example, there is a jump in the heat capacity of the metal, and its magnetic properties change sharply.

Other properties, on the contrary, change smoothly at the critical point. The transition takes place without the release or absorption of latent heat.

All this indicates that the transition to the superconducting state (just as the transition of liquid helium through the $\lambda$-point) is a phase transition of the second kind.

However, superconductivity can be destroyed not only by raising the temperature to its critical value, but also by a magnetic field.

Table 1

Element	Critical temperature	Year of discovery
Mercury Hg	4.12	1911
Tin Sn (white)*	3.69	1913
Lead Pb	7.26	1913
Thallium Tl	2.38	1922
Indium In	3.37	1923
Tantalum Ta	4.38	1928
Gallium Ga	1.07	1929
Thorium Th	1.43; 1.32	1929
Titanium Ti(?)	1.77; 1.81	1930
Niobium Nb (columbium Cb)	9.22	1930
Aluminum Al	1.14	1933
Zinc Zn	0.79	1934
Vanadium V	4.3	1934
Cadmium Cd	~0.6	1935
Zirconium Zr	~0.7	1935
Hafnium Hf	~0.3	1935
Lanthanum La	4.7	1937
Rhenium Re	~0.9	1942
Uranium U	~1.30	1946

field. The magnetic-field strength required for this, $H_k$, which has been given the name critical field, depends on temperature, falling to zero near $T_k$. What has been said is illustrated by Figs. 20 and 21, borrowed from the works of Shubnikov, Khotkevich, Shepelev and Ryabinin$^{100}$ and of Alekseevskii$^{6}$.

The transition from the normal state to the superconducting state and back in the presence of a magnetic field differs essentially from the transition carried out in the absence of a field; in particular, in this case there is a release of the latent heat of transformation, which corresponds to a phase transition of the first kind.

* Gray tin is not a superconductor.

The condition for two phases—the superconducting and the normal—to be able to coexist in equilibrium with one another is the equality \(^{17a}\)

\[ F_n = F_s + \frac{H_k^2}{2}, \]

where \(F_n\) and \(F_s\) are the free energies per unit volume of the normal and superconducting phases, respectively, with \(F_n\) corresponding to the case when the magnetic field is absent.

Fig. 20. Dependence of the critical field on temperature (Shubnikov, Khotkevich, Shepelev, and Ryabinin).

Since \(S = -\dfrac{\partial F}{\partial T}\), the heat absorbed in the transition of a unit volume of the metal from the superconducting to the normal state is equal to

\[ Q = T(S_n - S_s) = -\frac{T H_k}{4\pi}\frac{dH_k}{dT}. \]

A priori one may assert that the conductivity of a superconductor is effected by electrons. The well-known experiments carried out by Kikoin and Gubar \(^{45,46}\) lead to the same conclusion (see also \(^{85,26}\)). These authors investigated the Einstein–de Haas effect occurring in superconductors. The Einstein–de Haas effect consists in the fact that when a body is magnetized, a rotational moment appears in it. Kikoin and Gubar set up the following experiment: a superconducting sphere on a thin suspension was placed in a periodically varying magnetic field, with the period of the change in the direction of the field chosen equal to the period of the sphere’s own torsional oscillations on the thread. From the amplitude of the established oscillations, the damping

and other data, it was possible to obtain the rotational moment. The ratio of the magnetic moment to the mechanical moment was found to be equal to \(\frac{e}{2mc}\), where \(e\) and \(m\) are the charge and mass of the electron. This proved the existence of closed electron currents.

Since no experiments have been able to detect differences between the crystal lattices of superconductors above and below the critical point, it must be admitted that the phase transformation is undergone by the electron liquid. In this case, from a state characterized by a large value of the entropy, it passes into a state with a smaller value of the entropy.

Fig. 21. Dependence of the critical field of vanadium on temperature (Alekseevskii).

Landau\(^{56}\) suggested that, in the phase transition from the normal to the superconducting state, the electron liquid undergoes a transformation similar to the transition of liquid helium, i.e. the electron liquid acquires the property of superfluidity, by which superconductivity is explained. As indicated, helium-II is represented as consisting of two liquids—superfluid and normal; two types of motion are attributed to it—superfluid and normal. According to the proposed analogy, it was necessary to assume the existence of “normal” conductivity of a superconductor. In the stationary case the electric field in a superconductor must be equal to zero; otherwise the superconducting current would increase without bound. Therefore, in the stationary case the normal current is absent, and normal conductivity does not manifest itself in any way. If, however, a superconductor is placed in an alternating magnetic field, then in the surface layer of the superconductor, in which the external magnetic field is not equal to zero (on this see below), there will also be present an alternating electric field, which will cause an alternating normal current (of course, along with the alternating superconducting current),

An alternating normal current) will lead to Joule losses, i.e., to heating of the body. This heating in an alternating magnetic field (at a frequency of the order of \(10^9\) hertz) was experimentally discovered by H. London\(^{18a}\). The further development of Landau’s theory of the superfluidity of an electron liquid was carried out in the work of Ginzburg\(^{24}\) (see also Ginzburg’s book Superconductivity*, 1946\(^{28}\)). On the basis of this theory Ginzburg calculated the jump in heat capacity, the penetration depth (see § 4), and certain other quantities.

In view of the fact that no properties have yet been found by which superconducting metals in the normal state would differ from nonsuperconducting ones, the only way at present to draw a conclusion about the ability of a given metal to pass into the superconducting state is by directly studying it at the lowest possible temperatures. Many of the metals—Cr, Si, Sb, Be, W, Rn, U—were investigated by Alekseevskii and Migunov\(^{14}\) down to temperatures of \(0.06\)—\(0.15^\circ\mathrm{K}\). In this, only uranium (as was already indicated) exhibited superconducting properties.

2. Magnetic properties

As has already been noted, in the case where the resistance is equal to zero, it follows from Ohm’s law that the electric field \(\mathbf{E}\) inside a superconducting specimen must be equal to zero. Then from Maxwell’s equation

\[ \operatorname{rot}\mathbf{E}=-\frac{1}{c}\frac{\partial \mathbf{B}}{\partial t} \]

it follows that \(\partial \mathbf{B}/\partial t=0\) and \(\mathbf{B}=\mathrm{const}\); the magnetic induction in the superconductor is equal to its value at the moment of transition into the superconducting state and does not depend on the magnitude of the external field (of course, if it is less than the critical one). This conclusion would lead one to expect “frozen-in” magnetic moments in specimens transferred into the superconducting state in a magnetic field. Therefore the property of superconductors, discovered in 1934, of expelling the magnetic field upon transition into the superconducting state is a new fundamental property of superconductors, not a consequence of electrodynamics and of the fact that the resistance is equal to zero. This property was discovered by Meissner and Ochsenfeld\(^{19a}\), and somewhat later and independently by Rjabinin and Shubnikov\(^{74,75}\), and it received the name of the Meissner effect. The experiment of Rjabinin and Shubnikov consisted in measuring the induction in a tin rod placed along a magnetic field. Upon transition into the superconducting state, by means of decreasing the external magnetic field, the induction after the transition did not remain constant, equal to its value just before the

*) For other cases of manifestation of the existence of a normal current, see the work of Ginzburg\(^{23}\).

transition, and at the transition point it fell abruptly to smaller values. It was subsequently established that, in the case of single-crystal specimens of a shape closer to a cylinder of infinite length, the induction falls practically to zero (see Fig. 22).

Thus, at the present time it is established that inside a superconductor both the electric and the magnetic fields are equal to zero. This in turn leads, as is seen from Maxwell’s equation

\[ \mathbf{j}=\frac{c}{4\pi}\operatorname{rot}\mathbf{B}-\frac{1}{4\pi}\frac{\partial \mathbf{D}}{\partial t}, \]

to the disappearance inside the superconductor also of the current density \(\mathbf{j}\).

In order to explain the equality to zero of the magnetic induction \(\mathbf{B}\) inside a superconductor, it must be assumed that undamped currents flow over the surface of the specimen, whose magnetic field compensates the external magnetic field. The concept of surface currents is, of course, an idealization; what is meant are currents flowing in a very thin surface layer. Naturally, in this thin layer the magnetic field is different from zero.

Fig. 22. Dependence of the induction in a superconducting cylinder on the intensity of the external field (field parallel to the axis of the cylinder).

This conception can now be regarded as proved by a number of experiments (carried out both on massive and on microscopic specimens; see § 5). With the aid of these experiments it was possible to establish the presence of a thin layer within which there exists a magnetic field, and even to estimate its thickness, which proved to depend on the temperature. The distance over which the magnetic-field intensity falls by a factor of \(e\) has received the name “penetration depth.” The penetration depth is of the order of \(10^{-5}\) cm; in the immediate vicinity of the critical point it begins to increase without bound as the temperature approaches the critical one.

In § 5 works devoted to the determination of the penetration depth will be considered. Here, however, we shall deal with a new factor destroying superconductivity—namely, the electric current.

It was experimentally discovered that when the current through a superconductor is increased, superconductivity is destroyed. Silsbee\(^{20a}\) proposed that the destruction of superconductivity by a current reduces to the action of the magnetic field of this current, and therefore occurs when the field of the current becomes equal to the value of the critical field.

The verification of Silsbee’s hypothesis is the subject of a paper by Alekseevskii1. In this work, superconductivity was destroyed by an external magnetic field, by an electric current, and also by the simultaneous action of these factors. Fig. 23 gives the transition curve when the superconductivity of a single-crystal tin rod is destroyed by an external magnetic field, and Fig. 24 gives the curves when the superconductivity of the same rod is destroyed by current at various values of the external field.

Fig. 23. Destruction of the superconductivity of a single crystal of tin by an external field.

Fig. 24. Destruction of the superconductivity of a single crystal of tin by current.

Along the ordinate axis of both figures is plotted the ratio of the resistance to its value at a temperature of 3.8 K; along the abscissa axis of Fig. 23, the external magnetic field \(H\); along the abscissa axis of Fig. 24, the current \(I\) and the corresponding field \(H_i\). The experiments established the validity of Silsbee’s hypothesis. In Fig. 25 the external field \(H\) and the current field \(H_i\) are plotted along the axes. Since these fields are perpendicular, their sum is represented by the radius vector from the origin of coordinates to the given point. The points lay on a circle with its center at the origin of coordinates. This shows that superconductivity is destroyed when the sum of the external field and the current field reaches, at the surface of the specimen, the critical value.

Silsbee’s hypothesis is also confirmed by Shubnikov’s experiment with a superconducting ring2. If a metallic ring passes into the superconducting state in a magnetic field perpendicular to its plane, then after the transition the flux of magnetic induction through the ring differs from zero. When the external field is weakened, a current is induced along the ring, preventing the exit of the mag-

field lines from the ring outward. Obviously the sum of the field of the current and the external field must not exceed the value of the critical field; otherwise superconductivity will be destroyed. This was also demonstrated in Shubnikov’s experiment.

It should be noted, however, that the destruction of superconductivity by a current is not a simple phenomenon. A more careful analysis leads to the following contradiction.

Consider a superconductor through which a current flows. As was already noted, the current flows along the surface of the superconductor. As the current is increased, the field will be maximal precisely at the surface. Suppose the field at the surface has reached the critical value, and the layer belonging to the surface has passed into the normal state. Naturally, the current will flow not through the normal layer, but through the superconducting core of the conductor, since the resistance of the latter is zero. At the boundary between the normal and superconducting phases the field will be equal to \(H_k\) (by the definition of the concept of the critical field), while in the layer of the normal phase the field will be less than \(H_k\) (as is known, the field of a cylindrical conductor is inversely proportional to the distance from the axis of the cylinder). Then in the normal layer conditions will be created for its transition into the superconducting state \((H < H_k)\). It is impossible to devise any other simple process of destruction of superconductivity by a current without arriving at such a contradiction.

Fig. 25. Dependence of the critical field of the current on the magnitude of the external longitudinal field.

It turns out that one should assume that, when superconductivity is destroyed by a current, the conductor is neither in the superconducting nor in the normal state, but in a certain third—“intermediate”—state.* This state is not connected exclusively with the destruction of superconductivity by a current, but has a very general character. The next paragraph is devoted to consideration of this state.

3. Intermediate state

The complete expulsion of the magnetic field from the volume occupied by a superconductor means, in essence, that every superconductor possesses the maximum possible diamagnetic susceptibility, equal—

* The following result of Alekseevskii’s work is connected with the intermediate state: when superconductivity is destroyed by a current, the resistance rises abruptly only to 0.8 of its value in the normal state.

—\(-1/4\pi\), which is \(10^4\) times greater than the diamagnetic susceptibility of ordinary substances. It is therefore natural that the introduction of a superconductor into a magnetic field leads, as a rule, to a sharp distortion of the character of the field distribution. No distortion in the pattern of the distribution of the magnetic field in the surrounding space is introduced only by an infinitely long cylinder parallel to the direction of the field. Bodies of any other shape, as well as cylinders not situated parallel to the field, distort the field to one degree or another. Examples of the field distribution near a superconducting cylinder and sphere are shown in Fig. 26. The disturbance of the homogeneity of the field in many cases is a decisive circumstance in interpreting the results of experiments on the destruction of superconductivity by a magnetic field.

Fig. 26. Examples of the distribution of the magnetic field near superconducting specimens.

Let us consider how, for example, a superconducting sphere will behave under a gradual increase of the external magnetic field. If the intensity of the external homogeneous field is \(H_0\), then at the equator of the superconducting sphere the field turns out to be equal to \(3/2\,H_0\), whereas at the pole it is equal to zero. It is therefore natural that, as the intensity of the external field is increased, the critical value \(H_k\) is reached first of all at the equator. Thus, at the equator conditions will be created for the transition of the sphere into the normal state. It would seem that, with a further increase of the field, a region of the normal phase will form at the equator, adjoining along some surface the inner superconducting region (Fig. 27). But this leads to the following contradiction: at the boundary of the normal and superconducting regions the field, by definition, must be critical. The boundary is a convex surface; therefore the field inside the normal region will be smaller than at its boundary with the superconducting region, i.e. smaller than \(H_k\). Then inside the normal region the condition will arise for a transition back to the superconducting state. Consequently, the transition of the sphere from the superconducting state to the normal one cannot take place in the indicated manner.

Fig. 27. Path of transition from the superconducting to the normal state, not realized in reality.

Studies of this transition \(^{21a}\) established the character of the dependence of the magnetic moment of a superconducting sphere on the magnitude of the external field (Fig. 28). The curve consists of three rectilinear sections. Section \(AB\) indicates that, beginning with the external field \(H=\frac{2}{3}H_k\), a magnetic field penetrates into the superconducting sphere (Fig. 29), which grows linearly from zero at \(H=\frac{2}{3}H_k\) to the critical value \(H_k\) when the external field is equal to \(H_k\). This state of the superconducting sphere, differing both from the normal and from the superconducting state, and not having found a simple explanation, was given the name

Fig. 28. Dependence of the magnetic moment in a sphere on the magnitude of the external field.

Fig. 29. Dependence of the mean field in a sphere on the magnitude of the external field.

of the intermediate state. From what was said at the beginning of this paragraph it is evident that the lower boundary of the intermediate state depends substantially on the shape of the body. In the case of an ellipsoid of revolution whose axis of revolution coincides with the direction of the field, the lower boundary is equal to \((1-n)H_k\), where \(n\) is the so-called demagnetizing factor, depending on the ratio of the axes of the ellipsoid. In particular, for an infinite plate placed perpendicular to the field, \(n=1\), and the intermediate state occurs for all fields from \(H=0\) to \(H=H_k\).

The original phenomenological theory of the intermediate state, constructed by Peierls \(^{22a}\), assumed the existence of some new state of the substance, differing in its properties both from the superconducting and from the normal states. In addition, it was assumed that the field inside a specimen in the intermediate state is also homogeneous, equal to \(B=\dfrac{H}{n}-\left(\dfrac{1}{n}-1\right)H_k\). In other words, in the intermediate state the magnetic field was assigned some intermediate (between zero and \(H_k\)) value.

The modern theory of the intermediate state was created by Landau \(^{55,48}\), who showed that the intermediate state is not any new state of matter, but is a mixture of superconducting and normal layers alternating with one another.

As one approaches the surface of the specimen, the normal layers begin to branch, gradually becoming thinner, as is shown schematically in Fig. 30. At the very surface there already arises a finely dispersed mixture of two phases; moreover, the thickness of the layers becomes of the order of the penetration depth, and a macroscopic description of the superconducting state as a state with \(B=0\) loses its meaning. Naturally, under these conditions the concept of an intermediate state also loses its meaning, since at the surface of the specimen, instead of a mixture of normal and superconducting regions, a new macroscopically homogeneous state arises, which Landau called “mixed.” Landau succeeded also in determining the thickness of the layers existing inside the specimen. The sum of the thicknesses of the superconducting and normal layers, in the first approximation, proved to be independent of the field strength and approximately equal to

\[ a+b \simeq 2\left(\frac{8\pi\alpha}{H_k^2}L^2\right)^{1/3}, \]

where \(L\) is the size of the specimen, and \(\alpha\) is the coefficient of surface tension between the superconducting and normal phases.

Fig. 30. Layered structure of the intermediate state. At the surface of the specimen the normal layers begin to branch.

The first confirmation of Landau’s theory was provided by the experiments of Nakhutin \(^{64,65}\) (see also \(^{101}\)), which revealed anisotropy of the conductivity of a tin sphere in the intermediate state. Nakhutin found that in the direction along the field superconductivity disappears at an external field equal to \(H_k\), whereas in the direction perpendicular to the field superconductivity disappears at a field equal to \(\frac{2}{3}H_k\). The disappearance of superconductivity was determined from the disappearance of current in a superconducting circuit consisting of a tin sphere and a lead wire soldered to opposite points of the sphere. The observed anisotropy is fully explained by the layered structure of superconductors in the intermediate state, if one takes into account that the layers are arranged parallel to the external field and perpendicular to the current.

Direct detection of the layered structure at the surface seemed impossible because of the dispersity of the mixed state, which lies beyond the resolving power of any measuring instrument. However, Shalnikov \(^{92}\) nevertheless succeeded in obtaining direct experimental proof of the layered structure of superconductors in the intermediate state.

If a superconducting sphere is cut along a diametral plane into two halves (by a plane perpendicular to the external field) and the two halves are then put together again, such a sphere will not be

differ from a solid sphere. According to Landau’s theory, along the plane of the cut the magnetic field will vary, being successively equal either to zero in the superconducting layer or to the critical value \(H_k\) in the normal layer. The electrical contact between the hemispheres is immaterial, since there should be no currents along the meridians for reasons of symmetry; therefore, if the hemispheres are moved apart by a very small distance, the configuration of the magnetic field will scarcely change, and along the slit the field will be alternately equal either to zero or to \(H_k\). With an increase in the distance between the hemispheres, beginning with some critical distance, the layers at the surface of the cut will begin to branch, and for large slit widths at the surface of the slit the superconductor will be in a mixed state, characterized by a homogeneous mean magnetic field.

Fig. 31. Dependence of the field in the slit between the hemispheres on the magnitude of the external field.

Since the critical distance was unknown, Shal’nikov made the slit minimal (\(\sim 25\) microns); the dimensions of the slit were determined by the thickness of the magnetic-field meter—the length of a bismuth ribbon placed along the diameter of the cut—and by the insulation. Measurement of the inhomogeneity of the magnetic field by means of a bismuth wire or ribbon is based on the quadratic dependence of the resistance of bismuth on the field strength.

As was stated at the beginning of the paragraph, the mean field in the intermediate state increases linearly with increasing external field from 0 to \(H_k\). According to Landau’s ideas, this means that the total cross-section of the normal layers changes linearly. For a given external field one can find what fraction of the length of the bismuth meter is in a field of strength \(H_k\) and what fraction is in a field equal to 0; then, with the aid of the calibration curve, determine the total change of resistance thereby obtained and the corresponding magnetic field. The magnetic-field values calculated in this way are shown in Fig. 31 together with the experimental points. The agreement of the experimental points with the calculated ones is the principal result of Shal’nikov’s experiment—namely, proof of the basic idea of Landau’s theory concerning the layered structure of a superconductor in the intermediate state.

Next, an approximate value was determined for the critical slit width, equal, for a tin sphere of diameter 28.4 mm, to 50 microns.

Landau’s theory makes it possible to calculate the coefficient of surface tension between the superconducting and normal phases for tin. This quantity \(\alpha\) is related to the critical width of the slit \(d_0\), the dimensions of the body \(L\), and the magnitude of the critical field \(H_k\) by the following relation:

\[ \alpha=\frac{H_k^2}{8\pi}\sqrt{\frac{d_0^3}{512L}\frac{dH_k}{dM}}. \]

The calculated values of \(\alpha\) amount to \(1.7—4.4\cdot 10^{-3}\) dynes/cm, which is hundreds of thousands of times smaller than the value of \(\alpha\) for liquid tin with respect to vacuum.

Knowledge of the value of \(\alpha\) enabled the author to estimate the total thickness of the normal and superconducting layers with the aid of the relation given by Landau’s theory,

\[ \delta=a+b=2\sqrt[3]{\frac{8\pi\alpha}{H_k^2}L^2}\ \text{cm}. \]

For a sphere of diameter \(28.4\) mm the total thickness proved to be of the order of \(0.5\) mm. The study of the topography of the field in the slit was the subject of the next experimental investigation by Shal’nikov, carried out by him together with Meshkovskii\(^{62}\).

Fig. 32. Topography of the magnetic field in the slit between hemispheres. The transition to the intermediate state was effected from the superconducting state by increasing the magnetic field.

The experiment consisted in the following: in the slit between two hemispheres the distribution of the magnetic field was measured with the aid of a small bismuth meter—a “micro-meter,” which could be moved along the diameter of the section. The micro-meter was a thin strip of bismuth \(0.3\) mm long, \(10\) microns wide, \(5\) microns thick, soldered at its ends to copper ribbons. The very first experiments with micro-meters revealed a sharp inhomogeneity of the field. Fig. 32 is a photographic-film record of measurements of the resistance of the meter as it was moved along the diameter of the section from one edge to the other.

E. L. ANDRONIKASHVILI AND K. A. TUMANOV

The upper horizontal line corresponds to the change in resistance caused by the critical field; the lower one, to the absence of any change in resistance. Peaks which do not reach the upper line are explained either, in all likelihood, by insufficient “resolving power” of the meter, despite its smallness, or by a distortion of the magnetic lines of force in the slit (the so-called “barrel” effect). The authors then established that the configuration of the regions of the normal and superconducting phases at a given point of the intermediate state depends essentially on the manner of transition to this point. Fig. 32 corresponds to the transition from the superconducting state by increasing the magnetic field at constant temperature. The curves obtained in the case of a transition to the intermediate state from the normal state by lowering the temperature at constant magnetic field have a different character (Fig. 33). The characteristic features of the curves for transitions of these two types were preserved from experiment to experiment. The authors found that the curves are noticeably deformed if the magnetic field is introduced not smoothly, but in steps.

Fig. 33. Topography of the field in the slit between hemispheres. The transition to the intermediate state was effected from the normal state by lowering the temperature.

From what has been said it follows that the transition to the intermediate state always occurred very non-equilibrially; the state depended on the path and character of the transition into it. This makes a quantitative test of the theory impossible until a method is found for carrying out an equilibrium transition to the intermediate state.

The results of further experiments proved rather unexpected. When the width of the slit was increased to one and even 2.3 mm, the inhomogeneity of the field in the slit did not disappear. The measurement was made on one of the surfaces of the hemisphere; in the middle of the slit, however, the field proved to be fairly homogeneous, which is explained by the “barrel” effect. Thus the normal and superconducting regions extend onto the flat surfaces of the hemispheres even at very large distances between them—such that they must be regarded as certainly much greater than the critical one*). In connection with this the question arose whether it is not

*) If one assumes that the critical width is of the order of 2 mm, then the width of the layers in a sphere of diameter 39 mm will be of the order of 3 mm, which is in no way consistent with the curve in Fig. 32.

the emergence at the surface of superconducting and normal regions without the formation of a macroscopic homogeneous “mixed” state is a phenomenon inherent in superconductors of any shape. The following work of Shalnikov and Meshkovsky[^63] was devoted to clarifying this question.

These authors investigated, with the aid of a microprobe, the topography of the field on the surface of an entire sphere in the intermediate state. The microprobe could be moved along a meridian of the sphere from the equator through the pole and to the equator. The measurements revealed a sharp inhomogeneity of the field on the surface of the sphere. In Fig. 34 are shown curves of the deflections of the galvanometer, i.e., to a known approximation—the field strength; the abscissa axis represents the straightened path of the probe, i.e., half the arc of a great circle. In the curve shown, sections are visible where macroscopic superconducting regions \(AB\), \(CD\), and \(EF\) emerge at the surface. The authors explain the discrepancy between the results of their experiments and Landau’s theory by the fact that Landau’s theory is constructed for thermodynamic equilibrium states, whereas as a result of the nonequilibrium transitions that occur in a real experiment, “supercooled” regions of the normal phase arise. Evidence for the nonequilibrium character of the transition had been obtained by the authors in the preceding work.

Fig. 34. Field topography on the surface of a sphere in the intermediate state.

Before the investigations of Meshkovsky and Shalnikov were carried out, the nonequilibrium character of the transition had been established by the work of Alekseevsky[^9], in which the change of induction in a superconductor was studied during an extremely slow transition from the normal state to the superconducting state, and back again. The transition was effected both by changing the temperature and by changing the magnetic field. To carry out a very slow change of temperature, an automatic temperature regulator invented by Alekseevsky and Shalnikov[^12] was used,

which operated in a weakly nonstationary regime. The slow change of the magnetic field was achieved by moving, by means of a clockwork mechanism, the iron resistor connected in series with the field-producing solenoid in a muffle furnace. Since it was necessary to measure very small changes in the magnetic flux, which cannot be recorded by an ordinary ballistic galvanometer, a photorelay (photoamplifier) was used in this work. Investigation of the transition of a long cylindrical single-crystal tin specimen (length 100 mm, diameter 0.8 mm) in a homogeneous field showed that the transition occurs at once (one throw of the galvanometer); in the case of an inhomogeneous field

Fig. 35. Induction jumps in the intermediate state during a slow change of temperature.

(the inhomogeneity was produced by an additional coil with a variable number of turns per unit length) the induction curve consisted of a large number of separate peaks, i.e., the transition takes place over a certain range of temperature or magnetic field; in other words, an intermediate state occurred. Figure 35 shows part of the curve with induction jumps in the intermediate state. This jump-like change of induction is, in outward appearance, very similar to the Barkhausen jumps during the magnetization of ferromagnets. Simultaneously with the induction jumps, the author observed a jump-like change in the resistance of the specimen.

First of all, the work proved that long cylindrical specimens in a parallel magnetic field can be in an intermediate state if the field is inhomogeneous in magnitude. The jump-like change of induction means that the transition into the intermediate state is accompanied by supercooling or superheating, i.e., is a nonequilibrium process. From the magnitude of the induction jumps the author estimated the sizes of suddenly appearing or disappearing regions of the normal or superconducting phase; they turned out to be of the order of 0.01 mm³. The author compared the jump-like appearance of regions ...

is connected with the formation of vapor bubbles when a liquid boils. The results of the experiment, as well as the fact that there is a surface tension between the superconducting and normal phases, allowed the author to suggest that in the presence of a magnetic field the process of formation of a new phase begins in a manner similar to that which occurs in other first-order phase transformations, namely with the appearance of centers of phase transitions*), which then begin to grow.

In the following work¹¹ on the kinetics of transitions of superconductors, Alekseevskii succeeded in determining the velocity of motion of the phase boundary. To measure the velocity, he chose the transition of a long cylindrical specimen in a longitudinal homogeneous magnetic field under radial cooling or heating of the specimen.

Obviously, the boundary between the two phases also moved radially. The change in the flux of induction in the specimen and the electromotive force induced in the measuring coil placed on it are proportional to the velocity of displacement of the phase boundary. The electromotive force was recorded with an oscillograph. Determined for ordinary cooling and heating rates, the velocity of displacement of the phase boundary proved to be of the order of 0.1 cm/sec. It should be noted that, when a cylindrical specimen is cooled in a longitudinal homogeneous magnetic field, the specimen passes into an intermediate state, as indicated by jumps on the electromotive-force curve. On heating, however, the transition occurred without an intermediate state.

A more detailed study of the macroscopic properties of the intermediate state was undertaken by Shalnikov⁹¹, ⁹², who carried out experiments with hollow spheres, and by Alekseevskii⁷, ¹², who worked with superconducting disks.

The experiments with hollow spheres were intended to decide whether the cavity inside a superconductor is essential, or whether only the external shape of the specimen matters. It turned out that, when the magnetic field is increased from zero up to the onset of the intermediate state, hollow spheres are indistinguishable from solid spheres. With a further increase of the field, the surface can be divided into the following parts: a belt at the equator, the wider the thicker the wall of the hollow sphere, imitating the behavior of a solid sphere, and two regions at the poles, in which the magnetic field is smaller than the field in the case of a solid sphere (the so-called “field depression”). The author explains the appearance of the belt—the “imitation zone”—by a superconductivity current flowing along the equator. In the transition from the normal to the superconducting state upon weakening of the magnetic field, hysteresis and a “frozen-in” magnetic moment were observed.

Hysteresis (a loop on the curve of the dependence of the magnetic induction of a superconductor on the magnitude of the external field) and residual magnet—

*) The formation of nuclei is considered in ²¹, ⁷³; see also ¹¹.

in superconductors of imperfect, i.e. different from ellipsoidal,* shape is explained as follows. In the intermediate state, owing to the irregularity of the body’s shape, the mean field is different at different points; therefore the transition to the superconducting state from the intermediate state occurs at different points of the body for different values of the external field. Thus there will inevitably be cases in which regions where the magnetic-field strength is still nonzero are surrounded by superconducting rings, and, consequently, the magnetic moment will be “frozen in.” The consequence of this is hysteresis. Another cause of hysteresis, indicated by Alekseevskii\(^7\), is that, for a finite rate of transition to the normal phase, Foucault currents arise, whose density is different at different points. In this case the transition to the superconducting state will be determined by the sum of the external magnetic field and the field of the Foucault currents, and will take place at different points non-simultaneously. The first cause is connected only with the shape of the specimen, the second with the rate of transition; apparently, the occurrence of hysteresis is explained by both causes.

Studying experimentally the phenomenon of hysteresis, Alekseevskii\(^7\) investigated the “frozen” magnetic field in superconducting disks. The residual magnetic field is explained by a macroscopic superconducting current flowing along the edge of the disk. To verify this, the author divided the disk into small pieces, which he coated with a thin layer of paraffin for insulation and then put together again. In this composite disk the field was almost not “frozen.” In addition, the existence of a circular current in the disk was proved by the characteristic distribution of iron filings on a plane perpendicular to the plane of the disk. In his subsequent investigations Alekseevskii\(^ {12}\) measured the distribution of the “frozen” field along the surface of the disk. The measurement was carried out with the aid of a miniature coil of dimensions \(2.5 \times 1\) mm, which could be moved along the radius of the disk and also rotated through an angle of \(180^\circ\); by the latter means the magnetic flux through the coil was changed. Since in some cases it was necessary to measure very weak changes of magnetic flux, a photorelay was used in this work. Using the formulas given by Fock, the author calculated, for the case of thin disks, the distribution of circular currents. Next, the author calculated the percentage content of the normal phase along the radius of the disk. The greatest content proved to be at the center of the disk.

*) Specimens of ellipsoidal shape (including an infinite cylinder and an infinite plate, which are limiting cases of an ellipsoid) we call perfect in the sense that, if they are placed in a uniform field, then, according to the well-known problem of electrodynamics, the field inside them will be uniform. Therefore the intermediate state of ellipsoids will be macroscopically homogeneous, and they will pass into the superconducting state at all points at once.

4. Superconducting Thin Films and the Penetration Depth

From the electrodynamics of superconductors23a (see also 28) it follows that an external magnetic field penetrates into a superconductor, decreasing in proportion to \(\exp\left(-\dfrac{x}{\delta}\right)\), where \(x\) is the distance from the surface of the specimen, and \(\delta\) is the penetration depth, which, according to the theory, is a function of temperature and does not depend on the magnitude of the field. From the fact of the penetration of the magnetic field into superconductors it follows that the diamagnetic susceptibility of specimens whose dimensions are comparable with the penetration depth is smaller in absolute value than the diamagnetic susceptibility of massive specimens. Another consequence of the small size of superconductors is an increase in the critical field; this was analyzed theoretically in the works of Pomeranchuk60 and Ginzburg61. Ginzburg calculated the change in the magnitude of the critical field for thin films, caused both by penetration of the field and by the influence of surface tension.

The first attempts at an experimental investigation of superconductors of small dimensions, undertaken by a number of scientists in Leiden24a, Toronto25a, and Cambridge26a, did not yield reliable results, since the researchers were unable to obtain specimens of sufficient purity.

In 1938 the first report was published on Shalnikov’s83 work with thin films of tin, lead, and thallium. Shalnikov83 developed a technique that made it possible to obtain films with thicknesses from \(5\cdot 10^{-7}\) to \(3\cdot 10^{-5}\) cm and ensured a high degree of purity of the specimens. Its essence consisted in the condensation of metallic vapor in vacuum onto a degassed glass surface cooled to \(4.2^\circ\) K. In these works it was established for the first time that films of lead, tin, and thallium remain superconducting down to a thickness of \(5\cdot 10^{-7}\) cm, i.e., down to 15 atomic layers. Further, a substantial difference was discovered in the properties of freshly deposited films and of films that, between deposition and measurements, had been warmed to room temperature (“annealed” films). The author explains this difference by the fact that condensation at \(4.2^\circ\) K greatly hinders the possibility of “creep” of metal atoms over the surface and prevents the packing of atoms into crystalline aggregates. Therefore, at low condensation temperatures, a structure close to the amorphous structure of metals is obtained. Annealing of the films leads to their recrystallization.

Further, in this work the critical temperatures were determined for both freshly deposited and recrystallized films. It turned out that for the former \(T_k\) is higher than the normal value, the increase being of the order of a degree. For recrystallized films \(T_k\) is very close to the value of the critical temperature of a massive specimen. In the following paragraph the phenomenon of the increase in the cri-

critical temperature upon stretching the specimen. If the shift of the critical temperature is explained by an increase in the mean distance between atoms, then the observed increases in \(T_k\) would correspond to enormous negative pressures of the order of \(3 \div 5 \cdot 10^4\ \text{kg}/\text{cm}^2\).

Freshly deposited films have substantially lower values of the critical currents than recrystallized films.

The study of the destruction of superconductivity by a magnetic field parallel to the surface of the film showed that fields of several thousand oersteds destroyed superconductivity only when the current through the specimen was very close to the critical current. In Figs. 36 and 37 the dependence of the destructive field on the current \(i\), obtained for freshly deposited films, is shown; by extrapolating the curves to \(i = 0\), one can determine the order of magnitude of the critical field. The critical field of recrystallized films is smaller than that of freshly deposited films (Fig. 38. Compare this figure with the curve for tin in Fig. 20).

Fig. 36. Dependence of the critical field on current. Lead, thickness—\(4.39 \cdot 10^{-6}\ \text{cm}\); \(T = 4.2^\circ\text{K}\).

Fig. 37. Dependence of the critical field on current. Tin, thickness—\(6.4 \cdot 10^{-7}\ \text{cm}\); \(T = 2.02^\circ\text{K}\).

Thus, along with the expected increase in the value of the critical field, Shal’nikov’s work discovered a new phenomenon consisting in the influence of the structure of the film on its superconducting properties.

The results of Shal’nikov’s work were developed in subsequent studies both abroad (Appleyard \(^{27a}\) and others) and in our country. In partic—

ness, curves of the dependence of \(H_k\) on temperature were taken for films of various thicknesses. Using these results, Ginzburg\(^{27}\), by means of the formula he obtained, calculated the temperature variation of the penetration depth, taking account of the surface energy (Fig. 39).

Fig. 38. Dependence of the critical field of freshly deposited (\(C\)) and recrystallized (\(P\)) tin films on temperature.

The critical fields of tin films with thicknesses from \(2.36 \cdot 10^{-5}\) to \(1.55 \cdot 10^{-6}\) cm were investigated by Alekseevskii\(^{2,5}\). The films were obtained by Shal’nikov’s method and recrystallized at room

Fig. 39. Temperature variation of the penetration depth.

temperature. The criterion of superconductivity in this work was the mechanical moment of the forces experienced by a superconducting film in a magnetic field. The film was suspended on an elastic thread and, in the superconducting state, was set parallel to the field. When the field became equal to the critical field and superconductivity was destroyed, the film rotated sharply. Like Shal’nikov, Alekseevskii found that freshly deposited films have higher critical fields than films “annealed” to room temperature. Figure 40 gives curves of the dependence of the critical field on temperature for “annealed” films of various thicknesses. Figure 41 shows the dependence, measured by the author, of the critical

fields on thickness at \(2^\circ\) K. The author calculated the penetration depth at \(2^\circ\) K, which proved to be \(1.64 \cdot 10^{-5}\) cm.

Shoenberg\(^{28a}\) carried out studies of the magnetic susceptibility of mercury emulsions and, from the data of these studies, found the penetration depth and its temperature dependence.

Determination of the penetration depth from the study of the magnetic properties of superconductors has an advantage in comparison with the critical-field method used by Appleyard and others. Namely, in processing the results it is not necessary to use thermodynamic considerations, and the results do not

Fig. 40. Dependence of the critical field of tin films of different thicknesses on temperature.

Fig. 41. Dependence of the critical field on the thickness of tin films at \(T = 2^\circ\) K.

depend on the magnitudes of the surface energies of the superconducting and normal phases.

However, Shoenberg’s experiments have their shortcomings. These include the insufficient uniformity of the mercury particles in size and the unquestionable contamination of the emulsion.

All this led researchers to attempt to measure the penetration depth and its temperature dependence on massive specimens. Ordinary ballistic measurements in this case would lead to nothing, if the magnetic moment of the specimen were measured with an accuracy to the sixth digit, which makes the problem practically impossible.

The first successful attempt to circumvent this difficulty belongs to Shalnikov and Sharvin\(^{93}\). Shalnikov and Sharvin devised an ingenious and delicate method. The idea of this method consists in the fact that

Fig. 42. Dependence of \(\lg \dfrac{d\delta}{dT}\) on \(\lg (T_k - T)\) for various values of the external field.

A superconducting specimen with a measuring coil placed on it is put into a constant magnetic field, and oscillations are produced

of the temperature of the specimen. If the penetration depth depends on temperature, then, when the temperature oscillates, the magnetic flux through the surface layer of the superconductor must oscillate, as a result of which an electromotive force will be induced in the measuring coil. It must be emphasized that the sole cause of the induction of the electromotive force in the measuring coil is the dependence of the penetration depth on temperature; thus, the measured effect is directly proportional to the derivative of the penetration depth with respect to temperature. The temperature oscillations in the experiment were produced by pressure oscillations of gaseous helium in which the specimen was located. The magnetic field, whose strict constancy was essential for the experiment, was produced by a superconducting lead solenoid. Since it was necessary to measure very small changes in magnetic flux, the authors used a photo-relay.

Fig. 43. Temperature dependence of the penetration depth at different values of the external field.

In Fig. 42 the dependence of $\lg \dfrac{\partial \delta}{\partial T}$ on $\lg (T_k - T)$ is given for different values of the external field for tin. The points corresponding to different values of the magnetic field at temperatures close to $T_k$ fall well on one and the same straight line. This indicates that, in fields of the order of several oersteds, the penetration depth does not depend on the magnetic field. The linear dependence observed near $T_k$ of $\lg \dfrac{\partial \delta}{\partial T}$ on $\lg (T_k - T)$ corresponds to the asymptotic law

\[ \frac{\partial \delta}{\partial T} = \frac{C}{(T_k - T)^{3/2}}, \]

where $C = 5.4 \cdot 10^{-6}$, $T_k = 3.715^\circ\ \mathrm{K}$. From the data on $\dfrac{\partial \delta}{\partial T}$ one can obtain $\delta$ as a function of temperature, with accuracy up to an integration constant. As Landau pointed out, the integration constant can be determined by using the asymptotic law given above. The dependence of the penetration depth on temperature is shown in Fig. 43. Fields of the order of tens of oersteds have a noticeable influence on the magnitude of the penetration depth (dashed curves).

5. Changes in the Boundaries of Superconductivity under Compression and Tension

The dependence of the critical temperature and the critical field on mechanical deformations of specimens was discovered by Sizoo and Kamerlingh Onnes \(^{29a}\) in 1925–1926 in Leiden. These investigators found that, under a tensile stress of \(2.5\ \mathrm{kg/mm^2}\), the critical temperature of tin rises by \(0.007^\circ\ \mathrm{K}\). Since the shift of \(T_k\) was very small, in order to confirm the reality of the effect it seemed of interest to determine the shift of \(T_k\) and \(H_k\) under large loads. Alekseevskii’s work \(^{4}\) was devoted to this. Tantalum, possessing high mechanical strength, was chosen as the object of investigation. The dependence of the critical temperature, critical field, and critical current on tensile stress, which reached \(16\,000\ \mathrm{kg/cm^2}\), was measured. In this case the shift

Fig. 44. Dependence of the critical field on stress.
Fig. 45. Dependence of the critical field on temperature for different values of the tensile force.

of the critical temperature amounted to \(0.12^\circ\ \mathrm{K}\). In this range of stresses the dependences of \(T_k\), \(H_k\), and \(i_k\) on stress proved to be linear. Figure 44 shows the dependence of \(H_k\) on stress; Fig. 45 gives a family of curves \(H_k=f(T)\) for various values of the tensile force. In another of his works \(^{8}\), Alekseevskii determined \(dT_k/dp\) for a single crystal of tin, registering the transition into the superconducting state not only by the change in the resistance of the tin, but also by the jump in magnetic induction in the specimen. A stress of \(0.64\ \mathrm{kg/mm^2}\) corresponded to a shift of \(T_k\) by \(0.001^\circ\).

In addition to homogeneous tension, Sizoo and Kamerlingh Onnes subjected specimens to all-round compression and determined the influence of all-round compression on the transition to the superconducting state. In these investigations, the medium transmitting the pressure was used as

liquid helium, the pressure above it being brought up to 300 kg/cm². As was shown subsequently, helium under pressure solidifies. It turned out that over a large range of pressures in the Leiden investigations the helium was solid and could not transmit pressure. This difficulty was overcome by Lazarev and Kan,^52 who, in their investigation of the transition of tin and indium to the superconducting state, took water as the pressure-transmitting medium, making use of the fact that water expands on freezing and that, down to helium temperature at normal pressure, the density of ice is less than the density of water. The pressure was determined from the bursting of a bomb, which, through a system of levers, turned a mirror. The fact of superconductivity was established either by the potentiometric or by the induction method. The latter consisted in measuring the magnetic moment of a superconducting ring by the ballistic method. The measurements were made at a pressure of 1750 kg/cm²; in this case the critical temperature of tin is lowered by 0.098°, and the critical field at different temperatures by 12–14 gauss. Comparing their results with those of the Leiden studies, the authors established that the displacement of \(T_k\) and \(H_k\) is proportional to pressure, up to pressures of the order of 2000 kg/cm². The authors proposed to explain the fact that transitions in polycrystals are broadened by the occurrence in them of inhomogeneous deformations.

Fig. 46. Dependence of the critical current of an inhomogeneously strained tin wire on temperature.

The influence of inhomogeneous deformations on the superconductivity of tin was studied in the work of Lazarev and Galkin.^53 To obtain an inhomogeneously deformed metal, tin wires 0.06 mm thick were glued to glass with celluloid cement. The difference in the coefficients of thermal expansion of tin, glass, and celluloid created the stresses. As a control, the properties of similar wires, but not glued to glass, were studied. The free wires were characterized by a normal value of the critical temperature, a normal course of the critical magnetic fields with temperature, and obedience to Silsbee’s rule.^20a Investigating the stressed specimens, the authors found a considerable hysteresis upon the destruction of superconductivity by current. The critical temperature of the specimens proved to be much higher than in ordinary unstressed tin specimens; it can be estimated from the curve of the dependence of the critical current on temperature (Fig. 46).

For ease of comparison, the values of \(T_k\), \(H_k\), \(\dfrac{\partial H_k}{\partial T}\), and \(i_k\) for deformed and undeformed specimens are given in the table:

Specimen	$T_k^\circ$ K	$H_k$ gauss at 2° K	$dH_k/dT$ gauss/deg. at 2° K	$i_k$ amperes at 2° K
Undeformed	3.72	210	100	3
Deformed	9	15 000	1750	0.067

As can be seen, apart from the shift of $T_k$ by $\sim 5^\circ$, the deformed wire is characterized by an approximately 100-fold greater value of the critical field and a value of the critical current smaller by tens of times, which is in complete contradiction to Silsbee’s rule. To explain the shift of $T_k$, the authors admitted the possibility of local tensile stresses, which, for $\Delta T_k$ equal to $5^\circ$, must be equal to 90 000 kg/cm$^2$. Such stresses, however, cannot explain the shift of $H_k$ by $\sim 15\,000$ gauss (for this a stress of $2 \cdot 10^6$ kg/cm$^2$ would be required); therefore the authors, along with the stresses explaining $\Delta T_k$, assumed the existence of inhomogeneities in the crystal lattice, dividing the metal into small regions. In § 4 we have already mentioned Alekseevskii’s work with thin films$^5$, in which the dependence of $\Delta H_k$ on the film thickness was established (Fig. 41). On the basis of these data one can estimate the sizes of the regions. For $\Delta H_k = 14\,000$ gauss, the sizes obtained are of the order of $10^{-6}$ cm. If it is now assumed that the superconducting current flows along filaments whose thickness is $10^{-6}$ cm, then Silsbee’s rule will be fulfilled.

The results obtained in the work under consideration are in many respects similar to the results of Shalnikov’s experiments with thin films$^{88}$. All the anomalies in the behavior of freshly deposited films, observed in the experiments of Shalnikov, Lazarev, and Kan, are explained by strong local distortions of the crystal lattice and by the finely dispersed structure of the metal. The lowering of the value of the critical current in this case is explained by the occurrence of superconducting filaments.

6. Superconducting alloys and compounds

In the present section superconducting alloys will be considered, namely: 1) eutectic alloys of superconducting metals, 2) alloys that are solid solutions of metals, the metals being able to be either both superconductors, or one a superconductor and the other a nonsuperconductor, and also 3) certain intermetallic compounds.

As is known, a eutectic alloy is a mixture of fine grains of both metals; therefore, in investigating the properties of such a natural dispersed state of superconductors, one might have expected that they would resemble the properties of an emulsion (an increase of \(H_k\), a decrease of the magnetic moment per unit volume, etc.).

Fig. 47. Dependence of the magnetic moment of the alloy (Zn 90%, Sn 10%) on the magnitude of the external field.

Such an investigation was carried out by Andronikashvili \(^{15}\) in Moscow and by Lazarev and Nakhutin \(^{50}\) in Kharkov. Alloys of tin with zinc were taken as the object of study. In both works the dependence of the magnetic moment of samples of various concentrations on the external field was measured. The measurements were made by the ballistic method. It was found that the critical temperature of the samples coincides with the critical temperature of pure tin. Further, it turned out that the disappearance of the magnetic moment occurs in a field equal to the critical field of pure tin. In fields smaller than the critical field, the magnetic moment depends little on the concentration of tin (down to small—5%—concentrations of tin) and is equal to the magnetic moment of pure superconductors. All this indicates that the alloy does not consist of separate superconducting grains of tin, but that a peculiar shielding of the zinc grains by tin takes place. The difference between these alloys and pure tin manifests itself upon transition to the superconducting state by a decrease of the magnetic field. In this case the magnetic moment exhibits hysteresis, which is the greater the smaller the concentration of tin. In Fig. 47 the curve of the dependence of the magnetic moment on the field is given. From the figure it is seen that, as the field is increased from zero, it begins to penetrate into the specimen before the critical value has yet been reached. Obviously, the behavior of such alloys can be explained in the same way as at the end of § 3 we explained the behavior of superconducting hollow spheres, superconducting disks, and, in general, superconductors of imperfect shape. The premature penetration of the field is explained by the occurrence of an intermediate state, for which the essential factor is not the elongated cylindrical shape of the specimens, but the very irregular shape of the tin interlayers in the alloy.

Thus, eutectic alloys did not exhibit the properties of an emulsion, as we expected. It should, however, be noted that Andronikashvili points to the presence of “tails” in the curves of the dependence of the magnetic moment on the field strength; they should be explained

with the presence of very small particles of tin, for which the critical field is higher than for massive specimens.

Solid solutions possess a far higher degree of homogeneity than eutectic alloys. This affects the properties of superconducting solid solutions. Their critical temperature is not equal to the higher of the two critical temperatures of the components, as in the case of eutectic solutions, but may be either greater or less than them; in addition, the critical temperature depends on the concentration.

The first experiments revealed that in such superconducting alloys the resistance is restored by very large fields. Since the critical temperatures are of the same order of magnitude as in pure metals, it follows that the derivative \(dH_k/dT\) is especially large for alloys (of the order of \(5000\) gauss/degree, whereas for metals it is of the order of \(150\) gauss/degree).

In § 1, from the equality of the free energies, the entropy difference per unit volume of the superconducting and normal phases at the transition point was calculated\(^{17a}\)

\[ S_n - S_s = -\frac{H_k}{4\pi}\frac{dH_k}{dT}; \]

from this one can calculate the jump in the specific heat occurring at the transition, which in the absence of a magnetic field is equal to

\[ \Delta C = C_s - C_n = \frac{T_k}{4\pi d}\left(\frac{dH_k}{dT}\right)^2, \]

where \(d\) is the density of the metal. It was therefore natural to suppose that there would be a large jump in the specific heat of solid solutions. Shubnikov and Khotkevich\(^{99}\) (see also\(^{100}\)) investigated the course of the specific heat of a superconducting alloy (Pb—65%, Bi—35%). However, they found not only no large jump, but in general no jump at all. This led to the idea that the magnetic field penetrates into such superconducting alloys.

Studying the magnetic properties of the alloy (Pb—65%, Bi—35%) and of Pb—Tl alloys of various concentrations (including the intermetallic compound PbTl\(_2\)), Ryabinin and Shubnikov\(^{100}\) (see also\(^{101}\)) found that for alloys it is necessary to distinguish the field \(H_{k1}\), up to which the induction inside the specimen remains zero, and \(H_{k2}\), at which resistance appears (Fig. 48). As can be seen from the figure, the curve of the dependence of the magnetic induction on the external field has a small hysteresis and a residual magnetic moment.

Furthermore, the authors measured the critical currents for wires of various cross sections and found that the critical current is proportional to the first power of the radius, which is in agreement with Silsbee’s hypothesis. From this one can calculate the critical field of the current \(H_{ki}\), which is somewhat

less than \(H_{k1}\) (by \(\sim 30\%\)). Thus, the alloys are characterized by three critical fields (Fig. 49). It should be noted that the behavior of the intermetallic compound \(\mathrm{PbTl}_{3}\) possessed the same features as the behavior of solid solutions.

Similar properties were exhibited by the alloys lead—bismuth, lead—indium, mercury—cadmium.

The existence of three critical fields in the alloys is confirmed by the work of Alekseevskii,\(^3\) who studied the dependence of the magnetic moment of rings made of gallium–lead alloys of different concentrations on the magnitude of the external field.

Fig. 48. Dependence of the induction of a superconducting alloy on the magnitude of the external field.

The behavior of the alloys can be explained in the following way.\(^{3ca,3}\) The critical parameters at different points are different. Therefore superconducting filaments and regions may form, surrounded by the normal phase. The destruction of superconductivity of the main mass of the metal occurs in an external field equal to \(H_{k1}\). At the same time, superconducting filaments remain, along which current can flow. If one assumes that their cross section is very small, then the field necessary to destroy superconductivity in them will be very large, equal to \(H_{k2}\). From the curves obtained for the dependence of the magnetic moment on the field strength, Alekseevskii determined the upper limit of the radius of a superconducting filament, which proved to be \(5 \cdot 10^{-5}\ \mathrm{cm}\).

The destruction of superconductivity by current also fits into the developed picture of superconductivity of alloys.\(^3\) In a current field

of order \(H_{k1}\), only the filaments again remain superconducting. Since the current of the external force flows through filaments of very small cross-section, it creates at their surface a field apparently equal to \(H_{k2}\), which destroys the superconductivity of the filaments. It follows from this that the field of the critical current must be equal to \(H_{k1}\). In fact, they differ little from one another, and it may be that further, more accurate experiments will prove the equality of \(H_{k1}\) and \(H_{ki}\).

Fig. 49. Dependence of the critical fields of the alloy on temperature.

It was stated above that the behavior of the superconducting intermetallic compound \( \mathrm{PbTl}_2 \) is qualitatively no different from that of alloys of lead with thallium of other concentrations. Alongside such compounds there are others which behave like pure metals, are characterized by a single critical field, by the presence of the Meissner effect, etc. Among them, superconducting alloys of nonsuperconductors are of particular interest. In this field Alekseevskii worked\(^{10,11}\), discovering four new superconducting alloys whose components are not superconductors: \(\mathrm{Bi}_3\mathrm{Ni}\), \(\mathrm{Bi}_2\mathrm{Rh}\), \(\mathrm{Bi}_4\mathrm{Rh}\), and \(\mathrm{Bi}_3\mathrm{Ca}\).

E. L. ANDRONIKASHVILI AND K. A. TUMANOV

LIST OF WORKS BY SOVIET AUTHORS ON SUPERFLUIDITY AND SUPERCONDUCTIVITY

1. Alekseevskii N. E., Destruction of superconductivity by current, ZhETF. 8, 342 (1938).
2. Alekseevskii N. E., Magnetic properties of thin superconducting films, DAN. 24, 27 (1939).
3. Alekseevskii N. E., Dependence of the critical current on the external magnetic field in superconducting Pb—Tl alloys, ZhETF. 8, 1098 (1938).
4. Alekseevskii N. E., Displacement of the critical values for tantalum under tension, ZhETF. 10, 746 (1940).
5. Alekseevskii N. E., Behavior of thin superconducting films in a magnetic field, ZhETF. 10, 1392 (1940).
6. Alekseevskii N. E., Critical fields of superconducting vanadium, DAN. 31, 327 (1941).
7. Alekseevskii N. E., Hysteresis in pure superconductors, DAN. 32, 31 (1941).
8. Alekseevskii N. E., Displacement of the critical temperature of tin under tension, ZhETF. 15, 244 (1945).
9. Alekseevskii N. E., Jump of induction during the transition to the superconducting state, Journ. of Phys., 9, 217 (1945).
10. Alekseevskii N. E., Superconductivity of Bi₃Ni, Journ. of Phys., 9, 350 (1945).
11. Alekseevskii N. E., Investigation of the superconductivity of pure metals and binary alloys of nonsuperconducting components. Dissertation, IFT (1946).
12. Alekseevskii N. E., Residual currents in disks of superconducting metals and the hysteresis phenomena associated with them, ZhETF. 16, 870 (1946).
13. Alekseevskii N. E. and Shalnikov A. I., Automatic regulator of the pumping speed of cryostats with annealed gases, ZhETF. 16, 361 (1946).
14. Alekseevskii N. E. and Migunov L., Investigations of metals at temperatures below 1° K., Journ. of Phys., 11, 95 (1947).
15. Andronikashvili E. L., Superconductivity of eutectic Sn—Zn alloys, DAN. 31, 542 (1941).
16. Andronikashvili E. L., Direct observation of two kinds of motion in helium-II, ZhETF. 16, 780 (1946).
17. Andronikashvili E. L., Temperature dependence of the normal component of helium-II, ZhETF (in press).
18. Andronikashvili E. L., Investigation of the viscosity of the normal component of helium-II (in press).
19. Arkadeev V. K., On the forces acting on diamagnetic bodies, DAN. 47, 18 (1945).
20. Bogolyubov N. N., Theory of superfluidity, Izv. AN SSSR, ser. fiz. 11, 77 (1947).
21. Bulashevich Yu. P., Influence of a magnetic field on the formation of superconducting nuclei, ZhETF. 8, 1267 (1938).
22. Galanin A. D., Density fluctuations in an ideal Bose–Einstein gas, ZhETF. 10, 1267 (1940).
23. Ginzburg V. L., Scattering of light in helium-II, ZhETF. 13, 243 (1943).
24. Ginzburg V. L., Remarks on the theory of superconductivity, ZhETF. 14, 134 (1944).
25. Ginzburg V. L., Thermoelectric phenomena in superconductors, ZhETF. 14, 177 (1944).

1. Ginzburg V. L., Gyromagnetic and electron-inertial experiments with superconductors. Electrodynamics of moving superconductors, ZhETF. 14, 326 (1944).

2. Ginzburg V. L., Surface energy and the behavior of superconductors of small dimensions, ZhETF. 16, 87 (1946).

3. Ginzburg V. L., Superconductivity, Publishing House of the Academy of Sciences of the USSR (1946).

4. Dorfman Ya. G., Theory of superconductivity, Nature. 130, 166 (1932).

5. Dorfman Ya. G., Remarks on the theory of superconductivity, Sow. Phys. 3, 366 (1933).

6. Dorfman Ya. G., Superconductivity and the Hall coefficient, Metallwirtschaft 12, 221, 235 (1933).

7. Dorfman Ya. G. and Kikoin I. K., Physics of Metals, GTTI, 1934.

8. Kapitza P. L., Superconductivity and residual resistance, Nature. 123, 870 (1923).

9. Kapitza P. L., The nature of superconductivity and residual resistance, Proc. Roy. Soc. A 126, 683 (1930). Phys. Zeits. 31, 713 (1930).

10. Kapitza P. L., Viscosity of liquid helium at temperatures below the λ-point, DAN. 18, 21 (1938); Nature. 141, 74 (1938).

11. Kapitza P. L., Heat transfer in capillaries with helium-II, ZhETF. 11, 1 (1941).

12. Kapitza P. L., On the properties of liquid helium, Izv. AN SSSR, phys. ser. 5, 7 (1941).

13. Kapitza P. L., Heat transfer and superfluidity of helium-II, ZhETF. 11, 581 (1941).

14. Kapitza P. L., Report on the superfluidity of helium-II, UFN. 26, 133 (1944).

15. Kikoin A. K., Thermal conductivity of solid helium, ZhETF. 8, 840 (1938).

16. Kikoin A. K. and Lazarev B. G., Properties of helium-II films, Nature. 141, 912 (1938).

17. Kikoin A. K. and Lazarev B. G., Properties of helium-II films, Nature. 142, 489 (1938).

18. Kikoin I. K. and Lazarev B. G., Superconductivity and the Hall constant, Nature. 129, 57 (1932).

19. Kikoin I. K. and Lazarev B. G., Superconductivity and the Hall effect. ZhETF. 3, 44 (1933).

20. Kikoin I. K. and Gubar’ S. V., The Einstein—de Haas experiment with superconductors, DAN. 19, 248 (1938).

21. Kikoin I. K., Gyromagnetic effect in superconductors, ZhTF. 16, 129 (1947).

22. Conference on low-temperature physics in Moscow, ZhETF. 11, 573 (1941).

23. Lazarev B. G., Superconductivity and the Hall effect, Sow. Phys. 4, 567 (1933).

24. Lazarev B. G. and Esel’son B. N., Obtaining temperatures below 0.8° K by pumping helium vapor, ZhETF. 12, 549 (1942).

25. Lazarev B. G. and Nakhutin I. L., Magnetic behavior of superconducting tin—zinc alloys, ZhETF. 12, 43 (1942).

26. Lazarev B. G. and Kan L., Development of a method for producing high pressures at low temperatures. ZhETF. 14, 439 (1944).

27. Lazarev B. G. and Kan L., Superconductivity of tin and indium under hydrostatic compression at a pressure of 1750 kg/cm², ZhETF. 14, 463 (1944).

28. Lazarev B. G. and Galkin A. A., Effect on the superconductivity of deformed tin whiskers, ZhETF. 14, 474 (1944). DAN. 37, 107 (1942).

29. Landau L. D., On the theory of superconductivity, Sow. Phys. 4, 43 (1933).

30. Landau L. D., Nature of the intermediate state, ZhETF. 7, 371 (1937). Sow. Phys. 11, 129 (1937). Nature. 141, 688 (1938).

1. Landau L. D., Theory of the Superfluidity of Helium-II, ZhETF. 11, 592 (1941).

2. Landau L. D., Theory of the Intermediate State of Superconductors, ZhETF. 13, 377 (1943). Izv. AN SSSR, ser. fiz. 6, 79 (1942).

3. Landau L. D., Development of the Hydrodynamics of Helium-II, ZhETF, 14, 112 (1944).

4. Landau L. D., Theory of the Superfluidity of Helium-II, Journ. of Phys. 11, 91 (1947).

5. Landau L. D. and Khalatnikov I. M., On the Theory of the Viscosity of Helium-II (in press).

6. Lifshits E. M., Radiation of Sound in Helium-II, ZhETF. 14, 116 (1944).

7. Meshkovsky A. G. and Shalnikov A. I., Structure of Superconductors in the Intermediate State. Izv. AN SSSR, ser. fiz. 11, 39 (1947).

8. Meshkovsky A. G. and Shalnikov A. I., Surface Phenomena in Superconductors in the Intermediate State, ZhETF. 17, 851 (1947).

9. Nakhutin I. L., Superconductivity in the Intermediate State, ZhETF. 8, 713 (1938).

10. Nakhutin I. L., A Superconducting Ellipsoid in the Intermediate State, ZhETF. 9, 709 (1939).

11. Nakhutin I. L., Absence of the Influence of the Orientation of the Axes of a Single Crystal in a Magnetic Field on Superconductivity, ZhETF. 12, 41 (1942).

12. Peshkov V. P., Second Sound in Helium-II, DAN. 45, 385 (1944).

13. Peshkov V. P., Determination of the Velocity of Propagation of Second Sound in Helium-II, ZhETF. 16, 1000 (1946).

14. Peshkov V. P., Second Sound in Helium-II. Dissertation, IFP (1946).

15. Peshkov V. P. and Zinov’eva K. N., Determination of the Velocity of Second Sound in the Region of Elevated Pressures (in press).

16. Pomeranchuk I. Ya., Critical Field in Superconductors of Small Form, ZhETF. 8, 1096 (1938).

17. Rudnitsky V. K., Diamagnetism of Atomic Rings and the Problem of Superconductors, ZhETF. 8, 1082 (1938).

18. Rudnitsky V. K., On the Question of the Meissner-Ochsenfeld Effect, ZhETF. 11, 107 (1941).

19. Ryabinin Yu. N. and Shubnikov L. V., Dependence of the Magnetic Induction of Superconducting Lead on the Field, Nature. 134, 286 (1934); Sow. Phys. 5, 641 (1934).

20. Ryabinin Yu. N. and Shubnikov L. V., Magnetic Induction of Superconducting Lead, Sow. Phys. 6, 557 (1934); ZhETF. 5, 140 (1935).

21. Ryabinin Yu. N. and Shubnikov L. V., Magnetic Properties and Critical Current Strength in PbTl₂, Sow. Phys. 7, 122 (1935).

22. Savich P. P. and Shalnikov A. I., Coagulation of Fog in Helium-II, Journ. of Phys. 10, 299 (1946).

23. Strelkov P. G., Radiometric Effect in Liquid Helium, ZhETF. 10, 743 (1940).

24. Strelkov P. G., Optical Diwar and Some Observations on Heat Transfer in Helium-II, ZhETF. 10, 1225 (1940).

25. Strelkov P. G., Radiometric Forces in Liquid Helium-II, Dissertation, IFP (1940).

26. Tumanov K. A., Shalnikov A. I. and Sharvin Yu. V., On the Question of the Superconductivity of Solutions of Sodium in Ammonia, DAN. 54, 35 (1947).

27. Fakidov I. and Lazarev B. G., Superconductivity and the Hall Effect. Sow. Phys. 7, 677 (1935).

28. Frenkel Ya. I., A Possible Explanation of Superconductivity by the Magnetic Interaction of Electrons, ZhETF. 3, 101 (1933).

29. Frenkel Ya. I., An Attempt to Explain Superconductivity by Electric Polarization, Nature. 133, 730 (1934).

1. Frenkel Ya. I. and Rudnitskii V. Gyromagnetic effect of a superconductor, ZhETF. 9, 260 (1939).

2. Frenkel Ya. I., Capillary properties of helium-II, ZhETF. 10, 650 (1940).

3. Khotkevich V. I., On the question of the critical values of the field and current for superconducting tin, ZhETF. 8, 515 (1938).

4. Shalnikov A. I., Superconductivity of thin films, Nature. 142 74. (1938); ZhETF. 8, 663 (1938).

5. Shalnikov A. I., Superconducting properties of thin metallic layers, ZhETF. 10, 630 (1940).

6. Shalnikov A. I. On the transition of a hollow sphere from the superconducting to the nonsuperconducting state, I. ZhETF, 10, 414 (1940).

7. Shalnikov A. I., On the transition of a hollow sphere from the superconducting to the nonsuperconducting state, II. ZhETF. 12, 34 (1942).

8. Shalnikov A. I., Structure of superconductors in the intermediate state, Izv. AN SSSR, ser. phys. 6, 80 (1942); ZhETF. 16, 757 (1946).

9. Shalnikov A. I. and Sharvin Yu. V., Investigations of the depth of penetration of a magnetic field into a massive superconductor (in press).

10. Sharvin Yu. V., Nonsuperconductivity of gray tin down to 1.32° K, Journ. of Phys. 9, 351 (1945).

11. Shubin S., Theory of superconductivity and the distance between atoms in the lattice, C. R. 192, 1020 (1931).

12. Shubnikov L. V. and Khotkevich V. I. Heat capacity of superconducting alloys. Sow. Phys. 6, 605 (1934).

13. Shubnikov L. V., Destruction of superconductivity by current and field. Gisterezis kol'tsa iz olova, Nature. 138, 545 (1936).

14. Shubnikov L. V. and Kikoin A. K., Optical studies of helium-II. Sow. Phys. 10, 119 (1936).

15. Shubnikov L. V. and Alekseevskii N. E., Transition curve in the destruction of superconductivity by an electric current, ZhETF. 6, 1200 (1936).

16. Shubnikov L. V., Khotkevich V. I., Shepelev Yu. D. and Ryabinin Yu. N., Magnetic properties of superconducting alloys and metals, ZhETF. 7, 221 (1937).

17. Shubnikov L. V. and Nakutin I. L., Electrical conductivity of a superconducting sphere in the intermediate state, ZhETF. 7, 566 (1937).

18. Yakovlev I. A., Scattering of light in liquid helium, Journ. of Phys. 7, 307 (1943).

CITED LITERATURE (FOREIGN)

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