Current State of the Theory of Superconductivity

V. L. Ginzburg

II. Microscopic Theory*)

Contents

1. Introduction ................................................................ 26
2. Hypothesis of spontaneous currents ........................................ 31
3. Diamagnetic hypothesis .................................................... 38
   a) Initial considerations .................................................. 38
   b) Superconductivity of ideal charged Bose and Fermi gases contained in a vessel ................................................................ 47
   c) The role of interactions with a high dielectric constant. Some critical remarks ................................................................ 55
4. Quasi-microscopic approach to the theory of superconductivity .............. 62
   a) Spectrum of “elementary excitations” and the theory of superfluidity of helium II ................................................................ 62
   b) Spectrum of excitations of the electron liquid in a metal and superconductivity ........................................................... 74
5. Conclusion ................................................................ 89

Appendix. Further development of the macroscopic theory of superconductivity ................................................................ 90
a) Behavior of thin superconducting films, cylinders, and spheres in a magnetic field ................................................................ 91
b) Surface energy and some other questions ................................. 101

Cited literature ............................................................. 113

*) The first part of the article, devoted to the macroscopic theory of superconductivity, was published in UFN 42, 169, 333 (1950). Below this first part is cited as I, and references to the corresponding formulas are given with the numeral I added; for example, (I; 5.1) denotes formula (5.1) from I-

V. L. GINZBURG

1. INTRODUCTION

A sufficiently developed microscopic theory of superconductivity still does not exist, and in this direction one can point only to a number of hypotheses, observations, and attempts somehow to explain the mechanism of superconductivity. Despite this state of the question and, to a certain extent, precisely taking into account the existing situation, it seems expedient to us to consider the state of the microscopic theory of superconductivity and, in particular, to clarify the true value of a number of recent works in this field.

Let us first of all dwell on the tasks confronting the microscopic theory of superconductivity.

1. It is necessary to indicate, from the point of view of the electronic mechanism, by virtue of what causes and under what conditions a metal is superconducting, i.e. in it there can flow a superconducting current \(\mathbf{j}_s\) not connected with the release of heat. In weak magnetic fields (i.e. under the condition \(H \ll H_k\), where \(H_k\) is the critical field) the current density \(\mathbf{j}_s\) satisfies the equations:

\[ \operatorname{rot}\Lambda\mathbf{j}_s=-\frac{1}{c}\mathbf{H}, \tag{1,1} \]

\[ -\frac{\partial \Lambda\mathbf{j}_s}{\partial t}=\mathbf{E}, \tag{1,2} \]

where \(\Lambda=\dfrac{m}{e^2 n_s}\) and \(n_s\) is the concentration of superconducting electrons (for more detail see I). The theory must lead to these equations.

Consideration of strong fields, necessary for understanding the character of the destruction of superconductivity in a magnetic field and a number of other questions, is very important for the study of superconductivity as a whole. But from the point of view of the microscopic theory, the question of strong fields should naturally be moved to the background and one should strive above all to understand the mechanism of the phenomenon in the simpler case of weak fields.

1. At the same time, in the theory of metals explaining the phenomenon of superconductivity, there must be reflected the fact that in a sufficiently strong magnetic field any superconductor is in the normal (nonsuperconducting) state, whose properties do not differ from the properties of nonsuperconducting metals.

2. The specific thermal properties of superconductors must be understood, i.e. there must be obtained a temperature dependence of the free energy corresponding to experiment and, consequently, also of the entropy and heat capacity in the superconducting state (in first approximation the heat capacity varies according to the law \(c_s \sim T^3\); in the absence of a magnetic field the transition from the superconducting state to the normal one is a transition of the 2nd kind).

1. From the theory there should follow the characteristic temperature dependence of the normal conductivity and thermal conductivity in the superconducting state (see 1). At the same time it is necessary to understand what the microscopic picture is of the transition from the superconducting state to the normal one from the point of view of the peculiarities of heat motion in the metal. This also includes the question of the dielectric constant in the superconducting state, which apparently reaches enormous values
   \(\varepsilon_0 \sim 10^8 \div 10^{10}\) (see (1; 6,19) and below).

2. After the discovery of the isotope effect it became quite clear that it is impossible to construct a microscopic theory of superconductivity without taking into account the interaction of the conduction electrons with the vibrations of the crystal lattice. Taking this interaction into account must lead to the experimentally obtained dependence

\[ M^{1/2}T_\kappa=\mathrm{const}, \]

where \(M\) is the mass of the nucleus and \(T_\kappa\) is the critical temperature; and, of course, the constant introduced above is different for different superconducting metals.

1. Without dwelling on certain points which at present appear secondary (the question of the thermoelectric properties of superconductors, the influence of pressure, and so on), let us emphasize that one of the fundamental tasks of the theory of superconductivity is to clarify the reasons why some metals are superconducting, whereas others (such as Au, Ag, Cu, Pt, etc.) certainly do not become superconducting down to the lowest temperatures attained and, as there is reason to believe, remain nonsuperconductors even down to absolute zero itself.

The difficulties in constructing a microscopic theory of superconductivity are connected to a large extent with the fact that in this case one cannot proceed from a model of a metal in which the electrons are regarded as noninteracting (this will be discussed in more detail in Sections 3, 4). Meanwhile, all the successes of the electron theory of metals, at least as regards conductivity, have up to the present been connected precisely with such a “gas” or one-electron model—a model in which the interaction between electrons is either not taken into account at all or is taken into account in the most summary fashion*). Thus, the development of a microscopic theory of superconductivity is inevitably connected with an expansion and generalization of the metal model used, with a more detailed and complete accounting than has hitherto been made of the interelectron interaction and of the interaction of the electrons with the lattice.

*) In the model under discussion, the electrons are assumed to move in some periodic field, and the correlation between the motion of different electrons is ignored. Taking into account the summary interelectron interaction reduces only to the assertion that the aforementioned, exactly unknown periodic field is due not only to the ions, but also to all the electrons.

This problem is very difficult, but at the same time its investigation is important not only from the standpoint of the theory of superconductivity. The point is that the widely used “gas” model is still, theoretically, not substantiated to the proper extent. Electrons in a metal in fact form rather an “electron liquid,” and not a gas, since the interaction between them is strong and at the same time, in order of magnitude, the same as their interaction with the lattice. And if such a “liquid” behaves in a number of respects like a gas, this is connected with the fact that the electrons obey Fermi statistics. In a degenerate Fermi gas at temperature \(T\), as is known, an active role is played only by particles whose energy differs from the limiting energy for a completely degenerate gas \(E_0\) by an amount of order \(kT\); if the total electron concentration is \(n\), then the “active” ones among them are only electrons with concentration

\[ \Delta n \sim n \frac{kT}{E_0} = n \frac{T}{T_0}, \]

where \(T_0 = \dfrac{E_0}{k}\) is the degeneracy temperature, equal to \(10^4 \div 10^5\) degrees. Only these peripheral electrons, situated near the Fermi surface, contribute to the heat capacity of the gas and can change their energy in collisions with one another or with other particles (for example, lattice phonons). All the other electrons, lying sufficiently far from the limiting surface, obviously cannot transfer or receive energy of order \(kT\). Therefore, in particular, the strong Coulomb interaction between the peripheral (active) electrons and all the other electrons cannot usually manifest itself completely, and effective, generally speaking, turns out to be only a relatively weak interaction between the peripheral electrons themselves. In view of what has been said, the behavior of a degenerate electron gas proves to be similar to the behavior of an electron liquid in which the “elementary excitations” possess an energy spectrum similar to the spectrum of peripheral electrons in an ideal Fermi gas*). The success of the gas model of electrons in a metal precisely indicates that such a closeness of the excitation spectrum of the electron liquid in metals to the spectrum of peripheral (i.e. close to the limiting energy) electrons in an ideal Fermi gas actually takes place. However, this circumstance has still not been sufficiently understood theoretically and, most importantly, it remains unclear under what conditions and for what reasons the indicated closeness of the spectra of the liquid and the gas is violated. On

*) For more details on this, see \(^{1}\) §§ 66—69, as well as \(^{2,3}\). In addition, we shall return to this question below, in Section 4.

the same fact—that the kinship between the spectra of an electron liquid and a gas is indeed not complete—indicates the existence of superconductivity, which cannot be understood from the point of view of the gas spectrum (see Section 4).

Besides superconductivity, from the standpoint of the gas model there is also no explanation for the sharp increase in resistance observed in nonsuperconducting metals when the temperature is sufficiently lowered (usually at \(T<1^\circ\mathrm{K}\); see \(^{2}\) § 23 and \(^{4}\)). The need for further development of the theory of metals in the direction of a more complete account of the interaction among electrons and of electrons with the lattice is also dictated by problems arising in the consideration of a number of other questions (see, for example, \(^{5}\)). Let us also note that the achievements of the gas model of electrons in metals are often overestimated. In many cases the correct conclusions following from this model (such, for example, as the formula for thermionic emission, Ohm’s law, the thermoelectric relations of Thomson, etc.) merely testify to the fact that its properties do not contradict the principles of thermodynamics or a number of other very general conditions and requirements. The genuine achievements of this particular theory of metals include, essentially, only such points as the establishment of the temperature dependence of the electrical conductivity (from the gas model it follows that \(\sigma \sim \frac{1}{T}\) at high temperatures and \(\sigma \sim \frac{1}{T^{5}}\) at low temperatures) and of certain other kinetic coefficients, an estimate of these coefficients in order of magnitude, and also the account of the influence of an external magnetic field on them.

It is generally accepted that for monovalent metals, for example for gold, in the question of electrical conductivity the agreement of theory with experiment at \(T \gg 1^\circ\) is good (see, for example, \(^{6,7}\)). However, even in this best-studied case there is a certain ambiguity (see \(^{8}\)). As for the electrical conductivity of polyvalent metals and, in particular, the thermal conductivity, thermoelectric and galvanomagnetic effects in metals, here the question of the agreement of the existing theory with experiment at low temperatures remains to a large extent open. Moreover, in a number of cases the matter lies not only in the insufficient reliability and accuracy of the experimental data, but also in the absence of rigorous and clear theoretical conclusions even when the gas model is used (especially important in this respect is the question of the role of “Umklapp processes” and the doubts connected with it concerning the correctness of the existing theory of conductivity at low temperatures; see \(^{6,7}\)). In this field there is an acute need for a modern critical review—unfortunately still lacking—of all available theoretical and experimental data. We do not

seems out of the question that such an analysis, and, chiefly, further theoretical and experimental work and, in particular, the performance of certain experiments not yet carried out (for example, clarification of the question of the range of applicability of Ohm’s law; see ² § 23; in this connection see also the papers ¹²⁰) will compel us to abandon the opinion, now very widespread, that the theory of metals based on the gas model, on the whole—if superconductivity is not considered—is in good agreement with experiment.

In one way or another, a further substantial development of the theory of metals is necessary, and in this respect the phenomenon of superconductivity will serve as the touchstone of any new theory claiming any degree of universal significance.

The known attempts to construct a microscopic theory of superconductivity may, in general, be divided into three groups. The first of these includes theories proceeding from the “hypothesis of spontaneous currents”—the assumption that in the superconducting state some spontaneous currents always flow in a metal (even in the absence of an external magnetic field). In Section 2 we shall try to show the untenability of this approach, which has hitherto been rather widespread, to the explanation of superconductivity. Further, in Section 3, attempts to construct a theory of superconductivity on the basis of the “diamagnetic hypothesis” are discussed; according to this hypothesis superconducting currents are likened to the currents responsible for the diamagnetism of atoms and molecules placed in a magnetic field. To this line of thought, which seems to us correct, rather much attention is devoted. Here the question is discussed of the necessity of taking into account the interaction of electrons with the lattice, which follows from the magnitude of the isotope effect observed experimentally. The last part of the article (Section 4) is devoted to the quasi-microscopic approach to the explanation of superconductivity, which is related to the approach successfully applied by Landau in the case of the superfluidity of helium II. Here the question is discussed of what the excitation spectrum of the electron liquid in a metal must be in order that superconductivity be observed, which with known justification may be regarded as the superfluidity of a charged (electronic) liquid in a metal.

Unfortunately, along this path, which is the most flexible, it has not yet been possible to attain such a degree of clarity and definiteness as in the case of the superfluidity of helium II.

In a brief conclusion the conclusions obtained in the preceding sections are compared.

In addition to the article, it has seemed advisable to illuminate certain essential points relating to the first part of the article (i.e., to I) and clarified to a considerable extent

over the approximately year and a half that have passed since this first part was written.

The nature of the subject and the desire to make the article more accessible and convenient to use have led to its being written in rather great detail.

2. THE HYPOTHESIS OF SPONTANEOUS CURRENTS

The hypothesis of spontaneous currents, as has already been mentioned, consists in the assumption that, in a state of thermodynamic equilibrium and in the absence of an external magnetic field, certain spontaneous currents flow in a superconductor. In other words, it is assumed that the energetically lowest state*) of a superconducting metal is a state in which the mean current density $\bar{\mathbf{j}}\ne 0$, just as in a ferromagnet below the Curie point the spontaneous magnetization $\bar{\mathbf{M}}\ne 0$. It follows from experiment that, in the absence of an external magnetic field, the total current in the case of sufficiently pure (“ideal”) superconductors is always absent. This fact, if one starts from the hypothesis of spontaneous currents, can be explained by supposing that the metal breaks up into separate regions with $\mathbf{j}\ne 0$, but arranged in such a way that on the average over a macroscopic volume $\bar{\mathbf{j}}=0$ (here again the analogy is used with ferromagnets, which in the absence of an external field break up into separate domains, so that on the average over the specimen $\bar{\mathbf{M}}=0$).

The hypothesis of spontaneous currents was advanced in 1933 by Bloch and Landau (see $^{9}$ and $^{6a}$ § 44); in recent years Heisenberg $^{10}$ (a review of work in this direction $^{11}$), Born and Cheng $^{12}$, and several other authors have based their attempts to construct a theory of superconductivity on it.

The objection most frequently encountered against the hypothesis of spontaneous currents rests on the so-called Bloch theorem and is to a considerable extent based on a misunderstanding. Bloch’s theorem, discussed in detail in $^{13}$, consists in the assertion that a state of a system of electrons with a current different from zero is not the energetically lowest one, at least when only the interaction of the electrons with one another and with the potential field of the lattice is taken into account, but when the interaction of the electron spin with the orbital magnetic field is neglected. Here, by a state with current one means a state in which the total current is different from zero, i.e. the current through the entire cross-section of the superconductor. In other words, it is asserted that, in the absence of external fields and without taking into account magnetic interaction, a state in which, for example, along

*) A somewhat more general assumption reduces to the statement that the state with current has not the lowest energy, but the lowest free energy. Below, for simplicity, we shall speak of energy, all the more since as $T\to 0$ the free energy does not differ from the energy.

no current flows in a straight wire is energetically lower than the state with a current. But in this form Bloch’s theorem is completely trivial and at the same time has no relation to the theory of superconductivity. Indeed, in the state with a total current (in the indicated sense) the electrons possess momentum \(\mathbf p=\frac{m}{e}\mathbf j\) (\(\mathbf j\) is the current density; the momentum \(\mathbf p\) is referred, evidently, to unit volume) and a definite kinetic energy. Since interaction forces depending on the velocity (momentum) of the electrons relative to the lattice are not taken into account, it is clear that states are possible which differ from the one under consideration only by the value of the momentum \(p\). Of these states, the state with zero momentum obviously has the least kinetic, and consequently also the least total, energy. This assertion, i.e. precisely Bloch’s theorem, is in essence classical, and its quantum-mechanical proof\(^{13}\) adds nothing new here. At the same time, as was said, in this form Bloch’s theorem has no direct relation to superconductivity and to the hypothesis of spontaneous currents\(^*\), since in a singly connected superconductor, in the absence of an external magnetic field, the total current is always equal to zero. Therefore, in order to refute the hypothesis of spontaneous currents, it is necessary to prove that the lowest state cannot be any state with an orbital current \(\mathbf j \ne 0\).

But such an assertion is known in advance to be false. As an example it is sufficient to point out that for many atoms the ground (i.e. energetically lowest) states are \(P-\), \(D-\), \(F-\ldots\) states, i.e. states with nonzero orbital angular momentum and, consequently, nonzero electron-current density (for example, the \(P\)-state is the ground state of the atoms C, O, F, etc.)\(^ {**}\). The electron current in an atom may differ in no essential way from a current in a region of macroscopic dimensions and, consequently, the existence of spontaneous stable currents in a metal is in principle possible.

Nevertheless, despite the fact that the hypothesis of spontaneous currents, by virtue of what has been said, cannot be rejected from the very beginning, it encounters numerous serious objections.

In the absence of an external magnetic field and of a total current, spontaneous currents in a superconducting specimen must be arranged

\(^*\) The widely circulated converse assertion (see, for example, \(^{14}\) § 24) is, in our opinion, erroneous.

\(^ {**}\) In \(^{13}\) an attempt is nevertheless made to prove that a state with angular momentum different from zero cannot be the lowest. The error of this proof is connected with the neglect of the identity of electrons and their subjection to the Pauli principle. For nonidentical particles, or particles obeying Bose–Einstein statistics, a state without orbital angular momentum (an \(S\)-state) is indeed the lowest, but in the case of electrons this assertion is incorrect (for more detail see \(^{15}\), p. 250).

in such a way that the specimen has no magnetic moment and, in general, from the macroscopic point of view carries no currents.

Further, the spontaneous currents must obviously be closed, and their dimensions are limited by energetic considerations. Let the radius of an individual “filament” of spontaneous current be \(r\), its length be \(l\), and the density of the spontaneous current in the filament be \(j_0\). Regarding the filament, for definiteness, as quasilinear (i.e., assuming \(l \gg r\)), we find that the intensity of the magnetic field produced by the filament at its surface is

\[ H = \frac{2\pi r^2 j_0}{rc} = \frac{2\pi r j_0}{c}, \]

and the energy of the field associated with the filament is

\[ W_m \sim \frac{H^2}{8\pi}\,\pi r^2 l = \frac{\pi^2 r^4 l j_0^2}{2c^2}. \]

If there were no surface energy between the individual current filaments, then the dimensions of the filaments, in order to ensure a minimum of the energy, would decrease without bound. Taking the surface tension at the boundaries between the filaments to be \(\alpha\), for the surface energy of a filament we obtain the value \(W_n = 2\pi r l \alpha\). In a unit volume there are \(1/\pi r^2 l\) current filaments (it is assumed that everywhere in the metal \(j \ne 0\), just as in a ferromagnet everywhere \(M \ne 0\), if one does not speak of surfaces or layers separating individual current filaments or domains in ferromagnets). Thus, the energy density associated with the presence of spontaneous currents is equal to

\[ W = \frac{2\alpha}{r} + \zeta\,\frac{\pi j_0^2 r^2}{2c^2}, \]

where \(\zeta\) is a coefficient of order unity, or, more precisely, of order \(\ln(l/r)\) (it has been set that

\[ W_m = \zeta\,\frac{\pi^2 r^4 l j_0^2}{2c^2} \]).
The energy is minimal if

\[ r^3 = \frac{2\alpha c^2}{\zeta \pi j_0^2} \sim \frac{\alpha c^2}{j_0^2}; \tag{2.1} \]

consequently, the size of the current filaments, i.e. of those regions into which, by assumption, the superconductor is divided, in contrast to the case of ferromagnets, does not depend on the dimensions of the specimen.

Since above we have been speaking only of orders of magnitude, it is clear that all expressions remain valid also for filaments with \(l \sim r\). Further, the density of the spontaneous current \(j_0\) may be related to the critical magnetic field for a massive specimen \(H_{km}\). Indeed, within the framework of the adopted model, the current flowing in a conductor in the presence of an external magnetic field appears as a result of the redistribution and merging of individual current filaments. Therefore the experimentally observed density of the superconducting current \(j_s\) must in any case be no greater than the density of the spontaneous current \(j_0\), similarly to how the magnetization of a ferromagnetic specimen does not exceed the magnitude of the spontaneous magnetization \(M_0\) (it is assumed that the change, under the influence of an external field, of the current density within the boundaries of a single filament is negligible).

of a single filament, just as the change of magnetization within the domain is sufficiently small; in the model adopted there is every reason for such an assumption. But the superconducting current \(j_s\), in any case, can reach the value \(j_k = \dfrac{c}{4\pi\delta} H_{\mathrm{cm}}\), since precisely such is the current density on the flat surface of a massive superconductor placed in the critical field \(H_{\mathrm{cm}}\) (see (I; 2,26)). Thus, \(j_0 > \dfrac{c}{4\pi\delta} H_{\mathrm{cm}}\), where \(\delta \sim 10^{-5}\) is the depth of penetration of the field into the metal. The surface tension \(\alpha\), in turn, cannot be greater than the surface tension \(\alpha_{ns} \sim \delta \dfrac{H_{\mathrm{cm}}^2}{8\pi}\) at the interface between the normal and superconducting phases of the metal, since the quantity \(\alpha_{ns}\) is relatively very large, and it would even be natural to suppose that \(\alpha \sim 10^{-7} \dfrac{H_{\mathrm{cm}}^2}{8\pi} \ll \alpha_{ns}\) (see I, end of § 2). Taking into account (2,1) and the remarks made concerning the possible values of \(j_0\) and \(\alpha\), we arrive at the conclusion that

\[ r \lesssim \delta \sim 10^{-5}\ \mathrm{cm}. \tag{2,2} \]

The surface energy, referred to a unit volume, in the state when there is no resultant current, is equal to \(\dfrac{2\alpha}{r} \lesssim \dfrac{H_{\mathrm{cm}}^2}{8\pi}\), i.e., in order of magnitude it can reach the difference of the free energies of the normal and superconducting phases of the metal \(F_n - F_s = \dfrac{H_{\mathrm{cm}}^2}{8\pi}\) (see (I; 2,36)). The minimum value of the energy \(\dfrac{2\alpha}{r}\), of order \(0.1\,\dfrac{H_{\mathrm{cm}}^2}{8\pi}\), is attained for \(\alpha \sim 10^{-7} \dfrac{H_{\mathrm{cm}}^2}{8\pi}\).

Under the influence of an external magnetic field, the spontaneous currents must redistribute themselves and merge, forming those observed superconducting currents which screen the penetration of the magnetic field into the bulk of the superconductor and obey equation (1,1). The possibility of obtaining this equation on the basis of the model of spontaneous currents appears very problematic and, in any case, has not been shown by any of the proponents of this model. The corresponding attempt by Heisenberg\(^{10}\) is completely unconvincing. First, it remains unclear how, for spontaneous currents intertwined with one another into a tangle or forming some cellular structure, one can at once adopt, as is done in\(^{10}\), an equation of type (1,2). Secondly, even if equation (1,2) were obtained, as is well known (see I, § 2), equation (1,1) still does not follow from it without further assumptions. Therefore Heisenberg’s “derivation” of this equation, as has already rightly been noted—

founded in \(^{16}\), is based on an assumption which in fact still has to be justified\(^*\)). Still more substantial is another objection, directly based on known experimental facts. It is known from experiment that superconducting currents “respond” to the slightest changes of the external magnetic field, and that hysteresis in fields smaller than the critical one, in the case of “ideal” superconductors, is completely imperceptible or, in any case, tends to zero as the purity and homogeneity of the sample increase. An entirely different picture should be observed from the point of view of the model of spontaneous currents, since any merging and redistribution of the threads of these currents is connected with a change in surface energy. From the estimate made above it is clear that the surface energy per unit volume of the superconductor is rather considerable, as a result of which hysteretic phenomena would inevitably have to be observed, just as occurs in the magnetization of ferromagnets. Thus, the absence in superconductors of hysteresis in fields \(H < H_k\) testifies against the hypothesis of spontaneous currents\(^ {**}\)). A number of other considerations also speak against this hypothesis. Thus, it appears

\(^*\)) Equation (1.1) was obtained in \(^{10}\) from (1.2) and from the requirement that the state under consideration correspond to an energy minimum. But this latter requirement is not obligatory and, for example, is not fulfilled for an ideal conductor (in the bulk of the metal, in this case, a certain field \(H_0 \ne 0\) may be preserved, although such a metastable state is stable). The requirement of minimum energy is equivalent to the denial of the possible realization of metastable states with \(H_0 \ne 0\) and, as is clear from \(^{10}\) or from I § 2, leads to equation (1.1). See in this connection also \(^{16,17}\).

\(^ {**}\)) A model of regions (threads) of current, based on the hypothesis of spontaneous currents, with zero mean current and magnetic moment, corresponds from the outset rather to an antiferromagnet than to a superconductor. Indeed, antiferromagnets are, as it were, bodies with \(j \ne 0\), but with current and magnetic moment averaged over a cell equal to zero (see I § 129; in ferromagnets, in contrast to antiferromagnets, the mean magnetic moment of a cell

\[ \mu=\frac{1}{2c}\int_{\text{over cell}}[\mathbf{r}\mathbf{j}]\,d\upsilon \]

is different from zero). In real antiferromagnets, apparently, the chief role, as in ferromagnets, is played by currents connected with the electron spin. But in principle there are also possible antiferromagnets of the “orbital type,” in which the determining role is played by the orbital magnetic moment. Under the influence of an external magnetic field the distribution of currents (magnetic moments) in an antiferromagnet changes, but in accessible fields only to a very weak degree. One may think that the model of spontaneous currents will actually behave in just such a fashion, since it is very difficult to imagine the assumed merging of individual threads and their rotation in an external field \(H \lesssim H_k\), despite the presence of surface energy between the threads and rather strong local magnetic fields

\[ \left(\text{for } r\sim\delta \text{ the local fields } H\sim \frac{2\pi r j_0}{c}\sim H_{km}\right). \]

It is highly probable, if not simply beyond doubt, that the phenomenon of superconductivity is akin to the phenomenon of superfluidity—superconductivity is, as it were, the superfluidity of an “electron liquid” in a metal (for more detail see § 4 b). But, as applied to superfluidity, the hypothesis of spontaneous currents or, better, flows, according to all available data, plainly does not correspond to reality. And even to assume, purely theoretically, that helium II at rest as a whole consists entirely of separate filaments or regions in which the velocity of the liquid is nonzero seems simply incredible; such an assumption is all the less well grounded because the phenomenon of superfluidity is explained in Landau’s developed theory on an entirely different basis\(^{18,19}\). Thus, taking the hypothesis of spontaneous currents as the basis of the theory of superconductivity is connected with a seemingly quite unnatural refusal to consider superconductivity as a phenomenon related to superfluidity. Another objection to the hypothesis of spontaneous currents consists in the fact that, besides it, a diamagnetic hypothesis has also been put forward to explain superconductivity; this has not yet led to the construction of a true microscopic theory of superconductivity, but is free of all the serious shortcomings associated with the hypothesis of spontaneous currents. Therefore recourse to the latter cannot be justified even by the absence of other, more plausible assumptions.

Besides the critical remarks of a general character already made, it must be noted that concrete attempts\(^{10–12}\) to develop a theory of superconductivity on the basis of the assumption of the existence of spontaneous currents encounter additional difficulties. Thus, in\(^{10–12}\), spontaneous currents are formed only by electrons lying near the Fermi surface at distances of order \(kT_k\) from it (\(T_k\) is the critical temperature, equal, as is known, to \(\sim 1—10^\circ K\)). The concentration of electrons in this layer is of order \(nT_k/T_0 \sim 10^{-4} \div 10^{-5} n\), where \(T_0 \sim 10^5\) is the degeneracy temperature and \(n\) is the total concentration of electrons. It follows from this that no more than \(10^{-4}\) of the electrons participating in conduction in the normal state will take part in superconductivity. But it is known from experiment (see I, § 6) that the concentration of superconducting electrons is \(n_s \sim 0.1 n_0 \sim 10^{22}\) (\(n_0 \sim 10^{23}\) is the concentration of conduction electrons in the normal state). Thus there is here a discrepancy of several orders of magnitude. This difficulty is very hard to overcome, since it is completely unclear how, at \(T \sim T_k\), spontaneous currents can be formed with the participation of electrons lying at distances much greater than \(kT_k\) from the Fermi surface. Apparently acknowledging the remark made, Heisenberg\(^{10}\) nevertheless, without any grounds for doing so and limiting himself only to a reference to experiment (!?), asserts that in the equation he obtains (1,2) there will figure-

concentration of electrons, \(\sim n_0\), and not \(\sim 10^{-4} n_0\) (as has already been mentioned; even apart from this point the derivation of equation (1.2) given in \(^{10}\) remains completely unclear to us; F. London \(^{16}\) also points to this). Further, in a theory based on the hypothesis of spontaneous currents in a strong field \(H \sim H_{km}\), one should expect sharp nonlinear phenomena. Such a conclusion was indeed drawn in \(^{10,11}\), but it is completely refuted by experiment \(^{20}\) (see also the addendum on this). It should also be noted that in \(^{10-12}\) the vibrations of the crystal lattice are not taken into account at all, whereas after the discovery of the isotope effect it became unquestionable that no detailed microscopic theory of superconductivity can be constructed without taking lattice vibrations into account.

Finally, one must not forget that until now we have not touched at all on the question of the extent to which the existing works have really shown that, under certain assumptions, spontaneous currents exist. Above we only indicated that the existence of spontaneous currents is possible in principle and, on the other hand, emphasized the difficulties that arise if one accepts the existence of these spontaneous currents. Let us note that in \(^{10}\) the existence of spontaneous currents is certainly not proved, and the author of that work himself claims only heuristic considerations indicating the possibility of the appearance of spontaneous currents. The same may be said of all other known attempts in this direction, with the possible exception of work \(^{13}\) (in \(^{13}\) the calculations of the energy of a system of electrons with allowance for the exchange Coulomb interaction are not given and have not been repeated by us; there are certain grounds to doubt their correctness).

Summing up, it may be said that, first, there are no grounds for accepting the hypothesis of spontaneous currents and, second, this hypothesis encounters various serious objections. Therefore, in our opinion, attempts to develop a theory of superconductivity on the basis of the hypothesis of spontaneous currents should be decisively abandoned, at least until some fundamental ways are found of overcoming all the difficulties listed above*).

*) Despite the sharply negative attitude expressed above toward using the hypothesis of spontaneous currents to explain superconductivity, it should be noted that this hypothesis admits direct experimental verification. If spontaneous currents exist in a superconductor, then local magnetic fields must inevitably be present in the bulk of the sample, and stray fields will exist at its surface. The presence of these magnetic fields can, in principle, be observed by methods used to study the intermediate state \(^{21}\), by the scattering in the superconductor of charged particles and neutrons, or in some other way.

V. L. GINZBURG

3. DIAMAGNETIC HYPOTHESIS

a) Initial considerations

A superconductor with dimensions considerably greater than the penetration depth \(\delta \sim 10^{-5}\ \text{cm}\), when placed in a magnetic field, behaves like an ideal diamagnet, i.e., like a body with magnetic susceptibility

\[ \chi=\frac{\mu-1}{4\pi}=-\frac{1}{4\pi} \]

(in such a body the magnetic induction \(B=0\)).

The superconducting current flowing in the surface layer of the metal in an unchanged external field \(H\) does not decay and is not connected with the production of heat; in this respect it is also analogous to those currents which flow under the action of a magnetic field in atoms and molecules and determine their diamagnetism.

The diamagnetic susceptibility referred to one atom, as is known (see, for example, \({}^{15}\) § 126), is equal to:

\[ \chi_1=-\frac{e^2 Z}{6mc^2}\overline{r^2}=-4.7\cdot 10^{-14}Z\overline{r^2}, \tag{3,1} \]

where \(Z\) is the number of electrons in the atom and \(\overline{r^2}\) is the mean square of their distance from the nucleus (\(e\) and \(m\) are the charge and mass of the electron). In atoms \(\overline{r^2}\sim 10^{-16}\), whence for one gram-atom

\[ \chi_A=N_A\chi_1=6.02\cdot 10^{23}\chi_1\sim -10^{-6}Z, \]

as is indeed in agreement with experiment (for example, for helium with \(Z=2\) and atomic weight \(A=4\) in the gaseous state \(\chi_A=-1.9\cdot 10^{-6}\), and in helium II, where the density \(\rho\simeq 0.15\), the susceptibility recalculated from the data for the gas and referred to unit volume is

\[ \chi=\frac{\chi_A}{A}\rho\simeq 7\cdot 10^{-8} \]

).

Let us now suppose that diamagnetic properties are possessed not by the atom, but by the metal as a whole. In this case, putting \(\overline{r^2}\sim 1\) and \(Z\sim n_0\sim 10^{23}\) (\(n_0\) is the concentration of conduction electrons), from (3,1) we obtain the value \(\chi\sim -5\cdot 10^7\), whereas in fact, as will now be shown, the value of \(\chi\) is close to \(-\dfrac{1}{4\pi}\). The point is that the expression for \(\chi\) obtained from (3,1):

\[ \chi=N_a\chi_1=-\frac{e^2 ZN_a}{6mc^2}\overline{r^2} \tag{3,2} \]

(\(N_a\) is the concentration of atoms) is valid only under the condition that

\[ |\chi|\ll 1. \tag{3,3} \]

In the general case, however, expressions (3.1) and (3.2) determine, strictly speaking, not the quantities \(\chi_1\) and \(\chi\), but respectively the proportionality coefficients \(\chi'_1\) and \(\chi'\) between the magnetic moment of the atom and the magnetization \(M\) and the magnetic induction \(B\), and not the magnetic field \(H\) (this circumstance is connected with the fact that it is precisely the induction \(B\) that is the average magnetic field acting on the electron). Meanwhile, by definition \(\chi = \dfrac{M}{H}\), i.e.

\[ M=\chi H=\frac{\chi}{\mu}B=\frac{\chi}{1+4\pi\chi}B=\chi'B;\qquad \chi=\frac{\chi'}{1-4\pi\chi'}. \tag{3.4} \]

Under condition (3.3), which for all known diamagnetics is always fulfilled, indeed \(\chi \simeq \chi'\). But if \(|\chi'|\gg \dfrac{1}{4\pi}\), then \(\chi \to -\dfrac{1}{4\pi}\). Therefore, if by formulas (3.1)—(3.2), which, by what has been said, determine the quantities \(\chi_1\) and \(\chi'\), very large values are obtained for \(\chi'\), this means that \(\chi \to -\dfrac{1}{4\pi}\).*)

Thus, if we had a giant atom of macroscopic dimensions and with a correspondingly large number of electrons, such an atom would behave in a magnetic field like a superconductor. This analogy is not destroyed by the fact that in an atom the magnetic field may be regarded as homogeneous, whereas in a superconductor it decays in a surface layer of thickness \(\sim \delta\). Such a difference is explained only by the fact that in an atom the shielding action of the diamagnetic currents is small and is neglected. If, however, the susceptibility \(\chi\) is large, it is obviously impossible to do this, and it is necessary to solve the problem of the behavior of the system in the field with allowance for the shielding currents. Such an allowance, as is immediately clear even qualitatively, will lead to the decay of the field into the interior of the diamagnetic atom.

The essence of the diamagnetic hypothesis on the nature of superconductivity consists in the considerations set forth, in likening a superconductor to a macroscopic diamagnetic atom. Strictly speaking, it is even difficult here to speak of a hypothesis; rather, there is a statement, following from all the experimental data, of a profound analogy between superconducting and diamagnetic currents. This analogy can also be traced quantitatively and makes it possible, as F. London already pointed out in 1935 \(^{24}\) (see also \(^{14,16,18}\)), to understand from the point of view of quantum mechanics the content of the fundamental equation of the theory of superconductivity (1.1).

*) We have dwelt in detail on this elementary remark, which in application to superconductivity has already been emphasized in \(^{22}\), because it is ignored in a series of works by Bardeen \(^{23}\), which for this reason are grossly erroneous (on this see also the end of this section and § 3 B).

If there is a system of \(N\) electrons with wave function \(\Psi(\mathbf r_1,\mathbf r_2,\ldots,\mathbf r_N)\), then the mean density of the electric current at a point is equal to (let us recall that, in agreement with I, the sign of \(e\) is chosen so that for electrons \(e<0\)):

\[ \mathbf j(\mathbf r)=\sum_{\alpha=1}^{N}\int \left\{ \frac{ie\hbar}{2m}\bigl(\Psi\nabla_{\alpha}\Psi^*-\Psi^*\nabla_{\alpha}\Psi\bigr) -\frac{e^2}{mc}\mathbf A(\mathbf r_\alpha)\Psi^*\Psi \right\} \times \]

\[ \times\delta(\mathbf r-\mathbf r_\alpha)\,d\mathbf r_1\ldots d\mathbf r_N, \tag{3.5} \]

where \(\mathbf A\) is the vector potential of the electromagnetic field, \(\delta\) is the delta function,

\[ \nabla_{\alpha}=\frac{\partial}{\partial x_{\alpha}}\mathbf i+ \frac{\partial}{\partial y_{\alpha}}\mathbf j+ \frac{\partial}{\partial z_{\alpha}}\mathbf k, \]

and the integration is carried out over the whole metal, i.e. over the region in which the \(\Psi\)-function is assumed normalized \(\left(\int \Psi^*\Psi\,d\mathbf r_1\ldots d\mathbf r_N=1\right)\).

The vector potential \(\mathbf A\), as is known, is not uniquely determined, even if the condition

\[ \operatorname{div}\mathbf A=0, \tag{3.6} \]

which we shall adopt, is imposed upon it.

Under condition (3.6), all physical quantities and, in particular, the current density \(\mathbf j\), must not change under a gradient transformation, i.e. under replacement of the potential \(\mathbf A\) by the potential

\[ \left. \begin{aligned} \mathbf A'&=\mathbf A+\nabla\lambda,\\ \nabla^2\lambda&=0, \end{aligned} \right\} \tag{3.7} \]

where \(\lambda(\mathbf r)\) is some scalar function of \(\mathbf r\). In order that the indicated requirement be fulfilled under the transformation (3.7), the function \(\Psi\) must be replaced by the function

\[ \Psi'( \mathbf r_1,\ldots,\mathbf r_N) = e^{\frac{ie}{\hbar c}\sum_{\alpha}\lambda(\mathbf r_\alpha)} \Psi(\mathbf r_1,\ldots,\mathbf r_N). \tag{3.8} \]

This assertion is easily verified by direct substitution (see, for example, \(^{14}\S 26\)).

In the absence of a magnetic field one can always set \(\mathbf A=0\); the corresponding function \(\Psi\) we shall denote by \(\Psi_0\), and assume that in the state \(\Psi_0\) the current \(\mathbf j=0\) (this is precisely the situation that obtains in reality when the magnetic field is everywhere zero and there is no normal current). Let us now turn on the magnetic field and suppose that the function \(\Psi_0\) remains unchanged, i.e. that its perturbation in the field may be neglected. Then, as is clear from what has been said and from (3.5), the current density is equal to

\[ \mathbf j(\mathbf r)= -\frac{e^2}{mc}\sum_{\alpha=1}^{N} \int \mathbf A(\mathbf r_\alpha)\Psi_0^*\Psi_0 \delta(\mathbf r-\mathbf r_\alpha)\,d\mathbf r_1\ldots d\mathbf r_N = \]

\[ = -\frac{e^2 n(\mathbf r)}{mc}\mathbf A(\mathbf r), \tag{3.9} \]

where the quantity

\[ n(\mathbf r)= \sum_{\alpha=1}^{N}\int \Psi_0^*(\mathbf r_1,\ldots,\mathbf r_N)\Psi_0(\mathbf r_1,\ldots,\mathbf r_N)\, \delta(\mathbf r-\mathbf r_\alpha)\,d\mathbf r_1\ldots d\mathbf r_N \tag{3.10} \]

by its very definition is the concentration of electrons, i.e. that part of the \(N\) electrons under consideration which falls per unit volume*).

If the body consists of separate atoms with practically non-overlapping wave functions \(\psi_\alpha(\mathbf r_\alpha)\), the function \(\Psi_0\) is equal to the product of these functions and \(n(\mathbf r)=\psi_\alpha^*(\mathbf r)\psi_\alpha(\mathbf r)\) (for simplicity it is assumed that in the atom there is only one electron; \(\alpha\) is the number of the atom within whose limits the point \(\mathbf r\) is located). In this case the diamagnetic effect is small and one may put \(\mathbf A=\dfrac{1}{2}[\mathbf H\mathbf r]\), where \(\mathbf H\) is the intensity of the external magnetic field. Further, the magnetic moment of the atom is equal to

\[ \boldsymbol\mu=\frac{1}{2c}\int[\mathbf r\mathbf j]\,dv =-\frac{e^2}{4mc^2}\int[\mathbf r[\mathbf H\mathbf r]]\,\psi_\alpha^*\psi_\alpha\,dv= \]

\[ =-\frac{e^2}{6mc^2}\,\overline{r^2}\mathbf H=\chi_1\mathbf H, \]

where it is assumed that the function \(\psi\) corresponds to an \(S\)-state (therefore

\[ \int[\mathbf r[\mathbf H\mathbf r]]\,\psi^*\psi\,dv =\int\{r^2\mathbf H-\mathbf r[\mathbf r\mathbf H]\}\,\psi^*\psi\,dv= \]

\[ =\frac{2}{3}\mathbf H\int r^2\psi^*\psi\,dv =\frac{2}{3}\,\overline{r^2}\cdot\mathbf H \Big). \]

The result obtained is identical with (3.1), as indeed was to be expected (the generalization of the calculation just carried out to the case \(Z\ne1\) is trivial; expression (3.1) is suitable only for \(S\)-states). The situation changes completely if the function \(\Psi_0\) corresponds to electronic motion embracing large, macroscopic regions, i.e. if the function \(n(\mathbf r)\) is constant or, at least, changes little as a function of \(\mathbf r\) (as will be easy to see from what follows, the change of \(n(\mathbf r)\) may be regarded as small if \(\dfrac{\partial n}{\partial z}\delta\ll n\), where \(z\) is the direction in which the function \(n\)

*) Indeed, the quantity \(\int \Psi^*\Psi\,\delta(\mathbf r-\mathbf r_\alpha)d\mathbf r_1\ldots d\mathbf r_N\) is the probability density of finding the electron \(\alpha\) at the point \(\mathbf r\) for any positions of all the other electrons. After summation of such expressions over \(\alpha\), i.e. over all electrons, we obtain the density (concentration) of electrons. With the appropriate changes, this remark also explains the meaning of expression (3.5) for the mean current density.

changes most rapidly). If \(n(\mathbf r)=n_s=\mathrm{const}\), then, by virtue of the equality \(\operatorname{rot}\mathbf A=\mathbf H\), from (3.9), by applying the operation \(\operatorname{rot}\), we immediately obtain\(^*)\):

\[ \operatorname{rot}\,\frac{m}{e^2 n_s}\,\mathbf j=-\frac{1}{c}\,\mathbf H, \tag{3.11} \]

i.e., we obtain equation (1.1) with \(\mathbf j=\mathbf j_s\) and

\[ \Lambda=\frac{m}{e^2 n_s}, \tag{3.12} \]

which is in complete agreement with (I; 2.9).

The result obtained, in its physical essence, is in full agreement with the considerations, based on the idea of magnetic susceptibility, set forth earlier.

Expression (3.11) is also obtained if the function \(\Psi\) in a field has the form

\[ \Psi=e^{\frac{i e}{\hbar c}\sum_{\alpha}\lambda(\mathbf r_\alpha)}\Psi_0, \]

where in the state \(\Psi_0\) the current is absent. In this case

\[ \mathbf j(\mathbf r)=\frac{e^2 n(\mathbf r)}{mc}\,[\nabla\lambda(\mathbf r)-\mathbf A(\mathbf r)] \tag{3.13} \]

and, since \(\operatorname{rot}\nabla\lambda=0\), equation (3.11) for \(n=\mathrm{const}\) is indeed preserved.

From (3.9), for \(n(\mathbf r)=n_s=\mathrm{const}\), one also obtains equation (1.2):

\[ \frac{\partial \Lambda\mathbf j_s}{\partial t} = \frac{\partial \dfrac{m}{e^2 n_s}\mathbf j_s}{\partial t} = \mathbf E, \]

since

\[ \mathbf E=-\frac{1}{c}\frac{\partial \mathbf A}{\partial t} \]

(the electric field described by the scalar potential \(\varphi\) is assumed absent; the generalization to the case when \(\varphi\ne0\) or \(\lambda\ne0\), as is the case if one starts from (3.13), is a somewhat special side question, all the more so since there is no certainty that an additional gradient term is absent in (1.2); see (I, § 27)).

Thus, if the wave function of the electrons in a body corresponds to a concentration \(n\) varying only weakly in space and, under the influence of a magnetic field, despite thermal motion, remains unchanged, then the body under consideration is a superconductor.

\(^*)\) More precisely, \(\operatorname{rot}\mathbf A=\mathbf B\), and in (3.11) the induction should appear. This is not essential in practice, but in superconductors we do not introduce any magnetization and everywhere take \(\mathbf B=\mathbf H\) (for more detail see I), unless speaking of arguments having merely illustrative significance, such as those presented at the beginning of this section. It is also clear that the condition of constancy or slowness of variation of the function \(n(\mathbf r)\) is too stringent. These requirements should in fact be applied not to the exact function \(n(\mathbf r)\), but to some mean value of \(n(\mathbf r)\) over a “physically infinitesimal” volume, i.e. a volume with dimensions considerably larger than atomic dimensions and, at the same time, substantially smaller than \(\delta\sim10^{-5}\) (what has been said follows from the fact that in (1.1) there occur, in fact, the macroscopic quantities \(\mathbf j_s\) and \(\mathbf H\), obtained by averaging over a physically infinitesimal volume).

Solid bodies in which there is a considerable number of “collectivized” mobile electrons and, consequently, for these electrons \(n(\mathbf r)\simeq \mathrm{const}\), are precisely metals, and thus the connection of superconductivity with the metallic state is obvious*).

But in a metal in the normal state, the invariance (“rigidity”) of the wave function does not occur. On the contrary, to a good approximation one may even assume that the diamagnetic current in a metal, associated with the conduction electrons, is equal to zero.

In the classical theory, as is known, this latter assertion in the case of thermodynamic equilibrium is quite rigorous. If, however, one relies on quantum theory, then a certain diamagnetic effect for conduction electrons in a metal remains, as was shown in 1930 by Landau \(^{25}\). The corresponding diamagnetic effect is very small and, in the case of completely free electrons, is completely outweighed by a paramagnetic effect three times larger, connected with the electron spin. The diamagnetic part of the susceptibility of the free electrons forming a degenerate gas is equal to

\[ \chi_{\mathrm{cv}}=-\frac{n}{2E_0}\left(\frac{e\hbar}{2mc}\right)^2 =-\left(\frac{\pi}{3}\right)^{2/3}\frac{e^2 n^{1/3}}{4\pi^2 mc^2}, \tag{3,14} \]

where \(n\) is the electron concentration and

\[ E_0=\left(\frac{3}{\pi}\right)^{2/3}\frac{\pi^2\hbar^2}{2m}\,n^{2/3} \]

is the energy at the Fermi surface.

*) From this, however, it still does not follow that a superconductor must necessarily possess all the properties of metals. Indeed, a conceivable body that behaves in a magnetic field like known superconductors (the equations (1,1)—(3,11) are satisfied) but is not a metal (conductor) in the usual sense of the word, i.e. does not carry current when it is included in an ordinary electric circuit.

In other words, such a “dielectric superconductor” would differ from an ordinary superconductor only by the absence of the possibility of passage of a superconducting current into the normal current flowing in a normal conductor bordering the superconductor. To a known extent, the whole distinction between superconductors of both types, if one does not speak of their behavior outside the superconducting phase, is thus reduced to surface or, more precisely, contact phenomena.

There are no indications of the existence of “dielectric superconductors.” Such a possibility also seems theoretically unlikely. But in view of the uncertainties that exist in the region of very low temperatures, one should nevertheless not forget here about new possibilities and, in connection with this, about the non-equivalence of methods of detecting superconductivity by a change of resistance and by a change of the magnetic flux through the specimen.

Incidentally, we note that the question of contact phenomena in superconductors and, first of all, the question of the transition of a superconducting current into a normal one in a mixed electric circuit appears to us very interesting and still entirely unclear.

Taking \(n\sim 10^{23}\) and \(E_0\sim 10\ \text{eV}\sim 10^{-11}\), we obtain \(\chi\sim -10^{-6}\). If the electrons are regarded not as free, but as moving in the periodic field of the lattice, then, under known conditions, formula (3.14) is suitable for \(\chi\), but with the replacement of the electron mass \(m\) by the effective mass \(m_{\mathrm{eff}}\); moreover, this replacement of \(m\) by \(m_{\mathrm{eff}}\) must be made in the first expression (3.14), since in passing to bound electrons the ratio \(\dfrac{n}{E_0}\) should naturally be considered approximately unchanged. Thus, for electrons in the lattice field, the part of the susceptibility associated with the diamagnetism of the electrons is

\[ \chi \sim \left(\frac{m}{m_{\mathrm{eff}}}\right)^2 \chi_{\mathrm{free}} \]

(see 6, 7, 27); experimentally, for metals \(|\chi|\sim 10^{-5} — 10^{-6}\) (for Bi \(\chi=-10.6\cdot 10^{-6}\), for Pt \(\chi=21\cdot 10^{-6}\); for most other metals \(|\chi|\) is smaller).

The smallness of the diamagnetic effect is connected with the appreciable change of the wave functions of the electrons in a magnetic field, as a result of which the first term in (3.5) (the term containing \(\nabla\Psi\) and \(\nabla\Psi^*\)) is not equal to zero and approximately compensates the second “diamagnetic term,” proportional to \(\mathbf A\). In order for this compensation not to occur, it is necessary, as was said, that the wave function be “rigid.” In an atom such “rigidity” indeed takes place and is used in obtaining expression (3.1) for the susceptibility, as we shall now show.

The energy operator for an atom in the field \(\mathbf H\) has the form:

\[ \mathcal H=\mathcal H_0-\frac{e}{2mc}\mathbf{HL}+\frac{e^2}{8mc^2}[\mathbf{Hr}]^2,\quad \mathbf L=[\mathbf{rp}]=-i\hbar[\mathbf r\nabla], \tag{3.15} \]

where \(\mathcal H_0\) is the energy operator in the absence of the field; spin is not taken into account; \(\mathbf r\) is the radius vector of the electron, which for simplicity is taken to be the only one; and it is set that \(\mathbf A=\dfrac12[\mathbf{Hr}]\). If the discussion concerns an \(S\)-state, then the change in energy is equal to

\[ \Delta E=\frac{e^2}{8mc^2}\overline{[\mathbf{Hr}]^2} =\frac{e^3}{12mc^2}H^2\overline{r^2} \]

(the bar denotes quantum-mechanical averaging over the unperturbed state), and, since

\[ \chi_1=-\frac{\partial^2\Delta E}{\partial H^2}, \]

for \(\chi_1\) we obtain formula (3.1). In a state with nonzero orbital angular momentum, the change in energy contains a principal term proportional to the first power of the field strength \(H\). But in both cases, as is easy to see, the change of the wave function is proportional to \(H^2\) and is very small, at least as long as the change of energy under the influence of the field \(H\) is much smaller than the distance between the given level of the system and the nearest other level. Thus it is precisely the “rigidity” of the atomic wave function that leads to the usual expression for \(\chi\). The change of the \(\Psi\)-functions proportional to \(H^2\) in

in the ratio \(\chi\) gives an effect of higher order of smallness and, in any case, does not at all change the expression for \(\chi\) in a sufficiently weak field.

The same behavior of the \(\Psi\)-function, corresponding to some part of the conduction electrons in the metal, ensures the validity of equations (1.1)—(3.11). At the same time, the requirement that the change of the \(\Psi\)-function be proportional to \(H^2\) is, generally speaking, excessively strong. Indeed, even if the change of the \(\Psi\)-function is proportional to \(H\), but is sufficiently small, then the terms containing \(\nabla\Psi\) and \(\nabla\Psi^*\) in the expression for the current (3.5) will be proportional to \(\mathbf H\) (or, more precisely, \(\mathbf A\)), but will not be able to compensate the diamagnetic term \(-\dfrac{e^2}{mc}\Psi^*\Psi\mathbf A\). Thus, the condition of “rigidity” of the wave function of the “superconducting electrons” need not be understood in any absolute sense: in essence it means only that the wave function changes sufficiently little*) in a magnetic field \(H \ll H_k\). In a strong field (\(H\sim H_k\)) the wave function may already change noticeably, since equations (1.1)—(3.11) are not valid in this case (see \(^{26}\) and I, §§ 3, 4).

The connection between the requirement of “rigidity” of the electron system and equation (1.1) is also clear from the considerations set forth in I, § 2, based on Larmor’s theorem.

The “rigidity” of the wave function in superconductors may be interpreted in the spirit of an indication of “condensation of electrons in momentum space.” Indeed, the invariance of the \(\Psi\)-function means that in the state under consideration the mean momentum of the electrons remains unchanged and, in the case (3.9), is equal to zero. On the contrary, in the case of free electrons their momentum changes when a field is switched on (the momentum is proportional to the first term in (3.5), containing \(\Psi\nabla\Psi^*\) and \(\nabla\Psi^*\)). Thus, one may say of a superconductor that in it the electrons are “condensed” in a state with some average momentum identical throughout the body (in a simply connected specimen this momentum is always equal to zero), and moreover an external field smaller than the critical one cannot remove them from this state. The constancy of the mean momentum throughout the whole superconductor also permits one to speak of the presence—

*) If the change of \(\Psi\) is proportional to \(H^2\), then the first terms in the expression for the current (3.5) in a weak field are altogether inessential and one may directly base oneself on formula (3.9). In the case where the change of \(\Psi\) is proportional to \(H\), calculation of the current requires knowledge of the form of the perturbed \(\Psi\)-function, and one may only suppose that, under the assumption of isotropy of the metal, (3.5) reduces to (3.9) with some function \(n(\mathbf r)\) different from (3.10). If this new function \(n(\mathbf r)\) depends only weakly on \(\mathbf r\), then, as before, (3.9) yields (3.11).

Below, unless otherwise stated, the “rigidity” of the wave function is understood in the sense that its change in a magnetic field may generally be neglected.

of “long-range order” in the spatial distribution of the mean momentum of superconducting (i.e., participating in superconductivity) electrons. Further, by virtue of the diamagnetic character of the superconducting current and the quantum nature of diamagnetism, one may say that superconductivity is a quantum phenomenon on macroscopic scales.

It seems to us that all such talk of “condensation in momentum space,” “long-range order,” and a “macroscopic quantum phenomenon,” rather often encountered in the literature (see, for example, \(^{14,16}\)), adds nothing to what has already been said above regarding the connection of superconductivity with diamagnetism and regarding the quantum-mechanical meaning of the equation basic to the theory of superconductivity, (1,1), (3,11).*)

The principal value of the above derivation of equations (1,1), (3,11), proceeding from the expression for the current density (3,5), consists in the fact that the problem of the microscopic theory of superconductivity can now be formulated in the language of the \(\Psi\)-function and, to a considerable extent, reduces to finding the conditions under which the \(\Psi\)-function of the conduction electrons is “rigid.”

If one relies on the “gas” (one-electron) model of electrons in a metal, then superconductivity cannot be obtained, since in this case the condition of “rigidity” of the wave functions is plainly not satisfied.**) Therefore, in order to explain superconductivity, it is necessary to abandon the “gas” model or, at any rate, to modify it substantially, taking into account a certain interaction energy which is neglected in the existing electron theory of metals. We shall return to this question again. Here we shall only note that the great “cohesion” of electrons in the superconducting state in comparison with the normal state, where the electrons in a known respect may be likened to free electrons, is already clear simply from the fact that in superconductors the superconducting state is energetically lower than the normal one and, as it were, possesses a binding energy (similar to that possessed by nucleons in the nucleus).

*) What has been said does not apply to the attempt to find the peculiarities of the density matrix that correspond to the superconducting and superfluid properties of a system of particles (see \(^{1}\) § 129 and \(^{26}\)).

**) Attempts have repeatedly been made (for the latest of them see \(^{23}\)) to connect the phenomenon of superconductivity with “anomalous diamagnetism,” which may have a place within the framework of a one-electron model in the case of a sufficiently small effective mass of the electron in the metal \(m_{\mathrm{eff}}\). As already mentioned, under certain assumptions

\[ \chi \sim \left(\frac{m}{m_{\mathrm{eff}}}\right)^2 \chi_{\mathrm{sv}}, \]

where \(\chi_{\mathrm{sv}} \sim 10^{-6}\) is the diamagnetic susceptibility of free electrons. Taking \(m_{\mathrm{eff}} \sim 10^{-3}m\), we obtain the value

\[ \chi \sim -\frac{1}{4\pi}. \]

But in this case de-

The task of the further exposition consists in discussing the conditions under which the “rigidity” of the wave function is ensured and superconductivity must occur. In subsection b), for this purpose, a certain methodological problem will be considered. In subsection c) we shall return to electrons in a metal.

b) Superconductivity of ideal charged Bose and Fermi gases contained in a vessel

To clarify certain points essential for the microscopic theory of superconductivity, we shall now consider a purely methodological problem, namely the question of the behavior in a magnetic field of ideal charged Bose and Fermi gases contained in a vessel of finite dimensions. In order to be able, even in the crudest approximation, to neglect Coulomb forces, it is assumed that, in addition to the particles under consideration, there is an equal number of particles with charge of the opposite sign but with a considerably larger mass, as is the case in an electron-ion plasma; the motion of the heavy particles is neglected.

At first we shall assume that the vessel in question is a long circular cylinder of radius \(R\) and length [[unclear: continues on next page]]

This is an error, already discussed at the beginning of this section and connected with the confusion of \(B\) and \(H\). In fact \(\chi' \sim \left(\dfrac{m}{m_{\mathrm{eff}}}\right)^2 \chi_{\mathrm{sc}}\), and, consequently,

\[ \chi = \frac{\chi'}{1 - 4\pi\chi'} \to -\frac{1}{4\pi} \]

only when \(|\chi'| \gg \dfrac{1}{4\pi}\) and \(m_{\mathrm{eff}} \ll 10^3 m\).

At the same time, if one is to liken a superconductor completely to a diamagnet, it is necessary that the equalities \(\chi = -\dfrac{1}{4\pi}\) and \(\mu = 1 + 4\pi\chi = 0\) be satisfied strictly (otherwise \(B \ne 0\) in the bulk of the superconductor; a value \(\mu < 0\) in a static field is inadmissible on thermodynamic grounds, since in this case the minimum of the energy corresponds to an infinitely large field \(H\)). For the equality \(\mu = 0\) to be fulfilled, it is necessary that \(|\chi'| \to \infty\) and \(m_{\mathrm{eff}} \to 0\). But even for \(m_{\mathrm{eff}} \sim 10^{-3}m\) the ordinary theory of diamagnetism is already, generally speaking, inapplicable—it is enough to say that, if the momentum of an electron at the Fermi surface is taken to be the usual one, then for \(m_{\mathrm{eff}} \sim 10^{-3}m\) the velocity of the electron at this surface will be of the order

\[ v_0 = \frac{p_0}{m}\frac{m}{m_{\mathrm{eff}}} \sim 10^8 \frac{m}{m_{\mathrm{eff}}} \sim 10^{11}\ (!) \]

(this remark is due to I. Ya. Pomeranchuk). Therefore, when using the expression \(\chi' \sim \left(\dfrac{m}{m_{\mathrm{eff}}}\right)^2 \chi_{\mathrm{sc}}\) for small values of \(m_{\mathrm{eff}}\), great caution is needed at best.

Finally, it should be noted that, even if one relies on the formula \(\chi' \sim \left(\dfrac{m}{m_{\mathrm{eff}}}\right)^2 \chi_{\mathrm{sc}}\) and considers the question of the value of \(m_{\mathrm{eff}}\) without regard to the theory of superconductivity, in [23] the small value of \(m_{\mathrm{eff}}\) was in fact apparently not convincingly obtained, as was ultimately acknowledged by the author himself (see the last of works [23] and [27]).

of length \(L\). The magnetic field \(H_0\) is parallel to the axis of the cylinder; the concentration of the particles under consideration, with charge \(e\) and mass \(\mu\), is equal to \(n\), and their total number is \(N=\pi R^2Ln\).

In the absence of a field, the Schrödinger equation for the wave function of an individual particle has the form

\[ -\frac{\hbar^2}{2\mu}\Delta\psi = -\frac{\hbar^2}{2\mu} \left\{ \frac{\partial^2}{\partial z^2} + \frac{1}{r^2}\frac{\partial^2}{\partial \varphi^2} + \frac{1}{r}\frac{\partial}{\partial r} \left(r\frac{\partial}{\partial r}\right) \right\}\psi = E\psi , \tag{3.16} \]

where \(z,r,\varphi\) are the introduced cylindrical coordinates and \(E\) is the energy of the particle. We shall assume that on the surface of the vessel \(\psi=0\) (an infinite potential barrier), and put

\[ \psi=\frac{1}{\pi L}f(r)e^{im\varphi}\sin\frac{\pi n_z}{L}z \quad (m=0,\ \pm1,\ \pm2,\ldots;\ n_z=1,2,\ldots), \]

which ensures the single-valuedness of \(\psi\) and its vanishing at \(z=0\) and \(z=L\). Then for \(f\) we obtain the equation

\[ \frac{d^2 f}{dr^2} + \frac{1}{r}\frac{\partial f}{\partial r} + \left( \frac{2\mu}{\hbar^2}E - \frac{m^2}{r^2} - \frac{\pi^2 n_z^2}{L^2} \right)f = 0, \]

whose solution has the form \(f_m=CJ_m(\lambda r)\), where

\[ \lambda^2=\frac{2\mu E}{\hbar^2}-\frac{\pi^2 n_z^2}{L^2}, \]

\(C\) is a constant, and \(J_m=(-1)^mJ_{-m}\) is the Bessel function of order \(m\). The condition \(f(R)=0\), i.e. the condition that \(\psi\) vanish on the walls of the cylinder, means that the quantity \(\lambda R\) must be a zero of the Bessel function, whence the particle energy is determined:

\[ E_{mn_zk}(0) = \frac{\hbar^2}{2\mu} \left( \frac{\beta_{mk}^2}{R^2} + \frac{\pi^2 n_z^2}{L^2} \right), \tag{3.17} \]

where \(\beta_{mk}\) is the \(k\)-th zero of the function \(J_m(x)\). We note that \(\beta_{01}=2.4048\), \(\beta_{02}=5.5201\), \(\beta_{03}=8.6537\), \(\beta_{11}=3.8317\), \(\beta_{12}=7.0156\), \(\beta_{13}=10.1735\), etc.

In the presence of a magnetic field described by the vector potential \(\mathbf A\), the wave equation is as follows:

\[ \frac{1}{2\mu} \left( -i\hbar\nabla-\frac{e}{c}\mathbf A \right)^2 = \left\{ -\frac{\hbar^2}{2\mu} \left[ \frac{\partial^2}{\partial z^2} + \frac{1}{r^2}\frac{\partial^2}{\partial\varphi^2} + \frac{1}{r}\frac{\partial}{\partial r} \left(r\frac{\partial}{\partial r}\right) \right] + \frac{ie\hbar A_\varphi}{\mu cr}\frac{\partial}{\partial\varphi} + \frac{e^2}{2\mu c^2}A_\varphi^2 \right\}\psi = E\psi , \tag{3.18} \]

where, owing to the neglect of the edge effect (a long cylinder), the field \(\mathbf H=\operatorname{rot}\mathbf A\) is assumed to be directed along the \(z\)-axis and also to possess cylindrical symmetry (therefore we have:

\[ A_z=A_r=0,\qquad H_z=\operatorname{rot}_z A = \frac{1}{r}\frac{\partial}{\partial r}(rA_\varphi) \]

and

\[ \operatorname{div}\mathbf A = \frac{1}{r}\frac{\partial A_\varphi}{\partial\varphi} = 0 \Bigg). \]

We neglect the interaction of the particle spin with the magnetic field, since from the point of view of what follows it is not essential.

The potential \(\mathbf A\) may be represented as the sum of two terms

\[ \mathbf A=\mathbf A_0+\mathbf A_1,\qquad \mathbf A_0=\frac{1}{2}[\mathbf H_0\mathbf r], \]

\[ \Delta\mathbf A_1=-\frac{4\pi}{c}\mathbf j =-\frac{4\pi}{c}\sum_{\alpha=1}^{N}\mathbf j_\alpha; \]

\[ \mathbf j_\alpha =\frac{ie\hbar}{2\mu}\left(\psi\nabla_\alpha\psi^*-\psi^*\nabla_\alpha\psi\right) -\frac{e^2}{\mu c}\psi^*\psi\,\mathbf A(\mathbf r_\alpha), \tag{3.19} \]

where \(\mathbf A_0\) is the potential corresponding to the homogeneous external field \(\mathbf H_0\), and \(\mathbf A_1\) is the potential of the field associated with the motion of the particles under consideration; the current \(\mathbf j\) in the general case has the form (3.5), but for independent particles it can be written simply in the form (3.19), where \(\psi\) is the wave function of an individual particle. If the concentration of particles \(n\) is sufficiently small, then the field \(H_1=|\operatorname{rot}\mathbf A_1|\ll H_0\); in the first approximation

\[ A_z=A_{0z}=\frac{1}{2}H_0r \]

and equation (3.18) becomes analogous to (3.15), since

\[ -i\hbar\frac{\partial}{\partial\varphi}=L_z \]

is the projection of the momentum on the \(z\)-axis. In this case, when the term proportional to \(A_z^2\) is neglected, the functions \(\psi\) satisfying equation (3.16) remain eigenfunctions of equation (3.18), and

\[ E_{mnzk}(H)=E_{mnzk}(0)+\frac{e\hbar mH_0}{2\mu c}. \tag{3.20} \]

In the second order with respect to \(H_0\), the change in \(\psi\) is already different from zero, i.e. \(\delta\psi\sim H_0^2\).

The situation here is analogous to that which occurs in the case of an atom (see § 3a).

The current \(j_\alpha\) has only the component \(j_\varphi\) and is equal to

\[ j_{\alpha\varphi} =\frac{e\hbar m}{\mu r_\alpha}\psi^*\psi -\frac{e^3}{\mu c}\psi^*\psi A_{0z},\qquad A_{0z}=\frac{1}{2}H_0r. \tag{3.21} \]

The magnetic moment of the cylinder associated with the current \(j\) depends essentially on the field strength \(H_0\) and on the type of statistics obeyed by the particles. In the case of Bose statistics, at temperature \(T=0\), all particles at \(H_0=0\) are on the lowest level

\[ E_{011}(0)=\frac{\hbar^2}{2\mu}\left\{\frac{2.4048}{R^2}+\frac{\pi^2}{L^2}\right\} \]

with moment \(m=0\).

When the field \(H_0\) is switched on, so long as the level \(E_{011}\) remains the lower one, all particles remain on it and the full current is

\[ j=-\frac{e^2 n}{\mu c}A=-\frac{e^2}{2\mu c}H_0 rn,\quad \text{where } n=\sum_{\alpha=1}^{N}\psi^*(\mathbf r_\alpha)\psi(\mathbf r_\alpha) \]

is the particle concentration. Under these conditions the cylinder with the charged gas behaves as a superconductor, since equations (3.9)—(3.11) are valid, with

\[ \delta^2=\frac{\Lambda c^2}{4\pi}=\frac{\mu c^2}{4\pi e^2 n} \]

(the dependence of the concentration \(n\) on \(r\), which is clearly expressed in the ground state of the Bose gas, is here and in what follows neglected for simplicity; it is easy to see that this point has no essential significance).

The conclusion drawn remains valid also in the case where the screening field \(H_1\) cannot be neglected. In this case, as before, up to terms of order \(H_0^2\) the function \(\psi\) does not change and

\[ E_{m n_z k}(H)=E_{m n_z k}(0)+\frac{e\hbar m}{\mu c}\int_0^R f_0^2 A_\varphi\,dr, \tag{3.22} \]

since the function

\[ \psi=\frac{1}{\pi L}f_m(r)e^{im\varphi}\sin \frac{\pi n_z z}{L} \]

is normalized to unity and

\[ \int_0^R f_m^2 r\,dr=1. \]

Expression (3.21) is preserved with the replacement of \(A_{0\varphi}\) by \(A_\varphi\), and in the ground state \(m=0\) still

\[ \mathbf j=-\frac{e^2 n}{\mu c}\mathbf A. \]

So long as \(H_1\ll H_0\), the cylinder may be regarded as uniformly magnetized, and its moment per unit length is equal to \((dv=2\pi r\,dr)\):

\[ M_\ell=\frac{1}{2c}\int rj_\varphi\,dv =\frac{e^2 n R^2}{8\mu c^2}\pi R^2 H_0 =\pi R^2\chi H_0, \tag{3.23} \]

where \(\chi\) is the susceptibility.

Neglecting terms of order \(H_0^2\), the moment of the cylinder will begin to change only when, under the action of the field, the level \(E_{-111}\) nearest to the lower level descends to the lower level \(E_{011}\), whose position does not depend on the field. According to (3.17) and (3.20) this will occur in the field

\[ H_{kp}=\frac{E_{-111}(0)-E_{011}(0)}{\dfrac{e\hbar}{2\mu c}} =\frac{8.9\,\hbar c}{e^2R^2} =\frac{5.85\cdot10^{-7}}{R^2}. \tag{3.24} \]

At large concentration \(n\) the screening field \(H_1\) cannot be neglected even under conditions where the penetration depth of the field

\[ \delta=\sqrt{\frac{\mu c^2}{4\pi e^2 n}}\ll R, \]

we have:

\[ H=H_0 e^{\frac{R-r}{\delta}},\quad A_\varphi=\delta H,\quad E_{mn_zk}(H)-E_{mn_zk}(0)\sim \]

\[ \sim \frac{e\hbar m}{2\mu c}\left(\frac{\delta}{R}\right)^2 H_0, \tag{3.25} \]

since \(f_0^2\sim \dfrac{1}{R^2}\) by virtue of the condition

\[ \int_0^R f_0^2 r\,dr=1. \]

In the case under consideration, the destruction of superconductivity begins in the field

\[ H_{ks}\sim \frac{R^2}{\delta^2}H_{kn}\sim \frac{5\cdot 10^{-7}}{\delta^2}, \tag{3.26} \]

and for \(\delta\sim 10^{-5}\) \((n\sim 5\cdot 10^{21})\), \(H_{ks}\sim 5\cdot 10^3\) oersted.

Thus, if the electrons in a metal formed a Bose gas, the phenomenon of superconductivity, even in fairly strong fields, would occur simply by virtue of the discreteness of the levels in a vessel of finite dimensions. The situation, however, changes substantially for a degenerate Fermi gas, with which it still makes some sense to compare electrons in a metal. The point is that, for energy levels near the Fermi surface, the distance between levels, for values of \(n\) and \(R\) of interest, is quite negligible, as a result of which the fields \(H_{kn}\) and \(H_{ks}\) also prove negligible.

To estimate the average distance between levels at the Fermi boundary, let us take into account that at large momenta the shape of the vessel is of little importance and one may use the known expressions for the number of levels in a cubic vessel

\[ Z(E)\,dE=\frac{2^{1/2}\mu^{3/2}E^{1/2}V\,dE}{8\pi^2\hbar^3}, \]

where \(Z(E)\,dE\) is the number of levels in the energy interval \(dE\), and \(V\) is the volume of the vessel (in I, § 6 the quantity \(Z(E)\) is denoted by \(\dfrac{dN}{dW}\)). Further, at the Fermi boundary the energy

\[ E_0\sim \frac{\pi^2\hbar^2 m_{\max}^2}{2\mu R^2}, \]

where \(R\sim l\) is put, and \(m_{\max}\) is the maximum occurring value of the quantum number \(m\) \((m_{\max}\sim k_{\max}\sim n_{z\max}\sim n^{1/3}R\), where \(n\) is the concentration of particles). The average distance between levels is

\[ \Delta E\sim \frac{1}{Z(E)}\sim \frac{\hbar^2}{\mu R^2 m_{\max}}\sim \frac{\hbar^2}{\mu n^{1/3}R^3}. \tag{3.27} \]

In order of magnitude, as it should be, \(\Delta E\) is the ratio of the degeneracy energy

\[ E_0\sim \frac{\hbar^2 n^{2/3}}{\mu} \]

to the total number of particles in the vessel \(nR^3\).

The value \(E_{111}(0)-E_{011}(0)\) in (3.24) is of order \(\dfrac{\hbar^2}{\mu R^2}\), i.e., greater than (3.27) by \(m_{\max}\) times. Moreover, according to (3.20), the change of the levels at the Fermi boundary with \(m\sim m_{\max}\) is \(m_{\max}\) times greater than the displacement of the level with \(m=1\). Hence it is clear that the values \(H_{kn}\) and \(H_{ks}\) (see (3.24) and (3.26)) in the case of a Fermi gas must differ by a quantity of order \(m_{\max}^2\sim n^{2/3}R^2\sim 10^{14}\) (for \(n\sim 5\cdot 10^{21}\) and \(R\sim 1\)). In this case the field

\[ H_{ks}\sim \frac{5\cdot 10^{-7}}{\delta^2 n^{2/3}R^2}\sim 10^{-10}\ \text{oersted}. \]

However crude the estimate made may be, and even taking into account that the field \(H_{ks}\) corresponds only to the beginning of the destruction of the superconducting (ideally diamagnetic) state, it is quite clear that in the case of a Fermi gas with \(n\sim 10^{22}\) the destruction of this state will be achieved in negligibly small fields.

In a field that provides strong mixing of the levels, the Fermi gas will possess only weak diamagnetism (3.14). The smallness of the effect is then connected with the almost complete compensation of the diamagnetic current

\[ j=-\frac{e^2}{\mu c}\,\psi^*\psi A \]

by the paramagnetic current

\[ \frac{e\hbar n}{\mu r}\,\psi^*\psi \]

(see (3.21)).

In fact, the diamagnetic current gives rise in the cylinder to the moment (3.23), while the moment of the paramagnetic current \(j_\alpha\), associated with the \(\alpha\)-th electron, is, in order of magnitude, equal to

\[ M=\frac{1}{2c}\int j_\alpha r\,d\sigma\sim \frac{e\hbar m}{\mu c}. \]

In the absence of a field, each electron gives rise to the same moment (the moment does not depend on \(H_0!\)), but since electrons with values \(m'=-m\) occur in equal number with electrons having the quantum number \(m\), the total moment of the body is zero. When the field is switched on, levels with \(m<0\) are lowered, while levels with \(m>0\) are raised, and therefore in the state of minimum energy (under the assumption \(T=0\), so that the free energy coincides with the energy) there is an excess paramagnetic moment.* This moment is equal to the sum of the moments of the electrons that have passed to levels with \(m<0\); the number of such electrons is of the order of the number of levels in the energy interval

\[ \frac{e\hbar m_{\max}H_0}{\mu c}, \]

over which, under the action of the field, the levels with \(m_{\max}>0\) and \(m_{\max}<0\) diverge, i.e., it is of order

\[ \Delta n\sim Z(E)\frac{e\hbar m_{\max}H_0}{\mu c} \sim \frac{e\hbar m_{\max}H_0}{\mu c}\frac{nR^3}{E_0} \sim \frac{e\hbar m_{\max}}{\mu c}\cdot \frac{\mu R^3 m_{\max}}{\hbar^2}. \tag{3.27'} \]

\[ \text{*} \]

In this section we assume that \(e>0\); in the opposite case, i.e., for \(e<0\), the role of the levels with \(m>0\) is played by levels with \(m<0\), and conversely.

Thus, the total paramagnetic moment of the cylinder, referred to unit length, is, in order of magnitude, equal to

\[ M_{\mathrm{p}} \sim \frac{e\hbar m_{\max}^{2}}{\mu c}\,\frac{\Delta n}{R} \sim -\,\frac{e^{2}m_{\max}^{3}RH_{0}}{\mu c^{2}} \sim \frac{e^{2}nR^{2}}{\mu c^{2}}\cdot \pi R^{2}H_{0}, \tag{3.28} \]

since in our crude approximation it was assumed (in estimating \(m_{\max}\)) that \(R \sim L\), and, consequently, the moment per unit length is equal to the total moment divided by \(R\) (this also explains the appearance of \(R\) in the denominator in the first expression (3.28)). The paramagnetic moment obtained is, in absolute value, of the same order as the diamagnetic moment (3.23). A more accurate calculation (see \(^{28}\)) shows that, if the discreteness of the levels is not taken into account, the moment \(M_{\mathrm{p}}\) exactly compensates the moment \(M_{\mathrm{d}}\), as it should in the classical theory, to which one passes when the distance between levels is neglected. If the discreteness of the levels is taken into account, then the terms \(M_{\mathrm{p}}\) and \(M_{\mathrm{d}}\) do not compensate each other exactly, but give a small residual diamagnetic effect (spin is not taken into account!). The corresponding value of the susceptibility (3.14), in order of magnitude, is then obtained from the same considerations as above, but assuming that the excess energy and moment are equal to \(\dfrac{e\hbar H_{0}}{\mu c}\) and \(\dfrac{e\hbar}{\mu c}\), and not to \(\dfrac{e\hbar m_{\max}H_{0}}{\mu c}\) and \(\dfrac{e\hbar m_{\max}}{\mu c}\) (such a substitution has a quite clear meaning, since it corresponds to the residual effect associated with a change of the quantum number \(m\) by the smallest amount—by unity).

The model of an ideal gas considered illustrates the general arguments set forth in § 3 a). As long as the \(\Psi\)-function of the system of charged particles may be regarded as unchanged, a diamagnetic current appears which obeys the equation for the superconducting current

\[ \operatorname{rot}\Lambda\mathbf{j}_{s}=-\frac{1}{c}\mathbf{H} \]

(see (3.11)). Under the influence of the magnetic field, a Zeeman splitting of degenerate levels occurs, and when the levels begin to overlap, the “superconducting” state is gradually destroyed*). The case of the circular cylinder discussed above is not only especially simple, but is also distinguished by the fact that the eigenfunctions in this case, to first approximation, do not change when the field is switched on. In a vessel without axial symmetry (for example, in a vessel with a rectangular cross section), the \(\psi\)-functions change already in first order with respect to \(H_{0}\). However, qualitatively this circumstance cannot be essential, and until overlap occurs

*) We note that in the example analyzed, when terms in \(H_{0}^{2}\) are neglected, the \(\psi\)-functions of the individual particles do not change. But the \(\Psi\)-function of the whole system changes upon overlap of the levels by virtue of the redistribution of particles among states.

levels, the diamagnetic current will not be compensated by the paramagnetic effect.

The critical magnetic field is determined by the distance between levels, and for a degenerate Fermi gas with a concentration corresponding to the concentration of electrons in a metal, this field is negligibly small. Hence it is clear that superconductivity cannot be obtained in the model of free electrons; moreover, the usual allowance for the influence of the crystal lattice (i.e., allowance for the periodic potential field) cannot change anything here. For superconductivity to arise in a model based on the approximation of independent electrons, some kind of gap must appear in the energy spectrum at the Fermi boundary. Suppose, for example, that the energy spectrum of electrons in a cylindrical vessel is such that at the Fermi boundary, at \(E=E_0\), there is a gap with width equal to

\[ \Delta \sim kT_k \sim 10^{-16} \div 10^{-15}, \tag{3.29} \]

since the critical temperature \(T_k \sim 1 \div 10^2\), and if some gap exists and is connected with the phenomenon of superconductivity, then its width must, in order of magnitude, be determined precisely by the value \(kT_k\).

For \(E \geq E_0+\Delta\) we shall assume the spectrum to be Fermi-like, but for \(E \leq E_0\) we must assume that the magnetic field has no noticeable influence on the state of the system as a whole*). The destruction of superconductivity in such a scheme, which without further refinement can have only the crudest illustrative significance, begins in the field in which the Zeeman displacement of the level with \(E(0)=E_0+\Delta\) reaches the value \(\Delta\), i.e., without allowing for screening, in the field

\[ H_{kn} \sim \frac{\Delta}{\dfrac{e\hbar m_{\max}}{2\mu c}} \sim \frac{\Delta}{\dfrac{e\hbar}{2\mu c} R n^{1/3}} \sim \frac{10^4 T_k}{n^{1/3}R} \sim \frac{10^{-2}}{R}, \tag{3.30} \]

and, allowing for screening, in the field

\[ H_{ks} \sim H_{kn}\frac{R^2}{\delta^2} \sim \frac{\Delta R}{\dfrac{e\hbar n^{1/3}\delta^2}{2\mu c}} \sim \frac{10^4 T_k R}{n^{1/3}\delta^2} \sim 10^8 R, \tag{3.31} \]

* In other words, we assume that expressions (3.20) and (3.22) do not hold for \(E \leq E_0\), since otherwise diamagnetism would disappear because of the redistribution of electrons over levels which, in the absence of a field, lay below \(E_0\). Of course, such assumptions are completely inconsistent, but this applies to the same extent to the very introduction of a gap into the spectrum of free particles. In \(^{28,29,30}\) the assumption of the absence, for \(E \leq E_0\), of redistribution of electrons over levels is connected with the presence of a certain “energy of reversal of velocity” (Bahnumkehrenergie).

where \(m_{\max}\sim n^{1/3}R\), and in passing to the numerical estimate it was taken that \(n\sim 10^{21}\), \(\delta\sim 10^{-5}\), and \(T_k\sim 10^\circ\).

If superconductivity has been destroyed and the field is decreasing, i.e. if one starts from the normal phase, then the corresponding state can exist as long as the field \(H_0>H_{kn}\). If, however, one starts from the superconducting state, one arrives at the normal state in a field of order \(H_{ks}\). But phase equilibrium can occur only at a definite field \(H_k\), such that the region \(H_{kn}<H_0<H_k\) corresponds to a supercooled normal phase, while the region \(H_{ks}>H_0>H_k\) corresponds to a superheated superconducting phase. The field \(H_k\), as well as the exact values of \(H_{kn}\) and \(H_{ks}\), can be found only as a result of more detailed calculations and of finding the free energy of the system in a field. In view of the inconsistency of the adopted model, such calculations would hardly have any value, and therefore we restrict ourselves to the remark that, applying by analogy with the van der Waals equation the so-called “equal-area rule,” which determines the pressure at which an equilibrium liquid and gas coexist, one may conclude\(^{28-30}\) that the equilibrium field \(H_k\) is approximately equal to the geometric mean of \(H_{kn}\) and \(H_{ks}\), i.e.

\[ H_k\sim \sqrt{H_{kn}H_{ks}}\sim \frac{\Delta}{\dfrac{e\hbar}{2\mu c}\cdot n^{1/30}} \sim kT_k\sqrt{\frac{4\pi n}{E_0}}\sim 10^2\div 10^3, \tag{3.32} \]

where it has been taken into account that

\[ \delta=\sqrt{\frac{\mu c^2}{4\pi e^2 n}} \quad\text{and}\quad E_0\sim \frac{\hbar^2 n^{2/3}}{2\mu}. \]

The considerations presented can lay claim to only one thing: they show that the introduction, at the Fermi boundary, of a gap of a certain quite reasonable width (3.29) can lead to the preservation of superconductivity up to fields \(H_k\sim 10^2\div 10^3\) corresponding to experiment. The task of the theory is to refine substantially the model adopted and to substantiate it on the basis of considering the interaction of the electrons with one another and with the lattice of the metal.

c) The role of lattice vibrations and of a high dielectric constant. Some critical remarks

Attempts to find the spectrum of electrons in a metal and its connection with the phenomenon of superconductivity must be made not only with allowance for the remarks already made, but also taking into account two further essential points following from the experimental data.

First, in superconductors an isotope effect has recently been established (see I, § 1 and ^31,32), and the relation

\[ M^{1/2}T_{\mathrm{k}}=\mathrm{const},\qquad dT_{\mathrm{k}}=-\frac{T_{\mathrm{k}}}{2M}\,dM, \tag{3,33} \]

is observed, where \(M\) is the mass of the atom and \(T_{\mathrm{k}}\) is the critical temperature.

The existence of such a strong isotope effect means that superconductivity cannot be understood as a purely electronic phenomenon, as all grounds for hoping earlier had seemed to indicate (the lattice does not change appreciably at the superconducting transition, the influence of pressure and deformations on superconductivity is comparatively small, etc.). If superconductivity were a purely electronic phenomenon (more precisely, if its appearance were possible even with immobile nuclei), then the isotope effect would also exist, but it would be considerably smaller than the observed one. Thus, for atomic terms the change in energy associated with the finiteness of the nuclear mass, in comparison with the case of immobile nuclei, is of the order \(\Delta E\sim -\dfrac{m}{M}E\), and therefore one might expect that the change in the critical temperature \(dT_{\mathrm{k}}\) when the nuclear mass changes by \(dM\) would be of the order (taking \(E\sim kT_{\mathrm{k}}\), \(m\) the electron mass):

\[ dT_{\mathrm{k}}\sim -\frac{m}{M}\frac{T_{\mathrm{k}}}{M}\,dM \sim -(10^{-4}\div 10^{-5})\frac{T_{\mathrm{k}}}{M}\,dM. \tag{3,34} \]

Since the observed effect is several orders of magnitude greater than the value (3,34), it is clear that lattice vibrations, and first of all its zero-point vibrations, are essential for the very appearance of superconductivity. For example, if superconductivity is associated with a gap in the energy spectrum of the electrons, then the width of this gap must depend strongly on the atomic mass \(M\) (assuming the gap width \(\Delta\sim kT_{\mathrm{k}}\), we obtain that \(\Delta\sim \dfrac{\mathrm{const}}{M^{1/2}}\)).

The second point, which also appears to be very important, is connected with the fact that in the superconducting state the dielectric constant must apparently reach enormous values \(\varepsilon_0\sim 10^8\div 10^{10}\) (see I, § 6)*). This circumstance has not yet finally

* It should be noted that an exceptionally strong increase in the dielectric constant \(\varepsilon_0\) may be characteristic not only of superconductors, but also occurs when the resistance of nonsuperconducting metals increases as \(T\to 0\) (see, for example, ^4). Indeed, if the conductivity in this case falls owing to a decrease in the number of free electrons, then these electrons pass into a bound state, and the characteristic frequencies are \(\omega_0\sim kT_0\), where \(T_0\) is the corresponding temperature, and, as in the case of superconductivity, the value of \(\varepsilon_0\) must apparently be very large (see I, § 6).

experimentally proven. However, quite recently\(^{46}\) data were obtained which fully confirm the assumption about the indicated value of \(\varepsilon_0\) in a superconductor (according to\(^{46}\), for tin at \(T \sim 2^\circ\), \(\varepsilon_0 \sim 5 \cdot 10^9\), and as \(T \to T_k\), \(\varepsilon_0 \to 0\); an indication of the value \(\varepsilon_0 \sim 10^9\) is also found in work\(^{47}\), with which the author, unfortunately, could not become acquainted in detail).

Since the appearance of superconductivity is connected with some effect to which an energy \(\Delta \sim kT_k \sim 10^{-16} \div 10^{-15}\) corresponds, it is necessary to point out a possible source of such an energy, not taken into account in the usual “gas” model of electrons in a metal.

The energy of the Coulomb interaction between electrons in the gas model is in fact ignored (see § 1), and in order of magnitude is equal to (all energies, as well as the quantity \(\Delta\), are referred to one electron)

\[ E_k \sim \frac{e^2}{r} \sim e^2 n^{1/3} \sim 10^{-12} \div 10^{-11} \sim 1 \div 10\ \text{eV}. \tag{3.35} \]

In view of the fact that \(E_k \sim E_0 \gg \Delta \sim kT_k\), it is often considered that the Coulomb energy of the interelectronic interaction is irrelevant to superconductivity. Such an opinion, however, is by no means convincing. First, it may always turn out that only some small part of the energy \(E_k\) is effective, while the greater part of the energy does not manifest itself, for example, by virtue of the Pauli principle. From this point of view, only the Coulomb interaction of electrons in the smearing zone of the Fermi distribution is essential, i.e. the interaction of \(n \dfrac{kT_k}{E_0} \sim 10^{-5} \div 10^{-4}\, n\) electrons, whence

\[ E'_k \sim e^3 \left( n \frac{kT_k}{E_0} \right)^{1/3} \sim 10^{-14} \div 10^{-13}. \]

Second, the interaction energy may drop sharply because of screening, which in a superconductor, owing to the large value of \(\varepsilon_0\), will probably be especially significant (it is enough to say that even for \(n \sim 10^{23}\) and \(\varepsilon_0 \sim 10^8\),

\[ E''_k \sim \frac{e^2}{\varepsilon_0 r} \sim \frac{e^2 n^{1/3}}{\varepsilon_0} \sim 10^{-20}; \]

for rapidly moving electrons the value of \(\varepsilon_0\) will be less than the static one, and it may turn out that \(E''_k \sim kT_k \sim 10^{-16} \div 10^{-15}\), for which it is necessary that \(\varepsilon_0 \sim 10^3 \div 10^4\).

The energy of the magnetic interaction between electrons is of the order (\(v\) is the electron velocity)

\[ E_m \sim \frac{e^2 v^2}{r c^2} \sim e^2 n^{1/3} \frac{E_0}{m c^2} \sim \frac{e^2 \hbar^2 n}{m^2 c^2} \sim E_k \cdot \frac{E_0}{m c^2} \sim 10^{-17} \div 10^{-16}, \tag{3.36} \]

since \(E_0 \sim 1 \div 10\ \text{eV}\) and \(mc^2 \simeq 5 \cdot 10^5\ \text{eV}\).

Accounting for the energy of magnetic interaction appears, at first glance, especially attractive, since here without further ado \(E_m \sim kT_k\), and, most importantly, it is precisely the magnetic interaction that can naturally be “opposed” to the influence of a magnetic field, which destroys superconductivity. Indeed, from § 3 b) it is clear that the destruction of superconductivity or, more precisely, its failure to appear, is connected with a redistribution of electrons among levels when a magnetic field is applied. This redistribution amounts to changing the direction of motion of some electrons to the opposite one, so that instead of a state with magnetic quantum number \(m<0\), the electron passes into a state with \(m>0\). For superconductivity to appear in such a scheme it is sufficient that there be a certain energy \(E\sim kT_k\) which prevents the electron from changing the direction of its velocity (see \(^{28,30}\); in this respect the superconductor turns out to be related in a certain sense to an orbital antiferromagnet). The energy of magnetic interaction depends precisely on the direction of the velocity and therefore attracts special attention. However, it is easy to see that the usual (classical) magnetic interaction cannot lead to the required result, since antiparallel currents repel one another, i.e. magnetic interaction favors parallelism of the velocities of all electrons instead of preventing the reversal of the velocities of some electrons associated with the establishment of such parallelism. The exchange magnetic interaction between electrons with parallel spins leads to attraction of antiparallel currents, but in magnitude this interaction for free electrons is considerably smaller than the ordinary magnetic one\(^{29}\). If, however, one proceeds not from the model of free electrons, then the whole question is shifted onto an entirely different plane (this will be discussed below), and any advantages of taking into account specifically the magnetic interaction are no longer at all clear. Moreover, in view of the necessity noted earlier of taking the influence of the lattice into account from the very beginning, it is clear that both the magnetic and the Coulomb interactions between electrons cannot by themselves ensure the appearance of superconductivity.

In the “gas” electron model of a metal, the interaction of an electron with lattice vibrations is taken into account only in calculating the conductivity and other kinetic coefficients, and is reduced to considering the processes of emission, absorption, and scattering of lattice phonons. However, the presence of vibrations and, in essence, the finiteness of the mass of the atom (nucleus) \(M\) also leads to the fact that the energy levels of the electrons must be shifted in comparison with the levels in the absence of vibrations \((M\to\infty)\) by an amount \(|\Delta E|\sim \dfrac{m}{M}E\). Therefore one may say that the interaction with the lattice leads to the appearance for the electron (with energy

CURRENT STATE OF THE THEORY OF SUPERCONDUCTIVITY

at the Fermi surface \(E_0 \sim 10^{-12} \div 10^{-11}\)) of additional energy

\[ E_p \sim \frac{m}{M} E_0 \sim m u^2 \sim 10^{-17} \div 10^{-16} \sim kT_k, \tag{3.37} \]

where \(u \sim \sqrt{\dfrac{E_0}{M}} \sim 10^5\) is the speed of sound.

Introducing the interaction (3.37) into the discussion is important and interesting also because it immediately leads to a relation of the type (3.33*).

As Fröhlich\({}^{33}\) pointed out, the energy \(E_p\) may be interpreted as the electron’s own energy associated with the virtual emission and absorption of phonons. Taking into account the interaction of electrons with the lattice as applied to a free (in the initial approximation) completely degenerate Fermi gas, Fröhlich came to the conclusion that, with a sufficiently strong binding energy with the lattice—such as can occur in superconducting metals—the Fermi surface changes substantially. Thus, if for

* According to (3.37) \(M T_k = \mathrm{const}\), whereas experimentally \(M^{1/2}T_k = \mathrm{const}\) (see (3.33)); in both cases, however, \(dT_k \sim \dfrac{T_k}{M}\,dM\). It is precisely this last point that is especially significant (see (3.34)). As for the more detailed form of the relation between \(M\) and \(T_k\), the order-of-magnitude estimates made are insufficient to establish it. This is connected, in particular, with the fact that all energy considerations refer first of all to the case \(T=0\). Therefore it is more correct to compare not the interaction energy per electron with \(kT_k\), but the interaction energy per unit volume with

\[ \frac{H_{0k}^2}{8\pi}, \]

where \(H_{0k}\sim 5\cdot 10^2\) is the critical magnetic field at \(T=0\).

the change in the energy of the electrons associated with the finiteness of the atomic mass \(M\) is of order

\[ E'_p \sim \frac{m}{M}E_0 n \sim 10^5 \div 10^6 \left(n\sim 10^{22},\ \frac{m}{M}E_0 \sim 10^{-17}\div 10^{-16}\right), \]

i.e., somewhat larger than

\[ \frac{H_{0k}^2}{8\pi}. \]

However, in estimating orders of magnitude at this point one can hardly see any difficulty, since interaction with lattice vibrations, of course, also occurs in the normal phase, and the quantity

\[ \frac{H_{0k}^2}{8\pi} \]

(the energy difference of the two phases) must be compared with the difference of the energies \(E'_p\) in the two phases. Assuming that \(H_{0k}^2\) is proportional to \(E'_p\), we have \(H_{0k}\sim M^{-1/2}\), which in general agrees with experiment\({}^{32}\). In the first approximation \(H_{0k}\sim T_k\), whence \(M^{1/2}T_k=\mathrm{const}\).

At the same time it must be noted that, while the relation \(M^{1/2}T_k=\mathrm{const}\) has been established with great accuracy, the same cannot yet be said of the relation \(M^{1/2}H_{0k}=\mathrm{const}\)\({}^{33}\).

for free electrons the Fermi surface is a sphere with radius \(p_0=\sqrt{2mE_0}\), then in the case of a sufficiently strong coupling with the lattice, the equilibrium distribution at \(T=0\) will be the distribution shown in Fig. 1. The electrons fill a sphere of radius \(p'_0\), then for \(p'_0<p<p''_0\) there are no electrons and, finally, the region \(p''_0<p<p'''_0\) is again filled with electrons. The width of the gap

\[ \Delta'=\frac{p_0^{\prime\prime 2}}{2m}-\frac{p_0^{\prime 2}}{2m}<E_p. \]

The state for which the Fermi surface has such a form as in Fig. 1 was identified by Fröhlich with the superconducting state.

Fig. 1.

Further study of the problem posed showed \(^{34-36,\,27}\) that Fröhlich’s assertion about the possibility of the appearance of a multiply connected surface of Fig. 1 is completely unproved. The point is that the use of perturbation theory, which formed the basis of Fröhlich’s calculations, is unlawful in the presence of a strong interaction of the electrons with the lattice and therefore there can be no question of any proof of the appearance of a gap \(\Delta'\) by the method of perturbation theory. This conclusion, substantiated in \(^{34-36,\,27}\), is also clear from the following simple considerations. The appearance of a gap \(\Delta'\) means that near the Fermi surface the perturbation is not weak but, on the contrary, very strong, since it changes the distribution function by an amount of the order of the initial values of this function. At the same time perturbation theory is applicable only so long as the perturbation is small, and thus it is clear that by such a method one cannot obtain any reliable “splitting” of the Fermi surface.

From the point of view of the theory of superconductivity, another point is still more important: even if one were to admit for a moment that Fröhlich’s calculation is correct and that the Fermi surface really assumes the form of Fig. 1, it by no means follows from this that superconductivity is to be understood. In other words, the identification made by Fröhlich of the state with a gap (i.e. the state with a surface of the type shown in Fig. 1) with the superconducting state is absolutely unjustified and, moreover, erroneous. We have already said that for superconductivity to arise, apparently, the appearance of some gap in the energy spectrum of the electrons is necessary. But this does not at all mean that any gap will lead to superconductivity. In the case of Fig. 1, for \(p>p''_0\) an electron may have any energy and, when a magnetic field is switched on, if the corresponding perturbation is regarded as independent of the perturbation due to the lattice (in the first approximation this is legitimate), there will occur, seem-

there would be a redistribution of the electrons over the levels, and the usual Landau diamagnetism would result. How the interaction with the lattice, which has already played its role (it presumably caused the appearance of the gap in Fig. 1), can prevent the redistribution of the electrons and the compensation of the diamagnetic current

\[ \mathbf{j}=-\frac{e^2 n}{mc}\mathbf{A} \]

by a paramagnetic current remains unclear; no reasons for this are evident. In Fröhlich’s work[^33] no attempt is even made to relate the state with a gap to the diamagnetic current; instead, reliance is placed on the appearance, in his model, of certain spontaneous currents. This part of the work[^33] no longer withstands any criticism, and there is no point in dwelling on it, since the author himself rejects it in a subsequent communication[^37]. There he bases himself on the diamagnetic hypothesis and tries to show that, when the interaction with the lattice is taken into account, the redistribution of electrons among states in a magnetic field cannot compensate the diamagnetic current, as a result of which superconductivity sets in. The corresponding consideration, however, has no demonstrative force, and in it, in essence, what is to be proved is assumed. We shall not dwell on this question in greater detail, especially since Fröhlich’s concrete error has been identified in a special study by Schafroth[^38], which showed in a very general way that, when the interaction of the electrons with the lattice (the phonon field) is taken into account within perturbation theory, superconductivity cannot be obtained1. This conclusion is in complete agreement with everything said above: superconductivity requires a “rigidity” of the wave function, which may be provided by the appearance of a certain gap, but not by just any gap, and in particular not by the gap of Fig. 1; see also § 4 b) on this point. Within perturbation theory, applied to a free electron gas, one cannot obtain such a result, corresponding to a strong (and not weak) change in the properties of the system near the Fermi surface. Thus, starting from the gas model and somehow “correcting” it by taking into account various weak perturbations, one cannot obtain superconductivity. It follows from this, in agreement with what was said already in § 1, that the development of a microtheory of superconductivity is connected with a radical change in the adopted gas model of the electrons in a metal, or even

with the adoption of some other model of the metal*). But at the present time there have as yet been no successes along the path of a reasonably rigorous and consistent treatment of the many-body problem (for example, electrons in a metal) in the aspect of interest for the theory of superconductivity. Therefore, at least at the first stage in the development of a microtheory of superconductivity, attention should be attracted by the analysis even of very crude models of a metal (a plasma, a charged liquid coupled to a charged elastic continuum, etc.). Moreover, even the finding of the energy spectrum of electrons in a metal would not be a major success, but rather a sufficiently complete clarification of the question of what spectrum is sufficient for explaining the properties of superconductors and the mechanism of the destruction of superconductivity at a temperature above the critical one \((T>T_k)\) and in a magnetic field \(H\) greater than the critical field \(H_k(T)\). The importance and fruitfulness of a theory connected with postulating a definite spectrum is clear from the example of the microtheory of superfluidity of helium II, developed by Landau\(^{18,19}\), precisely along such a path, which may be called quasi-microscopic. In view of what has been said, as well as the undoubted kinship of the phenomena of superconductivity and superfluidity, we shall below briefly dwell on this latter phenomenon as well.

4. QUASI-MICROSCOPIC APPROACH TO THE THEORY OF SUPERCONDUCTIVITY

a) The spectrum of “elementary excitations” and the theory of the superfluidity of helium II

In gases and solids, thermal motion can, as is known, be traced in considerable detail and in general form—regardless of the specific properties of the given substance. On the contrary, in the case of liquids—aggregates of strongly interacting particles that do not form a crystalline lattice—thermal motion is very complex (since, unlike in solids, the displacements of atoms are not small), and to examine it

*) In Giss’s work\(^{54}\) an attempt is made to describe the ground state of superconductors as a certain superposition of atomic wave functions. However, we have not been able to find in this work any points of interest. The proof that, from the adopted model, equation (1.2) follows\(^{55}\) leads in fact to the conclusion that this equation is valid only for times

\[ t \ll \frac{10^{-31}}{E}, \]

where \(E\) is the electric-field strength. This means that, for any reasonable values of the field, equation (1.2) is always completely inapplicable, in complete contradiction with experiment and with the assertion that is “proved” in\(^{55}\).

enough in a sufficiently general way. However, at sufficiently low temperatures, near absolute zero, where thermal motion is least intense and relatively simplest, a certain progress in understanding the properties of liquids has nevertheless already been achieved. It is connected with the idea that every weakly excited state of any macroscopic body can be regarded as an aggregate of separate “elementary excitations”1, 18, 19, 39. From the quantum-mechanical point of view these excitations are similar to certain particles (or “quasiparticles”) moving in the body and possessing a definite energy \(E\) and momentum \(p\).

In a gas the “elementary excitations” are simply the individual atoms, and

\[ E=\frac{p^2}{2m}. \tag{4,1} \]

In the case of a solid body the “elementary excitations” are phonons; moreover, for the acoustic branches of the crystal spectrum, for not too large momenta,

\[ E=up, \tag{4,2} \]

where \(u\) is the speed of sound.

For the Born (optical) branches, in a first approximation \(E=\hbar\omega_0+ap^2\), where \(\omega_0\) is the limiting frequency of an optical oscillation and \(a\) is a constant*).

Near absolute zero, only helium He\(^4\) and helium He\(^3\), as well as their mixtures, are liquids. These liquids may be called “quantum liquids,” since according to the classical theory all bodies at \(T\to0\) should be solid, and the non-solidification of helium at ordinary pressures is connected with the presence of zero-point oscillations. Understanding the term liquid in an extended sense, as was done above, one may include among liquids the conduction electrons in a metal (“electron liquid”), as well as certain other objects (for example, the aggregate of interacting spins in ferromagnets and paramagnets may be regarded as a “spin liquid,” etc.). The spectrum of a quantum liquid cannot at present be obtained theoretically, and only on the basis of the properties of liquid helium and metals known from experiment can one point to the existence of spectra of at least two types: Fermi and Bose (see1 §§ 66–69).

The Fermi spectrum is similar to the spectrum of electrons at the boundary of a degenerate Fermi gas and is characterized by the fact that the elementar—

*) Here and in (4,2), for simplicity, anisotropy is not taken into account. For each branch the constants \(u\), \(\omega_0\), and \(a\) are, generally speaking, different. Moreover, no distinction is made between momentum and quasimomentum.

... excitations appear and disappear in pairs (“electron” and “hole”) and, apparently, have spin \(1/2\) and obey Fermi statistics. The momentum of one component of the pair (the “electron”) is always greater than a certain limiting momentum \(p_0\), and

\[ E=\frac{p^2}{2m}-\frac{p_0^2}{2m},\quad p>p_0. \tag{4,3} \]

The momentum of the “hole” is always less than \(p_0\), and

\[ E=\frac{p_0^2}{2m}-\frac{p^2}{2m}. \]

A liquid with such a spectrum behaves, at least from the point of view of its thermodynamic properties (the magnitude and temperature dependence of the heat capacity, etc.), in exactly the same way as a degenerate ideal Fermi gas. Since the thermodynamic properties of metals in the normal (non-superconducting) state are similar to the properties of such a gas (the electronic part of the heat capacity \(c^{(e)}=\gamma T\)), one may suppose that the spectrum of the electron liquid in the normal state is the Fermi spectrum (4,3); this was already discussed in § 1. Apparently, the spectrum of liquid \(\mathrm{He}^3\) is also Fermian.

Fig. 2.

In the case of a Bose-type spectrum, elementary excitations appear and disappear singly and obey Bose–Einstein statistics. At small momenta \(p\), as long as the wavelength

\[ \lambda=\frac{2\pi\hbar}{p}\gg a\sim 10^{-8} \]

(\(a\) is the interatomic distance), these excitations correspond to ordinary sound waves, i.e., are phonons, for which \(E=up\) (see (4,2)), where \(u\) is the velocity of sound in the liquid (here, in contrast to a solid, only longitudinal sound can be involved), and \(p\) is the momentum of the excitation. The change in the character of the spectrum with increasing momentum \(p\) can be determined only as a result of processing experimental data on the heat capacity of the given substance. In the case of helium II, whose spectrum belongs to the Bose type (see below), the experimental data agree with the spectrum shown in Fig. 2 (see \(^{18,19,40}\)).

In thermal equilibrium the elementary excitations have an energy corresponding to the vicinity of one of the minima of the energy on the curve in Fig. 2. Near the first minimum, i.e., near the energy \(E=0\), the excitations, as stated, are phonons with \(E=up\).

Near the second minimum, located at \(p=p_0\), the energy of the excitations, called rotons, has the form:

\[ E=\Delta+\frac{(p-p_0)^2}{2\mu}. \tag{4.4} \]

According to the latest data\({}^{40}\)

\[ \left. \begin{aligned} \Delta&=(8.9\pm0.2)^\circ, \qquad p_0=(2.1\pm0.05)\cdot10^{-19}\ \mathrm{g\,cm\,sec^{-1}},\\ \mu&=(1.72\pm0.68)10^{-24}\ \mathrm{g}. \end{aligned} \right\} \tag{4.5} \]

According to earlier determinations\({}^{19}\), \(\Delta=9.6^\circ\), \(p_0=2.06\cdot10^{-19}\) and \(\mu=5\cdot10^{-24}\). Such a large change in the value of \(\mu\) testifies to a rather great sensitivity, essentially of all the parameters \(\Delta\), \(p_0\), and \(\mu\), to changes in the quantities characterizing the liquid (heat capacity, density, etc.). It is not excluded that, in the future, more radical changes will have to be introduced into the spectrum as well, for example the introduction of one more minimum on the curve \(E(p)\). Apparently, the introduction of such a minimum, connected with the addition of three more free parameters, does not contradict the existing experimental data (in this case, of course, the parameters \(\Delta\), \(p_0\), and \(\mu\) for the first minimum will have to be changed). Thus, the spectrum of Fig. 2 is only the simplest of the possible ones. At the same time, since this spectrum was from the outset simply fitted to explain the experimental data and is quite sufficient for this purpose, at present there is no reason to consider more complicated spectra (this remark does not apply, however, to possible changes of the spectrum that are unimportant from the standpoint of calculating the thermodynamic properties of helium II in appreciable volumes, but essential for explaining the critical velocities and the properties of thin films of helium II; see\({}^{41}\) and below).

Knowing the dependence of the excitation energy on their momentum \(E(p)\), one can compute all the thermodynamic characteristics of the substance. Phonons obey Bose–Einstein statistics, and their number \(N_{\phi}\), free energy \(F_{\phi}\), and energy \(E_{\phi}\), referred to unit volume, are equal \(\left(\dfrac{p^2\,dp\,d\Omega}{(2\pi\hbar)^3}\right.\) is the volume element in \(r\)-space, \(d\Omega\) is the element of solid angle):

\[ \left. \begin{aligned} N_{\phi} &=\frac{1}{(2\pi\hbar)^3}\int \frac{p^2\,dp\,d\Omega}{e^{up/kT}-1} =\frac{\pi^2(kT)^3}{90\hbar^3u^3},\\[4pt] E_{\phi} &=\frac{1}{(2\pi\hbar)^3}\int \frac{u p^3\,dp\,d\Omega}{e^{up/kT}-1} =-3F_{\phi}=3kTN_{\phi},\\[4pt] F_{\phi} &=\frac{kT}{(2\pi\hbar)^3}\int \ln\!\left(1-e^{-up/kT}\right)p^2\,dp\,d\Omega\\ &=-kTN_{\phi} =-\frac{\pi^2(kT)^4}{90\hbar^3u^3}. \end{aligned} \right\} \tag{4.6} \]

Hence for the entropy \(S\) and the heat capacity \(c\) we have:

\[ S_{\phi}=-\frac{\partial F}{\partial T} =\frac{2\pi^{2}k^{4}T^{3}}{45\hbar^{3}u^{3}}, \qquad c_{\phi}=T\frac{\partial S}{\partial T} =\frac{2\pi^{2}k^{4}T^{3}}{15\hbar^{3}u^{3}}. \tag{4.7} \]

The expressions obtained essentially coincide, of course, with the Debye expressions for the free energy, entropy, and heat capacity of a solid at low temperature (the difference consists only in the need to take into account three acoustic branches in a solid).

Rotons must also obey Bose statistics, since they lie on the same energy branch as the phonons. However, in the temperature region of interest to us, \(\Delta \gg kT\), and therefore for rotons one may simply use classical statistics:

\[ \left. \begin{aligned} N_{\rho} &=\frac{1}{(2\pi\hbar)^{3}}\int e^{-E/kT}p^{3}\,dp\,d\Omega \simeq \frac{2(\mu kT)^{1/2}p_{0}^{2}}{(2\pi)^{3/2}\hbar^{3}}e^{-\Delta/kT},\\ F_{\rho}&=kTN_{\rho}, \end{aligned} \right\} \tag{4.8} \]

where in the calculations expression (4.4) was used (see \(^{19}\), where all quantities are referred to unit mass).

As already mentioned, the expressions obtained for \(F=F_{\phi}+F_{\rho}\) and for other derived quantities are in complete agreement with experiment almost up to the \(\lambda\)-point itself (\(T_{\lambda}=2.19^\circ\)), where helium II, as a result of a second-order phase transition, passes into helium I. On approaching the \(\lambda\)-point the number of excitations (phonons and rotons) in helium II increases so strongly that they can no longer be regarded as an ideal gas. It is precisely for this reason that the properties of helium II in the immediate vicinity of the \(\lambda\)-point and the properties of helium I over the whole range of its existence can no longer be obtained theoretically, even if a definite dependence \(E(p)\) is specified. The same applies to all other liquids (except He\(^3\) and mixtures of He\(^4\) and He\(^3\)), since they exist only at “high” temperatures, greater than several degrees (for example, liquid hydrogen solidifies at \(14^\circ\)).

Helium II is, as is known, superfluid. The phenomenon of superfluidity, discovered by Kapitsa in 1938, is manifested above all in the fact that helium II can flow through narrow capillaries and slits without any friction (more precisely, only the superfluid part of helium II moves without friction; its density \(\rho_s\) at \(T=0\) is equal to the total density of helium II, while at \(T_{\lambda}\) it is equal to zero)\(^*\). Since helium atoms adhere to the wall, superfluid flow is possible in which the superfluid liquid moves

\(^*\) We do not set ourselves the aim here of giving a survey of the theory of superfluidity (see \(^{19}\)), and shall dwell briefly only on the main points essential for what follows.

in a slit with constant velocity \(v_s\), is connected with the formation of a tangential jump (discontinuity) of the velocity at the wall. The character of this discontinuity has not yet been completely studied; its width is apparently of atomic dimensions and, in any case, no greater than \(10^{-6}\), since superfluidity is already observed in films of thickness \(\sim 3\cdot 10^{-6}\).

One of the principal achievements of Landau’s theory of superfluidity \({}^{18,19}\) is the clarification of the condition for the stability of this discontinuity from the point of view of the form of the function \(E(p)\). In the reference frame associated with the capillary, where helium is moving with velocity \(v\) (we consider the case \(T=0\)), the energy of an excitation that has appeared, as follows from the mechanical formulae for the transformation of energy in passing to another reference frame, is equal to

\[ E' = E(p)+\mathbf{pv}, \tag{4,9} \]

where \(E\) and \(\mathbf p\) are the energy and momentum of the excitation in the coordinate system associated with the liquid. In order that the excitation which has appeared should correspond to a retardation of the liquid, it is necessary that \(E'<0\). Hence it follows that the appearance of excitations and, consequently, the beginning of the destruction of the superfluid state is possible only under the condition

\[ v \geqslant \left[\frac{E(p)}{p}\right]_{\min}. \tag{4,10} \]

If \(E=\dfrac{p^{2}}{2m}\), as is the case in an ideal gas, condition (4,10) gives \(v \geqslant \left[\dfrac{p}{2m}\right]_{\min}=0\), i.e. superfluidity is impossible. The same applies to a system with the fermion spectrum (4,3), where the difference \(p_0^2-p^2\) can be equal to zero. In the case of the spectrum shown in Fig. 2, on the contrary, superfluidity is possible, since for phonons

\[ \frac{E(p)}{p}=u=2.35\cdot 10^{4}\ \text{cm/sec}, \]

whereas in the case of rotons

\[ \left[\frac{E(p)}{p}\right]_{\min} \simeq \frac{1}{p}\left(\sqrt{2\mu\Delta+p_0^2}-p_0\right) \simeq \frac{\Delta}{p_0}\simeq 6\cdot 10^{3}\ \text{cm/sec}. \]

The superfluidity of helium II observed experimentally thus compels one to assign the spectrum of this liquid to the Bose type.

According to (4,10), superfluidity should be destroyed at a critical velocity of helium II relative to the wall equal to \(v_c=\left[\dfrac{E(p)}{p}\right]_{\min}\), i.e. in the case of the spectrum of Fig. 2 at \(v_c\simeq 6\cdot 10^{3}\ \text{cm/sec}\). In fact, however, the value of \(v_c\) is sometimes several orders of magnitude smaller and depends on the width \(d\) of the slit. Reliable measurements of the critical velocity are still lacking, but the available data apparently agree with the relation

\[ v_c \sim \frac{10^{-3}}{d}\ \text{cm/sec}. \]

If the low value of \(v_c\) is not due to some incidental causes, then here we have a clear disagreement between theory and experiment. From the data on \(v_c\) it would seem to follow that in helium II there are some excitations with a value

\[ \left[\frac{E(p)}{p}\right]_{\min}, \]

substantially smaller than for phonons and rotons and, at the same time, depending on the width of the slit \(d\). Moreover, since the phonon-roton spectrum of helium II leads to values of the heat capacity \(c=c_\phi+c_\rho\) consistent with experiment, new excitations should make only a very small contribution to the heat capacity. One of the possible ways of satisfying these requirements consists in the assumption\({}^{41}\) that the additional excitations are connected with the surface of helium (similar to Rayleigh waves in a solid or capillary waves on the free surface of a liquid). In this case their contribution to the heat capacity of large volumes of helium II will be insignificant. At the same time, if the velocity of helium is greater than the critical velocity \(v_c\) for these excitations, the latter will be excited and will destroy the supercurrent. The dependence of \(v_c\) on \(d\) may in this case be connected with quantum effects. Indeed, suppose, for example, that the excitation behaves like a free particle, i.e. its Hamiltonian has the form \(\mathcal{H}=p^2/2\mu\). In a slit of width \(d\), assuming that at the wall the \(\psi\)-function of the excitation is equal to zero:

\[ E=\frac{p_y^2+p_z^2}{2\mu}=\frac{p_y^2}{2\mu}+\frac{\pi^2\hbar^2}{2\mu d^2}, \]

where the \(z\)-axis is perpendicular to the slit and the motion takes place along the \(y\)-axis (the momentum \(p_y\), obviously, is continuous). In this case, as is easy to see\({}^{41}\) (\(m_{\mathrm{He}}\) is the mass of a helium atom),

\[ v_c=\left[\frac{p_y}{2\mu}+\frac{\pi^2\hbar^2}{2\mu d^2p_y}\right]_{\min} =\frac{\pi\hbar}{\mu d} =\frac{5.0\cdot10^{-4}}{(\mu/m_{\mathrm{He}})d}\ \text{cm/sec}. \tag{4,11} \]

Of course, one should not attach special significance to the agreement of formula (4,11) with the experimental data for \(\mu\sim m_{\mathrm{He}}\), both because of the insufficient reliability of these data and because the assumption made about the form of the excitations is in no way connected with the requirement that they have a surface character (see above). However, from the example given it is clear that a quantum effect may be important for elucidating the question of the critical velocity. Clarification of the question of the critical velocity and, in general, further substantial progress in the study of helium II are impossible without a detailed experimental investigation both of the critical velocities themselves and of the behavior of thin films of helium II. The latter is clear from the fact that, if there are any surface excitations in helium II, this should lead to a difference between the heat capacity of thin films and the heat capacity of substantially

...volumes of helium II*); the same applies to the ratio \(\rho_n(T)/\rho\) (see below). As has already been indicated, the question of the structure of the velocity discontinuity of helium II at the wall has likewise not been studied at all.

Let us note that the widespread opinion about the absence of superfluidity for an ideal gas is, at the very least, inaccurate. Indeed, in the case of the spectrum \(E = \dfrac{p^2}{2m}\) the critical velocity \(v_c = 0\) only if the momentum \(p\) can take all values, i.e., is not quantized. But in a finite volume—and it makes sense to speak of superfluidity only in the presence of walls—the momentum (more precisely, the square of the momentum projections) is quantized. Therefore the answer to the question of the superfluidity of an ideal gas is not immediately clear. It is easy to see, however, that when a gas moves in a slit, superfluidity does not occur, since the momentum \(p_y\) in the direction of the velocity is continuous and its change can be arbitrarily small**).

The situation changes in the case of a gas contained in a cylindrical vessel (length \(L\), radius \(R\)). When the vessel is rotated about its axis with a not too large angular velocity \(\Omega\), the gas is not entrained by the vessel, i.e., it is superfluid (at \(T = 0\) helium II behaves precisely in this way—remaining entirely at rest while the vessel rotates). To prove the statement just made, we proceed in exactly the same way as in considering, in the theory of superfluidity,\(^{18,19}\) the question of the entrainment of excitations by a rotating vessel. In the reference frame rotating together with the vessel, the external conditions in which the gas is found are stationary, and in equilibrium a Gibbs distribution must obtain; in the case of an ideal Bose gas this is the distribution

\[ \frac{1}{e^{E'/kT}-1}, \]

where \(E'\) is the energy of a particle in the rotating reference frame, equal—

*) One work devoted to the measurement of the heat capacity of thin layers already exists.\(^{42}\) In this case the heat capacity proved to be substantially different from the heat capacity of “bulk” helium. However, measurements are needed over a larger temperature interval and under various conditions (let us note that, in the presence of a free surface of helium II, the contribution to the heat capacity made by capillary waves must be taken into account).

**) In the example given earlier, in considering excitations with \(\mathcal{H} = p^2/2\mu\), these excitations were created, and therefore \(E \ne 0\). In the case of a gas of particles, however, the latter merely change their state of motion and, consequently, the excitation must be taken to be the difference \(E - E_0\), where \(E_0\) is the energy in the initial state. For a slit in which the gas moves with velocity

\[ v_0 = \frac{p_{0y}}{m}, \]

we have:

\[ E_0=\frac{\pi^2\hbar^2}{2md^2}+\frac{p_{0y}^{\,2}}{2m} \quad\text{and}\quad E=\frac{\pi^2\hbar^2}{2md^2}+\frac{p_y^{\,2}}{2m}, \]

and the difference \(E - E_0\) can be arbitrarily small, and hence

\[ \left[\frac{E-E_0}{\Delta p}\right]_{\min}=0, \]

where \(\Delta p = |p - p_0|\). At the same time the quantity \(\Delta p \ne 0\) when \(E - E_0 \to 0\), if the momentum \(p\), while remaining in magnitude close to \(p_0\), strongly changes its direction, for example if \(p_y = -p_{0y}\).

equal to \(E' = E - M\Omega\) (\(E\) is the energy in the stationary system, \(M\) is the momentum of the particle, and \(\Omega\) is the angular velocity of the vessel). The momentum of the entire gas is equal to (\(dV\) is an element of volume, \(d\Omega'\) an element of solid angle):

\[ \overline{M} = \frac{1}{(2\pi\hbar)^3} \int \frac{M p^2\,dp\,d\Omega'\,dV} {e^{(E-M\Omega)/kT}-1}. \]

The energy levels in the vessel are determined by expression (3.17), and at \(T=0\) all particles of the ideal Bose gas are on the lowest level with energy

\[ E_{011}=\frac{(2.4048)^2\hbar^2}{2\mu R^2} \]

(for simplicity we assume the height of the vessel \(L\) to be very large) and with moment \(M_z=m\hbar=0\). When the vessel rotates, as \(T\to 0\), the moment \(\overline{M}\) can begin to increase only when the energy \(E-M\Omega\) for one of the higher levels (at \(\Omega=0\)) reaches the level \(E_{011}\). For the level nearest to \(E_{011}\),

\[ E_{111}=\frac{(3.8317)^2\hbar^2}{2\mu R^2} \]

with moment \(M_z=\hbar\), the equality \(E_{011}=E_{111}-M\Omega\) will hold at

\[ \Omega_c=\frac{8.9\hbar}{2\mu R^2},\qquad v_c=\Omega_c R=\frac{4.5\hbar}{\mu R} =\frac{6.7\cdot 10^{-4}}{(\mu/m_{\mathrm{He}})R}}\ \text{cm/sec}, \tag{4.12} \]

which, apart from the numerical coefficient, is in agreement with (4.11).

For \(\Omega>\Omega_c\) a redistribution of particles over the levels begins, and the moment \(\overline{M}\) grows; but for \(\Omega<\Omega_c\) the particles are not carried along by the rotating vessel, as occurs in the case of helium II.

A Fermi gas in a vessel will also be superfluid, but in this case, at not too small densities, the spacing between levels is small and the value of \(\Omega_c\) is likewise negligible. The situation here is completely analogous to that which occurs with regard to the “superconductivity” of ideal Bose and Fermi gases, which was discussed in § 3b). This is also understandable, since, if terms \(\sim H^2\) are neglected, a system in a magnetic field \(H\) behaves in the same way as in a vessel rotating with the Larmor angular velocity

\[ \Omega=\omega_L=\frac{eH}{2mc}. \]

Since an ideal gas is not superfluid when moving in a slit (see above), it is clear that, in order to explain the superfluidity of helium II, taking into account the interaction between atoms is, of course, absolutely necessary. The interatomic interaction in helium, as in any liquid, is very large, and at present no ways are apparent for a rigorous analysis of the question of the excitation spectra of liquids. Therefore, the consideration of the excitation spectrum of a nonideal gas has great, at least methodological, significance. But here too the problem, in a rigorous formulation, is very complicated, and apparently the only serious rigorous investigation in this area—

... is the work of Bogolyubov ($^{43}$ and subsequent calculations), who considered the question of the spectrum of a Bose gas with weak interaction. In doing so, using a definite method of perturbation theory, it was shown that for small excitations the spectrum of the system corresponds to a set of elementary excitations for which

\[ E=\sqrt{u^2p^2+\left(p^2/2m\right)^2}, \tag{4,13} \]

where $u$ is the speed of sound in the gas under consideration.

For small $p$, (4,13) goes over into (4,2), and for large $p$ into (4,1), i.e. the excitations behave like free particles, which is natural at large momenta. The spectrum (4,13) obviously satisfies the condition of stability of superfluid flow. However, this conclusion, as applied to a Bose gas with weak interaction, which according to $^{43}$ must correspond to repulsion between particles, will be valid only if excitations of the type (4,13) exhaust all possible ones. If, however, any other excitations are possible in the system, then, without knowing their spectrum, it is no longer possible to assert that the gas is superfluid. At the same time, in $^{43}$ it is shown that excitations (4,13) exist, but, as far as we can judge, it is not proved that there can be no other excitations. Acting by the method of perturbation theory, it is extremely difficult, if at all possible, to answer such a question. Moreover, the data of work $^{44}$ suggest that the spectrum (4,13) does not exhaust all the excitations of a Bose gas. The point is that in $^{44}$ the spectrum (4,13) is obtained from the quantum kinetic equation with a self-consistent field (in $^{44}$ the momentum of the excitation is $p=\hbar k$). But, acting by such a method, one can also arrive at the conclusion that the condition of superfluidity is satisfied for a wide class of systems, including a Fermi gas, ordinary electron plasma, and, in fact, all real gases*).

Such a conclusion cannot be regarded as obviously incorrect, since at low temperatures, when one can speak of a gas of elementary excitations, the systems under consideration do not exist

*) According to $^{44}$ and subsequent calculations by V. P. Silin, and also according to $^{44a}$, in a charged Fermi gas and in a classical plasma, at sufficiently small momenta

\[ v_c=\left[\frac{E(p)}{p}\right]_{\min}>0, \]

since as $p=\hbar k\to0$ the energy $E$ tends not to zero but to $\hbar\omega_0$, where

\[ \omega_0=\sqrt{\frac{4\pi e^2 n}{m}} \]

is the proper frequency of the plasma (see also § 46). The same conclusion, that $v_c>0$, is obtained also for other real gases with mutually interacting particles. In this case it is necessary only that the interaction be sufficiently strong on the average in space and that the value of the interaction energy between any two particles be positive (predominance of repulsive forces).

in reality and the possibility of their superfluidity does not contradict anything. However, the conclusion that real gases are superfluid is doubtful and cannot be considered proved (not to mention the inadmissibility of ignoring the role of the walls), because the self-consistent-field method makes it possible, generally speaking, to find only part of the excitations of the system. Therefore it may turn out that the systems under consideration in fact also have excitations for which \(v_c=0\). If this is so, then the coincidence of the spectrum obtained in \(^{43}\) with the “self-consistent” spectrum \(^{44}\) suggests the incompleteness of the spectrum obtained in \(^{43}\), although it is also possible that precisely in the case of a Bose gas the self-consistent-field method is more accurate than in other cases.

The question of the superfluidity of real gases evidently requires further investigation, in the course of which it is necessary also to take account of the influence of the walls, since a discussion of the problem of superfluidity for an unbounded medium is, strictly speaking, meaningless.

Returning to helium II, let us consider the motion of this liquid at \(T\ne0\). In this case there are excitations in helium, and in equilibrium these excitations are on the average at rest relative to the walls, i.e. they are not carried along by the moving liquid. The latter follows from the fact that the excitations interact with the wall and are slowed down by it, just as occurs in the case of a gas moving through a tube. It can be seen that a definite part of the mass of the liquid is associated with the excitations. Indeed, the distribution function for the excitations in a system connected with the walls, where the external conditions are stationary, has the usual form \(n(E')=\left[e^{E'/kT}-1\right]^{-1}\), where \(E'=E+\mathbf{p}\mathbf{v}\) and \(\mathbf{v}\) is the velocity of motion of helium relative to the walls. In a reference system connected with the liquid, the excitations possess momentum \(\mathbf{p}\) and move as a whole with velocity \(-\mathbf{v}\). The momentum of the excitation gas is then equal to

\[ \mathbf{P}=\int \mathbf{p}\,n\bigl(E(\mathbf{p})+\mathbf{p}\mathbf{v}\bigr)\,\frac{p^{2}\,dp\,d\Omega}{(2\pi\hbar)^{3}}. \]

In the case of phonons, when \(E(p)=up\), \(\mathbf{P}_{\phi}=-\rho_{n\phi}\mathbf{v}\), where

\[ \rho_{n\phi}=\frac{4E_{\phi}}{3u^{2}(1-v^{2}/u^{2})^{3}},\qquad E_{\phi}=\frac{\pi^{2}(kT)^{4}}{30\hbar^{3}u^{3}}. \tag{4.14} \]

Here \(\rho_{n\phi}\) is the density of the liquid associated with the phonons. The total density of the “normal part” of helium II is \(\rho_n=\rho_{n\phi}+\rho_{nr}\), where we shall not calculate the roton part of the density \(\rho_{nr}\) (see \(^{19,40}\)). Since the critical velocity \(v_c\ll u\), in helium II one may practically always use expression (4.14) with \(v=0\) (the velocity \(v\) in (4.14) is the velocity of the superfluid part of the liquid \(v_s\), while the normal part of the liquid is assumed to be at rest; in the most general case in (4.14) \(v=v_s-v_n\), where \(v_n\) is the velocity of the normal part of the liquid). The densities \(\rho_n\) and \(\rho_s=\rho-\rho_n\) depend

from the temperature—see (4.14)—and at \(T=0\), \(\rho_n=0\), while at \(T=T_\lambda\), \(\rho_n=\rho\), where \(\rho\) is the density of helium II.

In view of what has been said, helium II may be regarded as if it consisted of an interpenetrating mixture of two liquids (superfluid and normal), with the corresponding densities \(\rho_s\) and \(\rho_n\) and velocities \(\mathbf v_s\) and \(\mathbf v_n\). There is no transfer of momentum between the two parts of the liquid, and it must not be forgotten that the idea of two liquids (the two-fluid model) is not to be understood literally, but is intended only to reflect the peculiar situation that obtains in reality.

One of the essential, if not the most essential, achievements of Landau’s theory of superfluidity is the establishment of macroscopic (hydrodynamic) equations for both “liquids” of which helium II “consists.”

The motion of the superfluid part of helium II is irrotational:

\[ \operatorname{rot}\mathbf v_s=0. \tag{4.15} \]

The equation of motion for \(\mathbf v_s\) is as follows\(^{18,19}\):

\[ \frac{\partial \mathbf v_s}{\partial t} = -\operatorname{grad}\left[ \frac{\Phi}{\rho} + \frac{v_s^2}{2} - \frac{\rho_n}{2\rho}(\mathbf v_n-\mathbf v_s)^2 \right], \tag{4.16} \]

where

\[ \frac{\Phi}{\rho}=\frac{F}{\rho}+\frac{p}{\rho} \]

is the thermodynamic potential per unit mass of stationary helium II, \(p\) is the pressure, and \(F/\rho\) is the free energy per unit mass; it has also been taken into account that, under condition (4.15),

\[ \frac{d\mathbf v_s}{dt} = \frac{\partial \mathbf v_s}{\partial t} + \operatorname{grad}\frac{v_s^2}{2}. \]

No conditions are imposed on the tangential-to-the-surface component \(\mathbf v_s\). The component of \(\mathbf v_s\) normal to the surface must be equal to zero, unless at the surface the superfluid part of the liquid is transformed into the normal part, as occurs when there is a heat flux through the wall.

In addition to equations (4.15) and (4.16), and the obvious relation

\[ \rho=\rho_s+\rho_n, \]

where \(\rho\) is the total density of helium II, there must, of course, also be a continuity equation:

\[ \frac{\partial \rho}{\partial t}+\operatorname{div}\mathbf j=0,\qquad \mathbf j=\rho_s\mathbf v_s+\rho_n\mathbf v_n. \tag{4.17} \]

When the viscosity of the normal part of the liquid is neglected, the complete system of equations of motion of helium II also includes the equations:

\[ \left. \begin{aligned} \frac{\partial j_i}{\partial t} &= -\sum_{k=1}^{3}\frac{\partial \Pi_{ik}}{\partial x_k}, \qquad \Pi_{ik}=p\delta_{ik}+\rho_s v_{si}v_{sk}+\rho_n v_{ni}v_{nk},\\[6pt] \frac{\partial S}{\partial t} &+\operatorname{div}(S\mathbf v_n)=0, \end{aligned} \right\} \tag{4.18} \]

where \(S\) is the entropy of a unit volume of helium II. If compressibility is not taken into account, then equations (4.18) are greatly simplified\({}^{19}\). In the absence of heat exchange with the wall, on the latter \(v_n=0\). Allowance for viscosity and thermal conductivity leads to the appearance in (4.18), and also in the right-hand side of equation (4.16), of certain additional terms (see \({}^{19,40}\)).

The theory of the superfluidity of helium II, consisting of a quasimicroscopic part connected with the adoption of a definite spectrum and a macroscopic (hydrodynamic) part, made it possible to understand the whole complex of phenomena in helium II at velocities below the critical one.

b) The excitation spectrum of an electron liquid in a metal and superconductivity

The quasimicroscopic approach, elucidated in the preceding section as applied to helium II, is naturally to be extended also to superconductivity. There is all the more basis for this because the phenomena of superfluidity and superconductivity are deeply related. In both cases there is an absence of friction (resistance) for motions of a definite type—superfluid flow and superconducting current. In both cases, under definite conditions, there is observed the coexistence of flows without friction with “normal” flows—normal flow in helium II and normal current in superconductors. Moreover, the equations determining the superconducting current \(\mathbf{j}_s\), in weak fields, are completely analogous to equations (4.15)—(4.16) for the velocity of the superfluid part of helium II \(\mathbf{v}_s\), and, in fact, from the point of view of these equations, the whole difference of the superconducting case is connected with the fact that the electron “liquid” is charged.

Indeed, the basic equation for the superconducting current (1.1), if \(\mathbf{j}_s\) is written as \(\mathbf{j}_s=en_s\mathbf{v}\), has the form (see (I; 2.17)):

\[ \operatorname{rot}\mathbf{v}=-\frac{e}{mc}\mathbf{H}. \tag{4.19} \]

This equation is a generalized condition for the absence of vortices and, for \(e=0\), goes over into (4.15). The second condition for \(\mathbf{j}_s\)—equation (1.2)—is, in turn, analogous to (4.16) (see (I; 2.18)).

The motion of “normal” electrons in a superconductor is also, in essence, analogous to the motion of the normal part of helium II. The difference consists only in the fact that, in the case of a metal, the frictional forces are expressed much more strongly. This is connected, roughly speaking, with the fact that helium experiences friction chiefly at the walls, whereas normal electrons are slowed down, owing to their interaction with the lattice, throughout the whole volume. Therefore, for the normal current at frequencies of the electromagnetic field at which superconductivity is still observed, one may neglect the inertia of the electrons and, if

distract oneself from the anomalous skin effect and use Ohm’s law $\mathbf{j}_n=\sigma \mathbf{E}$. The equation of motion (4.18) for $\mathbf{v}_n$, when strong viscosity is taken into account, would also lead to an analogous relation—to proportionality between $\mathbf{v}_n$ and the pressure gradient (Poiseuille flow).

Thus, without any exaggeration, one may say that superconductivity is superfluidity of an electronic liquid in a metal (see § 25).

This, of course, does not mean that the concept of the velocity of the superconducting liquid

\[ \mathbf{v}=\frac{\mathbf{j}_s}{e n_s} \]

can be applied without restriction, or even as widely as the concept of the velocity of the superfluid part of helium II. But in view of what has been said it is clear that the hydrodynamic picture of the motion of the electronic liquid, for motions that are not too fast and not too short-wavelength, when equations (1.1)—(1.2) are valid, proves quite suitable. As for excitations of the electronic liquid, one may suppose that the hydrodynamic approach is applicable to them so long as $\lambda \gg a\sim 10^{-8}$, where $\lambda=\dfrac{2\pi\hbar}{p}$ is the wavelength of the excitation.

Turning to the question of the possibility of the existence of superconductivity from the standpoint of the form of the excitation spectrum, i.e., the form of the function $E(p)$, where $E$ is the excitation energy and $p$ its momentum, one may, exactly as for helium II, note that superconductivity is possible only if (see (4.10))

\[ v_c=\left[\frac{E(p)}{p}\right]_{\min}>0 . \tag{4.20} \]

Such a criterion is in complete agreement with the results of § 3, obtained by an entirely different route. Indeed, if there is an ordinary Fermi spectrum (4.3), then $v_c=0$ and superconductivity is impossible. Taking account of interaction with the lattice, if it does not lead to the appearance of a gap at the Fermi surface itself, but introduces only the changes evident from Fig. 1, likewise cannot lead to superconductivity, since in this case too $v_c=0$. Finally, the appearance at the Fermi surface of an energy gap $\Delta$ immediately leads to the fact that $v_c\sim \dfrac{\Delta}{p_0}>0$. All these conclusions were also obtained in § 3.

It is not difficult to indicate an excitation spectrum of the electronic liquid that satisfies the condition for the existence of superconductivity (4.20) and leads to an expression for the heat capacity and other thermodynamic quantities in the superconducting state that agrees with experiment. Below even two such spectra will be presented.

However, in the case of superconductivity, in contrast to the superfluidity of helium II, one cannot restrict oneself to the requirements on the spectrum stated above. The point is that in helium II, on approaching the \(\lambda\)-point, the number of excitations increases greatly; they cease to form a gas, and one has to abandon the description of the details of thermal motion in helium II near \(T_\lambda\), and in helium I throughout the whole region of its existence. A superconducting metal, on the other hand, at \(H=0\) and \(T>T_k\), or at any temperature but \(H>H_k(T)\) (\(H_k\) is the critical field, \(T_k\) the critical temperature at \(H=0\)), is in the normal state, where over wide limits the Fermi spectrum (4.3) is applicable; thus there are no grounds for refusing to consider thermal motion. It is therefore necessary to find out how the “superconducting” spectrum is replaced by the normal one.

Further, the transition from the superconducting state to the normal one must be such as to ensure, in particular, the continuity of the change in electrical and thermal conductivity (in the superconducting state, of course, what is meant is the “normal” conductivity). In addition, the theory must somehow reflect, if not explain, the existence of the isotope effect, the appearance in the superconducting state of an enormous dielectric constant \(\varepsilon_0\), and, finally, the nonuniversality of superconductivity, which is not observed for all metals. The presence of all these requirements leads to the fact that the construction even of a quasi-microscopic theory of superconductivity encounters considerably greater difficulties than in the case of helium II. The corresponding problem has not yet been solved. We shall therefore have to confine ourselves to a number of remarks, as a result of which the outlines of a quasi-microscopic theory of superconductivity nevertheless become somewhat clearer.

The simplest spectrum satisfying condition (4.20) and ensuring the correct temperature dependence of the heat capacity in the superconducting state is obtained by introducing a gap into the Fermi spectrum (4.3). As a result we obtain the spectrum\(^3\):

\[ \begin{aligned} E_- &\gg kT, && p<p_0,\\ E_- &= \Delta+\frac{p^2}{2m}-\frac{p_0^2}{2m}, && p>p_0,\\ E_+ &\gg kT, && p>p_0,\\ E_+ &= \Delta+\frac{p_0^2}{2m}-\frac{p^2}{2m}, && p<p_0, \end{aligned} \tag{4.21} \]

where \(E_-\) is the energy of a negatively charged excitation (an “electron”), and \(E_+\) is the energy of a positive “hole.”

For \(\Delta=0\), (4.21) goes over into the Fermi spectrum (see (4.3)). However, the spectrum (4.21), understood literally, is inadmissible, as

this follows from considering the behavior of excitations in an electric field \(\mathbf E\). Under the action of the field, the momentum, for example, of an “electron,” on which the force \(\mathbf F=e\mathbf E\) acts (of course, \(e<0\)), depending on the sign of the quantity \(\mathbf F\cdot\mathbf p\), either decreases or increases. But, having reached the value \(p_0\), the momentum, according to (4,21), can no longer decrease, while on the other hand the force continues to act. In the case of a Fermi spectrum, when \(\Delta=0\), the way out of this difficulty consists in assuming that, on reaching the limiting momentum, the excitation disappears (more precisely, two excitations disappear simultaneously—an “electron” and a “hole,” approaching the boundary \(p=p_0\) from different “sides”). In place of the vanished pairs, for the negative components of which \(\mathbf F\cdot\mathbf p<0\), the field creates new pairs with \(\mathbf F\cdot\mathbf p>0\). Such a picture is, evidently, simply a representation (in terms of creation and disappearance of pairs of excitations) of that change (displacement in \(p\)-space) which the Fermi distribution for an electron gas undergoes under the action of an electric field. But for the spectrum (4,21) the pair creation in a field constant in time is already impossible, at least for \(T\to0\) and for a gap \(\Delta\) that is not too small (see below). This difficulty can nevertheless be circumvented and the spectrum (4,21) can be given a real meaning if one recalls that in a crystal \(\mathbf p\) is not a momentum, but a quasimomentum of the excitation, and the energy \(E\) must be a periodic function of this quasimomentum with periods \(\dfrac{2\pi\hbar\mathbf b_i}{\hbar}\), where \(\mathbf b_i\) are the periods of the reciprocal lattice. For a cubic lattice, to which we shall restrict ourselves for simplicity, \(b_i=b=\dfrac1a\), where \(a\) is the period of the crystal lattice. Thus, for example, \(E_n\left(p_x+\dfrac{2\pi}{a}\right)=E(p_x)\), where the index \(n\) reflects the fact that, for a given \(p_x\), the energy \(E\) may take different values*.

In view of what has been said, the spectrum (4,21), where \(\mathbf p\) is the quasimomentum, must be imagined as it is shown in Fig. 3 for the motion of an “electron” along the \(x\)-axis. In this spectrum a substantial refinement has been introduced, connected with the fact that a discontinuous spectrum is inadmissible, and therefore the transition from the region \(p<p_0\) to the region \(p>p_0\) is smoothed and made continuous. The differences between the spectrum (4,21) and the spectrum of Fig. 3, from the point of view of calculating thermodynamic quantities, can be made arbitrarily small, and in what follows we shall neglect them. In view of the multivaluedness of the function

* The fact that in a crystal the energy \(E_n\) is a periodic function is, in books on the theory of metals, strangely enough, usually not emphasized (see, however, \(^{7\mathrm b}\)). The quasimomentum is customarily denoted by \(\hbar\mathbf k\), not by \(\mathbf p\), but we shall not do this, so as not to complicate all the notation.

energies, at the same values of \(p_x\), but for energies larger than those shown in Fig. 3, there are still other branches (not shown in Fig. 3) of the function \(E_n(p)\). In other words, the curve shown in Fig. 3 corresponds to the lower “zone” for the excitations. The excitation velocity, equal to \(v=\dfrac{\partial E}{\partial p}\), at the points of minimum and maximum of the function \(E(p)\) becomes zero and then changes sign,

Fig. 3.

which corresponds to reflection and the formation of standing waves\({}^{6,7}\). An analogous situation occurs in the usual scheme of the theory of metals upon reflection from the zone boundary and is free of internal contradictions.

Assuming the excitations with spectrum (4.21) to obey Fermi statistics for the number of excitations in the energy interval \(dE_\pm\), we have:

\[ dN_\pm=\frac{g\,dE_\pm}{e^{E_\pm/kT}-1},\qquad g(E_\pm)=\frac{V\sqrt{2}\,8\pi m^{3/2}E_\pm^{1/2}}{(2\pi\hbar)^3}. \]

For \(\dfrac{\Delta}{kT}\ll 1\), the spectrum (4.21), from the thermodynamic point of view, is equivalent to the Fermi spectrum (4.3), and the free energy of the excitations is equal to

\[ F_n=\frac{H_{k0}^{2}}{8\pi}-\frac{\pi^2}{12}\alpha_n(kT)^2,\qquad \alpha_n=\left(\frac{2mp_0}{\pi^2\hbar^3}\right)_n, \tag{4.22} \]

where the additive constant has been chosen in such a way that \(F_n(0)=\dfrac{H_{k0}^{2}}{8\pi}\), with \(H_{k0}\) the critical magnetic field at \(T=0\). Processing the data on the heat capacity of a superconductor shows that, if one relies on the spectrum (4.21), \(\dfrac{\Delta}{kT}\sim 3\). In this case, for the free energy of the superconducting phase, with an accuracy up to \(\sim 1\%\), the limiting expression obtained for \(\dfrac{\Delta}{kT}\gg 1\) and hav-

has the form\(^3\):

\[ F_s=a_s e^{-\Delta/kT},\qquad a_s=\left(\frac{2mp_0}{\pi^2\hbar^3}\right)_s . \tag{4.23} \]

From the requirement that at the transition point of the second kind, i.e. at \(T_k\), the free energy \(F\) and the entropy \(S=-\dfrac{\partial F}{\partial T}\) be continuous, one can express \(a_n\) and \(a_s\) in terms of \(H_{k0}\), \(\Delta\), and \(T_k\). In this case it turns out that \(a_s>a_n\), i.e. the limiting momentum \(p_0\) in the two phases is different. Taking into account that

\[ \frac{H_k^2}{8\pi}=F_n-F_s \]

as the result of calculating \(a_n\) and \(a_s\), we obtain the following expression \(\left(\alpha=\dfrac{\Delta}{kT_k}\right)\):

\[ \left(\frac{H_k}{H_{k0}}\right)^3 = 1-\left(\frac{T}{T_k}\right)^2 \left\{ \frac{2+\alpha}{\alpha} - \frac{2}{\alpha}e^{\alpha\left(1-\frac{T_k}{T}\right)} \right\}. \tag{4.24} \]

By choosing the value of the parameter \(\alpha=\dfrac{\Delta}{kT_k}\), one can achieve very accurate agreement of formula (4.24) with experimental data. This is clear from the table, where experimental data for Hg according to \(^{45}\) are given, as well as values calculated by formula (4.24) and by the often used formula

\[ \left(\frac{H_k}{H_{k0}}\right)^2 = \left[1-\left(\frac{T}{T_k}\right)^2\right]^2 . \tag{4.25} \]

Formula (4.25) corresponds to the assumption that in the superconducting state the heat capacity \(c_s\sim T^3\), while in the normal state \(c_n=aT^3+\gamma T\).

Table

Values of \(\dfrac{H_k}{H_{k0}}\) as a function of \(\dfrac{T}{T_k}\)

\(\dfrac{T}{T_k}\)	\(\dfrac{H_k}{H_{k0}}\) exp.	\(\dfrac{H_k}{H_{k0}}\) according to (4.24) with \(\alpha=2.75\)	\(\dfrac{H_k}{H_{k0}}\) according to (4.25)	\(\dfrac{T}{T_k}\)	\(\dfrac{H_k}{H_{k0}}\) exp.	\(\dfrac{H_k}{H_{k0}}\) according to (4.24) with \(\alpha=2.75\)	\(\dfrac{H_k}{H_{k0}}\) according to (4.25)
0.00	1.000	1.000	1.000	0.55	0.708	0.708	0.6975
0.05	0.998	0.998	0.9975	0.60	0.649	0.648	0.6400
0.10	0.992	0.991	0.9900	0.65	0.586	0.583	0.5775
0.15	0.980	0.980	0.9775	0.70	0.517	0.513	0.5100
0.20	0.965	0.965	0.9600	0.75	0.442	0.438	0.4375
0.25	0.944	0.945	0.9375	0.80	0.363	0.358	0.3600
0.30	0.918	0.919	0.9100	0.85	0.279	0.275	0.2775
0.35	0.886	0.888	0.8775	0.90	0.190	0.187	0.1900
0.40	0.850	0.852	0.8400	0.95	0.098	0.091	0.0975
0.45	0.808	0.810	0.7975	0.975	0.049	0.051	0.050
0.50	0.760	0.761	0.7500	1.000	0.000	0.000	0.000

In the calculation the values\(^{45}\) \(H_{k0}=412.6\) and \(T_k=4.167^\circ\) were adopted; the experimental data for \(\dfrac{T}{T_k}<0.25^\circ\) were obtained by extrapolation. Taking into account the magnitude of the experimental error and the possibility of a more accurate choice of the constant \(\alpha=\dfrac{\Delta}{kT_k}\), it is clear from the data in the table presented that formula (4.24) agrees with experiment in any case no worse than the often-used formula (4.25).

However, the spectrum (4.21) and the entire scheme based on it cannot, of course, be accepted without further qualification. First, the difference already noted between the values of \(\alpha_n\) and \(\alpha_s\) in (4.22)—(4.23) means that the limiting momenta \(p_0\) in the two phases are also different, and this difference turns out to be large. Thus the spectrum (4.21) as \(\Delta \to 0\) does not pass over into the Fermi spectrum (4.3) with the same limiting momentum, as one would expect. Secondly, the number of excitations of type (4.21) is very small: even for \(\Delta \to 0\) and \(T \to T_k\) it is of order \(\Delta n \sim n\dfrac{T_k}{T_0}\sim 10^{-4}n\) (see § 1), and for \(\Delta \ne 0\) and \(T<T_k\) it is still smaller. Therefore the electrical conductivity associated with the excitations will be considerably smaller than in the normal state. The point is that in the normal state, contrary to assertions sometimes encountered, all conduction electrons with concentration \(n\), and not \(\Delta n\), participate in carrying the current. This is clear from the formula for the electrical conductivity of a free electron gas (see, for example, (1; 6.28)), and, in the language of the electron liquid and its excitations, is connected with the mechanism already discussed for the creation and disappearance of “electron”—“hole” pairs under the action of an electric field. The presence of the gap \(\Delta\), generally speaking, disrupts this mechanism, the electrical conductivity falls sharply, and, in general, with respect to the behavior of the normal conductivity the metal becomes very similar to a semiconductor.

The rather detailed discussion undertaken of the spectrum (4.21), despite all the difficulties noted, is explained by the fact that allowance for the influence of the dielectric constant \(\varepsilon_0\) substantially changes the whole situation. As was pointed out already in 1945 by L. D. Landau, when the quantity \(\varepsilon_0\) is introduced, the velocity of light in the metal for frequencies for which \(\varepsilon \simeq \varepsilon_0\) is equal to \(u=\dfrac{c}{\sqrt{\varepsilon_0}}\), and for \(\varepsilon_0\sim 10^{10}\) is so small that thermal radiation in the metal should make a noticeable contribution to its heat capacity. In the simplest case, when \(\varepsilon=\varepsilon_0\) and does not depend either on frequency or on temperature, electromagnetic waves (photons in the medium) are quite similar to phonons, but, unlike phonons in a liquid, are transverse. The energy of the photons then has the form \(E=up\), \(u=\dfrac{c}{\sqrt{\varepsilon_0}}\). The free energy, entro-

tion and heat capacity associated with these photons (i.e., with thermal radiation in the metal) are as follows:

\[ \left. \begin{gathered} F_{\phi}=-\frac{\pi^{2}\varepsilon_{0}^{3/2}(kT)^{4}}{45\hbar^{3}c^{3}},\qquad S_{\phi}=\frac{4\pi^{2}\varepsilon_{0}^{3/2}k^{4}T^{3}}{45\hbar^{3}c^{3}},\qquad E_{\phi}=\frac{\pi^{2}\varepsilon_{0}^{3/2}(kT)^{4}}{15\hbar^{3}c^{3}},\\ c_{\phi}=\frac{4\pi^{2}\varepsilon_{0}^{3/2}k^{4}T^{3}}{15\hbar^{3}c^{3}} =3.06\cdot 10^{-14}\varepsilon_{0}^{3/2}T^{3}\ \frac{\mathrm{erg}}{\mathrm{cm}^{3}\,\mathrm{grad}}. \end{gathered} \right\} \tag{4,26} \]

These formulas are obtained from (4,6)—(4,7) as a result of replacing \(u\) by \(c/\sqrt{\varepsilon_{0}}\) and multiplying by 2, associated with the presence of two directions of polarization of transverse waves.

If one assumes that all the heat capacity in the superconducting state is associated with the electromagnetic field *), i.e., that \(c_s=c_{\phi}\), then we arrive at the conclusion that \(u\sim 10^{5}\) and \(\varepsilon_{0}\sim 10^{10}\). For example, for mercury \(u=1.5\cdot 10^{5}\) and \(\varepsilon_{0}=2.0\cdot 10^{10}\).

Having experimental data on \(\varepsilon_{0}\) and on the heat capacity in the superconducting state \(c_s\), one might think that it would be possible immediately to determine the specific weight of the photon part of the spectrum. Unfortunately, this is not so simple because of the dependence of \(\varepsilon_{0}\) on frequency and temperature \(T\), which has not been taken into account in (4,26).

The general form of \(\varepsilon\), when absorption is neglected—which is possible for \(T<T_k\) and not too high a frequency—is as follows (see (I; 5,9) and (I; 5,11)):

\[ \left. \begin{gathered} \varepsilon=\varepsilon_{0}-\frac{c^{2}}{\omega^{2}\delta_{0}^{2}} =\varepsilon_{0}-\frac{4\pi e^{2}n_s}{m\omega^{2}} =\varepsilon_{0}-\frac{\omega_s^{2}}{\omega^{2}},\\ \varepsilon_{0}=1+\frac{4\pi e^{2}}{m}\sum_{k\ne 0}\frac{n_{0k}}{\omega_{0k}^{2}-\omega^{2}}. \end{gathered} \right\} \tag{4,27} \]

The enormous value of \(\varepsilon_{0}\) at not too high frequencies is connected with the fact that among the eigenfrequencies \(\omega_{0k}\) there are low frequencies \(\omega_{0k}\sim kT_k/\hbar\). Such effective frequencies will be possessed by those electrons which were free in the normal state and did not become “superconducting” for \(T<T_k\). If the concentration of electrons in the normal state is \(n_0\) (this quantity is determined from infrared measurements; see (I; 6,17)), and the concentration of “superconducting” electrons is \(n_s\) (this quantity is determined from measurements of the penetration depth \(\delta_0\); see (I; 6,18)), then, by virtue of the sum rule, the concentration of bound electrons is

\[ n_c=\sum_{k\ne 0} n_{ck}=n_0-n_s, \tag{4,28} \]

where the summation is carried out over all initial states,

*) Such an assumption is, in principle, admissible, since from (4,26) formula (4,25) for \(H_k\) follows, which in general agrees with experiment.

corresponding to electrons that are free in the normal state, and over all final states \(k \ne 0\) (the same applies to the sum in (4.27), if one neglects the contribution to \(\varepsilon_0\) from strongly bound electrons).

The distribution of the frequencies \(\omega_{0k}\) is unknown, and therefore it remains only to introduce some average frequency \(\omega_0\) and, for \(\omega^2 \ll \omega_0^2\), to write the expression for \(\varepsilon_0\) at \(T=0\) in the form

\[ \varepsilon_0(0)=\frac{4\pi e^2 n_c}{m\omega_0^2} =3.18\cdot 10^9\,\frac{n_c}{\omega_0^2} =\frac{\omega_c^2}{\omega_0^2}, \tag{4.29} \]

where it has been taken into account that \(\varepsilon_0 \gg 1\). As was said, one may think that \(\hbar \omega_0 \sim kT_k\)*). For tin, for example, \(kT_k \simeq 5\cdot 10^{-16}\), and if one puts \(\omega_0=\dfrac{kT_k}{\hbar}\) and \(n_c=5\cdot 10^{22}\) (see (I; § 6)), then \(\omega_0=5\cdot 10^{11}\), \(\omega_c=1.25\cdot 10^{16}\), and \(\varepsilon_0(0)=6\cdot 10^8\). Experimentally\(^{46}\) \(\varepsilon_0 \simeq 5\cdot 10^9\), whence one may conclude that \(\omega_0 \simeq \dfrac{kT_k}{3\hbar} \simeq 1.7\cdot 10^{11}\).

At temperature \(T\), oscillations with frequencies of order \(\omega=\dfrac{kT}{\hbar}\) and smaller are excited. Hence, and from (4.27), it is clear that the dispersion of \(\varepsilon_0\) can be neglected only at temperatures \(T \ll T_k\), or, more precisely, under the condition

\[ T^2 \ll \left(\frac{\hbar\omega_0}{k}\right)^2 . \tag{4.30} \]

For tin this practically leads to the requirement that \(T<0.6 \div 0.8^\circ\). But at low temperatures and, consequently, low frequencies \(\omega\) essential for the heat capacity, as is clear from (4.27), it is no longer possible to assume \(\varepsilon=\varepsilon_0\) and the velocity \(u=\dfrac{c}{\sqrt{\varepsilon_0}}\). For example, for tin, taking from the data\(^{48,49}\) \(\delta_0=6.3\cdot 10^{-6}\) at \(T=0\), we obtain:

\[ n_s=\frac{mc^2}{4\pi e^2\delta_0^2} =\frac{2.83\cdot 10^{11}}{\delta_0^2} =7\cdot 10^{21} \]

and

\[ \omega_s=\sqrt{\frac{4\pi e^2 n_s}{m}}=4.75\cdot 10^{15}. \]

Further, taking \(n_c=5\cdot 10^{22}\) and \(\omega_0=1.7\cdot 10^{11}\) (see above), we find that, according to (4.27) and (4.29),

\[ \varepsilon=\frac{\omega_c^2}{\omega_0^2} -\frac{\omega_s^2}{\omega^2} =5\cdot 10^9-\frac{2.25\cdot 10^{31}}{\omega^2}, \]

and, consequently, one may put

\[ \varepsilon \simeq \varepsilon_0=\frac{\omega_c^2}{\omega_0^2} \]

only at frequencies satisfying the condition

\[ \omega^2 \gg \frac{\omega_s^2\omega_0^2}{\omega_c^2}. \tag{4.31} \]

*) It must be stipulated that the question of the values of the frequencies \(\omega_{0k}\) is not entirely clear, especially in connection with the remarks given at the beginning of p. 85.

Substituting the above values for tin into (4.31), we obtain the condition \(\omega^2 \gg 4,10^{21}\), or, practically, \(T \sim \dfrac{\hbar \omega}{k} \gg 1^\circ\).

We thus see that, in the case under consideration, the formulas (4.26), strictly speaking, have no range of applicability. However, if one neglects the dispersion and the temperature dependence of \(\varepsilon_0\), one can obtain other formulas whose applicability from the side of low temperatures is not restricted. The velocity of light in the metal, which is of interest to us for calculating thermodynamic quantities, is equal to \(u = \dfrac{c}{\sqrt{\varepsilon}} = \dfrac{\omega}{k}\), where \(k\) is the wave vector. Hence, from (4.27), we immediately have:
\[ \omega^2=\frac{\omega_s^2}{\varepsilon_0}+\frac{c^2 k^2}{\varepsilon_0} =\frac{\omega_s^2\omega_0^2}{\omega_c^2} +\frac{\omega_0^2 c^2 k^2}{\omega_c^2}. \tag{4.32} \]

In other words, the energy of the excitations (photons) has the form
\[ E(p)=\hbar\omega=\sqrt{\frac{\hbar^2\omega_s^2}{\varepsilon_0}+\frac{c^2p^2}{\varepsilon_0}}, \]
where \(p=\hbar k\). Starting from the spectrum (4.32) and taking into account that the excitations obey Bose statistics, one can calculate all thermodynamic quantities. In doing this, if the region of frequencies satisfying condition (4.31) is significant, then, of course, the former result (4.26) is obtained. In the other limiting case of very low temperatures, to which we shall confine ourselves, only long waves are excited, so that \(c^2k^2\ll\omega_s^2\), and
\[ \omega=\frac{\omega_s\omega_0}{\omega_c} +\frac{\omega_0 c^2 k^2}{2\omega_s\omega_c}. \tag{4.33} \]

In this case, for the excitations, just as for rotons in helium II, one may use classical statistics, and as a result one obtains without difficulty \((E=\hbar\omega,\ p=\hbar k)\)
\[ \left. \begin{aligned} F_\phi&=-2kT\int e^{-\frac{\hbar\omega}{kT}}\, \frac{4\pi p^2\,dp}{(2\pi\hbar)^3} \\ &=-2kT\left(\frac{\mu kT}{2\pi\hbar^2}\right)^{3/2} e^{-\frac{\Delta}{kT}} =-kTN_\phi, \\[6pt] c_\phi&=2\left(\frac{\mu}{2\pi\hbar^2}\right)^{3/2} \frac{k^{1/2}\Delta^2}{T^{1/2}}\, e^{-\frac{\Delta}{kT}} \left[1+\frac{3kT}{\Delta} +\frac{15}{4}\left(\frac{kT}{\Delta}\right)^2\right], \\[6pt] \Delta&=\frac{\hbar\omega_s\omega_0}{\omega_c}, \qquad \mu=\frac{\hbar\omega_s\omega_c}{\omega_0c^2}, \end{aligned} \right\} \tag{4.34} \]
where \(N_\phi\) is the number of excitations, which behave like particles with mass \(\mu\) and “rest energy” \(\Delta\) (indeed, according to (4.33),
\[ E=\Delta+\frac{p^2}{2\mu} \]
).

In the case of tin, formulas (4.34) may be used for

\[ T \ll \frac{\Delta}{k} \sim 0.5^\circ . \]

In addition to transverse waves, in a medium where \(\varepsilon\) can vanish, longitudinal oscillations are also possible with a frequency determined, for \(k \to 0\), from the condition \(\varepsilon(\omega)=0\) (such oscillations are known for a plasma and are often called plasma oscillations\(^*\)). In our case this frequency is equal to

\[ \omega=\frac{\omega_s \omega_0}{\omega_c}. \]

But one cannot restrict oneself to such an expression for \(\omega\), since in order to calculate the free energy one must have an expression for \(\omega\) in the next approximation—with allowance for its dependence on the wave vector \(k\). Such an expression can be found only for a concrete model, for example for a degenerate Fermi gas. It must be noted, however, that for transverse waves as well the expression (4.32) is valid only when the thermal and, generally speaking, internal motion in the system is neglected. Therefore in this case too formula (4.32) can be refined only for a concrete model. However, for small \(k\) the desired expression can be obtained by adding to (4.32) a term \(\sim v_0^2 k^2\), i.e. it takes the form

\[ \omega^2=\frac{\omega_s^2}{\varepsilon_0}+\frac{c^2 k^2}{\varepsilon_0}+v_0^2 k^2, \]

where \(v_0\) is a characteristic velocity of the internal motion; in the case of the longitudinal wave mentioned above the same expression is obtained, but without the term \(\dfrac{c^2 k^2}{\varepsilon_0}\) and with a somewhat different value of \(v_0^2\) than for transverse waves. For a classical charged gas (plasma)

\[ v_0^2 \sim \frac{kT}{m}, \]

for a degenerate Fermi gas

\[ v_0^2 \sim \frac{p_0^2}{m^2} \]

(for longitudinal oscillations\({}^{44}\)

\[ v_0^2=\frac{3}{5}\frac{p_0^2}{m^2}, \]

for transverse waves, as reported to the author by V. P. Silin,

\[ v_0^2=\frac{p_0^2}{5m^2} \]

).

Under ordinary conditions, for transverse waves the term \(v_0^2 k^2\) and possible terms \(\sim k^4\) may be neglected. But in our case the quantity

\[ \frac{c^2 k^2}{\varepsilon_0} \]

may turn out to be smaller than \(v_0^2 k^2\). For a Fermi gas

\(^*\) In papers\({}^{50}\) ideas about a plasma and its oscillations are made the basis of the theory of superconductivity. In doing so, however, the starting equations are taken to be equations equivalent to (1.1)—(1.2), which in fact must be justified. Further, the authors proceed from the completely erroneous opinion that the heat capacity in the superconducting state is smaller than in the normal one (!?) Therefore, taking for the plasma frequency the frequency \(\omega=\omega_s\) (they do not take into account the influence of \(\varepsilon_0\)), the authors erroneously believe that the resulting quite negligible electronic part of the heat capacity of a superconductor agrees with experimental data. In reality, electromagnetic excitations of both longitudinal and transverse character can be of interest for the theory of superconductivity only if the quantity \(\varepsilon_0\) is very large (of order \(10^9 \div 10^{10}\)).

this does take place. How matters stand for the electron liquid in a metal remains unclear, but in this case the value of \(\omega_0^2\) may be considerably smaller than in a gas. If this is not so, then the basic expression (4.27) for \(\varepsilon\), and to a large extent all subsequent reasoning, are incorrect. More precisely, if terms of order \(k^4\) and higher are not taken into account (under some conditions this may prove necessary), then one may still write

\[ \varepsilon=\varepsilon_0-\frac{\omega_s^2}{\omega^2}, \]

but the estimates of the quantities \(\varepsilon_0\) and \(\omega_s^2\) change\(*\). Additional difficulties are associated with taking damping (absorption) into account, which was neglected above, but which, beginning with some frequency, must occur also at \(T\to0\).

In addition to all that has been said, it must be borne in mind that, from considerations of the continuity of the transition from the superconducting state to the normal state, it follows that \(\varepsilon_0\) depends on temperature and, as \(T\to T_k\), tends to the value corresponding to the normal metal, which is relatively small and may be set equal to zero (despite the fact that in absolute magnitude \(\varepsilon\) in a normal metal may reach a value \(\sim 10^6\). This is confirmed experimentally\(^{46}\), where for \(T\to T_k\) practically \(\varepsilon_0\to0\). Meanwhile in (4.26) the dependence of \(\varepsilon_0\) on \(T\) in the transition from \(F\) to \(S=-\dfrac{\partial F}{\partial T}\) and \(c=T\dfrac{\partial S}{\partial T}\) was not taken into account. If one starts from expression (4.26) for \(F\), then, taking into account the dependence of \(\varepsilon_0\) on \(T\), we obtain:

\[ c_{\phi}= \frac{4\pi^2 k^4 \varepsilon_0^{3/2} T^3}{15\hbar^3 c^3} \left[ 1+\frac{T}{\varepsilon_0}\frac{d\varepsilon_0}{dT} +\frac{T^3}{16\varepsilon_0^2}\left(\frac{d\varepsilon_0}{dT}\right)^3 +\frac{T^2}{8\varepsilon_0}\frac{d^2\varepsilon_0}{dT^2} \right]. \tag{4.37} \]

However, there is no complete certainty that, in the case considered below, it is the expression for the free energy \(F\) that should be taken as the starting point. The latter is valid if \(\varepsilon_0\) is a dielectric constant having, so to speak, an external character with respect to the electromagnetic waves under consideration. In that case \(\dfrac{\varepsilon_0 E^2}{8\pi}\) is the free energy, and it is this that must be computed. But in our case the quantity \(\varepsilon_0\), apparently, is closely connected with the electromagnetic oscillations themselves. Indeed, just as in the case of superfluidity a part of the density of the liquid is connected with the excitations (see (4.14)), so also the excitations in a superconductor must apparently be attributed to the normal part of the electron liquid. Thus, if one assumes that the excitations are photons with velocity

\[ \text{\(*\) In this respect one may apparently conclude that the theoretical inference that in a superconductor \(\varepsilon_0\sim 10^4 \div 10^{10}\) (if \(n_c\sim n_0\); see (4.28)) is not conclusive. Therefore the experimental determination of \(\varepsilon_0\) is especially important.} \]

\(u=\dfrac{c}{\sqrt{\varepsilon_0}}\), where \(\varepsilon_0=\mathrm{const}\), then, by analogy with (4.14), one can establish that the effective concentration of electrons associated with the excitations is equal to (see also (4.26)*):

\[ n_{\mathrm{B}}=\frac{4E_{\phi}}{3mu^2} =\frac{4\pi^2(kT)^4\varepsilon_0^{3/2}}{45\hbar^3mc^5}. \tag{4.36} \]

For tin at \(T_{\mathrm{k}}=3.7^\circ\), putting \(\varepsilon_0=5\cdot10^9\), we find \(n_{\mathrm{B}}\simeq 4\cdot10^8\). This value is still considerably smaller not only than \(n_c\simeq 5\cdot10^{22}\), but also than \(n_s=7\cdot10^{21}\). However, the inaccuracy of formula (4.36) by no means excludes the possibility that \(n_{\mathrm{B}}\) at \(T_{\mathrm{k}}\) will be comparable with \(n_c\) at \(T=0\). Under such conditions the quantity \(\varepsilon_0(T)\), which depends on \(n_c(T)\), is not independent of the electromagnetic oscillations themselves, and it may turn out to be more correct to calculate first not the free energy, but the entropy (the difference is connected with the fact that only \(\varepsilon_0\) enters into the quantity being determined, whereas the derived quantities also depend on \(\dfrac{d\varepsilon_0}{dT}\); for this remark the author is indebted to L. D. Landau).

In summary, one may say that the quantitative aspect of the question of the contribution of thermal radiation in a superconductor to its heat capacity is still not clear. However, the need to take into account in superconductors the role of thermal radiation, or, in other words, excitations of the photon type, is beyond doubt (provided, of course, that further experiments confirm the conclusion that \(\varepsilon_0\simeq 10^9\text{--}10^{10}\)). The contribution of these excitations to the heat capacity may even prove to be the principal one. Thus, if for the heat capacity \(C\) one uses, for orientation, expression (4.26), then for tin at \(\varepsilon_0=5\cdot10^9\), for temperatures \(1\) and \(2^\circ\), the calculated value of the heat capacity is only about a factor of two smaller than the measured \(^{52}\) values.

Photon excitations satisfy the condition of superconductivity (4.20), and taking them into account, as has already been mentioned, makes it possible to avoid some of the difficulties associated with adopting the spectrum (4.21), which for brevity we shall call quasi-Fermi. Indeed, the noted difference in the values of \(a_n\) and \(a_s\), and, consequently, in the limiting momenta \(p_0\) in the normal and superconducting phases, was connected with the assumption that the excitations (4.21)

*) Just as in the case of helium II (see (4.14)), the density \(\rho_{\mathrm{B}}=mn_{\mathrm{B}}\) depends on the velocity of the liquid \(v\). Such a dependence should lead (see \(^{51}\) and the second addendum) to the nonlinearity of electrodynamic processes in a superconductor, since the velocity \(v\) is directly connected with the magnetic field \(H\) (see (4.19)). However, the nonlinear phenomena actually observed in superconductors are small (see the addendum to the present article). This indicates that even in a field \(H_{\mathrm{k}}\), \(v^2\ll u^2\), just as occurs in helium II, where \(v_c\ll u\). Therefore in (4.36) the dependence of \(E_{\phi}\) on \(v\) has not been taken into account.

are responsible for the entire heat capacity of the metal. Taking into account the presence of phonon excitations, such a requirement is already simply inadmissible, and one may put \(a_s = a_n\) (see (4.22)—(4.23)). Further, a considerable part of the “normal” electronic liquid may be associated with phonon excitations (see (4.36)). This point is essential from the standpoint of understanding the character of the continuous transition to the normal state. To this one must add that, as \(T \to T_k\), just as occurs in helium II on approaching the \(\lambda\)-point, the representation of a gas of excitations may prove unsuitable, and it is necessary to take into account the interaction of excitations, which weakens the acuteness of the question of the continuity of the change, for example, of the normal conductivity upon passage through the critical temperature.

On the other hand, the adoption of the quasi-Fermi spectrum (4.21) appears attractive to us for a whole number of reasons. First, the introduction of a certain gap separating the ground state from the excited states is dictated by all the considerations indicated in § 3. The gap is needed to ensure the “rigidity” of the wave function in the ground state, which leads to superconductivity. The destruction of superconductivity by a magnetic field in the presence of a gap is connected with the Zeeman shift of the levels, leading to “closing” of the gap. Second, the kinship of the quasi-Fermi spectrum (4.21) with the Fermi spectrum appears (along with the already noted increase in the number of excitations) to be a positive feature from the standpoint of understanding the transition of a superconductor to the normal state with a Fermi spectrum, although the question of this transition remains unclear. Finally, third, the introduction of the gap \(\Delta\) can also be connected with the appearance of a large \(\varepsilon_0\), and, consequently, with the appearance of a phonon spectrum. Indeed, as follows from the theory of dispersion of electromagnetic waves in metals (see\(^7\) § 4, 2) and is clear immediately, the part of \(\varepsilon\) inversely proportional to \(\omega^2\) (see (4.27)) is not connected with transitions to another state; the value of \(\varepsilon_0\) is determined by transitions of the electron to other bands. Since the distance between bands is large, i.e. the frequencies \(\omega_{0k}\) in (4.27) are large, the value of \(\varepsilon_0\) in the normal state is relatively small. The appearance of a gap at the Fermi boundary within the framework of the existing one-electron model of the metal may be connected with the appearance of a new band (for example, with a doubling of the number of bands; see below). In this case the new band is so close to the filled one that the transition frequencies are small (the lowest of them are of order \(\Delta/\hbar\)), and the value of \(\varepsilon_0\) becomes large. Without relying on purely limited band notions, especially from the standpoint of the theory of superconductivity, one may simply say that the appearance of a large \(\varepsilon_0\) is connected with the presence of low eigenfrequencies \(\omega_0 \sim kT_k/\hbar\), and so...

which frequencies, obviously, must also appear upon the introduction of the gap \(\Delta\), which according to § 3 is just of order \(kT_k\).

Thus, as it seems to us, there are grounds for thinking that the spectrum of a superconductor is combined and unites excitations of the type (4,21) and excitations of the photon type. In this connection the initial feature determining the appearance of the superconducting state with all its peculiarities is the existence of a certain gap \(\Delta \sim kT_k\), separating the ground state of the metal from its excitations of electronic type (for photon excitations, as is clear from (4,32), at \(k=0\) the gap is obtained automatically, but in this case it is in a certain sense inessential).

The mechanism of formation of the gap still remains entirely unclear. This was already discussed in § 3 c), and here we confined ourselves to pointing out the interesting work \(^{55}\), where the possibility is discussed of the formation of a gap connected with a change (for example, a doubling) of the period of the crystal lattice and, consequently, with a doubling of the number of zones in the crystal. If the change of the lattice is extremely small—and similar examples are known (antiferromagnets)—then the gap between the newly formed zones will also be negligible. In a metal the appearance of a small gap even at the Fermi surface itself will apparently be inessential until the uncertainty in the electron energy

\[ \Delta E \sim \frac{\hbar}{\tau}=\hbar \nu, \]

connected with the finiteness of the mean free time \(\tau=\frac{1}{\nu}\), becomes smaller than, or at least of the order of, the width of the gap \(\Delta\). Even in a very pure metal at low temperature, because of the residual resistance, \(\nu \sim 10^{10} \div 10^{11}\), and

\[ \frac{\Delta E}{k}\sim \frac{\hbar\nu}{k}\sim 0.1-1^\circ . \]

It remains unclear, however, whether by \(\tau\) one should understand only the mean free time associated with inelastic collisions. In general, the interesting question of the influence of a very narrow gap (in the limit, a gap tending to zero) on transport processes in a metal has not yet been investigated; but it is beyond doubt that a sufficiently narrow gap cannot exert any substantial influence on any processes.

In \(^{55}\) the appearance of a very narrow gap is invoked to explain the anomalous behavior of the resistance of nonsuperconducting metals as \(T \to 0\) (see, for example, \(^{2}\) § 23 and \(^{4}\)). The question of the possible identification of a gap of this type with the gap needed for the explanation of superconductivity deserves, in our opinion, close attention*). The connection of such a gap with the lattice, having—

*) From this point of view, an attempt at an experimental investigation of the question of the presence in the superconducting state of some kind, even very small, change of the lattice and especially of its period as compared with the normal state would be of particular interest.

... arising from the very beginning, appears natural from the point of view of the isotope effect; to explain it in the gap model one must assume that the width of the gap \(\Delta\) depends on the mass of the nucleus \(M\) (see § 3 c)).

The question of the nature of the gap belongs rather to the microscopic theory than to the quasi-microscopic one; but, of course, any opposition of the two approaches is entirely inappropriate, and the only correct course is their joint use. It is precisely this path that seems the most promising.

5. CONCLUSION

As a result of all that has been set forth, the situation in the field of the microscopic theory of superconductivity appears to us as follows. There are no grounds for the hypothesis of spontaneous currents (§ 2), and the only hypothesis deserving attention is the diamagnetic hypothesis (§ 3). According to the latter, the superconducting current is analogous to the diamagnetic current in atoms, and superconductivity can arise as a result of the “rigidity” of the wave function for a part of the conduction electrons in a metal. Such “rigidity” is most likely a consequence of the existence of a certain energy gap separating the ground state of a superconducting metal from its weakly excited state, corresponding to the presence in the metal of some number of “elementary excitations.”

In the language of excitations, the appearance of superconductivity is connected with a definite condition imposed on the dependence of the excitation energy on their momentum (see (4.20)). The question of the causes leading to the appearance of an energy gap in the spectrum, and in general of obtaining the spectrum of electrons in a metal in any sufficiently rigorous way, remains open.

All attempts undertaken in recent years to construct a microscopic theory of superconductivity (the works of Heisenberg and Koppe\(^{10-11}\), Born and Cheng\(^{12}\), Bardeen\(^{23}\), Fröhlich\(^{33,37}\), Tisza and Jüttinger\(^{54,55}\), Mellich and Rompe\(^{56}\), etc.) are, in our opinion, either erroneous or devoid of positive elements. This is explained by various reasons: by the adoption of the hypothesis of spontaneous currents, by the use of perturbation theory beyond its range of applicability and by the ignoring of the whole body of available experimental facts, as well as of the deep analogy between superfluidity and superconductivity. As one of the most essential examples in this respect we note that in none of the works listed is the presence in a superconductor of an enormous dielectric constant \(\varepsilon_0\) taken into account.

The development of the microscopic theory of superconductivity is connected with the construction of a new model of the metal in place of the existing one-

of the electronic (gas) model or, in any case, with an essential generalization or supplement to the latter. This, to a large extent, explains the exceptional difficulties encountered in creating a microscopic theory of superconductivity. It is rather difficult here to hope for the success of more or less rigorous methods of analyzing the spectra of a many-particle system. Therefore, the quasimiсroscopic approach to the theory of superconductivity and its combination with certain model considerations (§ 4) appears the most promising and fruitful.

In this respect, a continuous connection with experiment is especially important, as is the continuous testing of theoretical constructions against experiment. From the point of view of the microscopic theory of superconductivity, the following experimental investigations are most important: a comprehensive clarification of the question of the dependence of the dielectric constant \(\varepsilon_0\) on frequency and temperature; the most accurate possible determination of the heat capacity of superconductors in the superconducting and normal states, \(C_s\) and \(C_n\) (especially at low temperatures); determination by optical methods of the concentrations of free electrons in the normal state, \(n_0\); finding, for the same metals (and best of all for the same specimens for which \(\varepsilon_0\), \(C_s\), \(C_n\), and \(n_0\) are measured), the electrical conductivity \(\sigma(\omega, T)\), as well as the thermal conductivity \(\lambda(T)\); for the same specimens it is, of course, also necessary to know the penetration depth in a static field, \(\delta_0(T)\) (the determination of all three quantities \(\varepsilon_0(\omega,T)\), \(\delta(\omega,T)\), and \(\delta_0(T)\) reduces to finding the complex dielectric constant \(\varepsilon'(\omega,T)\)). Finally, X-ray (or some other, in methodology) attempts to detect even the very slightest changes in the crystalline structure of a metal upon its transition to the superconducting state would also be of great interest.

One may think that the rapid development of experiment and further theoretical work, taking into account the points already clarified in this article, will in the near future lead to the creation of a microscopic theory of superconductivity which, in its completeness and persuasiveness, will not be inferior to theoretical constructions successfully used in other areas of solid-state physics and low-temperature physics.

ADDENDUM. FURTHER DEVELOPMENT OF THE MACROSCOPIC THEORY OF SUPERCONDUCTIVITY

In roughly the year and a half that has passed since the writing of the first part of this article (cited as I), a number of questions concerning the macroscopic theory of superconductivity considered in I have been clarified. The aim of the present addendum is to illuminate these questions.

a) Behavior of thin superconducting films, cylinders, and spheres in a magnetic field

From the macroscopic theory of superconductivity developed in \(2^{6}\) (see I, §§ 3, 4) there follows a very peculiar behavior of superconducting films, as well as of superconducting specimens of other shapes (cylinders, spheres) with small dimensions. Recently, some conclusions of the theory have been confirmed experimentally \(^{48,49,20}\). In this connection it is expedient to discuss, in greater detail than in I, the properties of superconducting films \(^{56}\) and certain related questions \(^{57,58}\).

Both from the point of view of the transparency of the theoretical conclusions and from the point of view of an unambiguous interpretation of the experimental data, thin films are of special importance; for them

\[ \left( \frac{\chi d}{\delta_0} \right)^2 \ll 1, \tag{D1} \]

where \(d\) is the half-thickness of the film, \(\delta_0\) is the penetration depth in a weak field, and \(\chi\) is the characteristic parameter entering the theory (see I) and, for Hg and Sn, is equal to approximately \(0.15\), or even to a still smaller value.

In view of the smallness of \(\chi^2\), for films with \(d \lesssim \delta_0\) one may, to a good approximation, set \(\chi = 0\) altogether, i.e., as is clear from I, neglect the dependence of the concentration of superconducting electrons \(n_s = |\Psi|^2\) on the coordinates. In this case the whole theory assumes an especially simple form and can easily be developed independently of the more general consideration undertaken in \(2^{6}\) and I.

The density of the free energy of the normal phase, taking into account the energy of the magnetic field, is equal to

\[ F_{nH} = F_{n0} + \frac{H_0^2}{8\pi}, \tag{D2} \]

where \(F_{n0}\) is the free energy without taking into account the field energy, and \(H_0\) is the intensity of the external magnetic field, which below is assumed to be parallel to the film.

The density of the free energy of the superconducting phase for \(\chi = 0\) is equal to (see (I; 3,5)):

\[ F_{sH} = F_{n0} + \alpha |\Psi|^2 + \frac{\beta}{2} |\Psi|^4 + \frac{e^2}{2mc^2} A^2 |\Psi|^2 + \frac{H^2}{8\pi}, \tag{D3} \]

where \(H = \dfrac{dA}{dz}\) is the field strength in the metal.

The density of the superconducting current is

\[ \mathbf{j}_s = -\frac{e^2}{mc} |\Psi|^2 \mathbf{A}, \]

and the parameter usually introduced into the theory is

\[ \Lambda = \frac{m}{e^2 n_s} = \frac{m}{e^2 |\Psi|^2} \quad \text{(see (I; 2,9)).} \]

Therefore the density (D3) is equal to \(F_{sH}=F_{n0}+\alpha|\Psi|^{2}+\dfrac{\beta}{2}|\Psi|^{4}+\dfrac{N_s^2}{2}+\dfrac{H^2}{8\pi}\) and differs, at \(\chi=0\), from the energy density used in the old theory (see (1; 2,22)) only by the terms \(\alpha|\Psi|^{2}+\dfrac{\beta}{2}|\Psi|^{4}=\alpha n_s+\dfrac{\beta}{2}n_s^2\), which take into account the dependence of the free energy on the concentration of superconducting electrons and have the form usual in the theory of phase transitions of the second order.

In the absence of a magnetic field, in the state of equilibrium,

\[ \frac{dF_{sH}}{dn_s}=0, \]

i.e.

\[ n_s=|\Psi_\infty|^2=-\frac{\alpha}{\beta} \]

and

\[ F_{n0}-F_{s0}=\frac{H_{\mathrm{cm}}^2}{8\pi}=\frac{\alpha^2}{2\beta}, \]

where \(H_{\mathrm{cm}}\) is the critical magnetic field in the case of a massive metal. The magnetic field at \(\chi=0\) is determined by the equation (see (1; 3,8)):

\[ \Delta A=-\frac{4\pi j_s}{c}=\frac{4\pi e^2}{mc^2}|\Psi|^2 A. \tag{D4} \]

For what follows it is convenient to pass to the reduced concentration of superconducting electrons

\[ \Psi_0^2=\frac{\Psi^2}{\Psi_\infty^2} \]

(\(\Psi_\infty^2=-\dfrac{\alpha}{\beta}\)—the concentration at \(H_0=0\)), where, instead of moduli, the quantities themselves are used, since in the case of interest to us the function \(\Psi\) is real.

Equation (D4) and its solution for a plate of thickness \(2d\), situated in an external field \(H_0\), take the form

\[ \frac{d^2 A}{dz^2}-\frac{\Psi_0^2}{\delta_0^2}A=0,\qquad A=\frac{\delta_0 H_0\,\operatorname{sh}\dfrac{\Psi_0 z}{\delta_0}} {\Psi_0\,\operatorname{ch}\dfrac{\Psi_0 d}{\delta_0}},\qquad H=\frac{dA}{dz}= \frac{H_0\,\operatorname{ch}\dfrac{\Psi_0 z}{\delta_0}} {\operatorname{ch}\dfrac{\Psi_0 d}{\delta_0}}, \tag{D5} \]

where, let us recall once more, \(\delta_0\) is the penetration depth in a weak field \(H_0\to0\) (for \(H_0=0\), \(\Psi_0^2=1\)); the coordinate \(z\) is measured from the middle of the plate, and the field \(H\) and the potential \(A\) are directed respectively along the axes \(y\) and \(x\) (see 1, § 3).

We now find the free energy of the plate in the superconducting state \(F'_{sH}\), referred to unit volume. For this purpose one must integrate expression (D3) with respect to \(z\) from \(-d\) to \(+d\), where \(A\) and \(H\) have the values (D5), and divide the result by \(2d\). In addition, since we are interested in the total energy of the metal and the field, it is necessary to take into account the change in the energy of the field located outside the plate (this change appears explicitly if the plate is finite). For this purpose, as shown in 1 (see also \(^{5}\)), to the density \(F_{sH}\) one must add the density of the energy of “magnetization”

— \(M H_0=-\dfrac{H_a-H_0}{4\pi}H_0\). As a result the energy
\(F'_{sH}=\dfrac{1}{2d}\displaystyle\int_{-d}^{d}(F_{sH}-M H_0)\,dz\) is equal to*)

\[ F'_{sH}=F'_{nH}+\frac{H_0^2}{8\pi} \left(1-\frac{\operatorname{th}\dfrac{\Psi_0 d}{\delta_0}} {\dfrac{\Psi_0 d}{\delta_0}}\right) +\frac{H_{\mathrm{km}}^2}{8\pi}\left(\Psi_0^4-2\Psi_0^2\right), \tag{D6} \]

where the transition has already been made to the units \(\Psi_0^2=\dfrac{\Psi^2}{\Psi_\infty^2}\), and it has been taken into account that

\[ \frac{H_{\mathrm{km}}^2}{8\pi}=\frac{\alpha^2}{2\beta} \quad\text{and}\quad \Psi_\infty^2=-\frac{\alpha}{\beta}. \]

The quantity \(\Psi_0\) in the equilibrium state is determined from the condition of minimality of \(F'_{sH}\), i.e. from the condition

\[ \frac{\partial F'_{sH}}{\partial \Psi_0}=0, \]

which immediately leads to the equation

\[ \left(\frac{H_0}{H_{\mathrm{km}}}\right)^2 = \frac{4\Psi_0^2(\Psi_0^2-1)\operatorname{ch}^2\dfrac{\Psi_0 d}{\delta_0}} {1-\dfrac{\operatorname{sh}\dfrac{2\Psi_0 d}{\delta_0}} {2\Psi_0\dfrac{d}{\delta_0}}}. \tag{D7} \]

The dependence of \(F'_{sH}-F'_{nH}\) on \(\Psi_0\) according to (D6) is clear from Figs. 4 and 5, while the dependence of the extremal value \(\Psi_0\), determined from (D7), is shown in Fig. 6.

All curves are divided into two classes depending on the value of the ratio \(\dfrac{d}{\delta_0}\); moreover, as is shown below (see also I), the critical value \(d_k\), lying on the boundary between these classes, is equal to

\[ d_k=\frac{\sqrt{5}}{2}\,\delta_0=1.12\,\delta_0. \tag{D8} \]

If \(d<d_k\) (Fig. 4, curve \(b\), and in Fig. 6), equation (D7) has, for \(\Psi_0\ne0\), only the solution corresponding to a minimum (i.e. to a stable superconducting state); moreover, with increasing field \(H_0\), the value of \(\Psi_0\) at this minimum decreases and becomes zero at

\[ \frac{H_{k1}}{H_{\mathrm{km}}}=\sqrt{6}\,\frac{\delta_0}{d}. \tag{D9} \]

(This value is also the condition that the derivative \(\dfrac{\partial^2 F'_{sH}}{\partial \Psi_0^2}\) vanish at \(\Psi_0=0\).)

*) In the normal state the field in the plate is homogeneous and equal to \(H_0\). Therefore the energy per unit volume \(F'_{nH}\) is simply equal to the energy density

\[ F'_{nH}=F_{n0}+\frac{H_0^2}{8\pi}. \]

The normal phase, in which \(\Psi_0=0\), for \(d<d_k\) and \(H_0<H_k\), cannot exist even as a metastable one. Conversely, for \(H_0>H_{k1}\) the superconducting phase cannot exist. This case is that of a second-order phase transition—in the phase-transition point \(\Psi_0=0\). Hysteresis in this region is obviously impossible.

Fig. 4.

If \(d>d_k\), then for \(H_0<H_{k1}\) equation (D7) has only the solution corresponding to a minimum, but for \(H_{k2}>H_0>H_{k1}\) there is also a solution corresponding to a maximum (the region below points \(A\) on curves \(a\) and \(b\) of Fig. 6; see also Fig. 5).

In this region of fields both the normal and the superconducting phases can exist, one of them being metastable (for example, the normal phase is metastable if \(F'_{nH}>F'_{sH}\)). The phase transition under equilibrium conditions takes place where \(F'_{sH}=F'_{nH}\), i.e., according to (D6), in the field \(H_k\), determined from the condition

\[ \left(\frac{H_k}{H_{km}}\right)^3 = \frac{\Psi_0^2(2-\Psi_0^2)} {1-\dfrac{\operatorname{th}\Psi_0\dfrac{d}{\delta_0}} {\Psi_0\dfrac{d}{\delta_0}}}. \tag{D10} \]

To find the field \(H_k\) and the corresponding value \(\Psi_{0k}\) at the transition point, which is evidently a first-order transition,

one must solve jointly equation (D10) and equation (D7) with \(H_0=H_k\) (these equations then coincide with (I; 4.26), (I; 4.27)).

In a field \(H_0>H_{k2}\), where the field \(H_{k2}\) corresponds to the maximum on the curves of Fig. 6, the superconducting phase cannot exist.

Fig. 5.

It follows easily from (D7) and Fig. 6 that for \(\Psi_0=0\), \(\dfrac{\partial H_0}{\partial\Psi_0}=0\), and at the point \(d=d_k\), moreover, \(\dfrac{\partial^2 H_0}{\partial\Psi_0^2}=0\); from this last condition it is especially easy to arrive at formula (D8). The existence of a critical thickness \(d_k\), below which hysteresis is impossible and the critical field must be determined by expression (D9), was confirmed by the work of Refs. \(^{48,49}\).

For films with a half-thickness \(d\) greater than \(d_k=1.12\delta_0\), as was said, the transition from the superconducting to the normal phase, and conversely, is a first-order transition, and the existence of metastable normal and superconducting phases is possible, and hence also the presence of hysteresis. It is characteristic here that the regions of existence of metastable phases are bounded. Thus, the normal phase

Fig. 6.

In the figure: \(\psi\) (according to 27); curves \(a)\), \(b)\), \(c)\); \(d/\delta_0=2.05\), \(d/\delta_0=1.37\), \(d/\delta_0=1.0\); abscissa \(H_0/H_{\mathrm{KM}}\); marks \(\left(H_{k1}/H_{\mathrm{KM}}\right)_a\), \(\left(H_{k1}/H_{\mathrm{KM}}\right)_b\), \(\left(H_{k1}/H_{\mathrm{KM}}\right)_c\).

can exist only in a field \(H_0>H_{k1}\) (see (D9)). Therefore the region of “supercooling” (i.e., the region of existence of the metastable normal phase) corresponds to fields \(H_0\) in the interval \(H_{k1}<H_0<H_k\). The region of “superheating,” where a metastable superconducting phase may be observed, lies at \(H_k<H_0<H_{k2}\). The maximum amplitude of the hysteresis is thus bounded by the fields \(H_{k1}\) and \(H_{k2}\) (see Fig. 7, which gives the dependence of the fields \(H_{k1}\), \(H_k\), and \(H_{k2}\) on \(d/\delta_0\); note that, in comparing theory with experiment, it is more convenient to present the critical fields as a function of \(\delta_0/d\), and not \(d/\delta_0\), since the dependence of \(H_{k1}/H_{\mathrm{KM}}\) on \(\delta_0/d\) is represented by a straight line).

In experiment the hysteresis is very asymmetric in the sense that supercooling is observed without particular difficulty, whereas superheating is usually, apparently, not noticeable\(^{48}\); the same applies to massive specimens, where superheating has been observed only recently (there is an indication of this in \(^{58,104}\)). It is difficult to doubt that such an asymmetry is connected with the substantially different conditions in which, in the presence of a magnetic field, the nuclei of the normal and superconducting phases find themselves, possessing completely different “magnetic” properties. This question has not yet been examined in detail.

With increasing film thickness the hysteresis should at first increase (for \(d \leq d_k\) it is zero), but then it ceases to grow and, probably, even decreases. The point is that the phase transition in specimens of small dimensions is impeded, since in this case

Fig. 7.

the specific weight of the surface energy of the nuclei is especially large. The transition from the superconducting phase to the normal one is likewise not sharp, but occurs in the region with thickness \(\frac{\delta_0}{\chi} \sim 10^{-4}\ \mathrm{cm}\) (see below). Such boundaries in superconductors of small dimensions (\(d \sim \delta_0 \sim 10^{-5}\)) cannot exist at all; when the transition layer is thinned, the energy increases sharply. Therefore specimens of small dimensions must pass from one phase to another

as a whole, which will lead to an increase in the hysteresis as the size of the specimen increases. But when sufficiently large sizes are reached, the formation of nuclei is facilitated, and there are not only no longer grounds for an increase of the hysteresis, but it may even decrease.

Fig. 8.

In addition to the critical field, one can also measure the magnetic moment of the film as a function of the strength of the external field \(H_0\). The magnetic moment of the film, referred to a unit area of its surface, is equal to

\[ \mu=\int_{-d}^{d}\frac{H-H_0}{4\pi}\,dz=2d\overline{M}\quad \text{(see (I; 4,16) and} \]

\[ \text{(I; 4,32)),} \]

where the mean “magnetization” \(\overline{M}\) is equal to

\[ \overline{M}=-\frac{H_0}{4\pi} \left( 1-\frac{\operatorname{th}\dfrac{\Psi_0 d}{\delta_0}} {\Psi_0\dfrac{d}{\delta_0}} \right), \tag{D11} \]

where the function \(\Psi_0\) is determined by equation (D7).

The dependence of \(\overline{M}\) on \(H_0\) is clear from Fig. 8 and is in qualitative agreement with the results obtained in \({}^{30}\) (for a number of reasons, in the present case quantitative agreement could not be expected). According to the old theory, the dependence of \(\overline{M}\) on \(H_0\) should have had the form of a triangle \((\overline{M}=\mathrm{const}\cdot H_0\) for \(H_0 < H_k;\)

\(\overline{M}=0\) for \(H_0>H_k\)), shown by the dashed line on curve \(a\) in Fig. 8. If one disregards the technical difficulties, then formula (D11) is best checked on films with \(d<d_k\), when the effect of the dependence of \(\overline{M}\) on \(H_0\) is expressed most clearly and, with strict parallelism of the film and the field, there should be no hysteresis.

Above it was assumed that \(\chi=0\), whereas in fact in known cases \(\chi\simeq 0.15\). Therefore it is now necessary to consider the question of the influence of the magnitude of \(\chi\) on the critical field. For \(d\lesssim d_k\), the corrections connected with the fact that \(\chi\ne 0\) are determined by the parameter \(\left(\dfrac{\chi d}{\delta_0}\right)^2\), and, for example, instead of (D9) we have:

\[ \left(\frac{H_{k1}}{H_{kM}}\right)^2 = 6\left(\frac{\delta_0}{d}\right)^2 \left[ 1+\operatorname{const}\left(\frac{\chi d}{\delta_0}\right)^2 \right], \]

where the constant introduced is of the order of unity or smaller *). Even for \(d\sim d_k\), \(\left(\dfrac{\chi d}{\delta_0}\right)^2\sim 2\div 3\%\), and therefore in the region where a second-order transition takes place the corresponding correction is insignificant. However, in the region where \(\dfrac{\chi d}{\delta_0}\gtrsim 1\), the influence of \(\chi\) is already of first order, i.e., in terms of order \(\chi\), and not \(\chi^2\) (see \(2^{\mathrm{c}}\) and I). Unfortunately, this region is very difficult to investigate by analytical methods, and the formula for \(\dfrac{H_k}{H_{kM}}\) has so far been obtained only in the limiting case \(\dfrac{\chi d}{\delta_0}\gg 1\). This formula is formula (I; 4.20):

\[ \frac{H_k}{H_{kM}} = 1+\frac{\delta_0}{2d} \left(1+\frac{\chi}{8\sqrt{2}}\right), \qquad \frac{\chi d}{\delta_0}\gg 1, \tag{D12} \]

where it has been taken into account that \(\chi\ll 1\). In I and \(2^{\mathrm{b}}\), without sufficient grounds, the inequality \(d\gg\delta_0\) was indicated as the condition for applicability of formula (D12), instead of the stricter inequality \(\dfrac{\chi d}{\delta_0}\sim \dfrac{d}{6\delta_0}\gg 1\) (let us note that for \(\chi=0\) and \(d\gg\delta_0\), formula (D12) with \(\chi=0\) is valid).

In experiment the condition \(\dfrac{\chi d}{\delta_0}\gg 1\) is usually not fulfilled. Therefore the discrepancy \(^{48}\) in the value of \(\delta_0\) obtained from measurement of \(H_k/H_{kM}\) for

*) In \(2^{\mathrm{b}}\) and in (I; 4.33) the value of this constant is given as \(\mathrm{const}=7/60\). However, the corresponding calculation for \(\left(\dfrac{\chi d}{\delta_0}\right)^2\ll 1\), as has become clear, is not sufficiently correct, and therefore formulas (I; 4.33) and (I; 4.34) are apparently not correct. A more precise calculation of corrections of order \(\chi^2\) to the values of \(H_{k1}\) and \(d_k\) has not yet been carried out.

thick films using formula (D12), where the term

\[ \frac{\varkappa}{8\sqrt{2}}\sim 1\% \]

may be neglected, with the value \(\delta_0\) determined by other methods (from (D9) or (D11); see \(^{20,48,49}\)), may be connected with the unsuitability, for insufficiently thick films, of formula (D12). In addition, because of the presence of hysteresis and, in general, the possibility of supercooling and superheating, without further information it is not known with exactly what value of \(H_k\) we are dealing in the experiment. Formula (D12), of course, refers to the “equilibrium” critical field, for which \(F_{sH}=F_{nH}\). The question of the behavior in a magnetic field of thick films requires further experimental and theoretical study.

Besides thin films, samples of another form are also of interest and, first of all, superconducting cylinders and spheres. The behavior of such samples according to the old theory is indicated in I, § 2 and in the literature cited there. The solution of the problem of a superconducting cylinder and sphere in a magnetic field on the basis of the theory \(^{26}\) was carried out in \(^{57}\). The corresponding results, under the assumption that \(\varkappa=0\), are as follows:

Superconducting cylinder in a field parallel to its axis

\[ r_k=\sqrt[\,]{3}\,\delta_0,\qquad \frac{H_{k1}}{H_{km}}=\frac{4\delta_0}{r}. \tag{D13} \]

Superconducting cylinder in a perpendicular field

\[ r_k=\sqrt[\,]{3}\,\delta_0,\qquad \frac{H_{k1}}{H_{km}}=\frac{\sqrt{8}\,\delta_0}{r}. \tag{D14} \]

Sphere

\[ r_k=\frac{\sqrt{21}}{2}\,\delta_0,\qquad \frac{H_{k1}}{H_{km}}=\frac{2\sqrt{5}\,\delta_0}{r}, \tag{D15} \]

where \(r_k\) is the critical value of the radius of the sphere or cylinder. For \(r<r_k\), the destruction of superconductivity by the field occurs as the result of a second-order transition, the corresponding critical field being given by formulas (D13)—(D15). These formulas are analogous to expressions (D8)—(D9) for films.

In the case of a massive sphere or cylinder in a perpendicular field, complete destruction of superconductivity is preceded by a transition to an intermediate state. However, for samples of sufficiently small size the occurrence of an intermediate state is impossible, since the formation of a boundary between the superconducting and normal phases is, in any case, strongly hindered. Therefore a transition to the intermediate state can be expected only for samples for which \(r>10^{-4}\), and perhaps even \(r\gg 10^{-4}\). Further, for \(r<r_k\) the destruction of superconductivity of cylinders and spheres cannot be accompanied by hysteresis in

for the same reason as for films with \(d<d_k\) (see above). For \(r>r_k\) hysteresis is possible and, for the reasons indicated earlier, one may even expect that, as \(r\) increases, hysteresis phenomena will at first manifest themselves more and more distinctly, and then again decrease. Apparently, precisely such behavior is also characteristic of hysteresis in the case of small (colloidal) superconducting mercury spheres (in \(^{58}\) it is indicated that for \(r<5\cdot 10^{-6}\) hysteresis is very weakly expressed; then hysteresis grows up to values of \(r\) equal to several units times \(10^{-5}\), and with a further increase of \(r\) the hysteresis again falls*). In addition to the values \(r_k\) and \(H_k\), for the cylinder and the sphere in \(^{57}\) expressions were also obtained for the magnetic moment in these cases (more precisely, in \(^{57}\) not the magnetic moments are given, but the effective susceptibilities \(\chi=\)

\[ =\frac{\mu}{VH_0}, \]

where \(\mu\) is the magnetic moment, \(V\) the volume of the specimen, and \(H_0\) the external field; in the case of a cylinder, \(\mu\) and \(V\) are referred to unit length, i.e. \(V=\pi r^2\)).

b) Surface energy and some other questions

The question of the surface energy \(\sigma_{ns}\) at the boundary between the superconducting and normal phases is among the most important. According to \(^{26}\) (see I, § 4), the transition between the two phases does not occur abruptly, but in a certain layer of thickness of the order

\[ \frac{\delta_n}{\varkappa}\sim 5\div 10\,\delta_0, \]

in which the function \(\Psi_0\) changes smoothly from 0 (the normal phase) to 1 (the superconducting phase). For sufficiently small values of \(\varkappa\) the surface energy is positive and is equal to

\[ \sigma_{ns}=\Delta\frac{H_{km}^{2}}{8\pi},\qquad \Delta=\frac{\sqrt{2}\cdot 4\delta_0}{3\varkappa} =\frac{1.89\,\delta_0}{\varkappa}, \qquad \sqrt{\varkappa}\ll 1. \tag{D16} \]

For large values of \(\varkappa\), \(\sigma_{ns}<0\), which cannot occur in “ideal” superconductors. In I, however, the question remained unresolved of the value of \(\varkappa\) for which \(\sigma_{ns}=0\), and of the possibility of using formula (D16) at the value \(\varkappa=0.165\), corresponding according to some data to mercury. As a result of numerical integration it was found that for \(\varkappa=0.165\), \(\Delta=5.2\,\delta_0\), whereas according to (D16) in this case \(\Delta=11.4\,\delta_0\). Such a discrepancy cannot be surprising, since in the calcu-

\[ \text{*) We note that the theoretical considerations given in }^{58}\text{ are to a certain extent analogous to those used in }^{26}\text{ (the degree of order }\omega\text{ in }^{58} \]
\[ \text{is analogous to }\Psi^2\text{), but in a number of respects are cruder and, on the whole, appear to us unreliable and unfounded.} \]

in the example under consideration \(\sqrt{\varkappa}=0.407\), and the condition \(\sqrt{\varkappa}\ll 1\) is not fulfilled. For \(\varkappa=0.6\), \(\Delta=0.29\,\delta_0\), and for \(\varkappa=\dfrac{1}{\sqrt{2}}=0.707\), \(\Delta=0\). Of course, it is very difficult to prove by a numerical method that \(\Delta=0\) exactly at \(\varkappa=\dfrac{1}{\sqrt{2}}\), and this was done with an accuracy only to the third decimal place. Nevertheless, in our opinion, there can be no doubt that \(\Delta=0\) precisely at \(\varkappa=\dfrac{1}{\sqrt{2}}\), i.e., exactly at that value of \(\varkappa\) at which a peculiar instability of the normal phase arises (see \(^{26}\) § 2). This instability is apparently connected precisely with the fact that for \(\varkappa>\dfrac{1}{\sqrt{2}}\), \(\sigma_{ns}<0\). According to the idea of L. D. Landau,

Fig. 9a.

developed by A. A. Abrikosov, the values \(\varkappa>\dfrac{1}{\sqrt{2}}\) are realized in superconducting alloys, whose behavior, as is known (see, for example, \(^{2}\) Chap. V), differs substantially from the behavior of “ideal” superconductors.

The dependence of the function \(\Psi_0\) and of the magnetic field \(H\) in the transition region between the normal and superconducting phases is clear from Figs. 9a and 9b (in Fig. 9b the curve for the case \(\varkappa=0.6\) is not shown, since at the adopted scale it almost coincides with the corresponding curve for \(\varkappa=\dfrac{1}{\sqrt{2}}=0.707\)).

At the boundary of a superconductor with vacuum there is a certain surface energy, which may differ somewhat in value

depending on whether the metal is in the superconducting or in the normal state. The corresponding difference of surface energies \(\sigma'_n-\sigma'_s\) enters into the expression for the critical magnetic field at which superconductivity is destroyed. Until recently there existed a tendency (see, for example, \(^{3}\)) to use this circumstance, i.e. the difference of \(\sigma'_n-\sigma'_s\) from zero, to explain the dependence of the critical field on the thickness of the superconducting film. However, as was indicated in \(^{26}\) (see I, § 2), one may expect that

\[ \sigma'_n-\sigma'_s = \beta\,\frac{H_{\mathrm{km}}^{2}}{8\pi} \sim 10^{-8}\div 10^{-7}\,\frac{H_{\mathrm{km}}^{2}}{8\pi}, \]

but not

\[ \sigma'_n-\sigma'_s \sim \sigma_0\,\frac{H_{\mathrm{km}}^{2}}{8\pi} \sim 10^{-5}\,\frac{H_{\mathrm{km}}^{2}}{8\pi}, \]

as would be necessary for the influence of the difference of surface energies to be significant from the point of view of some influence on the magnitude of the critical field.

Fig. 96.

For this reason, in \(^{26}\) and I the difference of energies \(\sigma'_n-\sigma'_s\) is neglected. In this connection it is important to note that, as was recently pointed out \(^{59,20}\), the smallness of the difference \(\sigma'_n-\sigma'_s\) apparently follows directly from experimental data.

Let us consider a superconducting film situated in a magnetic field \(H_0\) parallel to it, and suppose that this external field \(H_0\) varies from the value \(H_0=0\) to \(H_0=H_k\), as a result of which the film isothermally passes into the normal state. Under such conditions, both at the beginning of the process \((H_0=0)\) and at its end (the metal in the normal state), the magnetic moment of the film \(\mu\) is equal to zero. The difference of free energies at the end

and at the beginning of the indicated process, without taking into account the energy of the field, is equal to

\[ \Phi_n-\Phi_s = \int_{\bar M=0,\;H_0=H_k}^{\bar M=0,\;H_0=0} H_0\,dM = -\int_0^{H_k} \bar M\,dH_0, \tag{D17} \]

where \(\bar M\) is the average magnetization of the film (i.e., its magnetic moment referred to unit volume), \(\Phi_n=F_{n0}+\frac{\sigma'_n}{d}\), \(\Phi_s=F_{s0}+\frac{\sigma'_s}{d}\), \(F_{n0}\) and \(F_{s0}\) are the volume free energies of the normal and superconducting phases, and \(\frac{\sigma'_n}{d}\) and \(\frac{\sigma'_s}{d}\) are the surface energies of the same phases, referred to unit volume of the film*). Further,

\[ F_{n0}-F_{s0}=\frac{H_{km}^2}{8\pi} \]

and

\[ \frac{ -\displaystyle\int_0^{H_k}\bar M\,dH_0 }{ \displaystyle\frac{H_{km}^2}{8\pi} } = 1+\frac{\beta}{d}, \qquad \beta= \frac{\sigma'_n-\sigma'_s}{ \displaystyle\frac{H_{km}^2}{8\pi} }. \tag{D18} \]

*) If one does not rely on the phenomenological theory of magnetics, formula (D17) can be obtained analogously to how this was done in \(^{2}\) § 17. In this way it is easy to show that the energy flux flowing into the film when the field \(H_0\) is changed from zero to \(H_k\) and the film is subsequently transformed into the normal state is equal to

\[ A_{12} = \frac{c}{4\pi}\int [EH_0]_n\,d\sigma\,dt = -\frac{1}{4\pi}\cdot\frac{1}{2d} \int_{-d}^{d}\int_0^{t_k} H_0\,\frac{\partial H(z)}{\partial t}\,dz\,dt \quad(\text{see }{}^{2}\S 17). \]

Further,

\[ F_{s0}+\frac{\sigma'_s}{d} - \left( F_{n0}+\frac{\sigma'_n}{d} +\frac{H_k^2}{8\pi} \right) = A_{12} \]

and, since

\[ \frac{H_k^2}{8\pi} = \frac{1}{4\pi}\cdot\frac{1}{2d} \int_{-d}^{d}\int_0^{t_k} H_0\,\frac{\partial H_0}{\partial t}\,dz\,dt \]

(at the beginning of the process, for \(t=0\), \(H_0=0\), and at the end of the process, for \(t=t_k\), \(H_0=H_k\)), we obtain:

\[ \Phi_n-\Phi_s = \frac{1}{2d}\int_{-d}^{d}\int_0^{t_k} H_0\,\frac{\partial}{\partial t} \left[ \frac{H(z)-H_0}{4\pi} \right]dz\,dt = \]

\[ = -\frac{1}{2d}\int_{-d}^{d}\int_0^{H_k} \left[ \frac{H-H_0}{4\pi} \right]dz\,dH_0 = -\int_0^{H_k}\bar M\,dH_0 \]

(see (D11); it has been taken into account that at the beginning and at the end of the process \(\bar M=0\), since at the beginning \(H_0=H=0\), while at the end \(H_0=H=H_k\)).

The quantity

\[ -\int_{0}^{H_k} M\,dH_0 \]

(the area of the magnetization curve) is measured directly in experiment, and thus from (D18) one can immediately find the value of the parameter \(\beta\). Processing the corresponding data for films of lead, indium, and tin led to the conclusion\({}^{30}\) that the value

\[ \beta=\frac{\sigma'_n-\sigma'_s}{H_{\mathrm{km}}^{2}/8\pi} \]

is substantially smaller than \(\delta_0\). In this case, however, \(\beta\) cannot be determined exactly because of the presence of hysteresis (in (D17) and (D18) it is, of course, assumed that the process proceeds reversibly). For a more accurate and reliable determination of \(\beta\) in the future it is necessary to work with films of semithickness \(d<d_k\), where hysteresis is impossible, at least if one disregards the nonparallelism of the film and the field and edge effects. Let us also note that determining \(\beta\) from experiments with spheres rather than films is also possible, but the use\({}^{59}\) of the simple formula (D17) does not seem sufficiently justified. The point is that for specimens with a demagnetizing factor different from zero, the work expended by the field is not expressed simply in the form

\[ \int_{\text{(over the specimen)}} H_0\,dM \]

(see, for example,\({}^{57}\)) and the transition to formula (D17) for a macroscopic specimen with superconducting spheres embedded in it is not justified without further argument. We shall not discuss this question in greater detail, since the case of films, discussed above, appears to be more interesting\({}^{*}\).

Let us now turn to the important question of the dependence of the depth of penetration of the magnetic field into a massive superconductor on the strength of the magnetic field. According to\({}^{26}\) and I, § 4,

\[ \delta(H_0)=\delta_0\left(1+\frac{\gamma}{4\sqrt{2}}\left(\frac{H_0}{H_{\mathrm{km}}}\right)^2\right) =\delta_0\left(1+\gamma'\left(\frac{H_0}{H_{\mathrm{km}}}\right)^2\right), \tag{D19} \]

where \(\delta(H_0)\) is the depth of penetration in the field \(H_0\) (\(H_0\) is the field at the surface of the flat boundary of the superconductor), and \(\delta_0\) is the depth of penetration in the field \(H_0=0\). Formula (D19), like the whole theory\({}^{26}\), should quantitatively be applicable only near \(T_k\), where

\[ H_{\mathrm{km}}=\left(\frac{dH_{\mathrm{km}}}{dT}\right)_{T_k}(T-T_k),\qquad \delta_0=\frac{\mathrm{const}}{\sqrt{T_k-T}}. \tag{D20} \]

\({}^{*}\) Formula (D17) is also applicable without further argument in the case of cylindrical specimens with an axis parallel to the field. In this case, for a circular cylinder of radius \(r\), in (D18) one must replace \(\beta/d\) by \(2\beta/r\), since the surface energy referred to a unit volume of the cylinder is equal to

\[ \frac{2\pi r\sigma}{\pi r^2}=\frac{2\sigma}{r}. \]

In an experiment one may also directly measure not the quantity itself, but the derivative \(\dfrac{\partial \delta}{\partial T}\), equal to

\[ \frac{\partial \delta(H_0)}{\partial T} = \frac{\partial \delta_0}{\partial T} \left\{ 1+\frac{\varkappa}{4\sqrt{2}} \left( 1- \frac{2\dfrac{dH_{\mathrm{cm}}}{dT}}{H_{\mathrm{cm}}} \frac{\delta_0}{\dfrac{\partial \delta_0}{\partial T}} \right) \left(\frac{H_0}{H_{\mathrm{cm}}}\right)^2 \right\} = \]

\[ = \frac{\partial \delta_0}{\partial T} \left( 1+\frac{5\varkappa}{4\sqrt{2}} \left(\frac{H_0}{H_{\mathrm{cm}}}\right)^2 \right) = \frac{\partial \delta_0}{\partial T} \left\{ 1+5\gamma\left(\frac{H_0}{H_{\mathrm{cm}}}\right) \right\}, \tag{D21} \]

where the transition to the last expression has been made taking (D20) into account. In formulas (D19) and (D21) the quantity \(\delta\), by definition, is equal to

\[ \delta=\frac{\displaystyle\int_{0}^{\infty} H(z)\,dz}{H_0}, \]

where \(z\) is the distance from the surface of the metal. In an experiment in which a weak alternating magnetic field \(H_{10}\) and, simultaneously, a strong field \(H_0\) are used, what is measured is not the indicated quantity \(\delta\), but the quantity

\[ \delta_1=\frac{\displaystyle\int_{0}^{\infty} H_1(z,H_0)\,dz}{H_{10}}, \]

i.e. the penetration depth of the weak field in the presence of a strong one (\(H_1(z,H_0)\) is the intensity of the weak field in the superconductor, \(H_{10}=H_1(0,H_0)\) is the intensity of the weak field at the surface of the metal). Relying on formula (32) from \(^{26}\), it is easy to show that

\[ \delta_1(H_0)= \delta_0 \left( 1+\frac{3\varkappa}{4\sqrt{2}} \left(\frac{H_0}{H_{\mathrm{cm}}}\right)^2 \right) = \delta_0 \left( 1+3\gamma\left(\frac{H_0}{H_{\mathrm{cm}}}\right)^2 \right). \tag{D22} \]

In recent experimental works \(^{60,61}\) the dependence of the penetration depth on the field strength can apparently be considered established. However, the quantitative side of the matter is still unclear.

In \(^{61}\) the quantity \(\dfrac{\partial \delta(H_0)}{\partial T}\) was measured for \(Sn\), and it was found that the coefficient \(\gamma\) in a formula of the type (D21) is approximately \(\gamma_{\mathrm{exp}}\simeq 0.06\), and this value in the interval \(2.5^\circ<T<T_{\mathrm{k}}\) does not depend on temperature. At the same time, if one uses the value

\[ \delta_0=\frac{4.6\cdot 10^{-6}}{\sqrt{T_{\mathrm{k}}-T}}, \]

obtained in the same work, then, using the formula (1; 3,14), (1; 3,14a):

\[ \varkappa=\sqrt{2}\,\frac{e}{\hbar c}\,\delta_0^2 H_{\mathrm{cm}} = 2.16\cdot 10^7 \delta_0^2 H_{\mathrm{cm}}, \tag{D23} \]

we obtain \(\varkappa=0.07\), and according to (D21) the theoretical value

$$ \gamma=\frac{\varkappa}{4\sqrt{2}}=0.012, $$

i.e., 5 times smaller than the experimental value: $\gamma_{\mathrm{exp}}\simeq 0.06$. On the other hand, in $^{60}$ the quantity $\delta_{1}(H_{0})$ was also measured on $Sn$, and it turned out that $\gamma_{\mathrm{exp}}$ depends on temperature and does not exceed the value $0.008$, i.e., it is even somewhat smaller than the theoretical value. The data of $^{61}$ were obtained under good conditions, but the observed dependence $\dfrac{\partial \delta}{\partial T}$ on $H_{0}$ lies at the limit of measurement accuracy; moreover, for certain reasons inflated values could have been obtained in $^{61}$. The accuracy of the method of $^{60}$ is higher, but in this case the field at the surface of the specimen is inhomogeneous and, most importantly, the field $H_{01}$ was of such high frequency ($\lambda\simeq 3\ \mathrm{cm}$) that comparison of the experimental results with the theory $^{26}$, which refers to statistical or slowly varying fields, may prove illegitimate, even though the strong field $H_{0}$ was constant in this case.

The question of the dependence of the depth of penetration of the magnetic field into a massive metal on the field strength obviously requires further investigation. Work in this direction is especially important because from the corresponding data one can determine the magnitude $\varkappa$ and compare it with the theoretical value (D23). At the same time it should not be forgotten that in (D23) the transition to the numerical coefficient was made under the assumption that the charge appearing in the theory is equal to the charge of a free electron. This assumption, within the framework of the theory $^{26}$, seems natural and no objections to it are apparent. Nevertheless, such an assumption does not seem to us altogether necessary. If formula (D23) with $e=4.8\cdot 10^{-10}$ turned out not to agree with the data on the penetration depth in a strong field (as has been said, there are as yet no grounds for such an assertion), then from measurements of $\delta$, $\delta_{1}$, or $\dfrac{\partial \delta}{\partial T}$ one should determine the parameter $\varkappa$ entering the theory, and the test of the theory would consist in measuring the surface energy $\sigma_{ns}$, which is determined by the same parameter $\varkappa$.

It should also be remembered that the theory developed in $^{26}$ directly applies only to isotropic materials (polycrystals) or to metals of cubic symmetry. In the case of metals with lower symmetry, among which strongly tetragonal white tin belongs, the theory requires a generalization $^{63}$. This generalization is quite obvious and reduces to the fact that, in the system of principal axes, the expression for the free energy (I; 3,5) takes the form:

$$ F_{sH}=F_{s0}+\frac{H^{2}}{8\pi}+\sum_{k=1}^{3}\frac{1}{2m_{k}}\left|-i\hbar\frac{\partial\Psi}{\partial x_{k}}-\frac{e}{c}A_{k}\Psi\right|^{2}, \tag{D24} $$

where the indices $k=1,2,3$ correspond to the axes $x,y,z$, as the ...

for which, as has been said, the principal axes are chosen (so that \(m_k\) are the eigenvalues of the mass tensor \(m_{kl}\)). If \(m_1=m_2=m_3\), then the material is isotropic and (D24) goes over into (I; 3,5). The expression (I; 3,8) for the current density now has the form

\[ j_k=-\frac{ie\hbar}{2m_k}\left(\Psi^*\frac{\partial\Psi}{\partial x_k} -\Psi\frac{\partial\Psi^*}{\partial x_k}\right) -\frac{e^2}{m_k c}\Psi^*\Psi A_k . \]

Without dwelling on the anisotropic case in greater detail, let us turn at once to the one-dimensional problem of the penetration of a magnetic field into a massive superconductor occupying the half-space \(z>0\); the current \(j\) and the potential \(A\) are taken, as in I, to be directed along the \(x\)-axis, whence \(H=H_y=\dfrac{dA_x}{dz}\).

Assuming that the chosen axes \(x,y,z\) are the principal axes, the anisotropic problem can be reduced to the isotropic one considered earlier if one introduces the variables (see (I; 3,12)):

\[ z'=\frac{z}{\delta_{0x}},\qquad \Psi'^2=\frac{\Psi^2}{\Psi_\infty^2},\qquad A'=\frac{A}{\sqrt{2}\,H_{\mathrm{km}}\delta_{0z}} =\sqrt{\frac{e^2}{2m_z c^2|\alpha|}}\,A, \]

\[ H'=\frac{dA'}{dz'}=\frac{\delta_{0x}H}{\sqrt{2}\,\delta_{0z}H_{\mathrm{km}}},\qquad \delta_{0x}^2=\frac{m_x c^2}{4\pi e^2\Psi_\infty^2}, \]

\[ \delta_{0z}^2=\frac{m_z c^2}{4\pi e^2\Psi_\infty^2},\qquad \varkappa_{xz}^2=\frac{2e^2}{\hbar^2c^2}\,\delta_{0x}^2\delta_{0z}^2H_{\mathrm{km}}^2 . \]

In these variables the equations (I; 3,13) are preserved, with \(\varkappa^2\) replaced by \(\varkappa_{xz}^2\).

The subsequent solution of the problem of the depth of penetration of the field into the superconductor leads to formulas (D19)—(D22), but with \(\delta_0\) and \(\varkappa\) replaced by

\[ \delta_{0x}=\sqrt{\frac{m_x c^2}{4\pi e^2\Psi_\infty^2}},\qquad \varkappa_{xz}=\frac{\sqrt{2}\,e}{\hbar c}\,\delta_{0x}^2\frac{\delta_{0x}}{\delta_{0z}}H_{\mathrm{km}} . \tag{D25} \]

Thus, in the general case, in order to test the theory one must know both quantities \(\delta_{0x}\) and \(\delta_{0z}\). In the case of tin, if the \(y\)-axis is the tetragonal axis, then \(\delta_{0x}=\delta_{0z}\), and for the calculation of \(\varkappa\) it is sufficient to carry out measurements for a single orientation.

When working at high frequency one must remember that the dielectric constant \(\varepsilon_0\) not associated with the superconducting current (see I, §§ 5, 6), when anisotropy is taken into account, is no longer a scalar, but a symmetric tensor of rank two with principal values \(\varepsilon_{0k}(\omega,T)\). For tin, of course, \(\varepsilon_{0x}=\varepsilon_{0z}\) (the tetragonal axis is the \(y\)-axis). High-frequency measurements should first of all

set with the aim of finding the functions \(\varepsilon_{0k}(\omega,T)^{*)}\). The study of the dependence of the penetration depth \(\delta\) on the strength of the magnetic field and on its orientation relative to the crystal axes must be carried out first of all in static or quasistatic fields (in practice, measurements at frequencies \(\omega \sim 10^7 \div 10^8\) may apparently be regarded as quasistatic).

In connection with the material under discussion, let us note that, according to the theory of superconductivity \(^{10,11}\), the dependence of the penetration depth on the field strength should be very pronounced—considerably stronger than according to \(^{26}\), which is in obvious contradiction with the answer \(^{20}\).

The next question on which we shall dwell is connected with the behavior of a superconductor in a high-frequency field.

In addition to I, § 6, we point out that, when treating experimental data relating to a massive superconducting metal, it is simplest to use interpolation formulas obtained by introducing an effective complex dielectric constant \(^{63}\):

\[ \begin{gathered} \varepsilon_{\mathrm{eff}} = \varepsilon - i\,\frac{4\pi\sigma}{\omega}\, \frac{\dfrac{2\pi}{\sqrt{3}}\,\delta_{\mathrm{sk}}}{l}, \qquad \delta_{\mathrm{sk}} = -\frac{ic}{\omega\sqrt{\varepsilon_{\mathrm{eff}}}}, \\[6pt] Z = \frac{4\pi}{c\sqrt{\varepsilon_{\mathrm{eff}}}}, \end{gathered} \tag{D26} \]

where \(Z=R+iX\) is the impedance, \(\sigma\) is the static conductivity, and \(l\) is the mean free path. The constant \(2\pi/\sqrt{3}\) in (D26) is chosen in such a way that for a normal metal one obtains formula (I; 6,21), derived from the kinetic equation under the assumption of diffuse reflection of electrons from the metal surface. By virtue of (D26), for \(y=\sqrt{\varepsilon_{\mathrm{eff}}}\) one obtains the equation

\[ y^3-\varepsilon y+\frac{8\pi^2 c\sigma}{\sqrt{3}\,\omega^2 l}=0, \]

from which it is easy to see that

\[ \frac{\sigma}{l} = \frac{16\pi\omega^2}{c^4}\, \frac{\sqrt{3}\,R}{(R^2+X^2)^2}, \qquad \varepsilon = -\frac{16\pi^2\,(X^2-3R^2)}{c^2(R^2+X^2)^2}. \tag{D27} \]

As \(\varepsilon \to 0\), these formulas lead to the relation \(X=\sqrt{3}\,R\) and to the expression for \(\sigma/l\) adopted in the theory of the anomalous skin effect in a normal metal (see (I; 6,27), where, owing to a misprint, \(R_n\) is written instead of \(R_n^3\)); as \(\sigma \to 0\), \(R \to 0\), and from (D27) one obtains completely

\({}^{*)}\) The value \(\varepsilon_{0k}\sim 10^8—10^{10}\) is so large that even in a comparatively weak electric field \(E\) the linearity of the problem (i.e. the independence of \(\varepsilon_0\) from \(E\)) may be violated. This point must always be kept in mind in the corresponding experimental investigations.

exact formula

\[ \varepsilon=-\frac{16\pi^2}{c^2X^2}\quad(\text{see }(1;\ 6,33)). \]

In the intermediate temperature region between \(T_0=0\) and \(T_k\), formulas (D27) have an interpolation character, but are probably quite reliable. In any case, without a detailed knowledge of the character of the excitations in a superconductor, any other formulas of this type can hardly lay claim to great accuracy. Experimentally, only the change in the magnitude \(X\) (i.e. the reactance) is measured when the temperature is lowered below \(T_k\). Therefore, in order to find the value \(X\) itself, one has to rely on the theoretical relation \(X_n=\sqrt{3}R_n\), which, to be sure, appears rather reliable. However, even more reliable in a number of respects are experiments with films\({}^{46}\), for which one may simply put \(X_n=0\) (the index \(n\) refers to the normal state). Indeed, in a film of thickness \(d\ll l\) and \(d\ll\delta_{\mathrm{sk}}\) (\(l\) is the free-path length, \(\delta_{\mathrm{sk}}\) is the skin-layer depth), the normal skin effect essentially takes place, i.e. \(\varepsilon_{\mathrm{eff}}=\varepsilon-i\frac{4\pi\sigma}{\omega}\), where \(\sigma\) is the conductivity of the film (obviously, \(\sigma\sim \frac{\sigma(0)d}{l}\), where \(\sigma(0)\) is the conductivity of the bulk metal). Under the conditions satisfied for thin films,

\[ \left|-\frac{\omega}{c}\sqrt{\varepsilon'}\,d\right|^2\ll 1,\qquad \left|\sqrt{\varepsilon'}\right|\gg 1,\qquad \left|\frac{\omega}{c}\sqrt{\varepsilon'}\,d\right|\gg \frac{1}{\left|\sqrt{\varepsilon'}\right|} \tag{D28} \]

the impedance of the film is equal to\({}^{63}\)

\[ \left. \begin{aligned} Z=R+iX&=-\frac{4\pi}{c\sqrt{\varepsilon'}\left(i\frac{\omega}{c}\sqrt{\varepsilon'}\,d\right)} =\frac{4\pi}{i\omega\varepsilon'd},\\ \sigma&=\frac{R}{d(R^2+X^2)},\qquad \varepsilon=-\frac{4\pi X}{\omega d(R^2+X^2)}. \end{aligned} \right\} \tag{D29} \]

In the normal state \(|\varepsilon|\ll \frac{4\pi\sigma}{\omega}\), and thus \(X\ll R\), by virtue of which one may put \(X=0\). As a result of measurements of \(X\) and \(R\) for superconducting films of \(Sn\), in\({}^{46}\) the value

\[ \varepsilon=\varepsilon_0-\frac{c^2}{\omega^2\delta_0^2}; \]

was found; then, using data on the static penetration depth \(\delta_0\), obtained for completely analogous films\({}^{48}\), it proved possible to determine also the quantity \(\varepsilon_0(T)\), which increases monotonically with decreasing temperature (conditionally one may put \(\varepsilon_0=0\) at \(T=T_k\)) and at \(T=2^\circ\) reaches the value \(\varepsilon_0=5\cdot10^9\) (\(\pm30\%\)). The further increase of \(\varepsilon_0\) as \(T\to0\) must already be small, since the derivative \(\left|\frac{d\varepsilon_0}{dT}\right|\) decreases with falling temperature.

decreases. The measurements were made at the frequency \(\omega \simeq 6\cdot 10^{10}\), and it turned out that

\[ \varepsilon_0 \sim \frac{c^2}{\omega^2\delta_0^2}, \]

but still \(\varepsilon < 0\). With increasing frequency, if one does not take into account the dependence of \(\varepsilon_0\) on \(\omega\), the quantity \(\varepsilon\) must become positive, i.e., the superconductor will become transparent. The question of whether this effect will be observed in some frequency interval or whether it is always masked by absorption remains still completely unclear. The study of the dependence of \(\varepsilon_0\) on the frequency \(\omega\), as was already indicated in § 4 b) and § 5, is one of the most interesting and important problems that must be solved for further progress in the understanding of superconductivity.

In conclusion, let us briefly dwell on two further questions: the anomalous thermal conductivity of some superconducting alloys and the question of the “anticipation” of superconductivity, i.e., the existence of certain peculiarities in the behavior of a superconducting metal at \(T > T_k\).

In some superconducting alloys, in contrast to pure superconductors, the thermal conductivity in the superconducting state proves to be higher than in the normal state (see I, § 5). At first glance it seemed that this fact testifies in favor of the existence in superconductors of a convective mechanism of heat transfer (see \({}^{3}\), § 16 and I, § 5). However, as was shown in \({}^{63}\) and (I, § 5), convective heat transfer in superconductors is, generally speaking, very small and insufficient to explain the observed effect of the increase in thermal conductivity, which thus remained completely unexplained. A very probable cause capable of explaining the effect under discussion was indicated in \({}^{64}\). As it turns out, in metals with an impurity content greater than \(0.1\%\), the thermal conductivity of the lattice is comparable with the electronic thermal conductivity. Further, in some cases the thermal conductivity of the lattice is determined by the scattering of phonons (waves traveling in the lattice) by conduction electrons. Upon transition to the superconducting state and lowering of the temperature, the number of conduction electrons decreases (see I, § 6), i.e., the number of scattering particles decreases, and the thermal conductivity increases. To test this hypothesis, one should, in particular, pay attention to the Wiedemann–Franz law, which applies only to the electronic part of the thermal conductivity and may help to isolate the latter (see \({}^{63}\)).

Turning to the question of the “anticipation” of superconductivity, let us recall that here the issue is whether, in the normal state near \(T_k\) (i.e., for \(T - T_k \ll T_k\)), there are any phenomena that foreshadow the superconducting transition occurring as the temperature is lowered. So far there has been only one positive indication in this direction, namely, it was asserted \({}^{65}\) that at a temperature \(0.1 \div 0.2^\circ\) higher than \(T_k\), the differential thermoelectromotive force of the metal begins to change rather strongly. However, in another work \({}^{66}\) indications of such an effect are absent.

At the same time, we would like to draw attention to the fact that there are certain grounds for expecting the presence of some phenomena preceding the superconducting transition[^67]. The point is that in the absence of a magnetic field this transition is a second-order transition, and for such transitions near the transition point (i.e., near \(T_{\mathrm{k}}\)) there must be relatively large fluctuations of the parameter characterizing the given transition. In the case of superconductivity this “ordering parameter” is the concentration of superconducting electrons \(n_s=\Psi_\infty^2\). For \(T>T_{\mathrm{k}}\), in the equilibrium state \(n_s=0\), and overheating of the superconducting phase is impossible (this is characteristic of second-order transitions, in which above the transition point there can exist neither stable nor even metastable nuclei of the phase stable below the transition point). However, as has been said, rapidly varying fluctuations of \(n_s\) for \(T>T_{\mathrm{k}}\) are not only possible, but as \(T\to T_{\mathrm{k}}\) must even be large. Thus, in the normal phase near \(T_{\mathrm{k}}\), fluctuation “nuclei” of the superconducting phase are being formed all the time. These “nuclei” may affect the value of the complex dielectric constant in the normal state, influence the thermoelectric properties of the metal, etc. The question of what effect can be expected here and whether it can be observed remains, for the time being, entirely open.

In light of what has been said, however, it seems unquestionable that the problem of the “anticipation” of superconductivity is of interest and should become an object of investigation.

In addition to the works discussed in the present article, a whole series of others has appeared recently, the contents of which we have not dwelt on. In order to facilitate acquaintance with the new literature, the bibliography lists, in addition to the works cited in the text, all works on superconductivity known to the author, both theoretical[^69]–[^87] and experimental[^88]–[^119], apart from those indicated in [^2], [^1] and in abstract collections[^68] *).

*) We take the opportunity to point out here the most important of the misprints noted in [^1]. On p. 173, in the phrase separated by Fig. 2, the comma should stand after the word “ferromagnetic,” and not after the parenthesis (for \(T>T_{\mathrm{k}}\)). In the note to p. 192, for the terms

\[ \frac{H_{\mathrm{KM}}^{2}}{4\pi} \]

one must put a minus sign. At the very end of p. 217 there should stand \(x\simeq 0.15\), and not \(x^2\simeq 0.015\). In formula (5,7) on p. 337 there should stand

\[ \left\{\frac{4\pi}{c}\left(\sigma+\frac{1}{i\omega\Lambda}\right)+\frac{i\omega}{c}\varepsilon_0\right\}. \]

At the bottom of p. 340 it should be

\[ R=\operatorname{Re} Z\sim \sigma^{-1/2}. \]

At the bottom of p. 342 one should read

\[ \frac{\partial f_1}{\partial z}\sim \frac{f_1}{\delta_{\mathrm{CK}}} \]

instead of

\[ \frac{\partial f_1}{\partial t}\sim \frac{f_1}{\delta_{\mathrm{CK}}}. \]

In formula (6,26) on p. 348 one must put \(R^3\) instead of \(R\). In formula (6,31) on p. 350, in the denominator there should stand \(3^{1/3}\), and not \(3^{1/2}\). Footnote \(32a\), which has fallen out of the bibliography of [^1], is footnote 65 of the present article.

References Cited

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2. V. L. Ginzburg, Superconductivity, Publ. House of the USSR Academy of Sciences (1946).

3. V. L. Ginzburg, Remarks on the theory of superconductivity, ZhETF 14, 134 (1946).

4. E. Mendoza and J. G. Thomas, The electrical resistance of Au, Ag and Cu at low temperatures, Phil. Mag. 42, 291 (1951).

5. S. V. Vonsovskii, Theory of the interaction of electrons in a crystalline lattice, Izv. AN SSSR (Physical Series) 12, 337 (1948).

6. a) G. Bethe and A. Sommerfeld, Electron Theory of Metals, ONTI (1938).
   b) F. Seitz, The Modern Theory of Solids, Gostekhizdat (1949).

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Literature Added in Proof

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1. In Bardeen’s papers[^23,^27] the interaction of electrons with lattice vibrations is likewise considered by the method of perturbation theory, and it is asserted that, as a result of this interaction, the effective mass of an electron near the Fermi surface turns out to be very small. Quite apart from the complete lack of proof of this result, even if the effective mass of an electron at \(E \sim E_0\) did turn out to be small, this still would not lead to superconductivity (see § 3a). A critique of Bardeen’s papers is also contained in the note that has just appeared[^121]. ↩