SUPERCONDUCTIVITY

D. Shoenberg, Moscow

I. Introduction

It has long been known that the resistance of all metals falls as the temperature is lowered: the precise course of this change depends, of course, on the particular metal, but in general it follows a curve like that shown in Fig. 1. Matthiessen1 was the first to point out that the resistance should be regarded as consisting of two parts: 1) a temperature-independent “residual resistance” \(\Delta R\), which increases with increasing mechanical stresses in the material, and also with its chemical contamination; 2) a temperature-dependent “ideal resistance,” on which mechanical deformations and chemical impurities have no appreciable effect. The work of Kamerlingh Onnes in Leiden showed that already at liquid-hydrogen temperatures this ideal resistance is usually very small, so that the greater part of the resistance is “residual” and does not depend on temperature. Soon after he succeeded in liquefying helium, Kamerlingh Onnes began to extend his investigations into this new region of low temperatures in order to determine whether this independence of temperature persists still closer to absolute zero. For some metals this indeed proved to be the case, but quite unexpectedly he found in 1911 that mercury suddenly loses its resistance when the temperature becomes sufficiently low, and that at still lower temperatures it no longer changes.2 This property appears suddenly at a quite definite temperature, as though it were characteristic of a certain new state of the metal—the “superconducting” state.

Fig. 1.

This discovery brought to the fore a whole series of problems concerning the limits and the nature of the new phenomenon. It was found that seventeen metals and a large number of alloys become superconducting, with each metal having a “transition temperature” characteristic of it, lying between 0.35° K (for hafnium) and 9.2° K (for niobium); some alloys have a somewhat higher “transition temperature.” The known superconducting elements fall roughly into two groups in the periodic system (Table 1). This suggests that superconductivity is perhaps not a universal property. On the other hand, however, every new success in lowering the temperature leads to the discovery of new superconductors; for example, the recent work of Kurti and Simon³ with cooling by adiabatic demagnetization. Moreover, some superconductors, say aluminum, do not quite fit into these two groups. In view of all this we still cannot be certain that superconductivity is inherent only in some metals and not in all metals. Since absolute zero cannot be reached, this question can be finally settled only by a theory of superconductivity, unless, of course, further progress in attaining low temperatures shows that all metals can indeed become superconducting.

Attempts have been made to connect superconductivity with some other properties of the metal, but so far most empirical rules of this kind have proved unsuitable for discovering new superconductors (with the possible exception of the rule mentioned above, which restricts superconductivity to certain groups of the periodic system). For example, it was assumed that only “soft” metals with a low melting point could become superconducting, until it was discovered that typical “hard” metals—tantalum and niobium—also become superconducting and even have fairly high transition temperatures. Similarly, superconductivity cannot be connected with any definite type of crystal lattice, since almost all types of lattices are represented among the known superconductors. By this, of course, we do not mean to assert that the crystal lattice is less important than the kind of atoms of the metal themselves, since, as in ordinary conductivity, any change of lattice for one and the same metal can have a substantial influence. An example of this is tin—the only superconductor with more than one modification: white tin is a typical superconductor, whereas gray tin, differing from white tin only in its crystal lattice, does not become superconducting down to the lowest temperatures tested (2.3° K)⁴.

An important question is whether, at the transition point, all the resistance disappears or only part of it, since it has sometimes been supposed that only the residual resistance disappears while the ideal resistance remains, but is too small to be observed.

TABLE 1. Superconductors in the periodic system¹)

H																		He
Li (1.19°)	Be (1.35°)												B (1.27°)	C (1.15°)	N	O	F	Ne
Na (1.23°)	Mg (0.05°)												Al 1.14° c. f. c.	Si (1.22°)	P	S	Cl	Ar
K (1.22°)	Ca (1.35°)	Sc	Ti 1.17° h.	V 4.3° b. c. c.	Cr (1.41°)	Mn (1.22°)	Fe (1.29°)	Co (1.36°)	Ni (1.34°)	Cu (0.05°)		Zn 0.78° h.	Ga 1.1° r.	Ge (0.05°)	As (1.13°)	Se (1.26°)	Br	Kr
Rb (1.13°)	Sr (1.32°)	Y	Zr 0.70° h.	Nb 9.2° b. c. c.	Mo (1.26°)	Ma	Ru (1.17°)	Rh (1.32°)	Pd (1.17°)	Ag (1.35°)		Cd 0.54° h.	In 3.37° t. f. c.	Sn 3.69° t.	Sb (1.16°)	Te (1.13°)	I	X
Cs (1.15°)	Ba (1.26°)	La 4.71° h.	Rare earths	Hf 0.35° h.	Ta 4.38° b. c. c.	W (1.31°)	Re (1.36°)	Os (1.6°)	Ir (1.29°)	Pt (1.35°)	Au (0.05°)	Hg 4.12° r.	Tl 2.38° h.	Pb 7.26° c. f. c.	Bi (0.05°)	Po	—	Em
—	Ra	Ac	Th 1.43° b. c. c.	Pa		U (1.41°)

¹ Numbers in parentheses denote the lowest temperatures, in °K, at which the corresponding metal was found to be superconducting.
h. — hexagonal lattice; c. f. c. — cubic face-centered lattice; b. c. c. — cubic body-centered lattice; r. — rhombohedral lattice; t. f. c. — tetragonal face-centered lattice; t. — tetragonal.

SUPERCONDUCTIVITY

in the Kammerlingh-Onnes experiments. This question was resolved only by later experiments,^5 in which indirect methods were used to detect a decrease in “persistent” currents in a superconducting ring (see Chapter IV); these experiments showed that the upper limit for any possible resistance in the superconducting state is less than \(10^{-15}R_0\), where \(R_0\) is the resistance at room temperature. Since the ideal resistance just above the transition point is, in any case, considerably higher (of the order of \(10^{-8}R_0\)), we see that all resistance disappears, and not merely the residual resistance. Up to the present time no experiment has detected any trace of resistance in a superconducting metal, and in what follows we shall always assume that a superconducting metal behaves as if its resistance were exactly zero. This is equivalent to the assertion that the component of the electric field \(\mathbf{E}\) in the direction of the current is zero; since, on the other hand, there is no indication of the existence of a nonzero component of the field in any other direction (for example, there is no Hall effect in a superconductor^6), we may say that an essential property of a superconductor is

\[ \mathbf{E}=0. \]

It is possible that the complete and sudden loss of resistance is the consequence of some more fundamental change in the electronic or atomic structure of the metal. The experiments that should reveal the nature of this change may be roughly divided into two classes: the first should establish whether these changes are reflected in other properties of the metal besides resistance; the second—whether this change can be influenced in any direction by any physical actions.

For a long time one of the most striking properties of a superconductor was considered to be the fact that all experiments following the first path gave negative results; in other words, it seemed that, apart from the loss of resistance, the metal has identical properties in the superconducting and nonsuperconducting normal states. More recent studies have found some exceptions, and at the same time thermodynamic considerations, as we shall see below, make the absence of changes in properties upon transition to the superconducting state more evident. Let us list here the various properties whose investigation has so far given negative results:

1. The X-ray diffraction patterns of the metal above and below the transition point are identical;^7 this shows that no change in the crystal lattice occurs. The absence of any noticeable change in the distribution of intensity further shows that the change in the electronic structure must be very weak.

2. No change occurs in the reflectivity of the metal either in the visible or in the infrared regions,^8 despite the fact that optical properties are usually closely connected with electrical resistance.

1. There is no change whatever in the absorption of fast or slow electrons[^9], nor in the photoelectric properties[^10].

2. The transition is not accompanied by the release or absorption of heat (in the absence of a magnetic field)[^11]; we shall see below that this is connected thermodynamically with the magnetic properties.

3. The elastic properties[^12] and thermal expansion[^13] are the same in the superconducting and normal states, and there is no noticeable change of volume at the transition.

The exceptions mentioned above (i.e., the properties that undergo a change upon transition to the superconducting state) are the following:

1. The magnetic properties undergo changes no less noticeable than the changes in the electrical properties; these changes will be considered in detail in the following chapters.

2. The heat capacity undergoes a jump at the transition point[^11], and in the presence of a magnetic field there is also a latent heat of transition[^14]. These features find a detailed explanation in the thermodynamic treatment.

3. All the thermoelectric properties of a metal disappear when it becomes superconducting[^15]. Keesom and Matthijs[^15] found that just above the transition point there is a noticeable increase in the coefficient of the Thomson effect (both in the presence and in the absence of a magnetic field); a detailed examination of the experimental data, however, indicates that this effect is caused, apparently, by the presence of impurities in the specimen and is secondary.

4. In the presence of a magnetic field the thermal conductivity changes discontinuously at the transition temperature, although its order of magnitude remains the same. Namely, the thermal conductivity becomes smaller in the superconducting state for pure metals and larger in the case of alloys[^16]. In the absence of a magnetic field no jump in the thermal conductivity is observed. This phenomenon, perhaps, will become comprehensible only when we have an electronic theory of superconductivity.

We now turn to the second path of investigation—to studies of the influence of various physical actions. The influence of high-frequency radiation, say, X-rays, apparently has not been investigated, but it is unlikely that it would produce any effect, since the processes accompanying their passage through a metal are, generally speaking, connected only with individual electrons or atoms, whereas conductivity is always a phenomenon connected with the collective behavior of many electrons.

The effect of increasing the frequency of the current in a superconductor was investigated by McLennan and his collaborators[^17] up to radio frequencies. Although their experiments did show that the transition temperature begins to fall at the highest frequencies they used ($10^7\ \mathrm{Hz}$), this effect was very small and could have been due to some extraneous causes, so that we may say that up to frequencies equal to $10^7\ \mathrm{Hz}$ the transition temperature does not depend noticeably on the frequency. We shall see in Chap. II that an essential prop—

that a property of a superconductor is that a magnetic field cannot penetrate into it, provided only that it does not exceed the “critical” field (see below). Consequently, in a superconductor in an alternating magnetic field there should be no absorption of energy, provided only that the frequency is not so high as to affect the very property of superconductivity. This has been confirmed experimentally^18 up to frequencies of the order of \(10^7\) Hz. The question of the order of magnitude of those frequencies at which absorption of energy could be expected will be discussed in Ch. VIII.

The transition temperature can be changed by deforming the body; moreover, a tension that increases the dimensions raises this temperature^19. This effect, however, is very small, possibly because the change in dimensions produced by practically attainable stretching or compression is also very small. Connected with this effect is also the slight influence of deformations on the critical value of the magnetic field at a given temperature. In Ch. V we shall see that the absence of any noticeable change in the coefficient of thermal expansion, compressibility, and volume of the metal at the transition point is thermodynamically connected with the small magnitude of the influence of deformation on superconducting properties.

A decrease in the dimensions of a specimen below \(10^{-4}\) cm changes the superconducting properties in many essential respects; it is more convenient, however, to consider these effects later (Ch. VII), after the properties of superconductors of normal dimensions have been considered.

The most significant of the physical factors now known to influence superconductivity is the magnetic field. If the magnetic field is applied parallel to the diameter of a superconducting wire, the normal resistance of the wire is suddenly restored at a definite field strength, depending on the temperature and characteristic of each given metal. This field is called the “critical” field^20. The restoration of resistance, however, occurs suddenly only in the case of a perfectly pure metal, in which there are no internal stresses, and if the current used to measure the resistance is vanishingly small. The absence of impurities and internal stresses is essential because they somewhat change the value of the critical field, so that different parts of the specimen have different critical fields and, in this way, the transition is “smeared out.” The influence of the measuring current and of certain geometrical conditions mentioned above will be explained further on. We may therefore say that the absence of abruptness in the restoration of resistance in a magnetic field at a given temperature (or, what is equivalent to this, upon increasing the temperature in a given magnetic field, which in a particular case may be equal to zero) is, generally speaking, an incidental feature; in our discussion we shall assume that the conditions are ideal and that at a given temperature there exists a quite definite critical field.

We shall also mention that, if these ideal conditions are fulfilled, which can be almost completely achieved in practice, then the transition from superconductivity to normal conductivity is reversible; i.e., if the field is decreased to a value below the critical one, then the resistance disappears at the same voltage at which it appears when the field is increased. In fact, in many early Leiden works,^20 complex hysteresis phenomena were observed, which, however, were considerably reduced when the wires used for measuring the potential were fused on instead of being soldered.^21 This confirms that the hysteresis was a secondary phenomenon, possibly connected with anomalous properties of alloys (ch. VI^1).

The dependence between the critical field, which we shall denote by \(H_c\), and the temperature is very important for characterizing the properties of any particular superconductor. The exact form of this dependence is different for different superconductors (these curves are given in Fig. 2), although it does reveal common characteristic features. It may be noted that the curves shown divide into two groups, which correspond to the two respective groups of superconductors in the periodic system: the “hard” superconductors of the left-hand group have very steep \(H_c—T\) curves, while the “soft” ones (the right-hand group) have considerably more gently sloping curves. It is not known, however, how generally applicable this rule is, since not all superconductors have been investigated in this respect. In this sense, measurements with vanadium and tantalum would be of interest.

It is often convenient to imagine the \(H_c—T\) curve as an equilibrium diagram, in many respects analogous to the \(p—T\) diagrams of ordinary phase transitions (say, melting or boiling). Thus, the states of the metal represented by points to the left of the curve are superconducting, and those to the right of it are normally conducting.

Near the normal transition temperature (i.e., at a field equal to zero) the \(H_c—T\) curve is approximately parabolic, intersecting the \(T\)-axis at an acute angle, i.e.

\[ H_c = a(T_0 - T)^2, \]

and it becomes flat on approaching absolute zero; we shall see below that Nernst’s theorem requires that \(\frac{dH_c}{dT}=0\) at

^1) Hysteresis could in part have been caused by contamination of the sample, since, as we shall see in ch. VI, superconducting alloys exhibit appreciable hysteresis phenomena; in some cases, however, the hysteresis had a character different from that which is inherent in alloys or contaminated samples, and perhaps was caused by some kind of “supercooling.” Although in our consideration of pure superconductors we shall always assume that hysteresis phenomena are of secondary nature and therefore should not be taken into account, it must nevertheless be borne in mind that from the experimental point of view the situation has not been fully clarified, and it is possible (although in our opinion rather improbable) that our assumption is not entirely admissible.

\(T = 0\). The order of magnitude of the maximum critical field lies between 100 and 1000 gauss for the known superconductors1.

An interesting consequence of the existence of a critical magnetic field is that there also exists a critical current strength which can flow in a superconductor2. The destruction of superconductivity by a current was discovered even earlier than destruction by a magnetic field.

Fig. 2.

by a magnetic field. It was discovered when Kamerlingh-Onnes attempted to obtain strong magnetic fields without expenditure of power by means of a superconducting solenoid. As soon as the current exceeded certain values, corresponding to a quite small field in the solenoid, resistance reappeared, and the current could be maintained only with a large expenditure of energy. After the discov-

...of the critical magnetic field Silsbee \({}^{23}\) pointed out that the destruction of superconductivity by a current may be directly caused by the magnetic field of the current; this hypothesis was subsequently confirmed experimentally, so that the current effect is, properly speaking, a secondary phenomenon.

We have enumerated most of the essential features and properties of pure superconductors \({}^{1}\) and shall now proceed to a more detailed consideration of how they are related to one another and to a discussion of the most recent results.

II. Magnetic Properties of an Ideal Conductor and of a Superconductor

In this chapter we shall first show what magnetic properties should be expected of a metal in the limiting case of infinitely large conductivity, and then compare them with the actual magnetic behavior of a superconductor. This comparison will show that describing a superconductor simply as an ideal conductor is, in any case, impossible and in some respects even incorrect.

Let us consider a metal with zero resistance and with magnetic permeability equal to unity. According to Maxwell’s equations,

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

\[ \operatorname{rot}\mathbf{H}=\frac{4\pi}{c}\mathbf{j}, \tag{2} \]

(\(\mathbf{E}\)—the electric-field strength, \(\mathbf{j}\)—the current density). Since the resistance is zero, we also have

\[ \mathbf{E}=0, \tag{3} \]

and from (1) we have \(\dfrac{\partial \mathbf{B}}{\partial t}=0\), i.e.

\[ \mathbf{H}=\mathbf{B}=\mathbf{H}_{0}, \tag{4} \]

where \(\mathbf{H}_{0}\) is the field that was in the metal at the moment when it lost its resistance. From (2) it then follows that

\[ \mathbf{j}=\frac{c}{4\pi}\operatorname{rot}\mathbf{H}_{0}=\mathbf{j}_{0}, \tag{5} \]

where \(\mathbf{j}_{0}\) is the current density that was in the metal at the moment of the loss of resistance. In particular (as we shall assume below in this chapter), if initially there was no current, then there will be no current in the metal subsequently either, no matter how we change the external field. The physical meaning of these results is that every change—

\({}^{1}\) The properties of superconducting alloys have a number of characteristic differences, but since these properties may possibly be of secondary origin, it is more convenient to consider them in a separate chapter (Ch. VI).

SUPERCONDUCTIVITY

a change of the external magnetic field induces currents on the surface of the metal, and their magnetic field just compensates the change of the external field, so that the field inside the metal remains constant. Since there is no resistance, the surface currents do not decay, and the field inside the metal remains constant in time.

The strength of the surface currents, as is known, is determined by the jump of the tangential component of the field at the surface of the metal (the surface curl of the field). Namely, the density \(\mathbf{g}\) of the surface current is equal to

\[ \mathbf{g}=\frac{c}{4\pi}\left(\mathbf{H}_{t}-\mathbf{H}_{0t}\right), \tag{6} \]

where \(\mathbf{H}\) and \(\mathbf{H}_0\) are vectors representing the tangential component of the external and internal fields \(\mathbf{H}\) and \(\mathbf{H}_0\) at the surface.

In order to calculate the magnetic moment of the metal due to these surface currents, one must determine the field and then integrate, over the whole surface, the vector products of the current in each surface element by the area vector of that element. It is simpler, however, to take into account the magnetic effect of the induced currents by considering the superconductor as a magnetic body, i.e., by assuming \(\mathbf{H}\ne \mathbf{B}\). The induced currents then do not appear explicitly, but are assumed to have a nature analogous to that of the atomic currents which produce ordinary magnetism; this is possible, of course, only because the induced currents do not change with time under stationary external conditions. The current density (which we shall continue to denote by \(\mathbf{j}\)) is determined as

\[ \mathbf{j}=\frac{c}{4\pi}\operatorname{rot}\mathbf{B}, \tag{7} \]

and, since \(\mathbf{B}=\mathbf{H}_0\), we conclude that the currents, as before, must be wholly superficial (if initially there were no volume currents), and that the surface density of the currents is determined by equation (6). The field\(^1\) \(\mathbf{H}\) is the gradient of a certain potential \(\varphi\), satisfying the equation \(\Delta^2\varphi=0\); and \(\varphi\) and the tangential component of \(\mathbf{H}\) must be continuous at the surface (surface currents are connected with discontinuity of the tangential component of \(\mathbf{B}\)); thus \(\mathbf{H}\) can be determined by ordinary mathematical methods. Then the magnetic moment per unit volume is determined from

\[ \mathbf{B}=\mathbf{H}+4\pi \mathbf{I}. \tag{8} \]

The problem of the distribution of the field can, as is known, be solved exactly for an ellipsoid in a homogeneous field (i.e., if the field were

\(^1\) The true magnetic field in the metal is \(\mathbf{B}\), not \(\mathbf{H}\). In the case when \(\mathbf{B}\) does not depend on the external field, \(\mathbf{H}\) is in reality only a mathematical notation, having no direct physical meaning; this will be explained in more detail in Ch. III.

homogeneous in the absence of the metal). In order to illustrate the magnetic properties, let us consider the simplest example—an infinitely long cylinder (the limiting case of an elongated ellipsoid) in a homogeneous field parallel to the axis of the cylinder. In this case the distribution of the field H is not changed by the presence of the specimen, so that H is equal to the external field. Thus

\[ I=-\frac{H-H_0}{4\pi}. \tag{9} \]

This holds so long as the resistance of the metal is zero, and we can draw the magnetization curve shown in Fig. 3a. If the cylinder has lost its resistance in the absence of a field, then \(H_0=0\), so that \(I=-\frac{H}{4\pi}\), i.e. the metal behaves as

Fig. 3a.　　　　　　　　　　　　　　　　Fig. 3b.

a diamagnet with volume susceptibility \(-\frac{1}{4\pi}\), that is, in other words, with zero permeability. This continues until the critical field \(H_c\) is reached, when, as we saw in Ch. I, the resistance appears again. The surface currents disappear, and the body remains without a magnetic moment. If the field is now again decreased, the resistance again disappears as soon as \(H\) becomes less than \(H_c\), but now with \(H_0=H_c\), so that the magnetization is determined by the formula

\[ I=-(H-H_c)\frac{1}{4\pi}. \tag{10} \]

We see, therefore, that a considerable hysteresis should have occurred and that, when the external field becomes equal to zero, the cylinder should have remained with a “frozen-in” paramagnetic moment \(H_c/4\pi\), corresponding to the “frozen-in” flux \(B=H_c\). The physical meaning of this hysteresis consists simply in the fact that, when the field is decreased, the surface currents

are induced in the direction opposite to that which occurs when the field is increased. The area \(ABCD\) is proportional to the energy irreversibly lost in the form of Joule heat when the surface currents decay (at point \(B\)).

Similarly, if the metal is cooled in some field \(H_0\), no magnetic moment is observed at the transition temperature or below it; but if, after this temperature has been passed, the field is reduced, then the magnetization is again determined by (9) and is in this case again paramagnetic (corresponding to the segment \(EF\) in Fig. 3a). In Fig. 3b the corresponding \(B, H\) curve is shown for an ideal conductor (i.e. \(B = H_c\) for \(H < H_c\) and \(B = H\) for \(H > H_c\)).

Until very recently these predictions were regarded as obvious and not requiring experimental verification, and in the literature on superconductivity one can often find references to a “frozen-in” magnetic moment, although such a moment had never been observed experimentally. In the course of some experiments on the distribution of the magnetic field around superconductors, Meissner and Ochsenfeld\(^{24}\) in 1933 first discovered that some of the predictions for an ideal conductor turn out to be entirely incorrect for an actual superconductor. Namely, they found that for a pure superconductor the field distribution always corresponds to a field equal to zero inside the superconductor, i.e. inside the superconductor always

\[ \mathbf{B} = 0 \tag{11} \]

instead of \(B = H_0\), independently of the initial conditions (i.e. of the field in which the metal became superconducting). Subsequent experiments showed that this result is quite general.

Let us now consider what this result—the “Meissner effect”—means in the simple case of a long cylinder.

If the cylinder was cooled below the transition temperature in the absence of a field, then the magnetization upon increasing the field will be, as in the case of an ideal conductor, \(I = -\dfrac{H}{4\pi}\), and at the critical field, again as before, the magnetization will disappear (in this case \(H_0 = 0\), and therefore \(B = 0\) is equivalent to the previous condition \(B = H_0\)). Only if the field is reduced again do the properties of the superconductor begin to differ from those which one would have expected for an ideal conductor. Since, according to the Meissner result, the magnetic field can never exist in a superconductor, when the field becomes smaller than \(H_c\), all lines of force are suddenly “expelled” from the cylinder, and the magnetization is again equal to \(I = -\dfrac{H}{4\pi}\), so that there is no hysteresis. Similarly, if the metal is cooled in a field \(H_0\), the magnetic lines of force ...

disappears and therefore it can “forget” what value it must acquire when it appears again on the other side; in particular, it can take the value corresponding to the existence of a total current on the surface. Another simple example is a long cylindrical wire; in this case it is again easy to see that, in the absence of an electric field, the conditions \(\operatorname{div}\mathbf{B}=0\) and \(\operatorname{rot}\mathbf{B}=0\) inside the body can be satisfied only under the condition \(\mathbf{B}=0\), if a current flows along the surface of the wire.

In reality, of course, such a total current can never appear in a simply connected superconductor, unless it is part of a circuit containing a source of electromotive force. But we shall see in Chap. IV that, in a multiply connected superconductor (for example, a ring), a change of the external field alone is already sufficient to induce a total current in the specimen. Let us emphasize once again that this ability of surface currents, which determine the magnetism of a superconductor, to form a total current in the absence of an electric field is characteristic only of bodies with permeability exactly equal to zero. This property sharply distinguishes a superconductor from all other bodies, in which the surface currents responsible for magnetism can never give rise to any total current.

Thus, although there can never be an electric field in a body with \(\mathbf{B}=0\), nevertheless a total current can flow in such a body; hence it is clear that the absence of resistance is indeed a secondary property of a superconductor, which cannot by any means be regarded as the limiting case of an infinitely conducting body. In fact, measurements of resistance confirm only the fact that no electric field can exist in a superconductor, and say nothing about the resistance. In this sense the very concept of resistance even loses its meaning in the superconducting state, since in the absence of an electric field the conductivity cannot be measured. However, without measurements of resistance and confirmation of the existence of non-decaying currents in a superconducting ring, we could not be certain that the small value of \(\mathbf{B}\) given by magnetic measurements can indeed be interpreted as exactly equal to zero.

As a result of these considerations we see that a superconductor must be described as a body into which the magnetic field does not penetrate. We see that describing superconductivity as the limiting case of high conductivity cannot explain the Meissner effect, but that the new description, proceeding from \(\mathbf{B}=0\) in a superconductor, can explain the resistance equal to zero and is therefore fundamental.

III. Intermediate State

properties will depend on the shape of the specimen, since any magnetic field will be altered in the presence of the specimen. For this reason, as the field is gradually increased, it will reach the critical value \(H_c\) at some points on the surface of the specimen earlier than at others (with the exception of the simplest case of an infinite cylinder parallel to the applied field). As soon as the field reaches \(H_c\) at some point, the metal will no longer be able to remain superconducting at that point, and superconductivity must begin to disappear. With a further increase of the external field, a certain transition must occur until, finally, the specimen becomes wholly normally conducting, when the external field reaches \(H_t\). In this chapter we shall consider how this transition takes place in connection with the magnetic properties of superconducting ellipsoids, and we shall also briefly describe the experimental methods by which these properties have been studied, and the results thereby obtained.

If the specimen is wholly superconducting (i.e., the applied magnetic field is not too large), then the magnetization of the specimen and the field distribution around it may be obtained by applying the usual methods to the case of magnetic permeability equal to zero. As is known, an exact solution can be obtained only for an ellipsoid in a homogeneous field; moreover, a non-ellipsoidal shape may introduce still other complications (irreversibility, Ch. IV). Therefore we shall now restrict ourselves to considering superconductors of ellipsoidal shape.

Consider an ellipsoid with demagnetizing coefficient \(4\pi n\) (relative to one of the principal axes), and let \(H\) be the external homogeneous magnetic field (i.e., the field in the absence of the specimen), directed parallel to the given principal axis. If the magnetization produced by this field is \(I\), then we may formally regard there as being, inside the ellipsoid, a “magnetizing” field \(H_i\), which is also homogeneous and parallel to the same principal axis,

\[ H_i = H - 4\pi n I . \tag{1} \]

Let us note that the quantity \(H_i\) is introduced here only as a mathematical notation and has no physical meaning, since the field inside a magnetized body is defined as \(\frac{\partial G}{\partial B}\), where \(G\) is the free energy; if, as in our case, \(B\) is constant (equal to zero), then this derivative has no meaning. We now have

\[ B = H_i + 4\pi I \]

or, since \(B = 0\),

\[ I = - \frac{H}{4\pi(1 - n)} . \tag{2} \]

Thus the magnetization curve of a superconducting ellipsoid is the steeper, the larger the demagnetizing coefficient, i.e. the more

which coincides with (2) [equation (2) is evidently applicable, since in this case the entire specimen is superconducting]. For \(H=H_c\), equation (5) also satisfies the obvious requirement that the specimen be in the normal state (\(I=0\)).

The values of \(B\) in the intermediate state are easy to determine, since \(B=H_i+4\pi I\), or \(B=H+4\pi(1-n)I\), so that, substituting (5), we find

\[ B=\frac{H}{n}-\left(\frac{1}{n}-1\right)H_c . \tag{6} \]

We see that as soon as the external field exceeds the value \((1-n)H_c\), the induction in the specimen ceases to be zero and increases linearly from 0 to \(H_c\); beyond this point the penetration of the lines of force becomes complete and \(B=H\). In the intermediate state the magnetic-field vector \(H_i(=H_c)\) already has a physical meaning, since \(B\) is now variable. This is the field that would actually be observed in a small channel cut through the body parallel to the direction of the external magnetic field. Let us emphasize once again that this is not so in the purely superconducting state, when \(B=0\), since, as is easy to see, the presence of a channel in this case changes the field distribution in such a way that the field observed in the channel has no relation whatever to \(H_i\) of equation (1). This is particularly clear in the case where the channel cuts all the way through the body, since the body is then no longer simply connected, and we shall see (Chapter IV) that in the case of a ring the field in the channel always remains constant. Even if the channel does not pass all the way through the body, the ends of the channel produce such a change in the field that the field inside the channel vanishes. In the case of the intermediate state, however, as in an ordinary magnetic body, the disturbance produced by the ends of the channel may be neglected, so that the field in the channel is indeed equal to \(H_i\).

These ideas about the intermediate state were introduced in order to explain the results of experiments on the magnetic properties of superconducting specimens during the transition from the superconducting to the normal state, and we shall now briefly show how these experiments fit into the theory described. It is convenient here to mention all the experimental methods that have been used to investigate the magnetic properties of superconductors, since some of them were applied to problems that will be considered below. At the same time there is no need to describe these methods in detail, since in most cases they are merely modifications of well-known magnetic methods.

1. In the original experiments of Meissner and Ochsenfeld\({}^{28}\), the field at different points around the specimen was measured by means of small test rotating coils. This method (also used by Tarr and Wilhelm\({}^{29}\) in experiments confirming

...field on the surface of the test coil and requires rather complicated devices for installing and rotating the coil. However, it is useful in those cases, for example, when one is dealing with a large specimen.

1. Mendelssohn and Babbitt[^30] measured the field at the equator of a tin sphere by means of a bismuth wire (at low temperatures the resistance of bismuth depends very strongly on the field1). Their results confirmed in general the behavior predicted above; thus, for \(H<\frac{2}{3}H_c\) the field at the equator was approximately equal to \(\frac{3}{2}H\), whereas for \(H_c>H>\frac{2}{3}H_c\) the field at the equator was approximately constant and equal to \(H_c\). The approximate character of this confirmation was due to the fact that the bismuth wire had a non-negligible thickness, so that the measured field was rather smaller than the true field at the equator, and also to the fact that the tin was somewhat contaminated (we shall see below that impurities have a strong influence on the magnetic properties). A more detailed investigation of the field around a tin sphere was later carried out by de Haas and Gorter[^32], using the same method; they also measured the fields inside various channels in the sphere. The results agree well with the theory described, in particular as regards the field outside the sphere (apart from a slight irreversibility when the field was decreased below \(H_c\), which may again be ascribed to slight impurities). Measurements of the field in a channel directly confirmed that the field \(H_i\) is equal to \(H_c\) in the intermediate state (i.e., for \(H_c>H>\frac{2}{3}H_c\)). In the superconducting state, however, this field, as already indicated, does not give the true \(H_i\). Indeed, when \(H\) was increased, the field in the channel was zero up to the attainment of the intermediate state, and when \(H\) was decreased it was constant and equal to \(H_c\) down to \(H=0\), which is precisely what was to be expected, since the channel passes right through the sphere and thus transforms it into a superconducting ring (Chap. IV).

2. Shoenberg[^33] used the direct method of measuring the force acting on a specimen in a weakly non-uniform field; this force, divided by the field gradient, then gives the magnetization. This is, of course, nothing other than the well-known Faraday method for measuring susceptibility. It is especially simple in the present case owing to the very strong diamagnetism of the superconductor, so that the balance by which the force is measured need not be especially sensitive. Fig. 7 shows the magnetization curve of a lead sphere obtained by this method. We see that it agrees well with the theory developed above. At first the slope of the curve

indeed very close to the calculated value \(\frac{3}{8\pi}\), and beyond \(\frac{2}{3}H_c\) (when the field begins to penetrate into the sphere) the curve turns sharply downward and is almost exactly linear; moreover, the magnetization vanishes at \(H_c\) and above. Upon subsequent reduction of the field the curve is retraced practically without hysteresis, thereby again confirming the Meissner effect (a slight hysteresis, as usual, can be explained by deviation from ideal conditions and, probably, is due to small impurities).

1. Another method for investigating the intermediate state, used by Shoenberg\(^{34}\) and Daunt\(^{35}\), consists in measuring the susceptibility of a specimen in an alternating field. This can be done by observing, by the alternating-current method, the change in the self-inductance of a coil when the specimen is inserted into it (in Daunt’s work the change in the mutual inductance of two coils was observed); the alternating field of the measuring current in the coil was superposed on

Fig. 7. The dashed curve refers to \(\dfrac{H_i}{H_c}\), the solid curve—to \(\dfrac{B}{H_c}\).

\[ \frac{1}{2}H_c, \]
i.e., as soon as the cylinder passes into the intermediate state. That this restoration of resistance at \(\frac{1}{2}H_c\) is in fact connected with the beginning of field penetration was shown by an experiment with a cylinder of elliptical, rather than circular, cross-section; corresponding to the different value of the demagnetizing factor, resistance begins to appear at a different value of the field. It is interesting to mention that Laue\(^{38}\) first pointed out that the restoration of resistance at a field \(\frac{1}{2}H_c\), in the case of a circular cylinder in a po-

The presence of resistance in the direction perpendicular to the external field indicates that it is precisely the second type of region that is realized. In this connection we mention the interesting experiments \(^{39}\), which showed that a sphere in the intermediate state has resistance perpendicular, but not parallel, to the external field, thereby confirming almost directly that the sphere actually consists of alternating superconducting and normal layers. Less direct confirmation of this type of structure is provided by experiments with a sphere in an alternating field, which also confirmed that a sphere in the intermediate state possesses anisotropic resistance.

Since there is no magnetic induction in the superconducting regions, and the induction in the normal regions is equal to \(H_c\), while the average induction is the value \(B\) observed for the whole specimen, it is clear that the fraction \(x\) of all the material that is in the normal state is equal to

\[ x=\frac{B}{H_c}. \tag{7} \]

However, in order to be able to say anything about the absolute thickness of each layer separately, it is necessary to examine the structure of the intermediate state in greater detail.

This question was investigated \(^{40,41}\) by a theoretical method analogous to that used in determining the structure of regions of spontaneous magnetization in ferromagnets. It turns out that, for a specimen with nonzero average value \(B\), the energetically most favorable distribution is in fact one in which the specimen consists of a large number of alternating superconducting and normal layers. A precise investigation shows that the number of layers (and, consequently, their thickness) changes from the interior of the body toward its surface: near the surface the layer is thinner than the penetration depth of the magnetic field into a superconductor, so that the usual macroscopic description \((B=0)\) is no longer applicable to superconducting layers at the surface of the specimen. By branching, the layers become thicker (and therefore fewer in number) toward the interior of the body, as shown in Fig. 9 for the case of a plane plate in a transverse magnetic field. The absolute value of the greatest thicknesses \(z_n\) and \(z_s\) of the layers (i.e. the thickness at the center of the body) is determined by the “surface tension” \(\alpha\) between the superconducting and normal phases (this quantity is at present unknown) and by the length \(d\) of the body in the direction of the magnetic field. Namely: in the case of a plane plate,

\[ z_n^2 z_s^2=\frac{32\pi\alpha}{H_c^2}(\sqrt{2}-1)^2 d^2;\qquad \frac{z_n}{z_s}=\frac{B}{H_c-B}, \tag{8} \]

where \(B\) is the average induction in the specimen [defined by (6)].

If we write (for \(B=\dfrac{1}{2}H_c\))

\[ z_n=d_0^{1/3}d^{2/3}, \tag{9} \]

\[ \left[d_0=\frac{32\pi d}{H_c^2}\left(\sqrt{2}-1\right)^2\right], \]

we see that the thickness of the layers is small in comparison with the dimensions of the body only in the case where the dimensions of the specimen are large in comparison with \(d_0\). Since \(d_0\) has the dimension of length, it is natural to suppose that this is the order of magnitude of the depth

Fig. 9.

of penetration of the magnetic field into a superconductor. Indeed, if the specimen is so small that its dimensions are comparable with the magnitude of the penetration of the magnetic field into the superconductor, the very concept of the intermediate state loses its meaning, and we shall not be able to say anything about how the superconducting transition in a magnetic field takes place until we understand the behavior of superconductors of small dimensions. We shall see in Ch. VIII that at present this question is unclear both from the experimental and from the theoretical side. However, the data presently available indicate that the penetration depth is of the order of \(10^{-5}\,\text{cm}\), so that from (9) we may conclude that the thickness of the layers in a plate of thickness \(1\,\text{cm}\) is of the order of \(2\cdot10^{-9}\,\text{cm}\). It is very difficult, however, to propose a method for measuring this thickness, since at the surface of the plate, owing to the branching described above, it becomes much smaller, and from the outside the specimen will appear completely homogeneous.

This theory is important in that it shows that the intermediate state is in reality not a new state, but merely a special kind of “mixture” of the ordinary superconducting and normal phases. It is true that it cannot explain certain other peculiarities of the intermediate state (which we shall describe below), some of which are perhaps connected with the properties of very thin layers; and these layers, as the theory shows, must be located on the surface of the specimen. Namely, when the thickness of the branching superconducting and normal layers becomes of the order of \(d_0\) (which occurs near the surface of the specimen), it is no longer possible to speak of separate superconducting and normal phases, and we have a kind of “mixed” phase, about whose properties very little can be said until we have a deeper theory (i.e. a theory capable of explaining the properties of superconductors of small dimensions).

We now turn to the properties of the intermediate state, for which at present there are hardly more than speculative explanations:

1. The most important of these properties is that the resistance of a metal in the intermediate state depends on the magnitude of the measuring current much more strongly than could have been expected from the influence of the current’s magnetic field on the surface of the specimen. Since the current must flow normally to the layers, and the fraction of the metal in the intermediate state is equal to

\[ \frac{B}{H_c} \]

(8), one might have expected that the resistance to the current would vary as

\[ \frac{B}{H_c}, \]

i.e. linearly from zero to its full normal value, independently of a weak measuring current (see the dotted line in Fig. 8). In reality, however, as is seen from Fig. 8, the resistance of a cylinder in a transverse magnetic field is the smaller the smaller the measuring current, even if its field is of the order of \(4 \cdot 10^{-3} H_c\); and in general the resistance turns out to be less than would be expected from a linear law. Similarly, in the experiments with an alternating field described on p. 468, it was found that the mean resistance of the sphere decreases when the amplitude of the alternating field is decreased (i.e. when the strength of the Foucault currents in the sphere is decreased) down to amplitudes of the order of \(H_c\). Over a comparatively large range of current strength, the resistance in a given external field decreases roughly as the logarithm of the current, so that very sensitive methods are required if one wishes to extrapolate to vanishingly small currents. The methods used up to the present have not permitted extrapolation to sufficiently small currents, but nevertheless both the experiments with an alternating field and certain recent works of Meissner\(^{28}\) with a wire in a transverse field (carried out by means of a very sensitive potentiometric method) indicate that the resistance may vanish completely at zero measuring current. This assumption, however, must be regarded—

should be regarded as speculative until further experimental confirmations are obtained.

Apparently, the only possibility of bringing these electrical properties into agreement with the layered picture of the intermediate state consists in ascribing the anomalous behavior to the “mixed” phase, which, as was mentioned, must be located on the surface of the specimen. Namely, we must suppose that at a very weak current the mixed phase has a resistance substantially smaller than the resistance from the layers inside the specimen (it is possible that this resistance even vanishes altogether in the limiting case of zero measuring current), and that this resistance increases as the current strength is increased. Then, at very weak currents the resistance of the specimen would be determined mainly by the properties of the mixed phase on the surface of the specimen, while at stronger currents it would be determined mainly by the properties of the layered structure in the middle of the specimen. Thus, according to this hypothesis we should expect that, as the current strength is increased, the dependence between the resistance and the field should approach a linear one, and this is in rough agreement with the observations. In order to investigate this hypothesis further, it would be necessary to study the effect of changing the diameter of the wire, but at present there are very few data in this respect.

1. Although the magnetic experiments in the case of a long cylinder in a magnetic field confirm that the field actually begins to penetrate at \(\frac{1}{2}H_c\), the resistance of the cylinder in some cases begins to appear only at somewhat larger values of the field. Thus, in the Leiden experiments the appearance of resistance often began at \(0.58H_c\) (this is seen in Fig. 8; to avoid confusion we did not indicate this circumstance earlier). Recent experiments by Burton and Mann\(^{43}\) and by Misener\(^{42}\) have shown, however, that the figure \(0.58H_c\) is obtained only if the temperature is comparatively much lower than the transition point, and that as the temperature approaches the transition point (i.e., at small fields) there is an approach to the theoretical value \(\frac{1}{2}H_c\)\(^{1}\).

The dependence of this property on temperature, and the fact that it is observed in very pure specimens, suggest that the nonappearance of resistance at \(\frac{1}{2}H_c\) (i.e., at the moment when field penetration begins) may be due to some effect analogous to supercooling; but it is also possible that the state with zero resistance between \(0.5\) and \(0.58H_c\) is stable and is due to some property of the mixed phase on the surface of the specimen, or is caused by the small thickness of normal layers in the initial stages of the transition. Let us note that the majority of experi-

\(^{1}\) This can also be seen from some Leiden data\(^{37}\).

ments was carried out with rather thin wires (radius of the order of \(10^{-2}\) cm), so that the thickness of the layers in the intermediate state is very small, especially at the beginning of the transition. It is possible that the anomaly at \(0.58 H_c\) is connected with the anomalous properties of such thin layers, the dependence of this property on temperature being due to the temperature dependence of \(d_0\) in equation (9) (i.e., to the change of the layer thickness with temperature). In this respect, systematic experiments with wires of different diameters would be of interest.

1. The preceding property may also be connected with the phenomenon of a time lag in the penetration of the magnetic field into a specimen in the intermediate state. De Haas and Engelkes\(^{44}\) showed that, after a sudden change of the external field, the field distribution of a sphere in the intermediate state assumes its final form only gradually (the time lag is of the order of several seconds near the onset of the intermediate state, i.e., about \(\frac{1}{2}H_c\), and decreases as the penetration of the field increases, becoming negligibly small when the field has fully penetrated). However, no lag at all was found in Shenberg’s experiments, in which the magnetic moment of a sphere was measured; perhaps this is connected with the fact that the sphere was several times smaller than the one used at Leiden, but we are more inclined to the view that the observed time effects were not of a fundamental nature. This is confirmed by the fact that the lag was found for specimens not of ellipsoidal form,\(^{45}\) for which the transition is more complicated than for ellipsoids (Ch. IV). It is therefore possible that the observed effects are due to irregularities of shape (for example, to holes drilled in the sphere for measuring the internal field).

2. In some cases, when the magnetic field is reduced below \(H_c\), a specimen which is in the normal state remains so until the field becomes \(2\)–\(3\%\) less than \(H_c\), after which the magnetic properties suddenly become those which they should be in the intermediate state.\(^{45}\) A related effect is perhaps the fact that the resistance of a long cylinder in a longitudinal field suddenly disappears at a field smaller than that at which the resistance appears when the field is increased.\(^{37}\) These effects are observed only in very pure specimens, but quantitatively they are never reproduced exactly, so that they are possibly analogous to supercooling in ordinary phase transitions. G. London\(^{46}\) pointed out that such supercooling may be connected with the surface tension between the superconducting and normal phases, but the exact dependence has not been clarified.

\(^{1}\) It was found, however, that this effect depends strongly on how the potential wires are attached to the specimen (and is much larger for soldered than for welded wires), so that it may partly have a secondary nature, due to some effect of the alloys in the connecting layers.

We have thus considered the majority of the phenomena connected with transitions of ellipsoids between the superconducting and normal states, caused by a magnetic field. Although the general nature of these transitions is clear, we see that there are many details that still require explanation. However, it is not yet clear which of them are fundamental and which are due to “nonideal” conditions. We shall see in Ch. IV that the intermediate state can also explain transitions caused by the current itself, and in Ch. V we shall become acquainted with the thermodynamics of the intermediate state.

IV. Superconducting Ring and the Destruction of Superconductivity by a Current

A multiply connected body, for example a ring, differs essentially from a simply connected body in that a “total” current can be induced in it, i.e. closed contours can be drawn around parts of the body (nowhere intersecting the surface of the body) which enclose a nonzero total current. If the body under consideration is not superconducting, then any such total current is always associated with an electric field, so that Joule heat is liberated, and the current cannot sustain itself. In a superconductor, however, the current, as we have seen, has the same nature as the surface currents responsible for ordinary magnetism, and is not associated with any electric field, so that it can flow for an indefinitely long time.

In order to demonstrate the original magnetic properties arising from the possibility of inducing such “nondecaying” total currents, we shall consider a case that can be investigated mathematically, namely a circular ring1 of radius \(R\), made of wire with circular cross-section and radius \(r\), small in comparison with \(R\). If the self-inductance of the ring is \(L\), then any change in the magnetic field \(H\), normal to the plane of the ring, causes a change in the total current \(i\) around the ring, namely:

\[ L \frac{di}{dt} - \pi R^2 \frac{dH}{dt} = 0 . \]

This equality expresses the fact that there is no electromotive force in the superconducting ring. Integrating, we find

\[ Li = \pi R^2 (H - H_0), \tag{1} \]

where \(H_0\) is the value of the field when there was no current in the ring. If there were no current in the ring, it would have a magnetic moment equal to the moment of a long wire in a transverse magnetic field (to first approximation the fact that the wire is bent into a ring is immaterial),

i.e. \(\pi Ri^{2}H\) would be the moment in the field \(H\). However, the total current \(i\) gives the ring an additional magnetic moment \(\pi R^{2}i\), which in order of magnitude is \(\left(\dfrac{R}{r}\right)^{2}\) times greater than the preceding one [this may be verified by substituting equation (3) for \(L\), see below, but qualitatively it is evident from the fact that the moment due to the current is of the order of the moment of a sphere of radius \(R\), whereas the moment of a ring without current is proportional only to the volume of the superconducting material in the ring]. Of course, such a division of the magnetic moment into two parts is in reality somewhat artificial, but it is convenient for describing the actual distribution of current, consisting of surface currents nonuniformly distributed over the cross-section of the wire. The current \(i\) represents the average current flowing in one direction, while the currents producing the magnetic moment for \(i=0\) are those which remain after subtracting \(i\); roughly speaking, equal currents in opposite directions, flowing along the inner and outer surfaces of the ring. Because these equal and opposite currents flow so close to one another, their magnetic moment is small in comparison with the moment of the current \(i\), flowing in one direction.

We see at once from (1) that the magnetic properties of the ring differ substantially from those of a simply connected body, since the magnitude of the magnetic moment is wholly determined by the initial conditions. If, for example, the ring is cooled below the transition temperature in the absence of a field, i.e. \(H_{0}=0\), then the current is determined by

\[ Li=\pi R^{2}H. \tag{2} \]

If, on the other hand, the ring was cooled in the field \(H_{0}\), then the current is determined by equation (1); in particular, if the field is reduced to zero after cooling, there remains in the ring a current \(\pi R^{2}\dfrac{H_{0}}{L}\), corresponding to a large “paramagnetic” moment. This non-uniqueness of the magnetic moment is analogous to that which we derived for a metal of infinite conductivity, and follows from the fact that the current around the ring is induced in such a way that the magnetic flux through the aperture of the ring remains constant. In the aperture of the ring (or inside a metal of infinite conductivity) \(B=H\), so that the induced current will depend on the initial value of \(H\); for a simply connected superconductor the surface currents are also induced in such a way that the magnetic flux through the section considered remains constant, but here this constant value is always equal to zero, independently of the value of the external field in which the body became superconducting (in a superconductor the magnetic permeability is equal to zero). In other words, the distinction between a simply connected superconductor and a superconducting ring or any merely infinitely conducting body consists in the fact that in the former the permeability is equal to zero along every section, regardless of what flux passed through it initially,—this flux is entirely “expelled”

“...is disrupted” when the body becomes superconducting. Conversely, in which the permeability in the greater part (or in all) of the cross-section (for example, the hole in the case of a ring) remains equal to unity, so that the field \(H_1\) can remain in the hole of the ring even after it has become superconducting (or, in the case of an infinitely conducting metal, inside the metal after it has become infinitely conducting).

The constancy of the current in a superconducting ring gives the most accurate confirmation of the equality to zero of the resistance of the metal. If some e.m.f. were associated with the current, energy would be lost in the form of Joule heat, and the current would decay according to the equation

\[ L\frac{di}{dt} + wi = 0 \]

(\(w\) is the resistance of the ring, i.e. the ratio of the e.m.f. to the current strength), or

\[ i = i_0 e^{-\frac{wt}{L}}. \]

The decay time of the current \(\tau = \frac{L}{w}\) for a metal at low temperature, but not superconducting, for any practically used dimensions of the ring is very small (usually a small fraction of a second).

For a superconductor, however, even for a ring made of very thin wire or of a thin layer deposited on a nonsuperconducting wire (in order to increase the presumed resistance), no decay of the current could be detected over several hours. In this way one can find the highest possible limit of the value that \(w\) could have, from the limit of sensitivity of the experimental method of measurement.

Fig. 10.

Returning to the discussion of the magnetic properties of a superconducting ring, we shall now describe the experimental results. Shoenberg \(^{33}\) investigated the magnetic properties by measuring the total magnetic moment of the ring by the force-measurement method; we note that this method has the disadvantage that only small rings can be used, and therefore it is very difficult to make a ring with a sufficiently constant cross-section, which is necessary for an exact comparison with theory. The field at various points around the ring was also measured \(^{47,48}\), from which the magnetic properties were inferred. Figure 10 shows how the magnetic moment of a ring, cooled in the absence of a field, changes during subsequent

for increases and decreases of the field, normal to the plane of the ring (this diagram is in fact a somewhat idealized representation of the experimental results, which in some of their details are distorted by extraneous causes of no interest to us here). At first the magnetic moment increases with the field, as was explained above. If one subtracts the moment of the ring in the absence of a persistent current, then the slope of the line \(OA\) agrees very well with equation (2); it is interesting to note that this agreement confirms that the current flows only over the surface of the ring (the absence of any current in the bulk of the superconductor is, of course, contained in the condition \(B=0\)). Thus the experiments show that (2) is confirmed quantitatively only if for \(L\) we take

\[ L=4\pi R\left(\lg \frac{8R}{r}-2\right), \tag{3} \]

i.e., the value that \(L\) has for a purely surface current, amounting for the rings used to about \(10\%\) of the value

\[ L=4\pi R\left(\lg \frac{8R}{r}-\frac{7}{4}\right) \]

for a current flowing over the whole cross-section of the ring1.

We see that as soon as point \(A\) is reached, the equation becomes inapplicable, and, with further increase of the field, the current begins to fall according to a linear law. The reason for this is that the total field on some surface of the ring (in the present case, on the outer edge) reaches the value \(H_c\), when the external field exceeds a certain value, which we shall calculate directly. As soon as this occurs, resistance appears in the ring, and the current decreases to a value at which the maximum field on the surface will be exactly equal to \(H_c\), i.e. such that the ring again becomes superconducting. Thus the current decreases along the part \(AB\) of the magnetization curve, while the ring remains superconducting throughout the entire region \(OAB\); this is proved by the fact that decreasing the field from any point \(x\) on \(AB\) causes a change of the current in accordance with (1), with the “surface” value of \(Z\), i.e. along a line parallel to \(OA\). The measurement of the self-inductance of an alternating—

also showed by direct current that the ring is entirely superconducting in the region \(AB\), as well as in \(OA\)¹).

It is easy to see that the field on the surface of the ring has, in the region \(OAB\), its maximum value at the outer edge; thus here the field \(H_l\) of the total current and the field \(H_e\), which would be present here in the absence of the total current, have the same direction. If we neglect correction terms of order \(r/R\), we have \(H_l=\dfrac{2i}{r}\) and \(H_e=2H\) (at the outer edge), so that the field corresponding to the point \(A\) is determined from

\[ 2H+\frac{2i}{r}=H_c, \tag{4} \]

or, substituting equation (2) for \(i\), we have

\[ H=\frac{Lr}{2\pi R^2}\,\frac{H_c}{1+\dfrac{Lr}{\pi R^2}} \tag{5} \]

for the field at which the current begins to fall.

Beyond \(A\), as has already been indicated, the current can no longer have its “full” value \(\dfrac{\pi R^2 H}{L}\), but becomes such as to satisfy (4). In other words, in the region \(BC\) of the magnetization curve the current is approximately determined by the equation

\[ i=r\left(\frac{1}{2}H_c-H\right). \tag{6} \]

We see that as soon as the external field \(H\) reaches the value \(\dfrac{1}{2}H_c\), the current disappears altogether (a better approximation shows that at \(\dfrac{1}{2}H_c\) there is still a weak current, which disappears discontinuously upon a further increase of the field). At the same time the ring passes into an intermediate state. With further increase of the field the ring behaves exactly like a long wire in a transverse field greater than \(\dfrac{1}{2}H_c\) (Chapter III), since there can no longer be any total current around the ring (any current, however small, would make the field on the surface of the ring exceed \(H_c\), and, moreover, owing to the resistance of the intermediate state any such current would die out). Indeed, in this region the multiple-connectedness of the ring is no longer essential; this was confirmed experimentally by repeating the measurements after cutting the ring; the experiment shows that the magnetization in the region \(BC\) is the same for cut and uncut rings, whereas the magnetization in

¹) Let us note here that the interpretation of the region \(BC\) on the magnetization curve originally given by the author is definitely erroneous,

the region \(OAB\) is greatly reduced when the ring is cut (\(OB\) refers to the cut ring).

As soon as the external field reaches \(H_c\), superconductivity is completely destroyed, and no magnetic moment remains. If the field is now decreased again, the region \(BC\) in Fig. 10 is repeated, the ring again becoming wholly superconducting, and a further decrease of the field induces in the ring a current in the direction opposite to that in which it was induced when the field was increased (a better approximation shows that this occurs at a field somewhat smaller than \(\frac{1}{2}H_c\), namely at

\[ H = H_c \left( 1 + \frac{r}{4R} \right). \]

Thus we should have expected that below \(\frac{1}{2}H_c\) the current is determined by equation (1) with \(H_0 = \frac{1}{2}H_c\), since there was no current in the ring at this value of the field. This current, induced when the field is decreased, can never, however, reach the full value given by (1), since in that case the field at the surface of the ring would exceed \(H_c\). The field at the surface of the ring is now greatest on the inner side of the ring (since the current now has the opposite direction), and the total field on the inner equatorial line is equal to \(2H - \frac{2i}{r}\). If the current were to attain its full value

\[ \frac{\pi R^2}{L}\left( H - \frac{1}{2}H_c \right), \]

then this total field would be equal to

\[ \frac{\pi R^2}{Lr} H_c - 2H\left( \frac{\pi R^2}{Lr} - 1 \right), \]

which is always greater than \(H_c\), if, as we have here, \(H < \frac{1}{2}H_c\) (it is easy to show that the dimensionless ratio \(\frac{\pi R^2}{Lr}\) is always greater than unity). Since the current cannot reach its full value, it will, as in the region \(AB\), be determined by the condition that the total field at the surface of the ring (now on its inner side) is exactly \(H_c\). In other words, in the region \(BD\) in Fig. 10 the current is determined by the equality

\[ I = -r\left( \frac{1}{2}H_c - H \right). \]

Let us note that the slopes \(AB\) and \(BD\) are equal and opposite (this too holds only in the approximation we have adopted and if the magnetization due to the incomplete current is neglected). Completely analogous considerations show that the region \(EF\) is determined by the condition that the field on the outer side is equal to \(H_c\), \(FG\) by the same condition for the inner side, and \(GAB\) again for the outer side. We see that the largest “persistent” current which can remain in the ring in a zero field is determined

from \(\dfrac{2i}{r}=H_c\) and can flow in both directions in the ring. At first glance it may seem unclear why the linear part \(BD\) continues to \(E\), since it would seem that, as soon as the field changes direction, the greatest value of the field should have been attained on the outer equatorial line, where \(H_I\) and \(H_e\) have identical directions. However, a better approximation shows that the values of \(H_I\) on the outer and inner equators differ by a small amount proportional to \(\dfrac{r}{R}\) (and similarly for \(H_e\)), so that the field still remains maximal on the inner side over a small interval after the change of the field direction (namely up to \(E\)). The position of the point \(G\) is explained analogously.

The line \(ABDEFG^{1)}\) may be regarded as a certain boundary curve limiting the possible values that the total current in the ring can have; everywhere inside the boundary the current changes, when the field is changed, according to (1), i.e. along lines such as \(DX\) or \(PQ\), parallel to \(OA\); as soon as the current reaches a point on the boundary curve, a further change of the field changes the current along the boundary curve itself. These considerations are of some practical interest, since they show that, in order to create the greatest “non-decaying” current in the ring (in the absence of a field), it is sufficient to cool the ring in the field corresponding to point \(Y\) in Fig. 10, and then switch off this field; if the ring was cooled in the absence of a field, it would be necessary to switch on and off a field at least as large as that which corresponds to point \(X\).

Let us note here that the magnetization of a singly connected superconductor may also exhibit hysteresis phenomena if the body has a complicated shape. As we shall explain shortly, it is possible that this hysteresis is due to the “freezing-in” of lines of force in superconducting rings. In some experiments with short cylinders\({}^{47}\) with sharp edges, a magnetization curve was obtained (Fig. 11) in a field parallel to the axis of the cylinder. First of all, we note that in increasing fields the curve no longer turns—

\({}^{1)}\) For completeness we give the equations for this curve, obtained in the best approximation. Namely:

\[ i=\mp r\left[\frac{1}{r}H_c-H(1\pm \alpha)\right]\frac{1}{1\pm \beta}, \]

where

\[ \alpha=\frac{r}{4R}\quad \text{and}\quad \beta=\frac{r}{2R}\left(\lg\frac{8r}{R}+1\right); \]

the lower signs should be taken where the field is equal to the critical field on the outer side, and the upper signs where the field is equal to \(H_c\) on the inner side. (\(H_c\) should be taken negative in the regions of the diagram on its left-hand side.) The observed magnetic moment contains, of course, also the term due to the magnetic properties of the ring in the absence of a total current.

changes abruptly, as in the case of an ellipsoid. This may be interpreted as the result of a nonuniform distribution of \(H_i\) in the cylinder, so that at sharp edges, for example, the field reaches \(H_c\) earlier than in the rest of the volume of the specimen.

Thus, during the transition from the superconducting to the normal state, the specimen may consist of a complex mixture of superconducting, intermediate, and normal regions. If the field is again reduced below \(H_c\), a hysteresis is observed\(^1\), somewhat similar to that observed for a ring (Fig. 10). A possible explanation is that, owing to the nonuniform distribution of the field, a ring on the surface of the specimen may, when the field is reduced, become superconducting earlier than the rest of the volume of the specimen.

Fig. 11.

With a further decrease of the external field, a current would be induced in this ring, maintaining the magnetic-field flux in the inner regions constant and thereby preventing them from becoming superconducting again.

It should be noted, however, that the possibility of such formation of superconducting rings in a simply connected body has not yet been proved theoretically; nevertheless, our qualitative explanation is to some extent confirmed by two other experimental facts:

1) it was found that the hysteresis in Fig. 11 can be considerably reduced by rounding the edges of the cylinder, which perhaps makes the field distribution more homogeneous and, according to our explanation, delays the formation of a superconducting ring;

\(^1\) The form of this hysteresis is independent of temperature (i.e., the magnetization curves at different temperatures can be brought into coincidence by a simple change of scale). This circumstance makes it possible to distinguish this hysteresis, caused by the shape, from hysteresis arising from impurities, which depends on temperature (Ch. VI).

2) in a field perpendicular, and not parallel, to the axis of the cylinder, there was practically no hysteresis at all (Fig. 11b). In this case there is no symmetry about the direction of the field, and therefore the specimen cannot become superconducting simultaneously everywhere around the cross-section normal to the field.

We have already pointed out that when the magnetic field of the current at the surface of a superconductor exceeds \(H_c\), resistance appears. In the case of a ring, in this situation the current simply adjusts itself so as to make the field exactly equal to \(H_c\), and the resistance disappears again; but if the current is kept constant (by means of an external source of e.m.f.), the resistance remains. We shall now consider the changes that occur when superconductivity is destroyed by a current in this way, and, for simplicity, we shall consider a long cylindrical wire (of radius \(a\)) along which a current \(i\) flows.

As soon as the current exceeds the value \(\frac{1}{2}aH_c\), the field at the surface will exceed \(H_c\), and, obviously, superconductivity will begin to disappear. If we suppose, as at first sight seems possible, that the transition to the normal state occurs by a gradual contraction of the internal superconducting region of the wire, we immediately encounter difficulties analogous to those considered in Chapter III in connection with the transition of an ellipsoid into the normal state under the influence of an external magnetic field. Indeed, if superconductivity were confined to an internal region of the cylinder, then the whole current would flow through this superconducting “core” and, consequently, would produce at its boundary a field even greater than that which was initially at the surface of the wire.

The destruction of superconductivity would have to continue until the whole wire had passed into the normal state; this, however, is impossible, since then the current would be distributed uniformly over the entire cross-section of the wire and the field would be less than \(H_c\) over the greater part of this cross-section (up to the radius \(\frac{H_c a^2}{2i}\)), so that this part could not be in the normal state. This paradox becomes still sharper if we consider what happens when a wire carrying a current \(i\) is cooled below the transition temperature. Since in the normal state the magnetic field of the current increases from zero as one moves away from the axis of the wire, superconductivity should first appear along the axis. But in that case the whole current would at once flow in this superconducting region and would create a very large field at its boundary, so that the superconductivity would have to disappear again.

As in the case of an ellipsoid in a homogeneous field, the paradox shows that the transition must occur in a more complicated manner. London \(^{49}\) showed that, since the “core”

the wire can be neither superconducting nor normal; it must be in the intermediate state with \(H=H_c\). It is easy to see how the resistance will then increase as the current increases. Let \(i=\frac{1}{2}aH_c\), and let \(x\) be the total current inside a cylinder of radius \(r\). Everywhere inside the “core” that is in the intermediate state the field is equal to \(H_c\), so that

\[ \frac{2x}{r}=H_c . \tag{7} \]

In particular, if \(r_0\) is the radius of the “core,” and \(x_0\) the current in it, then

\[ \frac{2x_0}{r_0}=H_c . \tag{8} \]

The current density \(j\) is determined by \(\frac{1}{2\pi r}\frac{dx}{dr}\), or, according to (7),

\[ j=\frac{H_c}{4\pi r}=\frac{x}{2\pi r^2}, \tag{9} \]

so that at the boundary of the “core” \(j_0=\frac{x_0}{2\pi r_0}\). But at the boundary between the normal and intermediate regions the two states pass continuously into one another, so that the current density must also be continuous at the boundary.

In the normal region outside the “core” the current density is constant and equal to \(\frac{i-x_0}{\pi(a^2-r_0^2)}\); equating this to \(j_0\), we find

\[ 2r_0^2\left(\frac{1}{x_0}-1\right)\frac{1}{(a^2-r^2)}=1. \tag{10} \]

Putting \(\frac{r_0}{a}=\rho\) and \(\frac{2i}{aH_c}=\lambda\) (\(\lambda\) is the ratio of the field at the surface of the wire to the critical field, so that \(\lambda>1\)), we rewrite (8) and (10) in the form

\[ \frac{x_0}{r}=\frac{\rho}{\lambda}=\frac{2\rho^2}{1+\rho^2}, \tag{11} \]

so that

\[ 1+\rho^2-2\lambda\rho=0, \tag{12} \]

whence

\[ \rho=\lambda-\sqrt{\lambda^2-1} \tag{13} \]

(with the other sign before the root one would have \(\rho>1\)).

If the total normal resistance of the wire is \(w_0\), and the resistance for the current \(i\) is \(w\), the resistance of the normal region around the “core” is \(\frac{w_0a^2}{a^2-r_0^2}\), or \(\frac{w_0}{1-\rho^2}\), so that for

in order that the electric field be constant over the entire cross-section of the wire, there must be

\[ w_0\,\frac{i-x_0}{1-\rho^2}=wi. \tag{14} \]

Substituting (11) and (13), we find

\[ \frac{w}{w_0}=\frac{1}{2}\left(1+\sqrt{1-\frac{1}{\lambda^2}}\right). \tag{15} \]

Thus the resistance increases by a jump to half of its full value as soon as \(i=\frac{1}{2}aH_c\) \((\lambda=1)\), and then continues to increase with a further increase of the current, reaching the full value only asymptotically (Fig. 13a). It will be more convenient to postpone discussion of the experimental results on the destruction of superconductivity by a current until we have developed some further theoretical considerations.

Fig. 12.

Let us note that equation (12), derived from the condition of continuity of the current density on the surface of the “core,” can also be derived from the condition of minimality of the Joule heat \(W\) released in the whole wire. The Joule heat \(W\) released per unit time is \(wi\), or, with the aid of (8) and (14),

\[ W=w_0 i\,\frac{i-x_0}{1-\rho^2} =\frac{w_0 i^2\left(1-\frac{\rho}{\lambda}\right)}{1-\rho^2}, \]

and, differentiating with respect to \(\rho\), it is easy to verify that the condition for the minimum of \(W\) does indeed coincide with (12).

We have so far said nothing about the structure of the metal in the intermediate state in the “core,” which is evidently different from that which we considered in Chap. III, since in view of the presence of a current the lines of force here are circular, not straight. From expression (9) for the current density inside the “core” we see that the intermediate state must consist of a mixture of superconducting and normal regions arranged in the manner shown in Fig. 12. Thus, since the current density increases on approaching the axis as \(1/r\), and the electric field is constant, the resistance of a filament parallel to the axis at a distance \(r\) must also increase proportionally to \(r\), so that the thickness of the layers must vary linearly with \(r\), with the normal layers becoming thicker as one moves away from the axis, until they fill the entire length of the wire at the boundary of the “core”; the superconducting layers, however, become thick-

becomes smaller as the axis is approached. On the axis itself the metal should, according to these considerations, have been wholly superconducting, but it is easy to see that along a very thin filament on the cylinder axis there must flow only a vanishingly small current, so that this leads to no contradictions.

As in the case of an ellipsoid in an external field, the scale of the structure, i.e. the number of layers per unit length of the wire, depends on the radius of the “core” and on the surface tension between the superconducting and normal phases.

The number of layers per unit length cannot be calculated exactly (because of mathematical difficulties), but it is of the order of magnitude

\[ \frac{1}{\sqrt{d r_0}}. \]

The value of \(B\) in the intermediate “core” varies from \(H_c\) at its boundary to 0 on the axis; in particular it is easy to show that at any point in the “core” \(B\) is equal to

\[ B=\frac{r}{r_0}H_c. \]

The result (15) also shows in what way the resistance of a wire carrying a constant current \(i\) should vanish on cooling. In this case the parameter changes owing to the change of \(H_c\) with temperature. Above the transition temperature \(T_0\), \(H_c\) is equal to zero and \(\lambda=\infty\), so that \(w=w_0\).

As soon as the temperature becomes lower than \(T_0\), \(\lambda\) becomes finite and decreases with further lowering of the temperature, so that the resistance falls according to equation (15). This continues, however, only down to \(\lambda=1\), i.e. until the critical field reaches the value

\[ \frac{2i}{a}, \]

after which the resistance abruptly falls from one half of its normal value to zero, and the wire becomes wholly superconducting.

During the process of cooling the intermediate core appears at the temperature \(T_0\) and then grows until it occupies the entire volume of the wire, precisely just before the sudden jump in resistance. Upon further lowering of the temperature such a structure becomes unstable and is replaced by the superconducting state throughout the wire.

The change of resistance with temperature for different currents, derived from these considerations, is shown in Fig. 13b (in this diagram it is assumed that \(H_c\) depends linearly on \(T_0-T\), which is true for sufficiently small \(T_0-T\)). It now becomes clearer why, in Chapter I, a vanishingly small measuring current was mentioned as one of the conditions for a completely sudden jump in resistance. Transition curves of the kind shown in Fig. 13b were in fact observed experimentally by de Haas and Voogd \(^{50}\), but the currents used were too small to provide more than qualitative confirmation of the theory.

In Fig. 13a we show how, according to (15), the resistance should be restored by the current at constant temperature.

This question is difficult for experimental investigation, since large currents are required for it; as soon as some resistance appears, a large amount of Joule heat is evolved, as a result of which it is difficult to maintain a constant temperature. Alekseevskii[^51] overcame this difficulty by an ingenious method of immersing the specimen in liquid helium below the \(\lambda\)-point (helium II), which, owing to its enormous thermal conductivity, can remove the Joule heat rapidly enough to prevent a rise in temperature.

Fig. 13a.
Axes: ordinate \(\frac{w}{w_0}\); abscissa \(\lambda = 2i/ahc\).

Fig. 13b.
Axes: ordinate \(\frac{w}{w_0}\); abscissa: temperature. Curves marked \(I=1\), \(I=2\), \(I=\frac{1}{2}\), \(I=0\); \(T_0\).

It is interesting to mention that if the same experiment is carried out above the \(\lambda\)-point (as in the Leiden work on the destruction of superconductivity by a current), then the wire is immediately insulated from the liquid by a layer of gas, and the Joule heat is often sufficient to melt the wire, accompanied by explosive boiling of the liquid helium. Alekseevskii’s experiments (the dashed line in Fig. 13a) showed that in a single-crystal tin wire there does indeed occur an abrupt restoration of the resistance at a current strength exactly equal to that predicted by Silsbee’s hypothesis; however, the abrupt increase gave a resistance equal to \(w = 0.8w_0\), instead of the theoretical value \(0.5w_0\).

This abrupt increase was followed by a slower rise, but not as slow as it should have been according to the theory, and \(\frac{w}{w_0}\) reached unity at a current approximately twice as large as the critical current, instead of approaching unity asymptotically. The reason for these discrepancies is not yet clear.

Experimental conditions apparently exclude the possibility of a noticeable increase in temperature (the diameter of the wire was only \(0.01\ \mathrm{cm}\)), so that, in view of the good thermal conductivity of tin, there could have been no appreciable difference in temperatures inside and outside the wire, and the wire was, perhaps, exactly cylindrical.

In view of this, it is possible that the observed discrepancy with theory has a profound origin, probably of the same kind as the difference between the experimental \(0.58\) and the theoretical \(0.50\) in the restoration of resistance by a transverse magnetic field (see above). As there, the theory perhaps ceases to be valid because the thickness of the layers in the intermediate state becomes too small (the thickness of the layers in Fig. 12 decreases with the diameter of the wire), and here again experiments with considerably thicker wires would be of interest.

Experiments were also carried out on the destruction of superconductivity by a current in the presence of a magnetic field parallel to the current. As was to be expected, the critical current decreases with increasing applied magnetic field, becoming zero at a field equal to \(H_c\). The magnitude of the step-like increase in resistance at first fell somewhat below \(0.8\), but for large fields increased above \(0.8\), approaching \(1\) when the field approaches \(H_c\).

The presence of a magnetic field greatly complicates the theoretical determination of the course of the restoration of resistance, since the field distribution can no longer be regarded as two-dimensional; however, some considerations indicate that the solution will probably contain the mixed state mentioned in Sec. III. If, instead of depicting the change in resistance as a function of current at a constant external magnetic field, one depicts it as a function of the field at various constant currents, the resulting curve is qualitatively similar to Fig. 13b (with the abscissa representing field instead of temperature). This was found by de Haas and Fogg for the case of weak measuring currents used in ordinary resistance measurements. The destruction of superconductivity by a strong current in the presence of a transverse magnetic field has not yet been investigated experimentally. Theoretically, in this case the question is complicated by the absence of circular symmetry of the field distribution and has also not yet been investigated.

Fig. 14.

In connection with the question of the current effect we shall mention the interesting experiments of Stark and Steiner \(^{52}\). In these experiments a coil connected to a ballistic galvanometer was wound on a hollow cylinder, as shown in Fig. 14, so that the lines of force of the current flowing through the cylinder are enclosed by the coil. In their experiments the cylinder, through which a current flows, was cooled, and a sudden deflection of the galvanometer was observed, corresponding to the sudden expulsion from the metal of all lines of force, when

it became superconducting (the deflection of the galvanometer was equal to that which was obtained when the current was switched off, when the cylinder was in the normal state). This is the most direct confirmation of the fact that the current becomes entirely a surface current when the metal becomes superconducting1. The same experimental method, with suitable modifications, could be used to verify the intermediate stages of the transition into the superconducting state, i.e. to show the redistribution of the field (namely, of course, of the distribution of \(B\)) at the time when the intermediate “core” grows in the cylinder2. This could give a more detailed test of the theory than is possible from measurements of resistance alone; in particular, one could verify the proposition that the field in the “core” is equal to the critical field.

(To be continued in the next issue)

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Superconductivity

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1. We mention that the surface character of the currents was also confirmed by Meissner and Heidenreich[^28], who studied the field distribution around two neighboring cylindrical superconductors through which a current flowed. The field distribution in this case depends on the distribution of the current in each cylinder (for a single cylinder, the distribution of the external field, of course, does not depend on the distribution of the current). ↩↩↩↩↩↩

2. Holes in the cylinder, however, change the structure considered by us for massive cylinders; since the field must vanish on the inner surface of the cylinder, there can be no intermediate state there. A detailed consideration of this question shows that on the inner side there should be a mixed state (Fig. III), beyond which follows the intermediate state and, finally, sometimes the normal state. ↩↩↩