A New Conception of Supraconductivity

By F. London, Institut Henri Poincaré, Paris

In the past few years, physicists have been much engageh by the phenomenon of supraconductivity. It is well known that various metals, when cooled below a certain very low temperature, characteristic of the metal in question, show the strange property of conducting electricity apparently without offering any resistance to the current. This curious phenomenon seems to contradict all our coustomary conceptions in physics. Particularly striking was the experiment of Kamerlingh Onnes and Tuyn, in which a current was induced in a supraconducting ring of lead and was found to persist there without any measureable decrease for many hours—so long as the low temperature could be maintained. This experiment seems to present a unique case of motion without any friction, whilst we have been accustomed to see in every mechanism an occasion for dissipation of kinetic energy into heat.

1. Attempts have been made to explain this phenomenon by various mechanisms. But in all of them the same type of difficulty is always encountered. As in an ordinary conductor, so in a supraconductor, it seems necessary to imagine an enormous number of different electronic states corresponding to the infinite number of different currents possible in it, different as regards direction and intensity. But on the other hand, it seems very difficult to comprehend why in these states the motion of the elctrons should not be damped, that is, why the electronic waves should not be dispersed. One would imagine that in any event the interaction with the ionic lattice would couse transitions between these numerous electronic states favouring the passage to states of less energy and less intensity of current. In a short time the irregularity of the thermal vibrations of the lattice should effect a complete dissipation of the intial current.

This difficulty still appeared aggravated when Bloch adduced a very general argument according to which the most stable state of a mechanism of electrons under rather general conditions cannot show any current if no external field is applied.

It can be said that all who have tried to construct a theoretical picture of a supraconductor have been completely baffled by this dilemma.

The new conception I have developed in different papers, partly in collaboration with H. London¹, differs essentially from the earlier attempts in so far as it exhibits the possibilty of representing all supracurrents realizable in a simply connected supracondutor by event one single electronic state alone; though to be sure, the presence of an external field has been found to be of fundamental importance.

A new experiment has given us the key to this possibility. Meissner and Ochsenfeld² found in 1933, that a supraconductor behaves not only like an ideal conductor, but in addition also like a very strongly diamagnetic metal. According to the Maxwell equations, an ideal conductor would not show any change of magnetic flux in its interior; this signifies that one should find, so to speak, 'frozen in', that magnetic field which was present at the moment when the supraconductivity was established. Meissner's experiment, however, has shown that in a supraconductor the magnetic flux is always equal to zero. It has been observed that those magnetic fields, present before the supraconductivity was established, are pushed out while the temperature is lowered below the transition point (provided the experiment is carried out under 'ideal' conditions; see further below).

According to Meissner's experiment, it looks as though the transition from the non-supraconducting to the supraconducting state in a magnetic field is reversible, so far as the magnetic flux can always be considered as equal to zero in any volume element in the supraconducting state independently of the way in which the transition temperature has been passed. That is quite different from the case of infinite conductivity. There the transition is not reversible and the supraconductor would show a kind of permanent memory of that magnetic field which was present when supracondiuctivity was last established. The point of view, that the transition into the supraconducting state is a reversible phase transformation, was already suggested by Rutgers and Gorder³, who, starting from this assumptioin of reversibility, derived certain thermodynamical relations between specific heat, magnetocaloric effect, etc., relations which have been verified in the meantime by many experimenters.

2. This state of affairs suggested an interpretation of supraconductivity which is entirely different from that which considers this phenomenon as a limiting case of ordinary conductivity. Though it is not possible to consider the diamagnetic phenomenon as a consequence of the infinite conductivity, the converse can to a certain extent be done.

A diamagnetic atom, as is well known, exhibits the possibility of permanent currents flowing in a system which is in its most stable state. These currents, indeed, do not appear except in the presence of a magnetic field, and that is precisely the reason why this mechanism is not covered by the theorem of Bloch mentioned above; for Bloch's theorem deals only with systems with no external field.

Let us for a moment consider the behaviour of a diamagnetic atom in a magnetic field. We may describe such an atom by the following properties:

1. Its lowest state is not degenerate and belongs to the discontinuous spectrum. Its wave function is real.
2. In a weak magnetic field h, the wave function y does not experience stronger perturbations than those proportional to the square of h or still higher powers of h: where y₀ is the wave function for h = 0.

In the (non-relativistic) wave mechanics, the density of current j of an electron in the state y is known to be given by the formula:

where h, m, e, c are the well-known universal constants, y^* is the conjugate complex value of y and A is the vector potential of the magnetic field h (h = curl A).

Substituting into this expression the above y of the diamagnetic atom, one obviously obtains as the greatest term, the only one proprotional to the field strength:

All the other terms are of the order h³ or still smaller. Calculating the moment of this current, one obtains the well-known expression for the induced diamagnetic moment of the atom.

It is perhaps of some interest to discuss in more detail how the diamagnetic atom succeeds in representing an infinite number of currents by one single state.

In a magnetic field the total momentum p of an electron is not simply proprotional to the velocity v; it is rather

This formula can be considered as the supplement to the well-known analogous resolution of the energy into 'kinetic' plus 'potential' energy, and accordingly the two terms mv and are sometimes distinguished as 'kinetic' and 'potential' momentum.

The formula (2) for the current is obviously based on the corresponding resolution of the velocity v equivalent to (4):

For the term (h/4pi) (y^* grad y − y grad y^*) in (2) represents the local density of the total momentum p in the state y. (This can easily be verified by putting, for example, a plane wave e^2pipx/h into this term.) It is a somewhat strange but quite characteristic feature of the wave-mechanical description that the wave-length of the de Broglie waves does correspond to the total momentum (p = h/l) and not to the kinetic momentum, whereas the latter, being proportional to v, is attached to the current. (Correspondingly the frequency is known to be attached to the total energy (E = hv) and not to the kinetic energy.)

Now, owing to equation (1), in a diamagnetic atom the term (h/4pi)(y^* grad y − y grad y^*) representing the mean total momentum p remains everywhere practically zero, even in a magnetic field. In this case the currents occurring are, so to speak, a kind of image of the actual magnetic field. The local kinetic momentum, that is, the local current, given by (3), is throughout equal but opposite to the local potential momentum, represented by the vector potential of the magnetic field, so that the sum of both, p, is everywhere zero. In such a manner a diamagnetic atom in its one lowest state can show an infinite variety of different currents corresponding to the infinite variety of different currents corresponding to the infinite variety of orientations and intensities of the applied magnetic fields, whereas its wave function does not show any appreciable reaction.

This mechanism of conduction is entirely different from that considered in the customary theories of conductivity: the transport of electricity is not based, as usually, on progressive waves (or progressive wave packets), but on stationary waves. By these a transport of electricity can only be effected in the presence of a magnetic field and this is precisely our assertion as to the nature of the supracurrents.

3. Let us now assume that in a simply connected supraconducting metal there may be one or several discrete electronic states of the same properties (a) and (b) below the continuum of ordinary (Bloch-) states. Since in all these states by a given magnetic field practically the same current is evoked, the transitions between these states caused by the interaction with the lattice vibrations will effect no dissipation of the diamagnetic currents. This is exactly the mechanism by which the interaction with the nuclear vibrations in a diamagnetic molecule is prevented from effecting any dissipation of the diamagnetic currents evoked by an external magnetic field.

Thus for a supraconducting electron also we will suppose the same equation (3) to be valid:

where signifies the probabilty of finding this electron, which we will suppose to be practically constant throughout the metal. Summing over all electrons, we therefore obtain for the density of the total current:

where n signifies the number of supraconducting electrons per cm.³. L=m/ne² is a constant of the dimensions [sec.²] characteristic of the supraconductor in question. As n≤10²³, one obtains L=3.2 × 10^-32 sec.².

The vector potential not being uniquely defined has yet to be normalized in a definite way in order to obtain in (5) an unambiguous statement. We can, however, get rid of this ambiguity by forming the curl of (5) and obtain

This is the fundamental macroscopic connexion between magnetic field h and current density J that we propose for the supraconducting state.

From our observations apropos of the diamagnetic atom, we may infer that in our model the notorious difficulties discussed above will not appear. Compared with the former conception of infinite conductivity the assumptions (a) and (b) certainly signify an appreciable reduction of the mechanism which remains to be explained by the theory of electrons. On the other hand, (a) and (b) form, of course, in no way a necessary basis of (6), and it is quite possible that the future development of the molecular theory will replace them by a still more reduced basis.⁴

4. In the following we shall discuss the macroscopic description furnished by (6). The currents which are admitted by this equation are very far from being identical with those which would correspond to an infinite conductivity. The variety of possible currents is considerably more restricted according to our interpretation, which admits only currents, which are correlated in a very special manner with a magnetic field. But it can be shown that it is really possible by just this restricted ensemble of currents to describe all the supracurrents which are actually observed.

Applying the Maxwell equation.

(neglecting here the displacement current) we can eliminate J in (6) and (7) and get

or since div h=0

This equation indicates that the magnetic field decreases exponentially from the surface to the interior of the supraconductor, in this way representing the Meissner effect. As in a diamagnetic atom, the induced currents behave like a screen; their magnetic field tends to diminish the original field. In a distance of the order of magnitude the field can be considered as practically zero.

In Meissner's experiment, it is obviously the applied external magnetic field which evokes the supracurrent as soon as the supraconducting state is established. In the case of the permanent current in a ring (and also in the case of an open wire which is fed by normal conducting leads), the magnetic field which maintains the current proves to be identical with that which is produced by the current itself. The most stable state of a ring has no current, unless an external magnetic field is applied. To be sure, the states in which the ring possesses a permanent flux through its central hole, are not states of lowest energy but are metastable under macroscopical conditions: only by a finite variation of the macroscopic parameters of the system (for example, by passing the transition temperature or by cutting the ring open) can the ring be brought into the obsolutely stable state which contains no flux.

To complete this theory it is necessary to add to (6) a further statement as to the behaviour of the electric field. In this regard the magnetic equation (6) as well as experience do not exclude a certain indeterminateness, and an experiment had, therefore, to be arranged in order to elucidate this point⁵. We cannot enter here into a detailed discussion of this question, and want only to state that as a result of this experiment the relation

(e being the electric field strength) seems now to be the most simple formulation of this supplementary electric equation. The electric fields possible according to (9) and (6) are reduced to just those which are inseparably attached by induction to the magnetic field. The equation (9) simply states that there are no other currents in the supraconductor than those which, according to (6), are evoked and maintained by the magnetic field.

It might be emphasized that our conception differs essentially from a description which has sometimes been given, according to which supraconductivity should be characterized by the particular value m=0 of the magnetic permeability. Though for simply connected isolated supraconductors both formulations give macroscopically identical results, they prove entirely different if one has to deal with supraconducting rings.

The essential characteristic of our theory can be seen in the following: The same relation (6), between current and magnetic field, which represents the Meissner effect and which for simply connected supraconductors is practically identical with the description m=0, is able, moreover, to describe the distribution of the permanent currents in supraconducting rings. The magnetic field of these rings, having a curl, requires, according to Maxwell's theory, the explicit introduction of the macroscopic current. It cannot, of course, be described by a particular value of the magnetic permeability only.

1. London, F. and H., Physica 2, 341 (1935). London, F., Proc. Roy. Soc., A, 152, 24 (1935). Comprehensive report: 'Une conception nouvelle de la supra-conductibilité', Actualités scientifiques et industrielles No. 458 (Hermann et Cie., Paris, 937.)
2. Meissner, W., and Ochsenfeld, R., Naturwissen., 21, 787 (1933); Meissner, W., and Heidenreich, T., Phys. Z., 37, 449 (1936).
3. Appendix to Ehrenfest, P., Leiden Comm. Suppl., 756 (1933). Gorter, C.J., Arch. Mus. Teyler, 7 378 (1933).
4. London, F., C.R., 205, 28 (1937).
5. London, H., Proc. Roy. Soc., A, 155, 102 (1936). See also v. Laue, M., London, F. and H., Z. Phys., 96, 359 (1935). Schrödinger, E., NATURE, 137, 824 (1936).

[To be continued.]

A New Conception of Supraconductivity

By F. London, Institut Henri Poincaré, Paris

5. According to these conceptions, there cannot exist any magnetic flux 'frozen' in the interior of pure supraconductors; a permanent flux should only be found confined to the hollows of supraconducting rings. The topological connectivity of a supraconductor, therefore, is a property extremely characteristic of its behaviour: the multiplicity of its connectivity, diminished by one, immediately indicates the number of independent conservative quantities, that is, of independent invariant magnetic fluxes.

Actually, however, in the classical experiments of Kamerlingh Onnes, already there have been found magnetic fields 'frozen' in even simply connected supraconductors. It was these permanent fluxes which seemed at that time directly to indicate the elementary phenomenon: an infinite conductivity. We, on the contrary, do not consider these experiments as representing the elementary case of the phenomenon, but rather as a relatively complicated affair which can be reduced to a still more elementary phenomenon.

According to our conceptions, we interpret these magnetic fluxes 'frozen' in the interior of the supraconductors as follows⁶: One knows that the presence of a magnetic field exceeding a certain critical value H_T (depending on the temperature T) destroys the supraconductivity. Now it can happen that some magnetic fluxes are confined in certain regions of the metal in such a manner that the critical magnetic field is there exceeded, whereas in the supraconducting regions the supraconductivity is maintained. Thus the appearance of the permanent fluxes should be conditioned by the formation of a complicated structure of the supraconducting and the normal phases in the metal in such a way that the supraconducting regions constitute rings embracing the magnetic fluxes in their non-supraconducting hollows.

6. It is easy to see that, even in very simple experiments, such a mixed structure of the two phases must automatically arise. This can be shown by considering, for example, a supraconducting sphere which is brought into a homogeneous magnetic field.

The sphere pushes back the magnetic lines of force and compresses them in the region near the equator. An elementary calculation shows that the intensity of the field immediately on the equator (H_e) is one and a half times greater than that (H_∞) at great distance from the sphere:

With an external field H_∞ = 2/3H_T, therefore, the field on the equator attains just the critical value H_T, whereas everywhere else it is smaller than H_T. When we now intensify the field a little, the supraconductivity will be destroyed in the sphere immediately behind the equator. But then the magnetic field can enter this region and the magnetic lines of force will be less compressed. As a consequence the magnetic field at the equator will be a little less than H_T, and the supraconducting state will here reappear. If now we intensify the field a little more, the supraconductivity will be destroyed anew immediately behind the equator, whilst the supraconducting layer just formed will move farther into the interior of the sphere.

7. At first sight it seems extraordinarily difficult to make such a microstructure of layers accessible to theoretical treatment. To do this it would be necessary to solve a very complicated boundary problem for which the shape of the boundaries has still to be determined, whilst even their number is not yet known. It is possible, however, to avoid this practically insoluble problem, if one abstains from determining that microstructure in detail and rather restricts oneself to considering the mean values of the field strengths taken over this microstructure of the phases. Actually it is these mean values of the fields which are above all the object of the experimenter.

The theory of this mixture of the two phases⁷, sometimes called 'intermediate state' is, therefore, nothing but a consistent application of the theory of the 'pure supraconducting' phase; but formally it forms for itself an independent whole⁸. Here we will only give some of the results.

The variables of the theory of this intermediate state are the averages of h and of e taken over the microscopic structure. These are the quantities which Lorentz identifies with the quantities B and E of Maxwell's theory:

Here we will restrict ourselves to the pure magnetostatic case. The theory can be completely characterized by indicating the free energy F which, it has been calculated, is given by:

By its derivatives with respect to B_x, B_y, B_z, the free energy defines the quantities H_x, H_y, H_z of the macroscopic Maxwell equations. One gets:

This equation can be simply interpreted by stating that in the intermediate state there is a diamagnetic permeability dependent on B which for B H_T is given by

Moreover, one has the equations

and the usual boundary conditions.

Although on account of equation (11) this theory is not a linear theory (like the theory of the pure supraconducting state or the ordinary Maxwell theory), it is nevertheless of extreme simplicity; (11) simply states that the magnetic field strength H is always parallel to the magnetic induction B, but that it has always the absolute value H_T, independently of the value of B. From this, among other things, it follows that, in the domain of the magnetostatics of the intermediate state, the magnetic lines of force are always straight lines.

For B = 0, however, according to (11) the field H is not defined as to its intensity or as to its direction. This comes from the fact that for B = 0 the pure supraconducting regions become unlimitedly large, which signifies that the description with the mean values B and H can no longer be legitimate and that one has now explicitly to apply the equations of the pure supraconducting state to the supraconductor as a whole. Obviously the case B = 0 cannot simply be considered as a limiting case of the non-linear theory.

8. We cannot enter here into a detailed discussion of the relation between theory and experiment. On the whole, one can say that the results of the theory agree fairly well with the experiments. With respect to the pure supraconducting state there is full agreement. Practically there exist three phenomena only: (1) the permanent current in a ring; (2) the current without electric field in an open supraconducting wire, which is fed by normal conducting leads; (3) Meissner's experiment. The consistent representation of these experiments was the basis of our theory. The greater part of the experiments (actually the Meissner effect also) concerns the transition between the normal and the supraconducting state and deals therefore with the intermediate state. Particularly striking in this respect are recent experiments of De Haas and Guinau, of Mendelssohn and of Shoenberg⁹ as to the transition, qualitatively discussed above, of a sphere in a magnetic field. These experiments are in very good agreement with the statements of our theory of the microstructure. In many cases, it is true, the experiments¹⁰ of the transition phenomena seem yet to be obscured by hysteresis and other retardation effects, which prevent the realization of thermal equilibrium and render difficult the theoretical discussion. The theory can also account qualitatively for these disturbing effects¹¹, though there still remains something to be done. But for a reasonable discussion of these questions we would have to occupy ourselves with much more detail than could be given here.

The macroscopic theory we have discussed shows that it is possible to interpret the phenomena in a way which avoids the paradoxes that seemed hitherto to render impossible any theory of supraconductivity. The new interpretation includes, moreover, a very simple description of the phenomenon in the language of wave kinematics. The next stage will have to be the development of the electronic basis of this theory. One might presume that the new aspect here presented of supraconductivity may also give an indication for the construction of a molecular model of the supraconductor¹².

6. The following interpretation seems first to have been given by Gorter, C. J., Nature, 132, 931 (1933). Gorter, C. J., and Casimir, H., Physica, 1, 305 (1934).
7. London, F., Physica, 3, 450 (1936); Nature, 137, 991 (1936).
8. The magnetostatic part of this theory has also been developed by Peierls, R., Proc. Roy. Soc., A, 155, 613 (1936), quite independently of our conceptions, as a pure phenomenological description of a new 'intermediate' state, different from both the pure supraconductive and the normal state. But it can be shown⁷ that, thermodynamically speaking, the intermediate state has not to be considered as a further independent phase but as a mixture of the two phases.
9. De Haas, W. J., and Guinau, A., Physica, 3, 182, 534 (1936). Mendelssohn, K., Proc. Roy. Soc., A, 155, 558 (1936). Shoenberg, D., Proc. Roy. Soc., A, 155, 712 (1936).
10. For example, De Haas, W. J., and Casimir-Jonker, M. J., Physica, 1, 291 (1934).
11. London, H., Proc. Roy. Soc., A, 152, 650 (1935). Keesom, W. H., and Van Laer, P. H., Physica, 4, 499 (1937). Grayson Smith, H., Trans. Roy. Soc. Canada, 31, 31 (1937). De Haas, W. J., Engelkes, A. D., and Guinau, O. A., Physica, 4, 595 (1937).
12. (Added in the proofs). In a paper just published (Phys. Rev., 52, 214 (1937), J. C. Slater has tried to sketch such a molecular model for our theory. See also Slater. J. C., Phys. Rev., 51, 195 (1937), and London, F., Phys. Rev., 51, 678 (1937).

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