IN the past few years, physicists have been much engaged 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 customary 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 measurable 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 electrons 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 cause 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 initial 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 possibility of representing all supracurrents realizable in a simply connected supraconductor by even 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-supraconduct-ing 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 supraconductivity was last established. The point of view, that the transition into the supra-conducting state is a reversible phase transformation, was already suggested by Rutgers and Gorter3, who, starting from this assumption 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:
(a) Its lowest state is not degenerate and belongs to the discontinuous spectrum. Its wave function is real.
(b) In a weak magnetic field h, the wave function fy does not experience stronger perturbations than those proportional to the square of h or still higher powers of h:
(1) where ^o is the wave function for h = 0.
In the (non-relativistic) wave mechanics, the density of current j of an electron in the state fy is known to be given by the formula:
where h, m, e, c are the well-known universal constants,)is the conjugate complex value of|> and A is the vector potential of the magnetic field h (h = curl A).
Substituting into this expression the above|> of the diamagnetic atom, one obviously obtains as the greatest term, the only one proportional 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 proportional to the velocity v; it is rather
The formula (2) for the current is obviously based on the corresponding resolution of the velocity v equivalent to (4):
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|A are sometimes distinguished as 'kinetic' and 'potential' momentum.
For the term (h/4ni) (\>∗ grad fy|> grad $∗) in (2) represents the local density of the total momentum p in the state|>. (This can easily be verified by putting, for example, a plane wave ezmpxih 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 = ft/X) 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^ni) (fy∗ grad [> 1> grad fy∗) 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 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 supra conducting electron also we will suppose the same equation (3) to be valid:
where ^02 signifies the probability 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. A= m/ne2 is a constant of the dimensions [sec.2] characteristic of the supra-conductor in question. As n 1023, one obtains A^ 3-2 x 1C'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 diamag-netic 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 (6) 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 basis4.
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
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 ^ C^/TL (^ 10"5 cm.) 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 absolutely 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 tha$ our conception differs essentially from a description which has sometimes been given, according to which supra -conductivity should be characterized by the particular value (ji = 0 of the magnetic permeability. Though for simply connected isolated supra-conductors 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 jz = 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.
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 scientiflques et industrielles No. 458 (Hermann et Cie., Paris, 1937.)
Meissner, W., and Ochsenfeld, R., Naturwissen., 21, 787 (1933); Meissner, W., and Heidenreich, T., Phys. Z., 37, 449 (1936).
Appendix to Ehrenfest, P., Leiden Comm. Suppl., 756 (1933). Gorter, C. J., Arch. Mus. Teyler, 7, 378 (1933).
London, F., C.R., 205, 28 (1937).
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).
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European Journal for Philosophy of Science (2019)
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Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences (2015)
Physica C: Superconductivity and its Applications (2015)