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 supra-conductors as follows6: One knows that the presence of a magnetic field exceeding a certain critical value HT (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 supra-conducting 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 (He) is one and a half times greater than that (H^) at great distance from the sphere:

With an external field Hw = 2/3HT, therefore, the field on the equator attains just the critical value HT, whereas everywhere else it is smaller than HT. 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 HT, 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:

of the macroscopic Maxwell equations. One gets: A,

By its derivatives with respect to Bx, By, Bz, the free energy defines the quantities Hx, Hy, Hz

This equation can be simply interpreted by stating that in the intermediate state there is a diamagnetic permeability dependent on B which for B 60; HT 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 HT, 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 supra conductivity. 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.

∗ Continued from page 796.