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An exact calculation is given of the upper critical field of a hollow isotropic type-ll superconducting cylinder. This fieid is the analogue of the surface-sheath critical field H,, . As the temperature decreases below To the critical field corresponds to increasing values of the azimuthal quantum number Im 1 , i.e. the fluxoid number. The results are compared with those for a flat surface, a film and a solid cylinder and with the approximate result for a thin-walled cylinder. For the last case, it is shown how the Little-Parks oscillations are modified as the cylinder wall thickness increases. The effects of normai-metai cladding on the inner and outer surfaces are calculated. The critical temperature is reduced and the Little-Park oscillations become less pronounced.

The analogue of the surface nucleation field Hc3 is calculated for a superconducting cylinder in a magnetic field parallel to its axis. For small radius, on the scale of the coherence length, superconductivity nucleates uniformly across the cylinder, while for large radius a surface sheath nucleates at the outer perimeter, the bulk of the cylinder remaining normal. The transition between these two limits is seen as a succession of flux entry points each corresponding to an increase by unity in the magnitude of the fluxoid quantum number.

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The subband energy levels of an electron confined in one dimension by a cylindrically symmetric square well potential (both infinite and finite) are investigated within the effective-mass approximation as a function of magnetic field applied along the cylinder axis. For small fields the doubly degenerate states ( mod m mod >0), where m is the azimuthal quantum number, are Zeeman split with subbands have m>0 shifting to higher energies whilst those with m<0 initially decrease in energy. A minimum in the energy of the negative-m states is predicted. This minimum occurs whatever the specific form of the confining potential, the only proviso being that it is cylindrically symmetric. This effect is intimately related to the number of elementary flux quanta Phi o(=h/e) contained within the electron's cyclotron orbit.

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