Pair-correlations and magnetic susceptibility of small Al-grains

Paircorrelations and the magnetic susceptibility of electrons in a spherical cavity are studied both for grand canonical and the canonical ensemble. The coupling constant of the $BCS$ Hamiltonian is adjusted to experimental values of the gap parameter. The gap parameter is found to increase for small grains as a consequence of the pronounced shell structure in the spectrum of the spherical cavity. The sharp phase transition at $T_c$ is smeared out for the canonical ensemble. The strong paramagnetic susceptibility of the normal electrons in the cavity is reduced by the superconductivity, but it remains positive.