LEVEL STATISTICS AND PARITY EFFECT ON SMALL SUPERCONDUCTING SYSTEMS

In this paper we have calculated the variations of the gap \Delta'(0,d) and transition temperature T'_c in small metallic grains as functions of grain size (or the level spacing d between discrete electronic states) for the cases of odd and even numbers of electrons by applying the random matrix theory to the mean field theory. We find the presence of enhancement of superconductivity and critical d_c, where the superconductivity of small grains breaks down. This agrees with Anderson's prediction (1959 J. Phys. Chem. Solids 11 28). We find that in the grains, as the size is lowered, the transition temperature T'_c decreases and \Delta'(0,d)/k_BT'_\rm c\leq \pi e^-\gamma in odd numbers of electrons, and for Gaussian orthogonal and unitary ensembles in some regimes \Delta'(0,d)/k_BT'_\rm c> \pi e^-\gamma in even numbers of electrons.