Exact study of the effect of level statistics in ultrasmall superconducting grains

The reduced BCS model that is commonly used for ultrasmall superconducting grains has an exact solution worked out long ago by Richardson in the conte xt of nuclear physics. We use it to check the quality of previous treatment s of this model, and to investigate the effect of level statistics on pairi ng correlations. We find that the ground-state energies are on average some what lower for systems with nonuniform than uniform level spacings, but bot h have an equally smooth crossover from the bulk to the few-electron regime . In the latter, statistical fluctuations in ground-state energies strongly depend on the grain's electron number parity.