Effect of Level Statistics on Superconductivity in Ultrasmall Metallic Grains

We examine the destruction of superconducting pairing in metallic grains as their size is decreased for both even and odd numbers of electrons. This occurs when the average level spacing d is of the same order as the BCS order parameter. The energy levels of these grains are randomly distributed according to random matrix theory, and we must work statistically. We find that the average value of the critical level spacing is larger than for the model of equally spaced levels for both parities, and derive numerically the probabilities $P_{o,e}(d)$ that a grain of mean level spacing d shows pairing.