Temperature Dependence of Even-Odd Effects in Small Superconducting Systems

Generalized BCS equations which consistently include the projection on the particle-number parity are derived from a systematic variational method. Numerical solutions are given that are illustrative of ultra-small metallic grains. Compared to the BCS approximation, the pairing is slightly enhanced in even and noticeably inhibited in odd systems. In the latter case, the gap increases with T near T=0; for a suitably small pairing this can result in a reentrance phenomenon. Paramagnetic effects are evaluated as functions of the temperature and the magnetic field.