Parity effect in ground state energies of ultrasmall superconducting grains

We study the superconductivity in small grains in the regime when the quantum level spacing delta epsilon is comparable to the gap Delta. As delta epsilon is increased, the system crosses over from superconducting to normal state. This crossover is studied by calculating the dependence of the ground state energy of a grain on the parity of the number of electrons. The states with odd numbers of particles carry an additional energy Delta(P), which shows nonmonotonic dependence on delta epsilon. Our predictions can be tested experimentally by studying the parity-induced alternation of Coulomb blockade peak spacings in grains of different sizes.