Superconducting correlations in ultra-small metallic grains

To describe the crossover from the bulk BCS superconductivity to a fluctuat ion-dominated regime in ultrasmall metallic grains, new order parameters an d correlation functions, such as "parity gap" and "pair-mixing correlation function", have been recently introduced. In this paper, we discuss the sma ll-grain behaviour of the Penrose-Onsager-Yang off-diagonal long-range orde r (ODLRO) parameter in a pseudo-spin representation. Relations between the ODLRO parameter and those mentioned above are established through analytica l and numerical calculations.