Exact canonical and grand canonical descriptions of the pairing in spherical mesoscopic systems

The pairing in spherical nanograins with free electron numbers N < 105 are treated in the exact manner on the basis of the single shell model (SSM) provided the Debye layer contains only one spherical shell. Such systems demonstrate the strengthening of the pairing gap (Δ) and critical temperature as compared with the bulk values. The temperature variations of Δ and χP (paramagnetic susceptibility) are calculated vs strength of the magnetic field for the canonical ensemble (CE) and grand canonical one (GCE). Increasing of the field stimulates stepwise decreasing Δ and oscillations in χP which are smoothed by the temperature. For the electron numbers N < 105 the BCS sharp temperature superconducting-normal phase transition in Δ and χP turns out to be essentially softened in CE and GCE. The temperature re-entrance of the superconductivity is considered for both CE and GCE at so strong magnetic fields which remove he pairing at T = 0.