Thermal pairing in Richardson model

The modified BCS theory (MBCS) is applied to the Richardson model for pairing with a level distance of 1 MeV at various values for the number of levels Ω and particles N. It is shown that the limitation of the configuration space sets a limiting temperature TM up to which the MBCS can be applied. Enlarging the space in the half-filled case (Ω=N) by one valence level (Ω=N+1) extends TM to a much higher temperature so that the predictions by the MBCS can be compared directly with the exact results up to T∼4–5 MeV even for small N. The MBCS gap does not collapse, but decreases monotonously with increasing T. The total energy and heat capacity predicted by the MBCS are closer to the exact results than those predicted by the BCS, especially in the region of the BCS superfluid-normal phase transition. The discontinuity in the BCS heat capacity at the BCS phase-transition temperature Tc is smoothed out within the MBCS, especially for small N, showing the disappearance of superfluid-normal phase transition in very light systems. With increasing N the peak at Tc in the heat capacity becomes more pronounced, showing a phase-transition-like behavior in heavy systems.