SPA+RPA approach to canonical and grandcanonical treatments of nuclear level densities

Using an exactly solvable pairing model Hamiltonian in the static path approximation together with small-amplitude quantal fluctuation corrections in the random phase approximation (SPA+RPA), we have analyzed the behaviour of the canonical (number projected) and grandcanonical treatments of the nuclear level densities as a function of temperature and the number of particles. For small particle numbers at a low temperature, we find that though the grandcanonical partition function in the SPA+RPA approach is quite close to its exact value, the small errors in its estimation causes significant suppression of the level density obtained using the number projected partition function. The results are also compared with the smoothed out exact values of the level density. Within this model study, it appears that due to the saddle point approximation to the multiple Laplace-back transform, the grandcanonical treatment of the level density at low temperature may be reliable only for a relatively large number of particles.