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