Improved Superconductivity Approximation for the Pairing Interaction in Nuclei

The pairing interaction in nuclei is investigated by using a model Hamiltonian $H-{\lambda }_{1}N-{\lambda }_2{N}^2$, where $H$ is the original Hamiltonian and $N$ is the nucleon-number operator. The introduction of the term ${\lambda }_2{N}^2$ enables one to suppress the effect of the nucleon-number fluctuation, which is inherent in the Bardeen-Cooper-Schrieffer (BCS) approximation and is a main source of its inaccuracy. A prescription is given to determine the parameters ${\lambda }_1$ and ${\lambda }_2$. The method is first illustrated in the case where all levels are degenerate, and then applied to realistic cases with nondegenerate levels. Quadrupole and other interactions are not considered. Excited states with seniority zero as well as those with nonzero seniority are discussed. The results obtained by this method are compared with those of Kisslinger and Sorenson and of Kerman, Lawson, and Macfarlane. For the ground-state energy an excellent accuracy is attained easily. It is observed that the BCS states obtained by Kisslinger and Sorenson are much better approximations to eigenstates of our model Hamiltonian rather than to those of their Hamiltonian. A new light is shed on the problem as to why the projected and renormalized BCS states are very good approximations to the true eigenstates.