Hartree-Fock Single-Particle Energy and Interaction Matrix Element for Density-Dependent Interactions in Superconducting Nuclei

Abstract

The density-dependent interactions have been successfully used as effective interactions in Hartree-Fock (HF) and RPA calculations. Among them the Skyrme type interactions are simple and easy to treat, and were employed by many authors to explain properties of giant resonances as well as ground state properties of magic nuclei. 1) It is well known that in the case of a density-dependent interaction there occurs an extra term in the HF field. This term vanishes in the case of a density-independent interaction and we call it the rearrangement field. For the RPA particle-hole matrix elements, the Landau prescription ) has been utilized to take account of the rearrangement effects. The density dependence brings the symmetric contributions to the particle-hole interaction. In a previous shortnote) a self-consistent evaluation of particle-hole type matrix elements for density-dependent interactions is presented. We evaluated the expectation value of the Hamiltonian with respect to a Slater determinant which is not orthogonal to a reference Slater determinant. The particle-hole type matrix elements which correctly include the rearrangement effect were obtained, and the Landau prescription came out as the simplest case. The purpose of this letter is to generalize the method to treat the superconducting nuclei. The rearrangement effects on the Hartree-Bogoliubov (HB) condition and interaction matrix elements in the quasi-particle TDA equation are obtained for densitydependent interactions. We now take 1<1» to be a quasi-particle vacuum