A core-particle coupled model of the effective charge: Theory and calculation in A = 39, 41 nuclei

In this paper we describe how effective charges may be calculated within the framework of the full-shell model by means of a core-particle coupling model (CPCM). We consider nuclei with a single particle (hole) outside (inside) a doubly magic core. The nuclear Hamiltonian is obtained from coupling the valence particle (hole) to the eigenstates of the diagonalized core. It is shown how the perturbation theory effective charge formalisms, as given by Brown and Shukla and by Siegel and Zamick, may be obtained from the more complete core-particle coupling formalism via various approximations to the eigenfunctions of the full-shell model matrix. The validity of these approximations is examined, and the formal equivalence of the two perturbation approaches is demonstrated. The formalism of the CPCM as a framework in which to calculate the effective charge, is applied to the single-particle and single-hole nuclei A = 41 and A = 39 with a ⁴⁰Ca core. The calculation of the TDA (RPA) core states and the valence-particle (hole) effective charges are done using the Kallio-Kolltveit matrix elements, with and without density dependence, as well as the "bare" Kuo-Brown matrix elements. The calculation is also performed, and comparisons made between the full-shell model, restricted shell model and the Siegel-Zamick effective charges, for the three sets of matrix elements.