Self-Consistent Treatment of the Independent-Particle Central-Field Nuclear Model

The purpose of the present work is, starting from a given assumed nucleon-nucleon interaction $V({\mathrm{r}}_i-{\mathrm{r}}_j)$ and the independent-particle, central-field approximation, to deduce all the consequences of this model by means of the Hartree-Fock-Slater theory. The $V({\mathrm{r}}_i-{\mathrm{r}}_j)$ is chosen to satisfy the saturation requirement and to be consistent with some of the properties of the two-nucleon system, but contains the strength factor $V_0$ as an undetermined parameter. This $V_0$ and the single-particle wave functions ${\psi }_n\left({\mathrm{r}}_i\right)$ are determined by the variational principle together with the requirement that the total binding energy of the nucleus be equal to its empirical value. The binding energies of the individual nucleons in the various shells are themselves approximately given by the Fock equations which also lead to a central field which is different for the different shells. The average central field implied in the usual treatment of the shell model is, however, the same for a nucleon in any shell, and this must be identified with some approximate average field obtained by a procedure such as that suggested by Slater for an electron in an atom or in a metal. On the other hand, the central field in the sense of Hartree (i.e., obtained from the Fock theory by neglecting the exchange terms) would be a very poor approximation as the exchange terms are not negligible compared with the direct terms. It is emphasized in the present work that the application of the variational principle to the problem rids the shell model of the inconsistent procedure in the usual treatments in which two independent assumptions concerning $V({\mathrm{r}}_i-{\mathrm{r}}_j)$ and the average central field $V\left(\mathrm{r}\right)$ are made. A comparison of the result of the present program with the empirical facts will form a correct basis on which to judge the fairness or failure of the central-field approximation in the shell model.