Theory of Finite Nuclei

Abstract

A theory of finite nuclei is formulated, based on the reaction-matrix theory of the nuclear many-body system. The reaction matrix appropriate to the finite nucleus is in the exact theory determined by the solution of coupled Hartree-Fock and reaction-matrix self consistency problems. This formal procedure is extremely difficult to carry out; the finite-nucleus reaction matrix has instead been approximated by the reaction matrix appropriate to the local density, which is a nonlocal coordinate space operator ($\mathrm{r}|K|{\mathrm{r}}^{\ensuremath{'}}$). It is shown that this approximation is equivalent to the assumption that a finite nucleus has the same short-range correlation structure as nuclear matter.The formalism used to determine ($\mathrm{r}|K|{\mathrm{r}}^{\ensuremath{'}}$) from the results previously obtained in the study of nuclear matter is derived, and the methods used in explicit evaluation are described. The numerical results discussed are based on the phenomenological two-body potentials of Gammel and Thaler which give an excellent description of all scattering data up to 300 Mev. The operator ($\mathrm{r}|K|{\mathrm{r}}^{\ensuremath{'}}$) obtained shows marked nonlocality for $r$ and ${r}^{\ensuremath{'}}$ less than ${10}^{\ensuremath{-}13}$ cm. That this is largely associated with the repulsive cores in the potentials is shown by a simple analytic approximation to ($\mathrm{r}|K|{\mathrm{r}}^{\ensuremath{'}}$). The nonlocality is further enhanced in the triplet states by the effects of the noncentral forces which lead to marked $l$ dependence in the even states.The reaction matrix so determined contains a large spin-orbit term. It is shown that this is almost entirely due to the spin-orbit two-body potential and that the tensor forces give only a very small contribution.Proceeding from the reaction matrix as an effective two-body interaction, the Hartree-Fock problem is formulated taking into account the complicated exchange and nonlocal character of the reaction matrix. A general result is obtained for the single-particle spin-orbit potential which is shown in the case of a local interaction to reduce approximately to the form of the Thomas interaction.An iteration method is proposed for solving the single-particle eigenvalue problem with a nonlocal potential which reduces the differentio-integral equation characteristic of the theory to an ordinary differential equation. This procedure requires the introduction of a linear derivative term in the differential equation. The method is exact in the limit of convergence of the iteration method or if the nonlocal potential is replaced by a local approximation.