Realistic Independent-Particle Model of the Nucleus

Abstract

A phenomenological central potential is explored which is characterized by a universal depth (${V}_{0}$), a universal surface extension ($d$), and an inner radius function $a={a}_{1}{A}^{\frac{1}{3}}+{a}_{0}$. The four parameters are adjusted to be in accord with experimental information obtained from low-energy neutron scattering and from neutron separation energies. The results ${V}_{0}=40$ Mev, $d=1$, ${a}_{1}=1.32$, and ${a}_{0}=\ensuremath{-}0.8$ (all in units of ${10}^{\ensuremath{-}13}$ cm), are in reasonable accord with what might be expected from other experimental and theoretical considerations. Since the experimental data used in the adjustment process bound the energy region in which the discreteness of nuclear energies is most evident, one might expect that the eigenvalues and eigenfunctions based upon this phenomenological model would furnish a highly realistic set for the theoretical study of low-energy nuclear transitions within the framework of the independent-particle model of the nucleus.The energy eigenvalues and the important parameters for the energy eigenfunctions are calculated by a procedure suitable for the restricted type of potential under consideration. A simple method is presented for handling small perturbations, particularly the Coulomb perturbation. In the latter case, it is shown that the derived proton eigenvalues agree with experimental proton separation energies, provided one introduces an anomalous attractive potential which cancels approximately one-half of the classical Coulomb potential acting upon an individual proton. The implications of these results are discussed.