Nucleon Energy Levels in a Velocity-Dependent Potential

Abstract

The neutron and proton level sequences in a diffuse, velocity-dependent potential have been investigated. A velocity-dependent interaction is used which manifests itself by attributing to a nucleon inside nuclear matter an "effective mass" which is a function of its position. Following Brueckner, Johnson and Teller, and Duerr, the effective mass in the center of the nucleus is chosen to be one-half the free-particle mass. The potential form was taken as $V(r)=\ensuremath{-}\frac{{V}_{0}}{{1+\mathrm{exp}[\ensuremath{\alpha}(r\ensuremath{-}a)]}}$, and for protons a Coulomb potential derived from a uniform charge distribution extending to $r=a$ was added. The proper neutron shell structure and level sequence was obtained with the parameters $\ensuremath{\alpha}=1.16\ifmmode\times\else\texttimes\fi{}{10}^{13}$ ${\mathrm{cm}}^{\ensuremath{-}1}$, $a=1.3{A}^{\frac{1}{3}}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm, ${V}_{0}=69$ Mev, and a spin-orbit coupling 33 times the Thomas term. For protons, using the same $\ensuremath{\alpha}$, radius, and spin-orbit coupling, it was found that the potential depth had to be increased by roughly 13 Mev to bind the correct number of protons in ${\mathrm{Pb}}^{208}$. If Pauli principle correlations are included, then a deeper proton potential is obtained. This correction depends critically on the form of the nucleon densities. Since a self-consistent treatment has not been made, this effect has been estimated in two ways: (1) A Fermi-Thomas approximation was used to compute the densities. In this case, the correct neutron-proton ratio is obtained but the correct proton level sequence is destroyed. (2) The neutron well depth was increased by an amount $\frac{(N+\frac{1}{2}Z)}{(Z+\frac{1}{2}N)}$. In this approximation the correct proton level sequence is obtained but it is not possible to bind the proper number of protons.