Semirealistic shell-model interaction for neutron hole states inPb208

The Brueckner $G$ matrix from the Reid soft-core potential is taken for the leading contribution to the effective two-neutron hole interaction in the ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ region. To accomodate the corrections expected from core polarization and other higher order effects, including truncation of the model space, we add phenomenological two-body pairing and multipole forces whose strengths are determined by fitting the ${}_{}{}^{206}{}_{}{}^{}\mathrm{Pb}$ spectrum. A satisfactory fit is obtained with an interaction where the average total contribution of the added terms is less than 45% of the interaction. The same total effective interaction is then shown to yield a satisfactory spectrum for ${}_{}{}^{204}{}_{}{}^{}\mathrm{Pb}$ which may indicate the neglected effective many-body forces are not important. A comparison with the results of McGrory and Kuo is presented to examine the impact on the effective interaction of limiting the model space. We conclude the prospects are good for extending this approach to other nuclei below ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ viewed as shell-model systems of valence holes in ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$.