Calculation of the Shell-Model Potential from the Optical-Model Potential

A dispersion-relation approach is developed for deriving the shell-model potential from the optical-model potential, i.e., for extrapolating the mean field from positive towards negative nucleon energies. This method is applied to neutrons in ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$. The average neutron-nucleus potential is assumed to have a Woods-Saxon shape; its depth, radius, and diffuseness are calculated for nucleon energies with range from -20 to +40 MeV.