The p${\mathrm{-}}^{40}$Ca mean field is derived from an optical-model (OM) analysis that explicitly incorporates the dispersion relation connecting the real and imaginary parts of the mean field. This analysis is based on differential cross-section, analyzing power, and reaction cross-section data available in the energy range between 20 and 180 MeV. The extrapolation of the OM potential from positive to negative energies provides the shell-model potential. This extrapolation is guided by known single-particle energies. The deeply bound 1p and 1s orbits clearly indicate the need for a linear rather than an exponential energy dependence of the Hartree-Fock potential at large negative energies. The analysis also provides root-mean-square radii, occupation probabilities, spectral functions, and absolute spectroscopic factors for proton single-particle orbits in ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$. Our calculated 15% depletion of the hole states in ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ is lower than that suggested for ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ from theoretical and experimental studies. We also find that a substantial amount of single-particle strength in ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ is located at rather high excitation energy.

• Received 26 March 1990

DOI:https://doi.org/10.1103/PhysRevC.42.693