Properties of double-closed-shell nuclei are calculated using the Hartree-Fock method. The effective nucleon-nucleon interaction is taken to be the density-dependent "modified $\delta$ interaction," which is closely related to the Skyrme interaction, but with only three free parameters. These parameters were chosen so as to reproduce, as well as possible, the binding energy and rms charge radii of ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ and ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$. This interaction leads to a binding energy 16.5 MeV/A and Fermi momentum 1.33 ${\mathrm{fm}}^{-1\mathrm{}}$ of nuclear matter. Good agreement is found with the empirical rms charge radii of the other double-closed-shell nuclei, ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$, ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}$, and ${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}$. The same holds for the single-particle removal energies, especially those from $1s$ levels. The Hartree-Fock calculations with the present spin-independent and $s$-state interaction leads to as good agreement with experiment as those made by others, in particular, those made by Vautherin and Brink, who used a Skyrme interaction with five parameters.

The differences between the Skyrme and modified $\delta$ interaction are discussed. It is pointed out that the effective mass of a nucleon depends strongly on the odd-state part of the interaction, while the asymmetry potential is also sensitive to its spin dependence.

• Received 2 March 1972

DOI:https://doi.org/10.1103/PhysRevC.6.217