Charge distributions ofPb208,Pb206, andTl205and the mean-field approximation

Charge distributions of ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$, ${}_{}{}^{206}{}_{}{}^{}\mathrm{Pb}$, and ${}_{}{}^{205}{}_{}{}^{}\mathrm{Tl}$ have been calculated within the Hartree-Fock and Hartree-Fock+BCS approximations with the Skyrme interaction. Using the force ${\mathrm{SkM}}^{\mathrm{*}}$ designed without any reference to this particular problem, we find good agreement with elastic electron scattering data. The role of pairing correlations beyond the mean field is studied by applying the Lipkin-Nogami method of variation after approximate projection on the good number of particles. We argue in this paper that, in our opinion, there is no significant discrepancy between the experimental data and the Hartree-Fock calculations using reasonable effective interactions. In particular, we do not see any compelling need for a large depletion of the occupation number of the proton 3$s_{1/2}$ orbital.