Realistic evaluation of Coulomb potential in spherical nuclei and test of the traditional approach

Abstract: A realistic evaluation of Coulomb potential has been made for some selected nuclei using the available model-independent data for the charge density and the recent development of Coulomb energy-density functional. Within the Woods-Saxon potential as a nuclear component, we are able to quantify the differences in proton single-particle energies due to the differences from the model-independent data of the uniform distribution, the two-parameter Fermi function, as well as the charge density obtained from a microscopic Hartree-Fock calculation using the effective Skyrme interaction. The obtained energy differences are generally small in magnitude, namely about 100~keV or less if the parameters of the charge density models are appropriately determined. Considerable larger differences appear when the last occupied state is highly filled and, at the same time, has a small orbital angular momentum. Sulfur isotopes ($Z=16$) are a perfect example of these nuclei. Unfortunately, despite its simplicity, the uniform distribution cannot be used for evaluating the Coulomb exchange term within a well-established method because it is not differentiable at the surface of a nucleus. Traditionally, the missing of exchange term is corrected for by simply excluding the last-proton contribution to the direct term. %% We also investigate this approach and find that its effect is simply an introduction of the factor $(Z-1)/Z$ into the Coulomb direct term. From medium to heavy nuclei (typically beyond the $sd$ shell) the resulting proton levels are 300-800~keV higher than those obtained with the exact Fock term. The result for lighter nuclei tends to be opposite because the factor $(Z-1)/Z$ decreases rapidly towards the limit of $Z\to 1$. Therefore, this traditional approach should be avoided for a precision nuclear structure calculation.