Coulomb energies and nuclear radii in the energy density formalism

The relation between Coulomb displacement energies, ΔE_c, and Δr=r_n-r_p, the difference between the rms radii of neutrons and protons in nuclei, is investigated within the energy density formalism (EDF). The variational equation, obtained by minimizing the Coulomb plus symmetry energies, is solved assuming the symmetry interaction is a simple functional of the local nuclear matter density. Varying parameters of the model, rather unique relation between ΔE_c and Δr is obtained (within ±50 keV). ΔE_c is independent of r_ex, the rms radius of the excess neutrons distribution. Using the experimental values of r_p and adjusting the model to reproduce the recent data on Δr (Δr{reversed tilde}~0.05 fm for⁴⁸Ca and²⁰⁸Pb), which are significantly smaller than those obtained from current Hartree-Fock calculations, the calculated ΔE_c agree with the experimental results. Using the value of Δr∼0.05 fm and the experimental values of r_ex, a small compression (<0.02 fm) of the proton core in the analogue state relative to its parent state emerges, thus contributing an additional electrostatic term to the Coulomb displacement energy. The size of this relative core-compression effect depends on the values assumed for Δr and r_ex, it increases with the decreasing of Δr and the increasing of r_ex. If Δr∼0.05 fm the effect is large enough to remove the long standing Coulomb energy anomaly. The main result of the present work is the correlation between ΔE_c and Δr, suggesting that the difficulties of current Hartree-Fock calculations in reproducing isotope shifts of r_p, the small value of r_n-r_p and the values of ΔE_c may all be different manifestations of some missing residual p n effective interaction.