Charge-dependent andA-dependent effects in isotope shifts of Coulomb displacement energies

Coulomb displacement energies in a series of isotopes generally decrease with $A$. This decrease can arise from an increase with $A$ of the average distance of interaction between pairs of protons. In the shell model a decrease can also result from charge-independence-breaking effects if the neutron-proton interaction for the valence nucleons is more attractive than the neutron-neutron interaction. Using the model recently proposed by Sherr and Talmi for the $1d_{\frac{3}{2}}$ shell, existing data for this shell and also the $1d_{\frac{5}{2}}$ and $1f_{\frac{7}{2}}$ shells were analyzed allowing all matrix elements to vary as $A^{-\frac{\lambda }{3}}$. Least squares calculations of the rms deviation $\sigma$ were carried out for varying values of $\lambda$ from -2 to +2. It was found that although there was a minimum in $\sigma$ vs $\lambda$ it was too shallow to exclude any $\lambda$ for -1 to +1 in the $1d_{\frac{3}{2}}$ and $1f_{\frac{7}{2}}$ shells or 0 to +1 in the $1d_{\frac{5}{2}}$ shell. It is therefore not possible to distinguish between $A$ dependence and charge dependence in this model. The magnitude of the latter as expressed in terms of $\mathrm{np}-\mathrm{nn}$ matrix elements depends strongly on the former. As $\lambda$ increases from -1 to +1, these $\mathrm{np}-\mathrm{nn}$ matrix elements decrease roughly linearly in absolute magnitude and eventually change sign. For $\lambda =0\mathrm{}$ they have appreciable and reasonable magnitudes for the $1d_{\frac{3}{2}}$ and $1f_{\frac{7}{2}}$ shells but for the $1d_{\frac{5}{2}}$ shell the values are too small to be considered significant.