Neutron and proton densities and the symmetry energy

The neutron/proton distributions in nuclei, in particular, the n-p difference, are considered in a "macroscopic" Thomas-Fermi approach. The density dependence F(rho) of the symmetry-energy density, where rho is the total density, drives this difference in the absence of Coulomb and density-gradient contributions when we obtain an explicit solution for the difference in terms of F. If F is constant then the n-p difference and, in particular, the difference deltaR between the neutron and proton rms radii are zero. The Coulomb energy and gradient terms are treated variationally. The latter make only a small contribution to the n-p difference, and this is then effectively determined by F. The Coulomb energy reduces deltaR. Switching off the Coulomb contribution to the n-p difference then gives the maximum deltaR for a given F. Our numerical results are for Pb-208. We consider a wide range of F; for these, both deltaR and the ratio chi of the surface to volume symmetry-energy coefficient depend, approximately, only on an integral involving F-1. For deltaRless than or similar to0.45 fm this dependence is one valued and approximately linear for small deltaR, and this integral is then effectively determined by deltaR. There is a strong correlation between deltaR and chi, allowing an approximate determination of chi from deltaR. deltaR has a maximum of congruent to0.65 fm.