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 $F\left(\rho \right)$ of the symmetry-energy density, where ρ 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 $\delta R$ 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 $\delta R.\mathrm{}$ Switching off the Coulomb contribution to the n-p difference then gives the maximum $\delta R$ for a given F. Our numerical results are for ${}^{208}\mathrm{Pb}.$ We consider a wide range of F; for these, both $\delta R$ and the ratio χ of the surface to volume symmetry-energy coefficient depend, approximately, only on an integral involving $F^{-1}.$ For $\delta R\lesssim 0.45\mathrm{fm}$ this dependence is one valued and approximately linear for small $\delta R,$ and this integral is then effectively determined by $\delta R.\mathrm{}$ There is a strong correlation between $\delta R$ and χ, allowing an approximate determination of χ from $\delta R.\mathrm{}$ $\delta R$ has a maximum of ≅0.65 fm.