Neutron and proton densities in nuclei and the semi-empirical mass formula

Abstract

The semi-empirical statistical method due to Wilets, with the use of trial nucleon densities, has been used to investigate the nucleon density distributions and their relation with the semi-empirical mass formula. In particular those features depending on the neutron excess have been isolated, especially the difference in extension δ between the neutron and proton densities and the surface symmetry energy E s τ . In addition to the usual Weiszäcker type gradient terms a new mixed gradient term has to be introduced. For both δ and E s τ the density dependence of the symmetry energy density ε τ plays an important role, while E s τ also depends on δ. The variational solution for δ together with the empirical value of E s τ then determine both δ and the density dependence of ε τ in terms of the magnitudes of the gradient terms. The general conclusions are independent of these magnitudes if these are consistent with the empirical surface energy and thickness. For stable nuclei, δ ∝ ( N− Z)/ A to a good approximation. δ is small (≲0.3×10 −13cm) even for the heaviest stable nuclei, the neutrons thus extending slightly beyond the protons. The density dependence of ε τ is found, tentatively, to be considerably smaller than in the independent particle approximation for nuclear matter. The empirically used dependence for E sτ ∝ (N−Z) 2/A 4 3 seems well justified for nuclei near the stable valley. The dependence of the total surface thickness on the neutron excess is found to be quite small.