Symmetry energy I: Semi-infinite matter

Energy for a nucleus is considered in the macroscopic limit, in terms of nucleon numbers. Further considered for a nuclear system is the Hohenberg–Kohn energy functional, in terms of proton and neutron densities. Finally, Skyrme–Hartree–Fock calculations are carried out for a half-infinite particle-stable nuclear-matter. In each case, the attention is focused on the role of neutron–proton asymmetry and on the nuclear symmetry energy. We extend the considerations on the symmetry term from an energy formula to the respective term within the Hohenberg–Kohn functional. We show, in particular, that in the limit of an analytic functional, and subject to possible Coulomb corrections, it is possible to construct isoscalar and isovector densities out of the proton and neutron densities, that retain a universal relation to each other, approximately independent of asymmetry. In the so-called local approximation, the isovector density is inversely proportional to the symmetry energy in uniform matter, at the local isoscalar density. Generalized symmetry coefficient of a nuclear system is related, in the analytic limit of the functional, to an integral of the isovector density. We test the relations, inferred from the Hohenberg–Kohn functional, in the Skyrme–Hartree–Fock calculations of half-infinite matter. Within the calculations, we obtain surface symmetry coefficients and parameters characterizing the densities, for the majority of Skyrme parameterizations proposed in the literature. The volume-to-surface symmetry-coefficient ratio, and the displacement of nuclear isovector relative to isoscalar surfaces, both strongly increase as the slope of symmetry energy, in the vicinity of normal density, increases.