Nuclear matter symmetry energy and the symmetry energy coefficient in the mass formula

Abstract

Within the Skyrme-Hartree-Fock (SHF) approach, we show that for a fixed mass number $A$, both the symmetry energy coefficient ${a}_{\mathrm{sym}}(A)$ in the semiempirical mass formula and the nuclear matter symmetry energy ${E}_{\mathrm{sym}}({\ensuremath{\rho}}_{A})$ at a subsaturation reference density ${\ensuremath{\rho}}_{A}$ can be determined essentially by the symmetry energy ${E}_{\mathrm{sym}}({\ensuremath{\rho}}_{0})$ and its density slope $L$ at saturation density ${\ensuremath{\rho}}_{0}$. Meanwhile, we find the dependence of ${a}_{\mathrm{sym}}(A)$ on ${E}_{\mathrm{sym}}({\ensuremath{\rho}}_{0})$ or $L$ is approximately linear and very similar to the corresponding linear dependence displayed by ${E}_{\mathrm{sym}}({\ensuremath{\rho}}_{A})$, providing an explanation for the relation ${E}_{\mathrm{sym}}({\ensuremath{\rho}}_{A})\ensuremath{\approx}{a}_{\mathrm{sym}}(A)$.Our results indicate that a value of ${E}_{\mathrm{sym}}({\ensuremath{\rho}}_{A})$ leads to a linear correlation between ${E}_{\mathrm{sym}}({\ensuremath{\rho}}_{0})$ and $L$ and thus can put important constraints on ${E}_{\mathrm{sym}}({\ensuremath{\rho}}_{0})$ and $L$. Particularly, the values of ${E}_{\mathrm{sym}}({\ensuremath{\rho}}_{0})=30.5\ifmmode\pm\else\textpm\fi{}3$ MeV and $L=$ $52.5\ifmmode\pm\else\textpm\fi{}20$ MeV are simultaneously obtained by combining the constraints from recently extracted ${E}_{\mathrm{sym}}({\ensuremath{\rho}}_{A}=0.1$ fm${}^{\ensuremath{-}3})$ with those from recent analyses of neutron skin thickness of Sn isotopes in the same SHF approach.