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

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 $E_{\mathrm{sym}}({\rho }_A)$ at a subsaturation reference density ${\rho }_A$ can be determined essentially by the symmetry energy $E_{\mathrm{sym}}({\rho }_0)$ and its density slope $L$ at saturation density ${\rho }_0$. Meanwhile, we find the dependence of $a_{\mathrm{sym}}(A)$ on $E_{\mathrm{sym}}({\rho }_0)$ or $L$ is approximately linear and very similar to the corresponding linear dependence displayed by $E_{\mathrm{sym}}({\rho }_A)$, providing an explanation for the relation $E_{\mathrm{sym}}({\rho }_A)\approx a_{\mathrm{sym}}(A)$. Our results indicate that a value of $E_{\mathrm{sym}}({\rho }_A)$ leads to a linear correlation between $E_{\mathrm{sym}}({\rho }_0)$ and $L$ and thus can put important constraints on $E_{\mathrm{sym}}({\rho }_0)$ and $L$. Particularly, the values of $E_{\mathrm{sym}}({\rho }_0)=30.5\pm 3$ MeV and $L=$ $52.5\pm 20$ MeV are simultaneously obtained by combining the constraints from recently extracted $E_{\mathrm{sym}}({\rho }_A=0.1$ fm${}^{-3})$ with those from recent analyses of neutron skin thickness of Sn isotopes in the same SHF approach.

Figure 1

$E_{\mathrm{sym}}({\rho }_A)$ with ${\rho }_A=0.09,0.10,$ and $0.11$ fm${}^{-3}$ as well as $a_{\mathrm{sym}}(A=208)$ and its approximation with Taylor expansion in Eq. (21) (with $S_0=30$  MeV and $L_0=60$  MeV) from SHF with MSL0 by varying individually $L$ (a), $G_V$ (b), $G_S$ (c), $E_0({\rho }_0)$ (d), $E_{\mathrm{sym}}({\rho }_0)$ (e), $K_0$ (f), $m_{s,0}^{*}$ (g), $m_{v,0}^{*}$ (h), ${\rho }_0$ (i), and $W_0$ (j).

Figure 2

Contour curves in the $E_{\mathrm{sym}}({\rho }_0)$-$L$ plane for $E_{\mathrm{sym}}(\rho =0.1$ fm${}^{-3})$. The region between the two thick solid lines represents the constraint obtained in the present work with 20.22 MeV$⩽E_{\mathrm{sym}}({\rho }_A=0.10$ fm${}^{-3})$ $⩽$ 24.74 MeV while the region between the two thick dashed lines is the constraint from the SHF analysis of neutron skin data of Sn isotopes within a $2\sigma$ uncertainty [18]. The shaded region represents the overlap of the two constraints.