Mass dependence of symmetry energy coefficients in the Skyrme force

Based on the semiclassical extended Thomas-Fermi approach, we study the mass dependence of the symmetry energy coefficients of finite nuclei for 36 different Skyrme forces. The reference densities of both light and heavy nuclei are obtained. Eight models based on the nuclear liquid drop concept and the Skyrme force SkM* suggest the symmetry energy coefficient $a_{\mathrm{sym}}=22.90\pm 0.15$ MeV at $A=260$, and the corresponding reference density is ${\rho }_A\simeq 0.1{\mathrm{fm}}^{-3}$ at this mass region. The standard Skyrme energy density functionals give negative values for the coefficient of the $I^4$ term in the binding energy formula, whereas the latest Weizsäcker-Skyrme formula and the experimental data suggest positive values for the coefficient.

Figure 1

Binding energy per particle as a function of $I=(N-Z)/A$. The Coulomb interaction is not involved in all calculations with the ETF2 approach. The squares denote the results with the Skyrme force SkM*, and the curves denote the fit with Eq. (3).

Figure 2

Symmetry energy coefficients as a function of mass number of nuclei.

Figure 3

Symmetry energy coefficient $a_{\mathrm{sym}}$ as a function of $A^{-1/3}$. The open circles and squares denote the results of ETF2 plus SLy4 and SkM*, respectively. The curves denote the fit with Eq. (4).

Figure 4

Reference density obtained from SkM* as a function of mass number of nuclei.

Figure 5

Symmetry energy coefficients of nuclei extracted from nuclear masses.

Figure 6

Binding energy per particle after removing the Coulomb energy, Wigner energy, and traditional symmetry energy.