$I^4$ dependence in nuclear symmetry energy

In this paper we investigate the possible $I^4$ dependence [$I=(N-Z)/A$] in “experimental” symmetry energy extracted from nuclear masses. Our results show that the inclusion of the $I^4$ term in mass formulas leads to sizable changes for symmetry energy coefficients of the $I^2$ terms.

Figure 1

Symmetry energy $e(Z,A)$ in the $I^2$ and $A^{-1/3}$ expansion.

Figure 2

Symmetry energy coefficients ($c_2^{\left(\mathrm{V}\right)}$ and $c_2^{\left(\mathrm{S}\right)}$) of the $I^2$ terms (in MeV). Here subscripts $``\left(1\right)\text{""}$ and $``\left(2\right)\text{""}$ of the coefficients represent the forms (1) and (2) for the Wigner energy, respectively. Panel (a): Results with different $(i,j$) for Case A. Panel (b): Results with all $(i,j)$ for three cases of $R_{i\mathrm{p}-j\mathrm{n}}$. In Case A $R_{i\mathrm{p}-j\mathrm{n}}=c_2^{\left(\mathrm{V}\right)}{X}_2^{\left(\mathrm{V}\right)}+c_2^{\left(\mathrm{S}\right)}{X}_2^{\left(\mathrm{S}\right)}$, in Case B $R_{i\mathrm{p}-j\mathrm{n}}=c_2^{\left(\mathrm{V}\right)}{X}_2^{\left(\mathrm{V}\right)}+c_2^{\left(\mathrm{S}\right)}{X}_2^{\left(\mathrm{S}\right)}+c_4^{\left(\mathrm{V}\right)}{X}_4^{\left(\mathrm{V}\right)}$, and in Case C $R_{i\mathrm{p}-j\mathrm{n}}=c_2^{\left(\mathrm{V}\right)}{X}_2^{\left(\mathrm{V}\right)}+c_2^{\left(\mathrm{S}\right)}{X}_2^{\left(\mathrm{S}\right)}+c_4^{\left(\mathrm{V}\right)}{X}_4^{\left(\mathrm{V}\right)}+c_4^{\left(\mathrm{S}\right)}{X}_4^{\left(\mathrm{S}\right)}$.