Robustness of the $I^4$ symmetry energy coefficient

In this paper we investigate the $I^4$ symmetry energy coefficient $[I=(N-Z)/A]$ extracted from a few popular mass models, and their corresponding databases improved, respectively, by the radial basis function (RBF) approach and the RBF with odd-even corrections. Our results show that the linear correlation between our resultant $I^4$ symmetry energy coefficients (denoted by $c_4^{\left(\mathrm{V}\right)}$) and the corresponding root-mean-square deviations from experimental masses of these theoretical databases is very robust. Interestingly, the extrapolated values of $c_4^{\left(\mathrm{V}\right)}$ using the above three versions of databases are consistent with each other.

Figure 1

$R_{i\mathrm{p}-j\mathrm{n}}/\left(ij\right)$ (MeV) versus mass number $A$ with $(i,j)=(2,2)$. (a), ${(\mathrm{a}}^{\prime }$), and ${(\mathrm{a}}^{\prime \prime }$) correspond to results of the WS4 based on the original database [12], the corresponding database improved, respectively, by the RBF [denoted by (1)], and the RBFoe [denoted by (2)], for the form (2) of Wigner energy. (b), ${(\mathrm{b}}^{\prime }$), and ${(\mathrm{b}}^{\prime \prime }$) [(c), ${(\mathrm{c}}^{\prime }$), and ${(\mathrm{c}}^{\prime \prime }$)]: same as (a), ${(\mathrm{a}}^{\prime }$), and ${(\mathrm{a}}^{\prime \prime }$) but for the FRDM [the HFB-17]. The blue and red lines refer to the average values of $R_{i\mathrm{p}-j\mathrm{n}}/\left(ij\right)$ based on Ref. [12] and the approach (1), respectively, (they are also plotted in the lowest panels for comparison), and the green lines correspond to those in the approach (2). The features in the plot would not change if one assumed the form (1) of Wigner energy.

Figure 2

Symmetry energy coefficients of the $I^4$ term ($c_4^{\left(\mathrm{V}\right)}$) versus the rmsDs ($\sigma$) from experimental masses [13] for different mass models. (a) and ${(\mathrm{a}}^{\prime }$): $c_4^{\left(\mathrm{V}\right)}$ for the whole database of mass model, and the selected database of known nuclei, respectively, by adopting the form (1) of Wigner energy. (b) and ${(\mathrm{b}}^{\prime }$): same as (a) and ${(\mathrm{a}}^{\prime }$) but for the form (2). Symbols of different shape stand for results based on different mass models. For a given mass model, three colors (blue, red, and black) are used to distinguish different versions of databases. Blue symbols represent results in Ref. [12] based on the original databases. Red ones refer to results based on the corresponding databases improved by the RBF, i.e., the approach (1). Black ones correspond to those based on the improved databases by the RBFoe correction, denoted by the approach (2). The lines are plotted by using linear polynomial fitting for different version of mass models.