Model dependence of the neutron-skin thickness on the symmetry energy

The model dependence in the correlations of the neutron-skin thickness in heavy nuclei with various symmetry-energy parameters is analyzed by using several families of systematically varied microscopic mean-field models. Such correlations show a varying degree of model dependence once the results for all the different families are combined. Some mean-field models associated with similar values of the symmetry-energy slope parameter at saturation density $L$, and pertaining to different families, yield a greater-than-expected spread in the neutron-skin thickness of the ${}^{208}\mathrm{Pb}$ nucleus. The effective value of the symmetry-energy slope parameter $L_{\mathrm{eff}}$, determined by using the nucleon density profiles of the finite nucleus and the density derivative $S^{\prime }\left(\rho \right)$ of the symmetry energy starting from about saturation density up to low densities typical of the surface of nuclei, seems to account for the spread in the neutron-skin thickness for the models with similar $L$. The differences in the values of $L_{\mathrm{eff}}$ are mainly due to the small differences in the nucleon density distributions of heavy nuclei in the surface region and the behavior of the symmetry energy at subsaturation densities.

Figure 1

Plots for the difference between the symmetry-energy coefficient for infinite nuclear matter $J$ and that for finite nuclei $a_{\mathrm{sym}}\left(A\right)$ as a function of the neutron-skin thickness (left panels) and of the bulk part of the neutron-skin thickness (right panels). The results are obtained by using five different families of mean-field models; namely, FSV (blue squares), TSV (red circles), DDME (green triangles), SAMi-J (orange diamonds), and KDE0-J (maroon inverted triangles). The correlation coefficients are $C(J-a_{\mathrm{sym}}\left(A\right),\mathrm{\Delta }{r}_{\mathrm{np}})=0.972$ (0.967) and $C(J-a_{\mathrm{sym}}\left(A\right),\mathrm{\Delta }{r}_{\mathrm{np}}^{\mathrm{bulk}})=0.988$ (0.979) for ${}^{208}\mathrm{Pb} \left({}^{132}\mathrm{Sn}\right)$ nuclei. The inner (outer) colored regions depict the loci of the 95% confidence (prediction) bands of the regression (see, e.g., Chap. 3 of Ref. [55]).

Figure 2

Plots for the ratio of the nuclear symmetry-energy coefficient for finite nuclei $a_{\mathrm{sym}}\left(A\right)$ to that for infinite nuclear matter $J$, as a function of the neutron-skin thickness (left panels) and of the bulk part of the neutron-skin thickness (right panels). The square shaded region in the upper-left panel corresponds to $a_{\mathrm{sym}}\left(A\right)=22.4\pm 0.3$ MeV and $J=32.5\pm 2.5$ MeV. The correlation coefficients are $|C(a_{\mathrm{sym}}\left(A\right)/J,\mathrm{\Delta }{r}_{\mathrm{np}})|=0.965$ (0.959) and $|C(a_{\mathrm{sym}}\left(A\right)/J,\mathrm{\Delta }{r}_{\mathrm{np}}^{\mathrm{bulk}})|=0.992$ (0.989) for ${}^{208}\mathrm{Pb} \left({}^{132}\mathrm{Sn}\right)$ nuclei. The inner (outer) colored regions depict the loci of the 95% confidence (prediction) bands of the regression (see, e.g., Chap. 3 of Ref. [55]).

Figure 3

Neutron-skin thickness (left) and its bulk (middle), and surface (right) contributions for the ${}^{132}\mathrm{Sn}$ nucleus plotted against the same quantities for the ${}^{208}\mathrm{Pb}$ nucleus. The shaded region corresponds to the values of the neutron-skin thickness in ${}^{132}\mathrm{Sn}$ determined from those estimated for the ${}^{208}\mathrm{Pb}$ nucleus (see also Fig. 2). The correlation coefficients obtained for the results presented in the left, middle, and right panels are 0.999, 0.993, and 0.995, respectively. The inner (outer) colored regions depict the loci of the 95% confidence (prediction) bands of the regression (see, e.g., Chap. 3 of Ref. [55]).

Figure 4

Plots for the symmetry-energy slope parameter $L$ as a function of the neutron-skin thickness (left), its bulk part (middle), and its surface part (right) for the ${}^{208}\mathrm{Pb}$ nucleus. The shaded region in the left panel projects out the values of $L=64\pm 23$ MeV obtained from $\mathrm{\Delta }{r}_{\mathrm{np}}=0.197\pm 0.047$ fm which, in turn, is obtained by using the empirical values of $J$ and $a_{\mathrm{sym}}\left(A\right)$ (see also Fig. 2). The arrow marks in the left panel indicate the points with the slope parameter $L\sim 65$ MeV. The values of the correlation coefficients are $C(L,\mathrm{\Delta }{r}_{\mathrm{np}})=0.950,C(L,\mathrm{\Delta }{r}_{\mathrm{np}}^{\mathrm{bulk}})=0.963$, and $C(L,\mathrm{\Delta }{r}_{\mathrm{np}}^{\mathrm{surf}})=0.469$. The inner (outer) colored regions depict the loci of the 95% confidence (prediction) bands of the regression (see, e.g., Chap. 3 of Ref. [55]).

Figure 5

The nuclear symmetry energy $S$ (lower panel) and its density derivative $S^{\prime }$ multiplied by $3{\rho }_0$ (upper panel) as a function of density for the five different models associated with the slope parameter for nuclear matter $L\sim 65$ MeV. Each of these models belongs to a different family (see also Table 1).

Figure 6

The variation of $r^2\rho \left(r\right)I^2\left(r\right)$ as a function of the radial coordinate $r$ in ${}^{208}\mathrm{Pb}$ for the five models that yield a symmetry-energy slope parameter $L\sim 65$ MeV.