Nuclear skin and the curvature of the symmetry energy

The effect of correlations between the slope and the curvature of the symmetry energy on ground-state nuclear observables is studied within the extended Thomas-Fermi approximation. We consider different isovector probes of the symmetry energy, with a special focus on the neutron skin thickness of ${}^{208}\mathrm{Pb}$. We use a recently proposed metamodeling technique to generate a large number of equation of state models, where the empirical parameters are independently varied. The results are compared to a set of calculations using 17 different Skyrme interactions. We show that the curvature parameter plays a non-negligible role on the neutron skin, while the effect is reduced in Skyrme functionals because of the correlation with the slope parameter.

Figure 1

Correlations between EoS parameters. Numbers on the left-hand side of each plot correspond to Pearson correlation coefficients between the parameters plotted on the axis. Upper (lower) values: meta-model (Skyrme).

Figure 2

ETF results corresponding to the ground state of Pb isotopes, for a representative EDF model (SLy4 [36]). Variational parameters $n_{\mathrm{bulk},n},n_{\mathrm{bulk},p}$ (top panel), $a_n,a_p$ (middle panel), and $R_n^{WS}$ and $R_p^{WS}$ (bottom panel) are plotted as a function of total isospin asymmetry. The results obtained by considering the diffusivity of the charge distribution [“self-consistent,” Eq. (12)] are confronted with those corresponding to the uniformly charged sphere approximation (“approx.”).

Figure 3

ETF results corresponding to the ground state of Pb isotopes and SLy4 [36]. Binding energy per nucleon (top panel), rms radii of neutron and charge distributions (middle panel), and neutron skin thickness are plotted as a function of total isospin asymmetry. When available, experimental masses [51] and charge radii [52] are plotted as well. For neutron skin thicknesses of ${}^{208}\mathrm{Pb}$, the following experimental data are illustrated: $0.1515\pm 0.0197$ fm [15], $0.{156}_{-0.021}^{+0.025}$ fm [16], and $0.3012\pm {\left(0.175\right)}_{\exp }\pm {\left(0.026\right)}_{\mathrm{model}}\pm {\left(0.005\right)}_{\mathrm{strange}}$ fm [13, 53]. As in Fig. 2, two methods for calculating the Coulomb energy are considered. Neutron skin thickness decomposition into bulk and surface contributions according to Eqs. ((17), (18)) is represented on the bottom panel for the case in which the Coulomb energy is calculated self-consistently (open symbols).

Figure 4

Correlations between neutron skin thickness in ${}^{208}\mathrm{Pb}$ and differences in the proton radii of mirror nuclei $R_p\left({}^{48}\mathrm{Ni}\right)-R_p\left({}^{48}\mathrm{Ca}\right)$ (top), electric dipole polarizability of ${}^{208}\mathrm{Pb}$ (middle), and IVGDR energy constant of ${}^{208}\mathrm{Pb}$ (bottom). Results corresponding to Skyrme and metamodeling are represented with open circles and, respectively, solid squares. Skyrme predictions corresponding to differences in the proton radii of mirror nuclei $R_p\left({}^{50}\mathrm{Ni}\right)-R_p\left({}^{50}\mathrm{Ti}\right),R_p\left({}^{52}\mathrm{Ni}\right)-R_p\left({}^{52}\mathrm{Cr}\right),R_p\left({}^{54}\mathrm{Ni}\right)-R_p\left({}^{54}\mathrm{Fe}\right)$ are also plotted in the top panel. Numbers on the left hand side of each plot correspond to Pearson correlation coefficients between the observables plotted on the axis. Upper (lower) values: metamodeling (Skyrme).

Figure 5

Correlations between $L_{\mathrm{sym}}$ and $\mathrm{\Delta }{r}_{np}\left({}^{208}\mathrm{Pb}\right)$ (top panel) and $L_{\mathrm{sym}}$ and $R_p\left({}^{48}\mathrm{Ni}\right)-R_p\left({}^{48}\mathrm{Ca}\right)$ (bottom panel). In addition to metamodeling EDF plotted in the previous figures, here we consider also metamodeling EDF with fixed values of $K_{\mathrm{sym}}=100,0,-100$ MeV. The numbers on the left-hand side mention the Pearson correlation coefficients in the following order: metamodeling with freely varying $K_{\mathrm{sym}}$, metamodeling with $K_{\mathrm{sym}}=100,0,-100$ MeV, and Skyrme.

Figure 6

Correlations between $L_{\mathrm{sym}}$ and ${\alpha }_D\left({}^{208}\mathrm{Pb}\right)$ (top) and $L_{\mathrm{sym}}$ and $D\left({}^{208}\mathrm{Pb}\right)$ (bottom). Numbers on the right-hand side of each plot correspond to Pearson correlation coefficients between the observables plotted on the axis. Upper (lower) values: metamodeling (Skyrme). As in Fig. 5, metamodeling EDF with fixed values of $K_{\mathrm{sym}}=100,0,-100$ MeV are plotted as well.