Constraining the density slope of nuclear symmetry energy at subsaturation densities using electric dipole polarizability in ${}^{208}\mathrm{Pb}$

Nuclear structure observables usually most effectively probe the properties of nuclear matter at subsaturation densities rather than at saturation density. We demonstrate that the electric dipole polarizibility ${\alpha }_{\text{D}}$ in ${}^{208}\mathrm{Pb}$ is sensitive to both the magnitude $E_{\text{sym}}\left({\rho }_{\text{c}}\right)$ and density slope $L\left({\rho }_{\text{c}}\right)$ of the symmetry energy at the subsaturation cross density ${\rho }_{\text{c}}=0.11$ ${\mathrm{fm}}^{-3}$. Using the experimental data of ${\alpha }_{\text{D}}$ in ${}^{208}\mathrm{Pb}$ from RCNP (Research Center for Nuclear Physics, Osaka University) and the recent accurate constraint of $E_{\text{sym}}\left({\rho }_{\text{c}}\right)$ from the binding energy difference of heavy isotope pairs, we extract a value of $L\left({\rho }_{\text{c}}\right)=47.3\pm 7.8$ MeV. The implication of the present constraint of $L\left({\rho }_{\text{c}}\right)$ to the symmetry energy at saturation density, the neutron skin thickness of ${}^{208}\mathrm{Pb}$ and the core-crust transition density in neutron stars is discussed.

Figure 1

The electric dipole polarizability ${\alpha }_{\text{D}}$ in ${}^{208}\mathrm{Pb}$ from SHF-RPA calculations with the MSL0 interaction by varying individually $L\left({\rho }_{\text{r}}\right)$ (a), $G_V$ (b), $G_S$ (c), $E_0\left({\rho }_0\right)$ (d), $E_{\text{sym}}\left({\rho }_{\text{r}}\right)$ (e), $K_0$ (f), $m_{s,0}^{*}$ (g), $m_{v,0}^{*}$ (h), ${\rho }_0$ (i), and $W_0$ (j) for ${\rho }_{\text{r}}=0.11$ and $0.16{\text{fm}}^{-3}$. The $E_{\mathrm{sym}}\left({\rho }_r\right)$ is shifted by subtracting 7 MeV for ${\rho }_r=0.16{\text{fm}}^{-3}$.

Figure 2

The electric dipole polarizability ${\alpha }_{\text{D}}$ in ${}^{208}\mathrm{Pb}$ as a function of $L\left({\rho }_{\text{c}}\right)$ for fixed $E_{\text{sym}}\left({\rho }_{\text{c}}\right)$. The open (solid) up- and down-triangles represent the results with $E_{\text{sym}}\left({\rho }_{\text{c}}\right)=26.45$ and $26.85$ MeV, respectively, from SHF-RPA calculations with the values of other parameters fixed in MSL0 (obtained in optimization). The band indicates the experimental value of ${\alpha }_D=20.1\pm 0.6$ ${\mathrm{fm}}^3$ from RCNP [30].

Figure 3

Same as Fig. 1 but for the core-crust transition density ${\rho }_t$ in neutron stars.