Constraints on Coulomb energy, neutron skin thickness in ${}^{208}\mathrm{Pb}$, and symmetry energy

The charge-symmetry-breaking (CSB) effect in a nuclear medium that gives rise to the first-order symmetry energy in finite nuclei is discussed in detail in the present paper. For heavy and superheavy nuclei with large neutron excesses, it should be nonnegligible in high-precision mass predictions, and importantly it affects the stability of these nuclei. Combined with this CSB effect, the Coulomb energy is constrained by using the experimental Coulomb displacement energy of mirror nuclei, and then the mass-dependent symmetry energy coefficients of heavy nuclei are reextracted with the experimental ${\beta }^{-}$-decay energies of heavy odd-$A$ nuclei and with the experimental mass differences. Based on these results, we probe the neutron skin thickness $\mathrm{\Delta }{R}_{np}$ of ${}^{208}\mathrm{Pb}$ and the density-dependent symmetry energy coefficient of nuclear matter. $\mathrm{\Delta }{R}_{np}$ in ${}^{208}\mathrm{Pb}$ is found to be $0.158\pm 0.014\mathrm{fm}$, and the slopes $L$ of the symmetry energy coefficient at densities of $\rho =0.16$ and $\rho =0.11{\mathrm{fm}}^{-3}$ are estimated to be $42\pm 8$ and $42\pm 3\mathrm{MeV}$, respectively. These results would be meaningful to discriminate between the models and the predictions that are relevant for the investigations on properties of nuclei and of neutron stars.

Figure 1

Calculated $E_{\mathrm{sym},1}\left(A\right)$ for some typical nuclei in different mass regions by employing the self-consistent Skyrme-Hartree-Fock approach with the SLy4 [26] interaction.

Figure 2

The calculated CDEs for $T=1/2$ mirror pairs compared with the experimental data [36] where the Coulomb energy formulas of Eqs. (4), (5), and (8) are used. The first-order-symmetry energy terms $E_{\mathrm{sym},1}^{\left(\mathrm{CSB}\right)}(A,T_z)$ induced by the CSB interaction are computed self-consistently by using the Skyrme-Hartree-Fock-BCS method with the SLy4 interaction where the empirical gaps from Ref. [37] are applied.

Figure 3

The established linear correlation between the neutron skin thickness $\mathrm{\Delta }{R}_{np}$ in ${}^{208}\mathrm{Pb}$ and the $J-a_{\mathrm{sym}}$ in our previous work [18]. The derived $J-a_{\mathrm{sym}}$ and $\mathrm{\Delta }{R}_{np}$ of ${}^{208}\mathrm{Pb}$ in the present paper are exhibited with the red symbol.