Density dependence of the symmetry energy probed byβ−-decay energies of odd-Anuclei

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The mass-dependent symmetry energy coefficients ${a}_{\text{sym}}(A)$ are extracted with the ${\ensuremath{\beta}}^{\ensuremath{-}}$-decay energies ${Q}_{{\ensuremath{\beta}}^{\ensuremath{-}}}$ of heavy odd-$A$ nuclei. The dominant position of this approach is that only the Coulomb energy survives in ${Q}_{{\ensuremath{\beta}}^{\ensuremath{-}}}$ to determine the unknown ${a}_{\text{sym}}(A)$. The obtained ${a}_{\text{sym}}(A)$ is employed to analyze the density dependence of the nuclear matter symmetry energy around the saturation density. The estimated ${a}_{\text{sym}}(A)$ of ${}^{208}\text{Pb}$ is 22.1--22.7 MeV. The slope parameter of the symmetry energy is found to be $50\ifmmode\pm\else\textpm\fi{}15$ MeV, with the symmetry energy ${S}_{0}=32.3\ifmmode\pm\else\textpm\fi{}1.3$ MeV at saturation density obtained from the analysis of pygmy dipole resonance (PDR) [Carbone et al., Phys. Rev. C 81, 041301(R) (2010)] as input. Furthermore, the corresponding neutron skin thickness in ${}^{208}\text{Pb}$ is estimated to be $\ensuremath{\Delta}{R}_{np}=0.174\ifmmode\pm\else\textpm\fi{}0.022$ fm.