Effect of Wigner energy on the symmetry energy coefficient in nuclei

The nuclear symmetry energy coefficient (including the coefficient $a_{\rm sym}^{(4)}$of$I^{4}$ term) of finite nuclei is extracted by using the differences of available experimental binding energies of isobaric nuclei. It is found that the extracted symmetry energy coefficient $a_{\mathrm{s}\mathrm{y}\mathrm{m}}^{\ast }(A,I)$ decreases with increasing of isospin asymmetry $I$, which is mainly caused by Wigner correction, since $e_{\mathrm{s}\mathrm{y}\mathrm{m}}^{\ast }$ is the summation of the traditional symmetry energy $e_{\mathrm{s}\mathrm{y}\mathrm{m}}$ and the Wigner energy $e_{\mathrm{W}}$. We obtain the optimal values $J=30.25\pm 0.10$ MeV, $a_{\mathrm{s}\mathrm{s}}=56.18\pm 1.25$ MeV, $a_{\rm sym}^{(4)}=8.33\pm1.21$MeVandtheWignerparameter$x=2.38\pm0.12$ through the polynomial fit to 2240 measured binding energies for nuclei with $20 \leq A \leq 261$ with an rms deviation of 23.42 keV. We also find that the volume symmetry coefficient $J\simeq 30$ MeV is insensitive to the value $x$, whereas the surface symmetry coefficient $a_{\mathrm{s}\mathrm{s}}$ and the coefficient $a_{\rm sym}^{(4)}$areverysensitivetothevalueof$x$intherange$1\leq x\leq 4$\mathrm{.}Thecontributionof$a_{\rm sym}^{(4)}$ term increases rapidly with increasing of isospin asymmetry $I$. For very neutron-rich nuclei, the contribution of $a_{\mathrm{s}\mathrm{y}\mathrm{m}}^{(4)}$ term will play an important role.