Effects of the symmetry energy on the isovector properties of neutron-rich nuclei within a Thomas-Fermi approach

We employ a variational method, in the framework of the Thomas-Fermi approximation, to study the effect of the symmetry energy on the neutron skin thickness and the symmetry energy coefficients of various neutron-rich nuclei. We concentrate our interest on ${}^{208}\mathrm{Pb}$, ${}^{124}\mathrm{Sn}$, ${}^{90}\mathrm{Zr}$, and ${}^{48}\mathrm{Ca}$, although the method can be applied in the totality of medium and heavy neutron-rich nuclei. Our approach has the advantage that the isospin asymmetry function $\alpha \left(r\right)$, which is the key quantity for calculating isovector properties of various nuclei, is directly related to the symmetry energy as a consequence of the variational principle. Moreover, the Coulomb interaction is included in a self-consistent way and its effects can be separated easily from the nucleon-nucleon interaction. We confirm, both qualitatively and quantitatively, the strong dependence of the symmetry energy on the various isovector properties for the relevant nuclei, using possible constraints between the slope and the value of the symmetry energy at the saturation density.

Figure 1

The nuclear symmetry energy $S\left(\rho \right)$, defined in Eq. (33), as a function of the density $\rho$ for various values of the slope parameter $L$ and the specific value $J=30$ MeV.

Figure 2

The total, neutron, and proton density distributions [(a) and (c)] for ${}^{208}\mathrm{Pb}$ and ${}^{48}\mathrm{Ca}$ for three values of $L$ [(a) and (c)] and the corresponding asymmetry functions $\alpha \left(r\right)$ for a variety of values of $L$ [(b) and (d)].

Figure 3

(a) The neutron skin $R_{\mathrm{skin}}$ for ${}^{208}\mathrm{Pb}$ as a function of the symmetry energy slope $L$ for various values of $J$. (b) The asymmetry coefficient $a_A$ for ${}^{208}\mathrm{Pb}$ as a function of the symmetry energy slope $L$ for various values of the parameter $J$.

Figure 4

(a) Plot of the pairs $L$ and $J$ that reproduce the empirical value of $a_A$ given by (45) for four nuclei. (b) The asymmetry coefficients $a_A$ as a function of $A$ for the relevant isotopes and for the sets $L=70$, $J=32$ and $L=65$, $J=34$. The solid line corresponds to the empirical formula (45) (for more details see text).

Figure 5

Regions of allowed values of pairs $J$ and $L$ (three bands) constrained from heavy-ion collisions [HIC(Sn + Sn) case] and nuclear structure observables ($a_D$ and nuclear masses) (for more details see Ref. [10]) in comparison with the corresponding results constrained from the present approach. The solid, dashed, and dotted arrows indicate constraints related to $a_D$, nuclear masses, and heavy-ion collisions, respectively. The four colored lines intersecting at the cross show the dependence of $L$ on $J$ according to Fig. 5 of our present work. The star and the cross correspond to the sets $L=70$, $J=32$ and $L=65$, $J=34$, respectively (for more details see text).

Figure 6

(a) The asymmetry coefficient $a_A$ as a function of the asymmetry parameter $I$ for various isotopes. (b) The corresponding neutron skin $R_{\mathrm{skin}}$ dependence on $I$.