Symmetry energy of hot nuclei in the relativistic Thomas-Fermi approximation

We develop a self-consistent description of hot nuclei within the relativistic Thomas-Fermi approximation using the relativistic mean-field model for nuclear interactions. The temperature dependence of the symmetry energy and other physical quantities of a nucleus are calculated by employing the subtraction procedure in order to isolate the nucleus from the surrounding nucleon gas. It is found that the symmetry energy coefficient of finite nuclei is significantly affected by the Coulomb polarization effect. We also examine the dependence of the results on nuclear interactions and make a comparison between the results obtained from relativistic and nonrelativistic Thomas-Fermi calculations.

Figure 1

Density distributions of neutrons (left panels) and protons (right panels) for ${}^{56}\mathrm{Fe}$ at $T=0$, 4, and 8 MeV obtained using the TM1 parametrization. The results of the nuclear liquid-plus-gas phase ($NG$), nucleon gas phase ($G$), and subtracted nuclear liquid phase ($N$) are shown in the top, middle, and bottom panels, respectively.

Figure 2

Same as Fig. 1, but for ${}^{208}\mathrm{Pb}$.

Figure 3

The rms radii of neutrons and protons, $R_n$ and $R_p$, and the neutron-skin thickness, $\Delta r_{np}=R_n-R_p$, as a function of the temperature $T$ for ${}^{56}\mathrm{Fe}$ obtained using four different RMF parametrizations.

Figure 4

Same as Fig. 3, but for ${}^{208}\mathrm{Pb}$.

Figure 5

Temperature $T$ as a function of the excitation energy per particle, $E^{*}/A$ (caloric curve), for ${}^{56}\mathrm{Fe},{}^{112}\mathrm{Sn},{}^{150}\mathrm{Sm}$, and ${}^{208}\mathrm{Pb}$ obtained using four different RMF parametrizations.

Figure 6

Symmetry energy coefficient $a_{\mathrm{sym}}$ as a function of $Z_1$ for $A=56$ and $A=208$ at $T=0$, 4, and 8 MeV obtained using the TM1 parametrization. The nuclear pair used for calculating $a_{\mathrm{sym}}$ is chosen to be ($A,Z_1$) and ($A,Z_1-2$). The results extracted from uncharged nuclei (without Coulomb) are compared with those calculated from charged nuclei by subtracting the Coulomb energy (with Coulomb).

Figure 7

Same as Fig. 6, but for the FSU parametrization.

Figure 8

Temperature dependence of $a_{\mathrm{sym}}$ for ${}^{56}\mathrm{Fe},{}^{112}\mathrm{Sn},{}^{150}\mathrm{Sm}$, and ${}^{208}\mathrm{Pb}$ obtained using the TM1 parametrization. The results obtained from uncharged nuclei (without Coulomb) are compared with those calculated from charged nuclei by subtracting the Coulomb energy (with Coulomb).

Figure 9

Same as Fig. 8, but for the FSU parametrization.

Figure 10

Temperature dependence of $a_{\mathrm{sym}}$ calculated from uncharged nuclei for ${}^{56}\mathrm{Fe},{}^{112}\mathrm{Sn},{}^{150}\mathrm{Sm}$, and ${}^{208}\mathrm{Pb}$ with four different RMF parametrizations.