Temperature effects on the nuclear symmetry energy and symmetry free energy with an isospin and momentum dependent interaction

Within a self-consistent thermal model using an isospin and momentum dependent interaction (MDI) constrained by the isospin diffusion data in heavy-ion collisions, we investigate the temperature dependence of the symmetry energy $E_{\mathrm{sym}}(\rho ,T)$ and symmetry free energy $F_{\mathrm{sym}}(\rho ,T)$ for hot, isospin asymmetric nuclear matter. It is shown that the symmetry energy $E_{\mathrm{sym}}(\rho ,T)$ generally decreases with increasing temperature while the symmetry free energy $F_{\mathrm{sym}}(\rho ,T)$ exhibits opposite temperature dependence. The decrement of the symmetry energy with temperature is essentially due to the decrement of the potential energy part of the symmetry energy with temperature. The difference between the symmetry energy and symmetry free energy is found to be quite small around the saturation density of nuclear matter. While at very low densities, they differ significantly from each other. In comparison with the experimental data of temperature dependent symmetry energy extracted from the isotopic scaling analysis of intermediate mass fragments (IMF's) in heavy-ion collisions, the resulting density and temperature dependent symmetry energy $E_{\mathrm{sym}}(\rho ,T)$ is then used to estimate the average freeze-out density of the IMF's.

Figure 1

Density dependence of the energy per nucleon $E$ (left panel) and free energy per nucleon $F$ (right panel) for symmetric nuclear matter at $T=0$ MeV, 5 MeV, 10 MeV, and 15 MeV with the MDI interaction.

Figure 2

Same as Fig. 1 but for pure neutron matter using the MDI interaction with $x=0$ (solid lines) and $-1$ (dashed lines).

Figure 3

$E(\rho ,T,\delta )-E(\rho ,T,\delta =0)$ as a function of ${\delta }^2$ at temperature $T=0$ MeV (a), 5 MeV (b), 10 MeV (c), and 15 MeV (d) for three different baryon number densities $\rho =0.5{\rho }_0,1.5{\rho }_0$ and $2.5{\rho }_0$ using the MDI interaction with $x=0$.

Figure 4

Same as Fig. 3 but for the free energy per nucleon $F(\rho ,T,\delta )$.

Figure 5

Density dependence of the symmetry energy $E_{\mathrm{sym}}(\rho ,T)$ (left panel) and the symmetry free energy $F_{\mathrm{sym}}(\rho ,T)$ (right panel) at $T=0$ MeV, 5 MeV, 10 MeV and 15 MeV using the MDI interaction with $x=0$ and $-1$.

Figure 6

Temperature dependence of the symmetry energy $E_{\mathrm{sym}}(\rho ,T)$ as well as its potential energy part and kinetic energy part using MDI interaction with $x=0$ at $\rho ={\rho }_0$ (a), $0.5{\rho }_0$ (b), and $0.1{\rho }_0$ (c).

Figure 7

Temperature dependence of the symmetry energy (solid lines) and symmetry free energy (dashed lines) using MDI interaction with $x=0$ (left panel) and $-1$ (right panel) at different densities from $0.1{\rho }_0$ to ${\rho }_0$. The experimental data from Texas A&M University (solid squares) and the INDRA-ALADIN Collaboration at GSI (open squares) are included for comparison.