Thermal effects on nuclear symmetry energy with a momentum-dependent effective interaction

The knowledge of the nuclear symmetry energy of hot neutron-rich matter is important for understanding the dynamical evolution of massive stars and the supernova explosion mechanisms. In particular, the electron capture rate on nuclei and/or free protons in presupernova explosions is especially sensitive to the symmetry energy at finite temperature. In view of the above, in the present work we calculate the symmetry energy as a function of the temperature for various values of the baryon density by applying a momentum-dependent effective interaction. In addition to a previous work, the thermal effects are studied separately both in the kinetic part and the interaction part of the symmetry energy. We focus also on the calculations of the mean-field potential, employed extensively in heavy-ion reaction research, both for nuclear and pure neutron matter. The proton fraction and the electron chemical potential, which are crucial quantities for representing the thermal evolution of supernova and neutron stars, are calculated for various values of the temperature. Finally, we construct a temperature dependent equation of state of β-stable nuclear matter, the basic ingredient for the evaluation of the neutron star properties.

Figure 1

The difference $E(n,T,x)-E(n,T,x=1/2)$ as a function of $(1-2x){}^2$ at temperature $T=0,T=20$, and $T=50$ MeV for three baryon number fractions $u=1,u=2$, and $u=3$.

Figure 2

Temperature dependence of the total nuclear symmetry energy and its interaction and kinetic energy part for various values of the baryon density $n$.

Figure 3

Temperature dependence of the symmetry energy for low values of the baryon density ($n=0.06,0.08,0.10,0.12,0.14$ fm${}^{-3}$). The experimental data are from Ref. [29] (solid squares) and Ref. [30] (open squares) are included for comparison.

Figure 4

Density dependence of the total nuclear symmetry energy $E_{\mathrm{sym}}^{\mathrm{tot}}(n,T)$ as well its kinetic $E_{\mathrm{sym}}^{\mathrm{kin}}(n,T)$ and interaction $E_{\mathrm{sym}}^{\mathrm{int}}(n,T)$ part for various values of the temperature $T$.

Figure 5

(a) The energy per particle of pure neutron matter as a function of the baryon density for various values of the temperature $T$. (b) The energy per particle of symmetric nuclear matter as a function of the baryon density for various values of the temperature $T$.

Figure 6

The single-particle potential of the pure neutron matter as a function of the momentum $k$ for various values of the temperature $T$ and for $n=0.1,0.3$, and $0.5{\mathrm{fm}}^{-3}$, respectively.

Figure 7

The single-particle potential of the symmetric nuclear matter as a function of the momentum $k$ for various values of the temperature $T$ and for $n=0.1,0.3$, and $0.5{\mathrm{fm}}^{-3}$, respectively.

Figure 8

The single-particle potential of β-stable matter for neutron [(a) and (b)] and for proton [(c) and (d)] as a function of the momentum $k$ for various values of the temperature $T$ for $n=0.1$ and $0.5{\mathrm{fm}}^{-3}$.

Figure 9

(a) The proton fraction $x$ in β-stable matter as a function of the density $n$ for various values of the temperature $T$. The straight line corresponds to the case $x=11\%$. (b) The electron chemical potential ${\mu }_e=\hat{\mu }={\mu }_n-{\mu }_p$ as a function of the density $n$ for various values of the temperature $T$.

Figure 10

The equation of state $P=P(\epsilon )$ of β-stable matter corresponding to the present momentum-dependent effective interaction model for various values of the temperature $T$.