arXiv2012
Within the nonlinear relativistic mean field model, we derive the analytical expression of the nuclear matter fourth-order symmetry energy $E_{sym,4}(ρ)$. Based on two accurately calibrated interactions FSUGold and IU-FSU, our results show that the value of $E_{sym,4}(ρ)$ at normal nuclear matter density $ρ_{0}$ is generally less than 1 MeV, confirming the empirical parabolic approximation to the equation of state for asymmetric nuclear matter at $ρ_{0}$. On the other hand, we find that the $E_{sym,4}(ρ)$ may become nonnegligible at high densities. Furthermore, the analytical form of the $E_{sym,4}(ρ)$ provides the possibility to study the higher-order effects on the isobaric incompressibility of asymmetric nuclear matter, i.e., $K_{sat}(δ)=K_{0}+K_{sat,2}δ^{2}+K_{sat,4}δ^{4}+\mathcal{O}(δ^{6})$ where $δ=(ρ_{n}-ρ_{p})/ρ$ is the isospin asymmetry, and we find that the value of $K_{sat,4}$ is generally small compared with that of the $K_{sat,2}$. In addition, we study the effects of the $E_{sym,4}(ρ)$ on the proton fraction $x_{p}$ and the core-crust transition density $ρ_{t}$ and pressure $P_{t}$ in neutron stars. Interestingly, we find that, compared with the results from the empirical parabolic approximation, including the $E_{sym,4}(ρ)$ contribution can significantly enhance the $x_{p}$ at high densities and strongly reduce the $ρ_{t}$ and $P_{t}$ in neutron stars, demonstrating that the widely used empirical parabolic approximation may cause large errors in determining the $x_{p}$ at high densities as well as the $ρ_{t}$ and $P_{t}$ in neutron stars within the nonlinear relativistic mean field model, consistent with previous nonrelativistic calculations.