Nuclear matter fourth-order symmetry energy in the relativistic mean field models

Abstract

Within the nonlinear relativistic mean field model, we derive the analytical expression of the nuclear matter fourth-order symmetry energy $E_{sym,4}(\ddot{I})$. Based on two accurately calibrated interactions FSUGold and IU-FSU, our results show that the value of $E_{sym,4}(\ddot{I})$ at normal nuclear matter density $\ddot{I}{}_0$ is generally less than 1 MeV, confirming the empirical parabolic approximation to the equation of state for asymmetric nuclear matter at $\ddot{I}{}_0$. On the other hand, we find that the $E_{sym,4}(\ddot{I})$ may become nonnegligible at high densities. Furthermore, the analytical form of the $E_{sym,4}(\ddot{I})$ provides the possibility to study the higher-order effects on the isobaric incompressibility of asymmetric nuclear matter, i.e., $K_{sat}(\hat{I}´)=K_0+K_{sat,2}\hat{I}{´}^2+K_{sat,4}\hat{I}{´}^4+\mathcal{O}(\hat{I}{´}^6)$ where $\hat{I}´=(\ddot{I}{}_n-\ddot{I}{}_p)/\ddot{I}$ 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 $E_{sym,4}(\ddot{I})$ on the proton fraction $x_p$ and the core-crust transition density $\ddot{I}{}_t$ and pressure $P_t$ in neutron stars. Interestingly, we find that, compared with the results from the empirical parabolic approximation, including the $E_{sym,4}(\ddot{I})$ contribution can significantly enhance the $x_p$ at high densities and strongly reduce the $\ddot{I}{}_t$ 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 $\ddot{I}{}_t$ and $P_t$ in neutron stars within the nonlinear relativistic mean field model, consistent with previous nonrelativistic calculations.

10 pages, 3 figures, 2 tables. Title changed a little, typos fixed. Published version in PRC