Basic quantities of the Equation of State in isospin asymmetric nuclear matter

Based on the Hugenholtz-Van Hove theorem, six basic quantities of the EoS in isospin asymmetric nuclear matter are expressed in terms of the nucleon kinetic energy $t(k)$, the isospin symmetric and asymmetric parts of the single-nucleon potentials $U_0(\rho,k)$ and $U_{\text{\text{sym,i}}}(\rho,k)$. The six basic quantities include the quadratic symmetry energy $E_{\text{sym,2}}(\rho)$, the quartic symmetry energy $E_{\text{sym,4}}(\rho)$, their corresponding density slopes $L_2(\rho)$ and $L_4(\rho)$, and the incompressibility coefficients $K_2(\rho)$ and $K_4(\rho)$. By using four types of well-known effective nucleon-nucleon interaction models, namely the BGBD, MDI, Skyrme, and Gogny forces, the density- and isospin-dependent properties of these basic quantities are systematically calculated and their values at the saturation density $\rho_0$ are explicitly given. The contributions to these quantities from $t(k)$, $U_0(\rho,k)$, and $U_{\text{sym,i}}(\rho,k)$ are also analyzed at the normal nuclear density $\rho_0$. It is clearly shown that the first-order asymmetric term $U_{\text{sym,1}}(\rho,k)$ (also known as the symmetry potential in Lane potential) plays a vital role in determining the density dependence of the quadratic symmetry energy $E_{\text{sym,2}}(\rho)$. It is also shown that the contributions from high-order asymmetric parts of the single-nucleon potentials ($U_{\text{sym,i}}(\rho,k)$ with $i>1$) cannot be neglected in the calculations of the other five basic quantities. Moreover, by analyzing the properties of asymmetric nuclear matter at the exact saturation density $\rho_{\text{sat}}(\delta)$, the corresponding quadratic incompressibility coefficient is found to have a simple empirical relation $K_{\text{sat,2}}=K_{2}(\rho_0)-4.14 L_2(\rho_0)$.