Basic quantities of the equation of state in isospin asymmetric nuclear matter

Abstract

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 $U_{\text {sym},i}(\rho ,k)$. 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 the 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 the 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 $K_{\text {sat,2}}=K_{2}(\rho _0)-4.14 L_2(\rho _0)$.