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

TL;DR

This paper derives six fundamental quantities of the nuclear equation of state in asymmetric matter using the Hugenholtz-Van Hove theorem, systematically calculating their properties with various effective interactions and analyzing their dependencies.

Contribution

It provides explicit expressions for key EoS quantities in asymmetric nuclear matter and highlights the importance of higher-order asymmetric potentials and the symmetry potential's role.

Findings

The symmetry potential significantly influences the density dependence of symmetry energy.

Higher-order asymmetric potentials are non-negligible in calculating basic quantities.

An empirical relation links the quadratic incompressibility coefficient to other EoS parameters.

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 $U_{\text{\text{sym,i}}}(\rho,k)$. 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 $K_{\text{sat,2}}=K_{2}(\rho_0)-4.14 L_2(\rho_0)$.