# Analytical and numerical assessment of accuracy of the approximated nuclear symmetry energy in the Hartree-Fock theory

## TL;DR

This paper evaluates the accuracy of approximating the nuclear symmetry energy using energy differences in Hartree-Fock theory, analyzing errors analytically and numerically across various effective interactions and densities.

## Contribution

It provides a detailed analytical and numerical assessment of the approximation's accuracy within Hartree-Fock theory, highlighting its density and interaction dependence.

## Key findings

- Errors are within 10% up to twice normal density.
- Accuracy varies significantly at higher densities depending on interactions.
- Analytical decomposition clarifies error sources.

## Abstract

The nuclear symmetry energy is defined by the second derivative of the energy per nucleon with respect to the proton-neutron asymmetry, and is sometimes approximated by the energy difference between the neutron matter and the symmetric matter. Accuracy of this approximation is assessed analytically and numerically, within the Hartree-Fock theory using effective interactions. By decomposing the nuclear-matter energy, the relative error of each term is expressed analytically; it is constant or is a single-variable function determined by the function type. The full errors are evaluated for several effective interactions, by inserting values of the parameters. Although the errors stay within $10\%$ up to twice of the normal density irrespective of the interactions, at higher densities accuracy of the approximation significantly depends on the interactions.