Analytic Solutions of the Ultra-relativistic Thomas-Fermi Equation

TL;DR

This paper derives exact solutions for a generalized ultra-relativistic Thomas-Fermi equation with Wood-Saxon-like proton distributions, revealing overcritical electric fields near the surface and analyzing how these fields vary with distribution parameters.

Contribution

It introduces new exact solutions for a generalized ultra-relativistic Thomas-Fermi equation using Wood-Saxon-like distributions, extending previous constant-proton models.

Findings

Exact solutions with Wood-Saxon-like distributions are obtained.

Overcritical electric fields are found near the surface.

Electric field variations depend on distribution parameters.

Abstract

It is well known that the ultra-relativistic Thomas-Fermi equation, amply adopted in the study of heavy nuclei, admits an exact solution for a constant proton distribution within a spherical core of radius Rc. Here exact solutions of a generalized ultra-relativistic Thomas-Fermi equation are presented, assuming a Wood-Saxon-like proton distribution and its further generalizations. These solutions present an overcritical electric field close to their surface. The variation of the electric fields as a function of the generalized Wood-Saxon parameters are studied.