Thomas-Fermi equation revisited: A variation on a theme by Majorana

TL;DR

This paper revisits Majorana's method for simplifying the Thomas-Fermi equation, extending it to weakly ionized atoms and recalculating key atomic physics quantities with improved efficiency.

Contribution

It extends Majorana's scaling approach to weakly ionized atoms and provides recalculated atomic quantities compared to older numerical methods.

Findings

Recalculated integrals relevant for atomic physics.

Extended the Thomas-Fermi solution to weakly ionized atoms.

Compared new results with 1980s numerical data.

Abstract

Majorana found a way to exploit the scaling properties of the Thomas-Fermi equation for converting this second-order differential equation into one of first order. We explore his method for the familiar neutral-atom solution and extend it to the solution that is relevant for weakly ionized atoms. Various integrals and other quantities with importance for atomic physics are recalculated and their values compared with the ones obtained in the 1980s by more tedious numerical procedures.