The large-Z behaviour of pseudo-relativistic atoms

TL;DR

This paper investigates the asymptotic behavior of the ground state energy of large atomic systems with relativistic electrons, showing it aligns with non-relativistic predictions under certain conditions.

Contribution

It establishes that, for large atomic number Z, the relativistic ground state energy matches the non-relativistic Thomas-Fermi energy, extending understanding to relativistic regimes.

Findings

Ground state energy asymptotically matches non-relativistic Thomas-Fermi energy.

Relativistic effects become negligible at large Z under fixed Z*alpha.

Results hold when Z*alpha is less than or equal to 2/pi.

Abstract

In this paper we study the large-Z behaviour of the ground state energy of atoms with electrons having relativistic kinetic energy sqrt(p^2c^2+m^2c^4)-mc^2. We prove that to leading order in Z the energy is the same as in the non-relativistic case, given by (non-relativistic) Thomas-Fermi theory. For the problem to make sense, we keep the product Z*alpha fixed (here alpha is Sommerfeld's fine structure constant), and smaller than, or equal to, 2/pi, which means that as Z tends to infinity, alpha tends to zero.