Asymptotics for Two-dimensional Atoms

TL;DR

This paper derives the asymptotic behavior of the ground state energy of two-dimensional atoms with large nuclear charge, revealing a logarithmic term and showing that their radius becomes unbounded as Z increases.

Contribution

It provides the first detailed asymptotic expansion for the ground state energy of 2D atoms and demonstrates the unbounded growth of their radius with increasing Z.

Findings

Ground state energy asymptotics include a logarithmic term in Z.

The atom's radius becomes unbounded as Z approaches infinity.

The energy expansion involves a Thomas-Fermi type variational problem.

Abstract

We prove that the ground state energy of an atom confined to two dimensions with an infinitely heavy nucleus of charge $Z>0$ and $N$ quantum electrons of charge -1 is $E(N,Z)=-{1/2}Z^2\ln Z+(E^{\TF}(\lambda)+{1/2}c^{\rm H})Z^2+o(Z^2)$ when $Z\to \infty$ and $N/Z\to \lambda$, where $E^{\TF}(\lambda)$ is given by a Thomas-Fermi type variational problem and $c^{\rm H}\approx -2.2339$ is an explicit constant. We also show that the radius of a two-dimensional neutral atom is unbounded when $Z\to \infty$, which is contrary to the expected behavior of three-dimensional atoms.