Why charges go to the surface: a generalized Thomson problem

TL;DR

This paper investigates how particles confined to a sphere distribute themselves under different interaction potentials, revealing a transition point at the Coulomb law where charges shift from surface to bulk occupation.

Contribution

It generalizes the Thomson problem to arbitrary power-law interactions and identifies the conditions under which charges prefer surface or bulk configurations.

Findings

For g ≤ 1, charges are expelled to the surface.

For g > 1 and n > n_c(g), charges occupy the bulk.

The Coulomb law (g=1) marks the transition point.

Abstract

We study a generalization of a Thomson problem of n particles confined to a sphere and interacting by a 1/r^g potential. It is found that for g \le 1 the electrostatic repulsion expels all the charges to the surface of the sphere. However for g>1 and n>n_c(g) occupation of the bulk becomes energetically favorable. It is curious to note that the Coulomb law lies exactly on the interface between these two regimes.