Defect free global minima in Thomson's problem of charges on a sphere

TL;DR

This paper proves that for charges on a sphere, the lowest energy configurations can be defect-free for all numbers of charges, challenging previous beliefs that defects are necessary for large N.

Contribution

The authors demonstrate that defect-free global minima exist for all N in Thomson's problem, providing a comprehensive catalog of such configurations.

Findings

Defect-free configurations are global minima for all N.

Adding dislocation defects is not always necessary for energy minimization.

Complete catalog of defect-free minima for various N.

Abstract

Given $N$ unit points charges on the surface of a unit conducting sphere, what configuration of charges minimizes the Coulombic energy $\sum_{i>j=1}^N 1/r_{ij}$? Due to an exponential rise in good local minima, finding global minima for this problem, or even approaches to do so has proven extremely difficult. For \hbox{$N = 10(h^2+hk+k^2)+ 2$} recent theoretical work based on elasticity theory, and subsequent numerical work has shown, that for $N \sim >500$--1000 adding dislocation defects to a symmetric icosadeltahedral lattice lowers the energy. Here we show that in fact this approach holds for all $N$, and we give a complete or near complete catalogue of defect free global minima.