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

Abstract

Given N unit points charges on the surface of a unit conducting sphere, what configuration of charges minimizes the Coulombic energy PN i>j=1 1/rij? 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 N = 10(h2 + hk + k2) + 2 recent theoretical work based on elasticity theory, and subsequent numerical work has shown, that for N > 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.