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 ${\sum }_{i>j=1}^{N}1/r_{ij}$? 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.

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