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

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