Energies and spacings of point charges on a sphere

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Abstract
Numerical methods are in general required for the determination of the stable configurations of N point charges on a sphere. The stable configurations for N up to 50 have previously been ascertained and the authors extend the calculations for values up to 101. They report for the first time some remarkable global features of these configurations. They show that the minimum energy accurately follows a simple half-integral power law in 1/N over the full range they have investigated. This power law is explicable in terms of the idealization of mapping a planar Wigner lattice onto the surface of the unit sphere; the pair distribution functions of the larger-N configurations indicate predominant hexagonal coordination. The coefficients of the observed power law are closely straddled by values calculated on the basis of hexagonal and square Wigner lattices. This highly accurate description of the energy permits them to remark on the detailed deviations of the individual structures from the general trend. For N(30, they note that structures with N prime are relatively less stable, while structures with N equal to 6, 12, 32, 44, 48 and 60 seem more stable.