Possible Global Minimum Lattice Configurations for Thomson's Problem of Charges on a Sphere

Abstract

What configuration of $N$ point charges on a conducting sphere minimizes the Coulombic energy? J. J. Thomson posed this question in 1904. For $N\ensuremath{\le}112$, numerical methods have found apparent global minimum-energy configurations; but the number of local minima appears to grow exponentially with $N$, making many such methods impractical. Here we describe a topological/numerical procedure that we believe gives the global energy minimum lattice configuration for $N$ of the form $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}10({m}^{2}{+n}^{2}+mn)+2$ ( $m$, $n$ positive integers). For those $N$ with more than one lattice, we give a rule to choose the minimum one.