Crystalline Order on a Sphere and the Generalized Thomson Problem

We attack the generalized Thomson problem, i.e., determining the ground state energy and configuration of many particles interacting via an arbitrary repulsive pairwise potential on a sphere via a continuum mapping onto a universal long range interaction between angular disclination defects parametrized by the elastic (Young) modulus Y of the underlying lattice and the core energy E_core of an isolated disclination. Predictions from the continuum theory for the ground state energy agree with numerical simulations of long range power law interactions of the form 1/r^γ (0<γ<2) to four significant figures. The generality of our approach is illustrated by a study of grain boundary proliferation for tilted crystalline order and square lattices on the sphere.