Modified Thomson problem

Abstract

The modified Thomson problem, which concerns an assembly of N particles mutually interacting through a Coulombic potential and subject to a Coulombic-harmonic confinement, is introduced. For sufficiently strong confinement strengths M, properties of its solutions (such as the energy and the particle positions at the minimum, and the corresponding zero-point vibrational energy) are accurately estimated by expressions dependent on only a few quantities pertaining to the original Thomson problem [such as the energy E_{Th}(N) ] and the reduced confinement strength xi=NM/E_{Th}(N). For N<or=12, this regime of the perturbed Thomson problem persists for all non-negative values of xi. On the other hand, the perturbed spherical Coulomb crystal regime emerges for xi<xi_{crit}(N) and larger numbers of particles. For 13<or=N<or=22, the transition that delineates these two regimes is due to the existence of two energy minima, the crystal-like one becoming global for sufficiently weak confinements. For N>or=23, the transition involves a catastrophe brought about by the vanishing of one of the Hessian matrix eigenvalues, the value of xi_{crit}(N) being related to the magnitude of radial instability in the corresponding solution of the original Thomson problem.