Crystalline Order on a Sphere and the Generalized Thomson Problem

TL;DR

This paper develops a continuum formalism to analyze crystalline order on spheres, accurately predicting ground state energies and exploring defect structures, including grain boundaries and square tilings, for various long-range interactions.

Contribution

It introduces a universal continuum approach to the generalized Thomson problem, extending understanding of defect arrangements on curved surfaces with diverse interaction laws.

Findings

Predicted ground state energies match simulations within four significant digits.

Analyzed grain boundary regimes in tilted crystalline structures.

Extended the approach to square tilings on spheres.

Abstract

We attack generalized Thomson problems with a continuum formalism which exploits a universal long range interaction between defects depending on the Young modulus of the underlying lattice. Our predictions for the ground state energy agree with simulations of long range power law interactions of the form 1/r^{gamma} (0 < gamma < 2) to four significant digits. The regime of grain boundaries is studied in the context of tilted crystalline order and the generality of our approach is illustrated with new results for square tilings on the sphere.