# On Born's conjecture about optimal distribution of charges for an infinite ionic crystal

## TL;DR

This paper proves Born's conjecture on the optimal charge distribution in ionic crystals, demonstrating the optimality of specific charge arrangements on cubic and triangular lattices for Coulomb-like interactions.

## Contribution

It establishes the optimal charge configurations for ionic crystals on cubic and triangular lattices, confirming Born's conjecture and connecting the problem to lattice theta function minimization.

## Key findings

- Proves the optimality of rock-salt charge distribution on cubic lattices.
- Shows the honeycomb charge distribution is optimal on triangular lattices.
- Connects the charge distribution problem to lattice theta function minimization.

## Abstract

We study the problem for the optimal charge distribution on the sites of a fixed Bravais lattice. In particular, we prove Born's conjecture about the optimality of the rock-salt alternate distribution of charges on a cubic lattice (and more generally on a d-dimensional orthorhombic lattice). Furthermore, we study this problem on the two-dimensional triangular lattice and we prove the optimality of a two-component honeycomb distribution of charges. The results holds for a class of completely monotone interaction potentials which includes Coulomb type interactions. In a more general setting, we derive a connection between the optimal charge problem and a minimization problem for the translated lattice theta function.