# Finite crystallization and Wulff shape emergence for ionic compounds in the square lattice

## TL;DR

This paper studies two-dimensional ionic crystal formation on a square lattice, proving ground state configurations, Wulff shape emergence, and effects of net charge on crystallization, revealing shape sensitivity and critical charge thresholds.

## Contribution

It introduces a detailed analysis of energy minimizers and shape emergence in ionic systems with net charge constraints on the square lattice, including sharp scaling laws and shape transitions.

## Key findings

- Ground states are connected subsets with alternating atomic types.
- Emergence of a square Wulff shape for large particle numbers.
- Crystallization fails beyond a critical net charge, with a diamond-like shape at the threshold.

## Abstract

We present two-dimensional crystallization results in the square lattice for finite particle systems consisting of two different atomic types. We identify energy minimizers of configurational energies featuring two-body short-ranged particle interactions which favor some reference distance between different atomic types and contain repulsive contributions for atoms of the same type. We first prove that ground states are connected subsets of the square lattice with alternating arrangement of the two atomic types in the crystal lattice, and address the emergence of a square macroscopic Wulff shape for an increasing number of particles. We then analyze the signed difference of the number of the two atomic types, the so-called net charge, for which we prove the sharp scaling ${\rm O}(n^{1/4})$ in terms of the particle number $n$. Afterwards, we investigate the model under prescribed net charge. We provide a characterization for the minimal energy and identify a critical net charge beyond which crystallization in the square lattice fails. Finally, for this specific net charge we prove a crystallization result and identify a diamond-like Wulff-shape of energy minimizers which illustrates the sensitivity of the macroscopic geometry on the net charge.