Calculus of Variations and Geometric Measure Theory

L. Kreutz - M. Friedrich

Crystallization in the hexagonal lattice for ionic dimers

created by kreutz on 28 Feb 2019
modified on 27 Apr 2019


Accepted Paper

Inserted: 28 feb 2019
Last Updated: 27 apr 2019

Journal: Math. Mod. Meth. Appl. Sci. (M3AS)
Year: 2018


We consider finite discrete systems consisting of two different atomic types and investigate ground-state configurations for configurational energies featuring two-body short-ranged particle interactions. The atomic potentials favor some reference distance between different atomic types and include repulsive terms for atoms of the same type, which are typical assumptions in models for ionic dimers. Our goal is to show a two-dimensional crystallization result. More precisely, we give conditions in order to prove that energy minimizers are connected subsets of the hexagonal lattice where the two atomic types are alternately arranged in the crystal lattice. We also provide explicit formulas for the ground-state energy. Finally, we characterize the net charge, i.e., the difference of the number of the two atomic types. Analyzing the deviation of configurations from the hexagonal Wulff shape, we prove that for ground states consisting of n particles the net charge is at most of order n to 1 over 4 where the scaling is sharp.