Effect of periodic arrays of defects on lattice energy minimizers

Abstract

We consider interaction energies $E_f[L]$ between a point $O\in {\mathbb{R}}^d$, $d\ge 2$, and a lattice $L$ containing $O$, where the interaction potential $f$ is assumed to be radially symmetric and decaying sufficiently fast at infinity. We investigate the conservation of optimality results for $E_f$ when integer sublattices $kL$ are removed (periodic arrays of vacancies) or substituted (periodic arrays of substitutional defects). We consider separately the non-shifted ($O\in kL$) and shifted ($O\notin kL$) cases and we derive several general conditions ensuring the (non-)optimality of a universal optimizer among lattices for the new energy including defects. Furthermore, in the case of inverse power laws and Lennard-Jones type potentials, we give necessary and sufficient conditions on non-shifted periodic vacancies or substitutional defects for the conservation of minimality results at fixed density. Different examples of applications are presented, including optimality results for the Kagome lattice and energy comparisons of certain ionic-like structures.

arXiv comment21 pages. 6 figures

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Cited by in corpus (3)

• On energy ground states among crystal lattice structures with prescribed bonds
• Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies
• Lattice ground states for Embedded-Atom Models in 2D and 3D