Bétermin Laurent
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Ann Henri Poincare. 2021;22(9):2995-3023. doi: 10.1007/s00023-021-01045-0. Epub 2021 Mar 27.
We consider interaction energies between a point , , and a lattice containing , where the interaction potential is assumed to be radially symmetric and decaying sufficiently fast at infinity. We investigate the conservation of optimality results for when integer sublattices are removed (periodic arrays of vacancies) or substituted (periodic arrays of substitutional defects). We consider separately the non-shifted ( ) and shifted ( ) 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.
我们考虑一个点((x,y,z))与一个包含(N)个点的晶格之间的相互作用能，其中相互作用势(\Phi)假定为径向对称且在无穷远处衰减足够快。我们研究当整数子晶格被移除（周期性空位阵列）或被替换（置换缺陷周期性阵列）时(N)的最优性结果的守恒性。我们分别考虑非平移（(x = 0)）和平移（(x \neq 0)）的情况，并推导了几个一般条件，这些条件确保了在包含缺陷的新能量的晶格中通用优化器的（非）最优性。此外，在反幂律和 Lennard-Jones 型势的情况下，我们给出了关于非平移周期性空位或置换缺陷在固定密度下保持极小性结果的充分必要条件。给出了不同的应用示例，包括 Kagome 晶格的最优性结果以及某些类离子结构的能量比较。
