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 (\(O\in k L\)) and shifted (\(O\not \in k L\)) 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.

我们考虑点\(O\in {\mathbb {R}}^d\)、\(d\ge 2\)和包含O的晶格L之间的相互作用能\(E_f[L]\)，其中假设相互作用势f是径向对称的，并且在无穷远处衰减得足够快。我们研究了当整数子晶格kL被移除（空位的周期性阵列）或替换（替代缺陷的周期性阵列）时\(E_f\)的最优性结果的守恒。我们分别考虑非移位（\(O\in k L\)）和移位（\(O\not \in k L\)) 情况，我们推导出几个一般条件，确保新能源的晶格中通用优化器的（非）最优性，包括缺陷。此外，在逆幂律和 Lennard-Jones 型势的情况下，我们给出了非位移周期性空位或替代缺陷的充要条件，以实现固定密度下的极小性守恒。介绍了不同的应用示例，包括 Kagome 晶格的优化结果和某些类离子结构的能量比较。