We investigate one-dimensional periodic chains of alternate type of particles interacting through mirror symmetric potentials. The optimality of the equidistant configuration at fixed density—also called crystallization—is shown in various settings. In particular, we prove the crystallization at any scale for neutral and non-neutral systems with inverse power laws interactions, including the three-dimensional Coulomb potential. We also show the minimality of the equidistant configuration at high density for systems involving inverse power laws and repulsion at the origin. Furthermore, we derive a necessary condition for crystallization at high density based on the positivity of the Fourier transform of the interaction potentials sum.

中文翻译：

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交替粒子链结晶的注意事项

我们研究了通过镜像对称势相互作用的交替类型粒子的一维周期链。固定密度下等距构型的最优性（也称为结晶）显示在各种设置中。特别是，我们证明了具有逆幂律相互作用的中性和非中性系统在任何尺度上的结晶，包括三维库仑势。我们还展示了涉及逆幂律和原点排斥力的系统的高密度等距配置的最小值。此外，我们基于相互作用势和的傅立叶变换的正性推导出高密度结晶的必要条件。