离子化合物在正方晶格中的有限结晶和Wulff形涌现
Finite crystallization and Wulff shape emergence for ionic compounds in the square lattice
摘要：
我们提出了两个不同的原子类型组成的有限粒子系统的正方形晶格中的二维结晶的结果。我们确定了能量极小的构型能量具有两体短程粒子相互作用，有利于不同原子类型之间的一些参考距离，并包含相同类型的原子排斥的贡献。我们首先证明，基态是连接的子集的正方形晶格交替排列的两种原子类型的晶格，并解决出现一个正方形的宏观武尔夫形状的粒子越来越多。然后，我们分析了两种原子类型的数量，所谓的净电荷，我们证明了急剧缩放${\rm O}(n^{1/4})$的粒子数$n$的有符号的差异。然后，我们研究了给定净电荷下的模型。我们提供了一个表征的最小能量，并确定一个关键的净电荷超过该结晶在正方形晶格失败。最后，对于这个特定的净电荷，我们证明了结晶的结果，并确定一个钻石般的伍尔夫形状的能量最小化，这说明了净电荷的宏观几何形状的敏感性。
英文摘要：
We present two-dimensional crystallization results in the square lattice for finite particle systems consisting of two different atomic types. We identify energy minimizers of configurational energies featuring two-body short-ranged particle interactions which favor some reference distance between different atomic types and contain repulsive contributions for atoms of the same type. We first prove that ground states are connected subsets of the square lattice with alternating arrangement of the two atomic types in the crystal lattice, and address the emergence of a square macroscopic Wulff shape for an increasing number of particles. We then analyze the signed difference of the number of the two atomic types, the so-called net charge, for which we prove the sharp scaling ${\rm O}(n^{1/4})$ in terms of the particle number $n$. Afterwards, we investigate the model under prescribed net charge. We provide a characterization for the minimal energy and identify a critical net charge beyond which crystallization in the square lattice fails. Finally, for this specific net charge we prove a crystallization result and identify a diamond-like Wulff-shape of energy minimizers which illustrates the sensitivity of the macroscopic geometry on the net charge.
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