On the optimality of the rock-salt structure among lattices with charge distributions
On the optimality of the rock-salt structure among lattices with charge distributions
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DOI：
10.1142/s021820252150007x
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发表时间：
2021-02-01
影响因子：
3.5
通讯作者：
Knuepfer, Hans
中科院分区：
文献类型：
--
作者：
Betermin, Laurent;Faulhuber, Markus;Knuepfer, Hans
本文的目标是研究d维岩盐结构的最优性，即体积V的立方晶格V1/dDOUBLE-STRUCK CAPITAL Zd，在晶格点上的电荷交替+/- 1，在周期性分布中电荷和晶格结构。我们假设电荷通过两种径向对称的相互作用势相互作用，根据它们的符号。我们首先限制我们的研究类的正交格。我们证明，对于我们的能量模型，d维岩盐结构总是一个临界点之间的周期性结构的固定密度。这适用于一大类势。然后，我们研究了逆幂律和高斯相互作用势的电荷交替的正交晶格之间的最小化问题。对于这两种类型的势，我们得到了高密度极小性结果和低密度非最优性结果。在数值上，我们研究了2维、3维和8维的几种特殊情况。数值结果支持了岩盐结构是所有晶格和周期电荷中的全局最优结构的猜想，并满足一些自然约束。对于d = 2，我们观察到一个相变的类型“三角形-菱形-正方形-矩形”的最小值的形状作为密度降低。
The goal of this paper is to investigate the optimality of the d-dimensional rock-salt structure, i.e. the cubic lattice V1/dDOUBLE-STRUCK CAPITAL Zd of volume V with an alternation of charges +/- 1 at lattice points, among periodic distributions of charges and lattice structures. We assume that the charges are interacting through two types of radially symmetric interaction potentials, according to their signs. We first restrict our study to the class of orthorhombic lattices. We prove that, for our energy model, the d-dimensional rock-salt structure is always a critical point among periodic structures of fixed density. This holds for a large class of potentials. We then investigate the minimization problem among orthorhombic lattices with an alternation of charges for inverse power laws and Gaussian interaction potentials. High density minimality results and low-density non-optimality results are derived for both types of potentials.Numerically, we investigate several particular cases in dimensions 2, 3 and 8. The numerics support the conjecture that the rock-salt structure is the global optimum among all lattices and periodic charges, satisfying some natural constraints. For d = 2, we observe a phase transition of the type "triangular-rhombic-square-rectangular" for the minimizer's shape as the density decreases.