High-dimensional generalizations of the kagomé and diamond crystals and the decorrelation principle for periodic sphere packings
High-dimensional generalizations of the kagomé and diamond crystals and the decorrelation principle for periodic sphere packings
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Kagomé 和金刚石晶体的高维推广以及周期性球体堆积的去相关原理
DOI：
10.1088/1742-5468/2011/10/p10017
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发表时间：
2011
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影响因子：
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通讯作者：
Salvatore Torquato
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文献类型：
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作者：
Chase E. Zachary;Salvatore Torquato
在本文中，我们介绍了高维概化的构造，这些高维概化是由金刚石晶体和金刚石晶体组成的。在反铁磁材料的几何挫折的背景下，二维kagom<s:1>晶体及其三维对立物，焦绿石晶体，已经被广泛研究。同样，单质碳的多晶态包括石墨烯所采用的金刚石晶体和相应的二维蜂窝状结构。d欧几里得维的kagom<s:1>晶体由顶点共享的d维简化体组成，其中所有的点在拓扑上是等价的。然后，金刚石晶体的d维泛化可以从每个简单体的质心得到，并且我们证明了金刚石晶体的这种自然结构在所有维度上都与Dd +家族的晶体不同。我们分析了这些高维晶体的结构特性，包括堆积密度、配位数、空隙排除概率函数、覆盖半径和量化误差。我们的结果证明了所谓的去相关原理，即无约束关联在渐近高维中消失的形式，显著地适用于具有固有长程序的周期点模式的情况。我们认为，通过与高斯核卷积得到的“平滑”对相关函数，在低维周期晶体中已经展示了去相关原理。这些观察结果支持了去相关原理对任何高维点模式的普适性，无论是否无序。这一普遍性质反过来表明，Torquato和Stillinger （2006 Expt. Math. 15 307）推导出的高欧几里得维的最大球体堆积密度的最佳推测下界实际上是最优的。
In this paper, we introduce constructions of the high-dimensional generalizations of the kagomé and diamond crystals. The two-dimensional kagomé crystal and its three-dimensional counterpart, the pyrochlore crystal, have been extensively studied in the context of geometric frustration in antiferromagnetic materials. Similarly, the polymorphs of elemental carbon include the diamond crystal and the corresponding two-dimensional honeycomb structure, adopted by graphene. The kagomé crystal in d Euclidean dimensions consists of vertex-sharing d-dimensional simplices in which all of the points are topologically equivalent. The d-dimensional generalization of the diamond crystal can then be obtained from the centroids of each of the simplices, and we show that this natural construction of the diamond crystal is distinct from the Dd + family of crystals for all dimensions . We analyze the structural properties of these high-dimensional crystals, including the packing densities, coordination numbers, void exclusion probability functions, covering radii and quantizer errors. Our results demonstrate that the so-called decorrelation principle, which formally states that unconstrained correlations vanish in asymptotically high dimensions, remarkably applies to the case of periodic point patterns with inherent long-range order. We argue that the decorrelation principle is already exhibited in periodic crystals in low dimensions via a ‘smoothed’ pair correlation function obtained by convolution with a Gaussian kernel. These observations support the universality of the decorrelation principle for any point pattern in high dimensions, whether disordered or not. This universal property in turn suggests that the best conjectural lower bound on the maximal sphere-packing density in high Euclidean dimensions derived by Torquato and Stillinger (2006 Expt. Math. 15 307) is, in fact, optimal.