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.
Source:IOPscience
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