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Toward the jamming threshold of sphere packings: Tunneled crystals
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22.As the name implies, the RCP state is traditionally thought of as the highest density that a large random packing of spheres can attain. Besides the important fact that randomness was never quantified, it was shown that one can create packings that are progressively denser by decreasing the degree of order, and hence the concept of the RCP state is ill defined. See Refs. 3 and 15 for further details.
24.Reference 23 focuses on a single jamming point called the point. However, it has been shown that there are in fact an uncountably number of different jammed packings with varying degrees of order (Refs. 3, 15, and 16), as indicated in Fig. 1. In the three dimensions, the point, which is determined from most probable configurations, is similar to the MRJ point for strict jamming. However, in two dimensions, the point is more problematic, as explained in the following paper: A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, Phys. Rev. E 70, 043301 (2004).
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29.Any useful order metric will not assign a value of unity (perfect order) to all periodic packings, but rather assign a value generally less than unity to each according its symmetry, vacancy distribution, etc. This has been shown to be the case in practice (Refs. 15 and 16). Therefore, since we expect that the jamming-threshold structures are periodic, point A in Fig. 1 will likely have a large value of but less than unity.
30.A. Donev, Ph.D. dissertation, Princeton University, Princeton, NJ, 2006.
31.The -dimensional Kagomé packing contains spheres per fundamental cell, i.e., it has a -particle basis. The centroids of the simplices of this structure are the sites of the -dimensional diamond crystal that possesses a two-particle basis and placing the largest nonoverlapping hypersphere at each of these sites produces the densest -dimensional diamond packing. The “two-dimensional diamond” packing is nothing more than the honeycomb packing, which is the basic building block used to create the tunneled three-dimensional crystals that are the focus of this paper. Placing the largest nonoverlapping hypersphere at each of the midpoints of the “bonds” joining the sites of the -dimensional diamond packing yields the densest -dimensional Kagomé packing. Detailed geometrical characteristics of the latter packing for arbitrary will be reported in a future study.
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