Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra

The determination of the densest packings of regular tetrahedra (one of the five Platonic solids) is attracting great attention as evidenced by the rapid pace at which packing records are being broken and the fascinating packing structures that have emerged. Here we provide the most general analytical formulation to date to construct dense periodic packings of tetrahedra with four particles per fundamental cell. This analysis results in six-parameter family of dense tetrahedron packings that includes as special cases recently discovered "dimer" packings of tetrahedra, including the densest known packings with density $\phi= 4000/4671 = 0.856347...$. This study strongly suggests that the latter set of packings are the densest among all packings with a four-particle basis. Whether they are the densest packings of tetrahedra among all packings is an open question, but we offer remarks about this issue. Moreover, we describe a procedure that provides estimates of upper bounds on the maximal density of tetrahedron packings, which could aid in assessing the packing efficiency of candidate dense packings.Comment: It contains 25 pages, 5 figures

Modeling Heterogeneous Materials via Two-Point Correlation Functions: II. Algorithmic Details and Applications

In the first part of this series of two papers, we proposed a theoretical formalism that enables one to model and categorize heterogeneous materials (media) via two-point correlation functions S2 and introduced an efficient heterogeneous-medium (re)construction algorithm called the "lattice-point" algorithm. Here we discuss the algorithmic details of the lattice-point procedure and an algorithm modification using surface optimization to further speed up the (re)construction process. The importance of the error tolerance, which indicates to what accuracy the media are (re)constructed, is also emphasized and discussed. We apply the algorithm to generate three-dimensional digitized realizations of a Fontainebleau sandstone and a boron carbide/aluminum composite from the two- dimensional tomographic images of their slices through the materials. To ascertain whether the information contained in S2 is sufficient to capture the salient structural features, we compute the two-point cluster functions of the media, which are superior signatures of the micro-structure because they incorporate the connectedness information. We also study the reconstruction of a binary laser-speckle pattern in two dimensions, in which the algorithm fails to reproduce the pattern accurately. We conclude that in general reconstructions using S2 only work well for heterogeneous materials with single-scale structures. However, two-point information via S2 is not sufficient to accurately model multi-scale media. Moreover, we construct realizations of hypothetical materials with desired structural characteristics obtained by manipulating their two-point correlation functions.Comment: 35 pages, 19 figure

Precise Algorithm to Generate Random Sequential Addition of Hard Hyperspheres at Saturation

Random sequential addition (RSA) time-dependent packing process, in which congruent hard hyperspheres are randomly and sequentially placed into a system without interparticle overlap, is a useful packing model to study disorder in high dimensions. Of particular interest is the infinite-time {\it saturation} limit in which the available space for another sphere tends to zero. However, the associated saturation density has been determined in all previous investigations by extrapolating the density results for near-saturation configurations to the saturation limit, which necessarily introduces numerical uncertainties. We have refined an algorithm devised by us [S. Torquato, O. Uche, and F.~H. Stillinger, Phys. Rev. E {\bf 74}, 061308 (2006)] to generate RSA packings of identical hyperspheres. The improved algorithm produce such packings that are guaranteed to contain no available space using finite computational time with heretofore unattained precision and across the widest range of dimensions ($2 \le d \le 8$). We have also calculated the packing and covering densities, pair correlation function $g_2(r)$ and structure factor $S(k)$ of the saturated RSA configurations. As the space dimension increases, we find that pair correlations markedly diminish, consistent with a recently proposed "decorrelation" principle, and the degree of "hyperuniformity" (suppression of infinite-wavelength density fluctuations) increases. We have also calculated the void exclusion probability in order to compute the so-called quantizer error of the RSA packings, which is related to the second moment of inertia of the average Voronoi cell. Our algorithm is easily generalizable to generate saturated RSA packings of nonspherical particles

Statistical properties of determinantal point processes in high-dimensional Euclidean spaces

The goal of this paper is to quantitatively describe some statistical properties of higher-dimensional determinantal point processes with a primary focus on the nearest-neighbor distribution functions. Toward this end, we express these functions as determinants of $N\times N$ matrices and then extrapolate to $N\to\infty$. This formulation allows for a quick and accurate numerical evaluation of these quantities for point processes in Euclidean spaces of dimension $d$. We also implement an algorithm due to Hough \emph{et. al.} \cite{hough2006dpa} for generating configurations of determinantal point processes in arbitrary Euclidean spaces, and we utilize this algorithm in conjunction with the aforementioned numerical results to characterize the statistical properties of what we call the Fermi-sphere point process for $d = 1$ to 4. This homogeneous, isotropic determinantal point process, discussed also in a companion paper \cite{ToScZa08}, is the high-dimensional generalization of the distribution of eigenvalues on the unit circle of a random matrix from the circular unitary ensemble (CUE). In addition to the nearest-neighbor probability distribution, we are able to calculate Voronoi cells and nearest-neighbor extrema statistics for the Fermi-sphere point process and discuss these as the dimension $d$ is varied. The results in this paper accompany and complement analytical properties of higher-dimensional determinantal point processes developed in \cite{ToScZa08}.Comment: 42 pages, 17 figure