Regular Packings on Periodic Lattices
Abstract
We investigate the problem of packing identical hard objects on regular lattices in d dimensions. Restricting configuration space to parallel alignment of the objects, we study the densest packing at a given aspect ratio X. For rectangles and ellipses on the square lattice as well as for biaxial ellipsoids on a simple cubic lattice, we calculate the maximum packing fraction φ_{d}(X). It is proved to be continuous with an infinite number of singular points X_{ν}^{min},X_{ν}^{max}, ν=0,±1,±2,…. In two dimensions, all maxima have the same height, whereas there is a unique global maximum for the case of ellipsoids. The form of φ_{d}(X) is discussed in the context of geometrical frustration effects, transitions in the contact numbers, and numbertheoretical properties. Implications and generalizations for more general packing problems are outlined.
 Publication:

Physical Review Letters
 Pub Date:
 November 2011
 DOI:
 10.1103/PhysRevLett.107.215503
 arXiv:
 arXiv:1110.4775
 Bibcode:
 2011PhRvL.107u5503R
 Keywords:

 61.50.Ah;
 64.70.kt;
 82.70.Dd;
 Theory of crystal structure crystal symmetry;
 calculations and modeling;
 Molecular crystals;
 Colloids;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics
 EPrint:
 5 pages, 4 figures, accepted for publication in Physical Review Letters