Geometrical frustration: A study of four-dimensional hard spheres

The smallest maximum-kissing-number Voronoi polyhedron of three-dimensional (3D) Euclidean spheres is the icosahedron, and the tetrahedron is the smallest volume that can show up in Delaunay tessellation. No periodic lattice is consistent with either, and hence these dense packings are geometrically frustrated. Because icosahedra can be assembled from almost perfect tetrahedra, the terms “icosahedral” and “polytetrahedral” packing are often used interchangeably, which leaves the true origin of geometric frustration unclear. Here we report a computational study of freezing of 4D Euclidean hard spheres, where the densest Voronoi cluster is compatible with the symmetry of the densest crystal, while polytetrahedral order is not. We observe that, under otherwise comparable conditions, crystal nucleation in four dimensions is less facile than in three dimensions, which is consistent with earlier observations [ M. Skoge et al. Phys. Rev. E 74 041127 (2006)]. We conclude that it is the geometrical frustration of polytetrahedral structures that inhibits crystallization.