Geometrical Frustration: A Study of 4d Hard Spheres

TL;DR

This study investigates the role of geometric frustration in 4d hard sphere packings, revealing that polytetrahedral order inhibits crystallization more in 4d than in 3d due to incompatibility with crystal symmetry.

Contribution

It provides the first computational analysis of 4d hard sphere freezing, highlighting how geometric frustration affects crystallization.

Findings

Densest Voronoi cluster in 4d aligns with crystal symmetry.

Polytetrahedral order is incompatible with 4d crystal structures.

Crystallization is less facile in 4d than in 3d due to geometric frustration.

Abstract

The smallest maximum kissing-number Voronoi polyhedron of 3d spheres is the icosahedron and the tetrahedron is the smallest volume that can show up in Delaunay tessalation. 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 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 4d is less facile than in 3d. This suggest that it is the geometrical frustration of polytetrahedral structures that inhibits crystallization.