Densest ternary sphere packings

TL;DR

This study exhaustively explores the densest packings of three different-sized spheres, revealing diverse structures and their relation to real crystals, advancing understanding of complex packing arrangements.

Contribution

It introduces an improved search method for identifying dense ternary sphere packings across various compositions and radii ratios, discovering 37 new structures and linking them to real crystal prototypes.

Findings

Identified 38 putative densest ternary sphere packings (DTSPs).

Discovered structural trends depend on small sphere radii.

Found high-symmetry semi-DTSPs correspond to real crystals.

Abstract

We present our exhaustive exploration of the densest ternary sphere packings (DTSPs) for 45 radius ratios and 237 kinds of compositions, which is a packing problem of three kinds of hard spheres with different radii, under periodic boundary conditions by a random structure searching method. To efficiently explore DTSPs we further develop the searching method based on the piling-up and iterative balance methods [Koshoji et al., Phys. Rev. E 103, 023307 (2021)]. The unbiased exploration identifies diverse 38 putative DTSPs appearing on phase diagrams in which 37 DTSPs of them are discovered in the study. The structural trend of DTSPs changes depending especially on the radius of small spheres. In case that the radius of small spheres is relatively small, structures of many DTSPs can be understood as derivatives of densest binary sphere packings (DBSPs), while characteristic structures specific to the ternary system emerge as the radius of small spheres becomes larger. In addition to DTSPs, we reveal a lot of semi-DTSPs (SDTSPs) which are obtained by excluding DBSPs in the calculation of phase diagram, and investigate the correspondence of DTSPs and SDTSPs with real crystals based on the space group, showing a considerable correspondence of SDTSPs having high symmetries with real crystals including $\mathrm{Cu}_2 \mathrm{GaSr}$ and $\mathrm{ThCr}_2 \mathrm{Si}_2$ structures. Our study suggests that the diverse structures of DBSPs, DTSPs, and SDTSPs can be effectively used as structural prototypes for searching complex crystal structures.