Diverse densest ternary sphere packings

Abstract

The exploration of the densest structures of multi-sized hard spheres under periodic boundary conditions is a fundamental problem in mathematics and a wide variety of sciences including materials science. We present our exhaustive computational exploration of the densest ternary sphere packings (DTSPs) for 451 radius ratios and 436 compositions on top of our previous study [Koshoji and Ozaki, Phys. Rev. E 104, 024101 (2021)]. The unbiased exploration by a random structure searching method discovers diverse 22 putative DTSPs, and thereby 60 putative DTSPs are identified in total including the 38 DTSPs discussed by the previous study. We classify the 60 DTSPs into seven groups based on how the structural framework is comprised of small, medium, and large spheres. Since the radius ratio of small spheres is not so small relatively, the discovered DTSPs are well-ordered; for example, the (9-7-3) structure is comprised of the cubic unit cell constituted by medium spheres, and the DTSP has the P m 3m symmetry if the structural distortion is corrected. The correspondence of DTSPs with real crystals is found based on the space group, and similarities of structural features in DTSPs are discussed for crystals synthesized experimentally or predicted computationally under high pressure. Our study suggests that the diverse structures of DTSPs can be effectively used as structural prototypes for searching complex crystal structures.