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 pilingup 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 Cu2GaSr and ThCr2Si2 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.

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The exploration of the densest spheres packings 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. Some of the discovered DTSPs are well-ordered, for example, the medium spheres in the (9-7-3) structure are placed in a straight line with comprising the unit cell, and the DTSP has the Pm-3m symmetry if the structural distortion is corrected. At a considerable number of radius ratios, the highest packing fractions are achieved by the phase separations consisting of only the FCC and/or the putative densest binary sphere packings (DBSPs) for all compositions. The trend is becoming more evident as the small and medium spheres are getting larger, which suggests either the binary packings are actually the densest packings or that the dense ternary packings have unit cells larger than those in this study. On the other hand, the number of the DTSPs increase as the particle size of small spheres gets small. The structural diversity indicates that many unknown DTSPs may hide in a very narrow range of radius ratio where the size of small spheres is smaller due to the competition with respect to the packing fractions of many structural candidates. Finally, we discuss the correspondence of the DTSPs with real crystals based on the space group. Our study suggests that the diverse structures of DTSPs can be effectively used as structural prototypes for searching multinary crystal structures.

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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.