Space fullerenes: computer search for new Frank–Kasper structures II

Abstract

A Frank–Kasper structure is a 3-periodic tiling of the Euclidean space E 3 by tetrahedra such that the vertex figure of any vertex belongs to four specified fullerenes with, respectively, 12, 14, 15, and 16 faces. Frank–Kasper structures occur in the crystallography of metallic alloys, clathrates, zeolites, and in geometrical optimization. 27 such physical structures are known. In Dutour et ai. (Acta Crystallogr A 66:637–639, 2010) we obtained, by computer enumeration, all 84 such structures with up to 20 cells in a reduced fundamental domain; 13 among them were known physical structures. In the present follow-up study, we managed, by improving the computation, to get all 37 new structures with 21 cells in a reduced fundamental domain. Those structures are described, using six invariants: group, the size of fundamental domain, mean coordination number $$ \bar{f} $$ , fraction sequence, cell orbits and major skeleton. We found pairs of distinct structures having all six invariants equal. So, we devised a new invariant, zigzag vector, and computed it for all 135 structures known from now; all have this invariant different. New bounds for $$ \bar{f} $$ and new directions (computational perspectives, number of Kekule structures, space octahedrites, space cubites) are also discussed.