Hexagonal and trigonal sphere packings. IV. Trivariant lattice complexes of trigonal space groups

Abstract

The 13 trivariant lattice complexes with trigonal symmetry are compatible with 218 types of homogeneous sphere packings, 7 types of interpenetrating sphere packings and one type of interpenetrating layers of spheres. Altogether, the lattice complexes with trigonal characteristic space group (with 0, 1, 2 or 3 degrees of freedom) give rise to 225 types of sphere packing. 110 of these types are compatible exclusively with one of the 13 trivariant lattice complexes, 31 in addition with some of the invariant, univariant or bivariant lattice complexes, whereas 6 types occur solely in such a lattice complex. 65 types encompass special sphere packings that can also be generated with hexagonal symmetry [Sowa, Koch & Fischer (2003). Acta Cryst. A59, 317-326; Sowa & Koch (2004). Acta Cryst. A60, 158-166; Sowa & Koch (2005). Acta Cryst. A61, 331-342]; cubic inherent symmetry occurs for certain sphere packings [Fischer (2004). Acta Cryst. A60, 246-249] belonging to 13 types. The maximal inherent symmetry is trigonal for 147 of the 225 types. The sphere packings of 61 types can be subdivided into connected layer-like subunits, those of 86 types into connected rod-like subunits. Such subunits may be used to construct some kind of ;descriptive symbols' that reflect certain properties of the sphere packings. Interpenetrating sphere packings with cubic inherent symmetry belong to one of the 7 types. All interpenetrating sphere layers that belong to the only type occurring in the trigonal crystal system show hexagonal inherent symmetry. Some examples depict crystal structures that can be described by means of sphere packings.