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

All types of homogeneous sphere packing and interpenetrating sphere packings and layers were derived that correspond to point configurations of the 15 trivariant hexagonal lattice complexes. The respective sphere packings are assigned to 147 types. In total, sphere packings of 170 types can be realized with hexagonal symmetry. 103 types of sphere packing refer exclusively to trivariant hexagonal lattice complexes. For 23 of these types, the corresponding sphere packings can be generated only in hexagonal lattice complexes with less than three degrees of freedom or with trigonal or lower symmetry. In addition, seven types of interpenetrating sphere packings and two types of interpenetrating sphere layers were found. Interpenetrating 4.8(2) nets of spheres with 120 degrees angles between the nets were assumed to be not possible, so far. The sphere packings belonging to 85 of the 170 hexagonal types can be split up into parallel layers of spheres with mutual contact and can be characterized by symbols derived from those for the Shubnikov nets. The sphere packings of 135 hexagonal types may be subdivided into rod-like subsets of spheres with mutual contact. Such rods may be described by rolling up a plane net. Only 23 types of sphere packing cannot be symbolized on the basis of layers or rods of spheres with mutual contact. Examples are given for crystal structures that can be described by means of sphere packings.