Geometric units in hexagonal and rhombohedral space groups

Hexagonal space groups, i.e. those with an hP lattice, are classified from the geometric-unit viewpoint by considering hexagonal crystal structures as combinations and permutations of some basic hexagonal prisms. Geometric units are the Dirichlet domains of the Wyckoff positions with the highest point-group symmetry in the space group. In this classification, there are six types of hexagonal space groups. Type h1 consists of two independent geometric units of the same symmetry per crystallographic cell; in type h2, the two units are identical, but differently oriented. Type h3 has six independent geometric units, again of the same point-group symmetry, but the six units can be made up of three pairs, each consisting of two identical units, thus giving rise to type h4. There are subclasses in types hl and h3. Centers of geometric units in hl(a) and h3(a) are uniquely defined by intersections of point-group symmetry elements, whereas those in h1(b) and h3(b) are not because the space groups in these subtypes are hemimorphic. Therefore, the two units along the polar axis may be combined as one. Type h5 consists of three units, each turned 120° from its neighbors owing to the screw axis 31, 32, 62 and 64. Similarly, type h6 has six units due to screw axes 61 and 65, and adjacent units are 60° apart. Rhombohedral space groups show two types of patterns: type r1 has two independent, and type r2 two identical, units. The h.c.p. and related structures are used to demonstrate the application of geometric units to crystal-structure descriptions.