Geometric units in tetragonal crystal structures

Abstract

Tetragonal space groups are classified from the geometric-unit view point by considering crystal structures as a result of combinations and permutations of some basic polyhedral units. There are nine patterns among two categories represented by four units packed on the (1{\bar 1}0) and (100) planes. Category (I) consists of five types with four units packed on the (1{\bar 1}0) plane. The centers of these units are 0,0,0; 0,0,½; ½,½,0 and ½,½,½. In that order, the patterns can be represented by ABCD, AA'BB', ABA'B', ABB'A' and AA 'A"A". Each letter here represents an independent unit: primes are used to indicate one of the following orientation relationships: identity, fourfold rotation, mirror plane parallel to (110), and mirror plane parallel to (100). These units have the shape of tetragonal prisms and they stack in the same way as the crystallographic unit cells. Category (II) has four types packed on the (100) plane and the centers of these units are at 0,0,0; 0,½,¼; 0,0,½ and 0,½,¾. In that order, the patterns can be represented by ACBD, ABA 'B', AA 'BB' and AA'A"A". The ideal polyhedra for category (II) are truncated tetragonal prisms or flattened truncated octahedra depending on the axial ratio c/a. For simplicity, these polyhedra are transformed into tetragonal prisms so that all geometric units have the same shape. Units in category (II) stack in an interlocking fashion, like the work of a bricklayer. The overlap displacements for the interlocking are in the (001) direction. The symmetries of the geometric units in some space groups depend on the choice of origin, but a shift to equivalent origins changes neither the packing patterns nor the symmetries of the geometric units.