Section 9.2.1.3. Symmetry of close-packed layer stackings of equal spheres

It can be seen from Fig. 9.2.1.2(a) that a stacking of two or more layers in the close-packed manner still possesses all three symmetry planes but the twofold axes disappear while the sixfold axes coincide with the threefold axes (Verma & Krishna, 1966). The lowest symmetry of a completely arbitrary periodic stacking sequence of close-packed layers is shown in Fig. 9.2.1.2(b). Structures resulting from such stackings therefore belong to the trigonal system. Even though a pure sixfold axis of rotation is not possible, close-packed structures belonging to the hexagonal system can result by virtue of at least one of the three symmetry axes parallel to [00.1] being a 6₃ axis (Verma & Krishna, 1966). This is possible if the layers in the unit cell are stacked in special ways. For example, a 6H stacking sequence has a 6₃ axis through 0, 0. It follows that, for an nH structure belonging to the hexagonal system, n must be even. A packing with n odd will therefore necessarily belong to the trigonal system and can have either a hexagonal or a rhombohedral lattice (Verma & Krishna, 1966).

Other symmetries that can arise by restricting the arbitrariness of the stacking sequence in the identity period are: (i) a centre of symmetry at the centre of either the spheres or the octahedral voids; and (ii) a mirror plane perpendicular to [00.1]. Since there must be two centres of symmetry in the unit cell, the centrosymmetric arrangements may possess both centres either at sphere centres/octahedral void centres or one centre each at the centres of spheres and octahedral voids (Patterson & Kasper, 1959).