Section 9.2.2.2.1. Close packing of spheres

Polytypism of structures based on close packing of equal spheres (note this idealization) is explained by the fact that the spheres of any layer can be placed either in all the voids of the preceding layer, or in all the voids – not in both because of steric hindrance (Section 9.2.1, Fig. 9.2.1.1).

A closer look reveals that the two voids are geometrically (but not translationally) equivalent. This implies that the two possible pairs of adjacent layers, say AB and AC, are geometrically equivalent too – this equivalence is brought about e.g. by a reflection in any plane perpendicular to the layers and passing through the centres of mutually contacting spheres A: such a reflection transforms the layer A into itself, and B into C, and vice versa. Another important point is that the symmetry proper of any layer is described by the layer group P(6/m)mm,² and that the relative position of any two adjacent layers is such that only some of the 24 symmetry operations of that layer group remain valid for the pair. It is easy to see that 12 out of the total of 24 transformations do not change the z coordinate of any starting point, and that these operations constitute a subgroup of the index [2]. These are the so-called τ operations. The remaining 12 operations change any z into −z, thus turning the layer upside down; they constitute a coset. The latter are called ρ operations. Out of the 12 τ operations, only 6 are valid for the layer pair. One says that only these 6 operations have a continuation in the adjacent layer. Let us denote the general multiplicity of the group of τ operations of a single layer by N, and that of the subgroup of these operations with a continuation in the adjacent layer by F: then the number Z of positions of the adjacent layer leading to geometrically equivalent layer pairs is given by Z = N/F (Dornberger-Schiff, 1964, pp. 32 ff.); in our case, Z = 12/6 = 2 (Fig. 9.2.2.1 ). This is the so-called NFZ relation, valid with only minor alterations for all categories of OD structures (9.2.2.2.7). It follows that all conceivable structures based on close packing of equal spheres are built on the same symmetry principle: they consist of equivalent layers (i.e. layers of the same kind) and of equivalent layer pairs, and, in keeping with these stipulations, any layer can be stacked onto its predecessor in two ways. Keeping in mind that the layer pairs that are geometrically equivalent are also energetically equivalent, and neglecting in the first approximation the interactions between a given layer and the next-but-one layer, we infer that all structures built according to these principles are also energetically equivalent and thus equally likely to appear.

Figure 9.2.2.1| top | pdf | Symmetry interpretation of close packings of equal spheres. The layer group of a single layer, the subgroup of its τ operations, and the number of asymmetric units N per unit mesh of the former, are given at the top right. The τ operations that have a continuation for the pair of adjacent layers, the layer group of the pair, and the value of F are indicated at the bottom right.

It is important to realize that the above symmetry considerations hold not only for close packing of spheres but also for any conceivable structure consisting of two-dimensionally periodic layers with symmetry P(6/m)mm and containing pairs of adjacent layers with symmetry P(3)m1. Moreover, the OD theory sets a quantitative stipulation for the relation between any two adjacent layers: they have to remain geometrically equivalent in any polytype belonging to a family. This is far more exact than the description: `the stacking of layers is such that it preserves the nearest-neighbour relationships'.