An alternative representation of close-packed structures

In the foregoing treatment of structures emphasis is placed on the coordination of the atoms occupying the interstices in the c.p. assemblies. Except for certain of the tetrahedral structures of compounds M2X and M3X2 of Table 4.5 these are metal atoms, the c.p. assembly being that of the anions. As we point out elsewhere the determining factor in some structures appears to be the environment of the anion rather than that of the cation, and this is emphasized in an alternative representation of c.p. structures. 

Crystal structures may be described in terms of the coordination polyhedra MX of the atoms or in terms of their duals, that is, the polyhedra enclosed by planes drawn perpendicular to the lines M-X joining each atom to each of its neighbours at the mid-points of these lines. Each atom in the structure is then represented as a polyhedron (polyhedral domain), and the whole structure as a space-filling assembly of polyhedra of one or more kinds. We can visualize these domains as the shapes the atoms (ions) would assume if the structure were uniformly compressed. For example, h.c.p. and c.c.p. spheres would become the polyhedra shown in Fig. 4.29. These polyhedra are the duals of the coordination polyhedra illustrated in Fig. 4.5. These domains provide an alternative way of representing relatively simple c.p. structures (particularly of binary compounds) because the vertices of the domain are the positions of the interstices. The (8) vertices at which three edges meet are the tetrahedral interstices, and those (6) at which four edges meet are the octahedral interstices. Table 4.9 shows the octahedral positions occupied in some simple structures c.p. structures in which tetrahedral or tetrahedral and octahedral sites are occupied may be represented in a similar way. (For examples see JSSC 1970 1 279.) 

The polyhedral domains for (a) hexagonal and (b) cubic closest packing, showing the six positions of octahedral coordination around a c.p. sphere. The remaining vertices of the domains are tetrahedral sites. 

Two close-packed AX3 layers (full and dotted circles) showing the positions, midway between the layers, for metal atoms (small black circles) within octahedra of X atoms. 

Structures built from close-packed AX 3 layers 

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