by points at the corners of the polyhedra. In spite of this obvious misrepresentation, the space‐filling polyhedron approach shows the topology or connectivity of a framework structure and indicates clearly the location of empty interstitial sites.
A complete scheme for classifying polyhedral structures has been developed by Wells and others. The initial problem is a geometric one: what types of network built of linked polyhedra are possible? The variables are as follows. Polyhedra may be tetrahedra, octahedra, trigonal prisms, etc. Polyhedra may share some or all of their corners, edges and faces with adjacent polyhedra, which may or may not be of the same type. Corners and edges may be common to two or more polyhedra (obviously only two polyhedra can share a common face). An enormous number of structures are feasible, at least theoretically, and it is an interesting exercise to categorise real structures on this basis.
The topological approach to arranging polyhedra in various ways takes no account of the bonding forces between atoms or ions. Such information must come from elsewhere. Also, the description of structures in terms of polyhedra does not necessarily imply that such entities exist as separate species. Thus, in NaCl, the bonding is mainly ionic, and physically distinct NaCl6 octahedra do not occur. Similarly, SiC has a covalent network structure and separate SiC4 tetrahedral entities do not exist. Polyhedra do have a separate existence in structures of (a) molecular materials, e.g. Al2Br6 contains pairs of edge‐sharing tetrahedra, and (b) compounds that contain complex ions, e.g. silicate structures built of SiO4 tetrahedra which form complex anions ranging in size from isolated monomers to infinite chains, sheets and 3D frameworks.
In considering the polyhedral linkages that are likely to occur, Pauling's third rule for the structures of complex ionic crystals (see Chapter 3 for other rules) provides a useful guideline. This states that shared edges and, especially, shared faces decrease the stability of a structure, particularly for cations of high valence and small coordination number, i.e. for small polyhedra, especially tetrahedra, that contain highly charged cations. When polyhedra share edges and, particularly, faces, the cation–cation distances (i.e. the distances between the centres of the polyhedra) decrease and the cations repel each other electrostatically. In Fig. 1.28 are shown pairs of (a) corner‐sharing and (b) edge‐sharing octahedra. The cation–cation distance is clearly less in the latter and for face‐sharing octahedra, not shown, the cations are even closer. Comparing edge‐shared octahedra and edge‐shared tetrahedra (b, c), the cation–cation distance M–M (relative to the cation–anion distance M–X) is less between the tetrahedra because the MXM bridging angle is 71° for them and 90° for the octahedra. A similar effect occurs for face‐sharing tetrahedra and octahedra. In Table 1.6, the M–M distances for various polyhedral linkages are given as a function of the M–X distance. The M–M distance is greatest for corner‐sharing tetrahedra or octahedra and is least for face‐sharing tetrahedra. For the sharing of corners and edges, the M–M distances given are the maximum possible. Reduced distances occur if the polyhedra are rotated about the shared corner or edge (e.g. so that the M–X–M angle between corner‐shared polyhedra is less than 180°).
Figure 1.28 Cation–cation separation in octahedra which share (a) corners and (b) edges and in (c) tetrahedra which share edges.
From Table 1.6, the M–M distance in face‐shared tetrahedra is considerably less than the M–X bond distance. This therefore represents an unstable situation, because of the large cation–cation repulsions which would arise, and does not normally occur. A good example of the non‐occurrence of face‐sharing tetrahedra is provided by the absence of any crystal whose structure is the hcp equivalent of the fluorite (or antifluorite) structure. In Na2O, which has the antifluorite structure, the oxide ions are in a ccp array and the NaO4 tetrahedra share edges. However, in a structure with an hcp anion array and all the tetrahedral cation sites occupied, the MX4 tetrahedra would share faces. Edge sharing of tetrahedra, in which the M–M separation is only 16% larger than the M–X bond distance (Table 1.6) is energetically acceptable, as shown by the common occurrence of compounds with a fluorite structure, but face sharing of tetrahedra is generally unacceptable.
Table 1.6 The M–M distance between MX4 or MX6 groups sharing X atom(s)
| Corner sharing | Edge sharing | Face sharing | |
|---|---|---|---|
| Two tetrahedra | 2.00 MX(tet.)a | 1.16 MX(tet.)a | 0.67 MX(tet.) |
| Two octahedra | 2.00 MX(oct.)a | 1.41 MX(oct.)a | 1.16 MX(oct.) |
a Maximum possible value.
For tetrahedra containing cations of high charge, edge sharing may be energetically unacceptable and only corner sharing occurs. For example, in silicate structures which are built of SiO4 tetrahedra, edge sharing of SiO4 tetrahedra never occurs. (Note: In an ideally ionic structure, the charge on Si would be 4+, but the actual charges are considerably less due to partial covalency of the Si–O bonds.) In SiS2, however, edge sharing of SiS4 tetrahedra does occur; the Si–S bond is longer than the Si–O bond and consequently the Si–Si distance in edge‐shared SiS4 tetrahedra is also greater and appears to be within the acceptable range for interatomic separations.
1.17 Some Important Structure Types
1.17.1 Rock salt (NaCl), zinc blende or sphalerite (ZnS), fluorite (CaF2), antifluorite (Na2O)
These structures are considered together because they all have ccp/fcc anions and differ only in the positions of the cations. In Fig. 1.24 are shown the anions in a fcc unit cell with all possible O, T+ and T– sites for the cations. There is no rule as to which sites should be labelled T+ and T–; the choice is yours. The choice of origin is also arbitrary; for present purposes, it is more convenient to place the anions at the origin and also, therefore, at face centre positions. The different structures are generated as follows:
rock salt: O occupied; T+ and T– empty
zinc blende: T+ (or T–) occupied; O, T– (or T+) empty
antifluorite: T+, T– occupied; O empty.
Unit cells are shown in Fig. 1.29, in oblique projection (a–c) and as projections on the ab face of the unit cell in (d–f). Each is described in more detail later.
Figure
