Anion-centred polyhedra

The stracture of BaS04 itself, and our interpretation of it, is shown in Fig. 3. The Ba -f- S array is that of a well-known alloy stracture, FeB(B 15), shown in Fig. 4. Thus, in terms of the complex ion SO4 , the stracture may be described as FeB type with SO4 in place of B and Ba in place of Fe (as in the centre of Fig. 3) or, better, in terms of anion-centred polyhedra it is simply a BaS array of B15 type with the oxygens inserted into SBa3 tetrahedra. These anion-centred tetrahedra are irregular - as is to be expected, because the sizes of Ba and S are different. 

Figure 7.18 The anion-centred polyhedron (rhombic dodecahedron) found in the cubic closest-packed structure (a) oriented with respect to cubic axes, the c-axis is vertical (b) oriented with [111] vertical (c) cation positions occupied in the halite, NaCl, structure (d) cation positions occupied in the zinc blende (sphalerite), ZnS, structure (e, f) the two anion-centred polyhedra needed to create the spinel, MgAl204, structure. Cations in tetrahedral sites are small and cations in octahedral sites are medium-sized. Adapted from E. W. Gorter, Int. Cong, for Pure and Applied Chemistry, Munich, 1959, Butterworths, London, 1960, p 303...

Just as the edges of cation-centred polyhedra represent anion diffusion paths in a crystal, the edges of anion centred polyhedra represent cation diffusion paths. The polyhedra shown in Figures 7.18 reveal that cation diffusion in cubic close-packed structures will take place via alternative octahedral and tetrahedral sites. Direct pathways, across the faces of the polyhedron, are unlikely, as these mean that a cation would have to squeeze directly between two anions. There is no preferred direction of diffusion. For ions that avoid either octahedral or tetrahedral sites, for bonding or size reasons, diffusion will be slow compared to ions which are able to occupy either site. In solids in which only a fraction of the available metal atom sites are filled, such as the spinel structure, clear and obstructed diffusion pathways can easily be delineated. 

The use of anion-centred polyhedra can be particularly useful in describing diffusion in fast ion conductors. These materials, which are solids that have an ionic conductivity approaching that of liquids, find use in batteries and sensors. An example is the high temperature form of silver iodide, a-Agl. In this material, the iodide anions form a body-centred cubic array, (Figure 7.20a). 

In a solid represented by packing of anion-centred polyhedra, the polyhedron edges represent possible ... 

The analogous anion-centred polyhedron for an hexagonal close-packed anion array, (Figure 7.19a), is similar to a rhombic dodecahedron, but has a mirror plane normal to the vertical axis, which is perpendicular to the close packed planes of anions and so forms the hexagonal c-axis. The... 

Figure 7.20 Perspective view of the structure of a-Agl (a) the body-centred cubic arrangement of iodide ions in the unit cell (b) the anion centred polyhedron, (a truncated octahedron), around the iodide ion at the unit cell centre. [Note that the ions in part (a) are depicted smaller than ionic radii suggest, for clarity]...

Thus, the difference between these structures and the apatites is simply in the choice of tetrahedral interstices in the alloy which are used to accommodate the anions. This determines the type of cation-centred polyhedron for the minority cation. (We ignore the trivial variations in anions inserted into the [OOz] tunnels.) This suggests that the substitution of C03 for P04 in apatites may well be a less radical change than is usually assumed (when anion packing is accepted as being dominant). 

Although the polyhedral representations described are generally used to depict structural relationships, they can also be used to depict diffusion paths in a compact way. The edges of a cation centred polyhedron represent the paths that a diffusing anion can take in a structure, provided that anion diffusion takes place from one normal anion site to another. Thus anion diffusion in crystals with the fluorite structure will be localised along the cube edges of Figure 7.13. 

The positional parameters of the heavy atoms provided by DM are submitted to automatic Rietveld refinement to improve their accuracy. The distances between the heavy atoms are analyzed to derive (or confirm) the cation connectivity (tetrahedral or octahedral). Let us suppose that two cations, say Cl and C2 (see Figure 8.10), have been located. The bridge anion Al, bonding Cl to C2, is expected to lie on the circle intersection of the two coordination spheres, centred in Cl and C2. A random point on the circle is chosen as a trial location of Al it is a feasible atomic position. The positions of the other anions A2, A3, A4 may be (randomly) fixed by a random rotation of the Cl polyhedron about the... 

Figure 11.15 The variation of potential energy with cation displacement, r, from the centre of a surrounding anion polyhedron...

In the complex rhodium carbonyl anion Rh,o(CO)22(M8-P) the P atom, (formally P ) lies at the centre of a bicapped square antiprism of metal atoms and the CO groups are attached to the comers and edges of this polyhedron [12] (8.244a). The P atom is believed to stabilise the complex and the Rh-P distances indicate it is covalently bound to eight of the metal atoms. The related anion Rh9(CO)2i([t8-P) has a similar structure with an apex Rh atom and a CO group missing. 

Fig. 14 Examples of polyhedral main group molecules. The hydrocarbon series shown at the top retains a connection between the number of C-C two-centre two-electron bonds and the edges of the polyhedron. This connection is lost for the deltahedral borane anions shown in the bottom line...

Because the cuboctahedral clusters are the most widespread and the largest scale ones, we will therefore consider them in detail. All eight normal anion positions (Fp) in the fluorite unit cell containing such a cluster are vacant. Twelve atoms F form the cubocta-hedron (8 12 0). The position in the centre of the unit cell could be occupied by one additional fluorine atom F" (8 12 1). The anion cluster is surrounded by an octahedron of R " " cations located in face centres (Figure 14.12). The coordination polyhedron of R " " cations is a square antiprism (coordination number equal to 8). 

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