Cation-centred polyhedron

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. 

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 Laporte selection rule is weakened, or relaxed, by three factors first, by the absence of a centre of symmetry in the coordination polyhedron second, by mixing of d and p orbitals which possess opposite parities and third, by the interaction of electronic 3d orbital states with odd-parity vibrational modes. If the coordination environment about the cation lacks a centre of symmetry, which is the case when a cation occupies a tetrahedral site, some mixing of d... 

In some cases, the CFSE attained by a transition metal ion in a regular octahedral site may be enhanced if the coordination polyhedron is distorted. This effect is potentially very important in most silicate minerals since their crystal structures typically contain six-coordinated sites that are distorted from octahedral symmetry. Such distortions are partly responsible for the ranges of metal-oxygen distances alluded to earlier, eq. (6.6). Note, however, that the displacement of a cation from the centre of a regular octahedron, such as the comparatively undistorted orthopyroxene Ml coordination polyhedron (fig. 5.16), also causes inequalities of metal-oxygen distances. 

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...

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). 

The coordination number of the cation is 11. The irregular coordination polyhedron can be described as a trigonal prism with additional fluorine atoms at some distance from the centre of each face above its plane. In case of Lao.5Ceo.5F3, La,Ce atoms have the following coordination environment two FI atoms at 2.353 A, three F2 fluorine atoms at 2.395 A and six more distant FI atoms at 2,736 A. All these fluorine atoms form an irregular polyhedron. 

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