In coordination polyhedra

Robinson, K, Gibbs, G. V. Ribbe, P. H. (1971) Quadratic elongation A quantitative measure of distortion in coordination polyhedra. Science, 172,567-70. 

The absence of a sharp diffraction pattern characterizing a catalyst is usually an indication of nanocrystalline rather than amorphous structures. Such patterns may change as a result of changes in the environment without a transformation of the material into a crystalline phase. In these cases a change occurs in the molecular motif leading to changes in coordination polyhedra (e.g., tetrahedral to octahedral upon oxidation or hydration) without a growth in crystallite size. 

Table II. Metal-Oxygen Distances in Coordination Polyhedra ...

Bookin AS, Smoliar BB (1985) Simulation of bond lengths in coordination polyhedra of 2 1 layer silicates. 5th Meet Eur Clay Groups, Prague, 1983, p 51-56... 

In a cube the fig parameter is negative and the 5g is positive (in the octahedron both parameters are positive). It is possible to discriminate between an octahedron (CN=6) and a cube (CN=8) in a cubic symmetry on the basis of the sign of the parameters. B will be in general larger than B. This may give the impression that the fourth-rank parameters will dominate in coordination polyhedra with CN = 8. 

It is furthermore to be anticipated that the cation-cation repulsion will operate in some cases to displace the cations from the centers of their coordinated polyhedra. This action will be large only in case the radius ratio approaches the lower limit for stability, so that the size of the polyhedron is partially determined by the characteristic anion-anion repulsive 21 Linus Pauling, Z. Krist., 67, 377 (1928). 

The Fe—O distances in hematite are 1.99 and 2.06 A. The (Mn,Fe)—O distances in bixbyite are expected to be the same in case that (Mn, Fe) has the coordination number 6, and slightly smaller, perhaps 1.90 A, for coordination number 4. The radius of 0= is 1.40 A, and the average O—O distance in oxide crystals has about twice this value. When coordinated polyhedra share edges the O—O distance is decreased to a minimum value of 2.50 A, shown by shared edges in rutile, anatase, brookite, corundum, hydrargillite, mica, chlorite, and other crystals. Our experience with complex ionic crystals leads us to believe that we may... 

Fig. 3. The four different coordination polyhedra in the Mg32(Al, Zn)49 structure.

The coordination polyhedron results when the centers of mutually adjacent coordinated atoms are connected with one another. For every coordination number typical coordination polyhedra exist (Fig. 2.2). In some cases, several coordination polyhedra for a given coordination number differ only slightly, even though this may not be obvious at first glance by minor displacements of atoms one polyhedron may be converted into another. For example, a trigonal bipyramid can be converted into a tetragonal pyramid by displacements of four of the coordinated atoms (Fig. 8.2, p. 71). 

Important structural principles for ionic crystals, which had already been recognized in part by V. Goldschmidt, were summarized by L. Pauling in the following rules. First rule Coordination polyhedra... 

The composition of a compound is intimately related to the way of linking the polyhedra. An atom X with coordination number c.n.(X) that acts as a common vertex to this number of polyhedra makes a contribution of l/c.n.(X) to every polyhedron. If a polyhedron has n such atoms, this amounts to n/c.n.(X) for this polyhedron. This can be expressed with Niggli formulae, as shown in the following sections. To specify the coordination polyhedra, the formalism presented at the end of Section 2.1 and in Fig. 2.2 (p. 5) is useful. 

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