Edge octahedra

Referring to Problem 3, what should the ratio 710/711 be if the equilibrium crystal is to be a regular octahedron, that is, to have (10) and (11) edges of equal length ... 

Structures of heteropolytungstate and isopolytungstate compounds have been determined by x-ray diffraction. The anion stmctures are represented by polyhedra that share corners and edges with one another. Each W is at the center of an octahedron, and an O atom is located in each vertex of the octahedron. The central atom is similarly located at the center of an XO tetrahedron or XO octahedron. Each such polyhedron containing the central atom is generally surrounded by octahedra, which share corners, edges, or both with it and with one another. Thus, the correct total number of... 

Complex carbides are very numerous. Many newer compounds of this class have been discovered and their stmctures elucidated (20). The octahedron M C is typical where the metals arrange around a central carbon atom. The octahedra may be coimected via corners, edges, or faces. Trigonal prismatic polyhedra also occur. Defining T as transition metal and M as metal or main group nonmetal, the complex carbides can be classified as (/)... 

The Niccolite Structure. The substances which crystallize with the niccolite structure (B8) are compounds of transition elements with S, Se, Te, As, Sb, Bi, or Sn. The physical properties of the substances indicate that the crystals are not ionic, and this is substantiated by the lack of agreement with the structural rules for ionic crystals. Thus each metal atom is surrounded by an octahedron of non-metal atoms but these octahedra share faces, and the edges of the shared faces are longer than other edges (rather than shorter, as in ionic crystals). Hence we conclude that the bonds are covalent, with probably some metallic character also. 

Fig. 2.—The unit of structure of anatase. The titanium octahedron shares the four edges marked with arrows with adjoining octahedra.

Fig. 1. The structure of rutile. Large circles represent the centers of titanium atoms, small circles those of oxygen atoms. One octahedron with oxygen atoms at its cornex and a titanium atom at its center is shown two of its edges, those indicated by arrows, are shared with adjoining octahedra.

Fig. 2. The structure of anatase. Four edges of the octahedron shown, indicated by arrows, are shared with adjoining octahedra.

The coordination theory, which has been applied successfully2 to brook-ite, the orthorhombic form of titanium dioxide, is based on the assumption that the anions in a crystal are constrained to assume positions about the cations such that they indicate the corners of polyhedra of which the cations mark the centers. These polyhedra are the fundamentally important constituents of the crystal retaining their form essentially, they are combined by sharing corners, edges, and faces in such a way as to build up a crystal with the correct stoichiometrical composition. Thus in rutile, anatase, and brookite there occur octahedra of oxygen ions about titanium ions in rutile each octahedron shares two edges with adjoining octahedra, in anatase four, and in brookite three. 

Fig. 6. A photograph of a model representing one half of the unit cube. The arrangement of the six 24 e octahedra sharing edges with an 8e octahedron is clearly shown.

Interestingly, when the particle size of metal nanoparticles becomes less than 2 nm, terraces become so small that they carmot anymore support the presence of step-edge site metal atom configurations. This can be observed from Figure 1.15, which shows a cubo-octahedron just large enough to support a step-edge site. 

Figure 1.15 Cubo-octahedron with step-edge sites [18].

Fig. 6.1b) in which twelve inner ligands bridge the edges of the Me octahedron, and six outer ligands occupy apical positions, predominate. These units are found in reduced zirconium, niobium, tantalum, and rare-earth halides, and niobium, tantalum, molybdenum and tungsten oxides [la, 6, 10]. 

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