Octahedra sharing only vertices

We noted earlier that the number of regular octahedra that can share a common vertex without sharing edges or faces is limited to two, assuming that the distance between any pair of non-bonded X atoms of different AXg groups is not less than the edge-length, X-X, taken to be the minimum van der Waals distance. If each vertex that is shared is common to two octahedra only, there is a simple relation between the formula of the structure and the number of shared vertices (X atoms)  

The finite A2X11 group, in which one vertex is shared, has been illustrated in Fig. 5.8(c). Examples are not known of structures in which all octahedra share three vertices for a layer of this kind see Fig. 4.35, p. 154. The only example of octahedra sharing five vertices is the double layer of TiOg octahedra in Sr3Ti207. 

Octahedra sharing cis vertices to form (a) cyclic tetramer, (b) the cis chain, (c) Octahedra sharing trans vertices to form the trans (Re03) chain, (b) and (c) also represent end-on views (elevations) of the cis and trans layers. The actual configuration of the cis layer in BaMnFa is shown at (d), where Fj and Fjj are the two non-equivalent F atoms referred to on p. 158. The bond angle M-Fj-M is 139° the bonds from Fj are approximately perpendicular to the paper (M-F2-M, 173°). 

The cyclic tetramers in crystals of a number of metal pentafluorides M4F20 are of two kinds, with collinear or non-linear M—F—M bonds  

Other pentafluorides form one or other of the two kinds of chain shown in Fig. 5.15  

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We shall see later that (i) and (ii) correspond to families of related structures while (iii) and (iv) produce a single structure in each case. In addition to these structures in which octahedra share only vertices, edges, or faces, there are structures in which vertices and edges are shared in which all the octahedra and all the X atoms are equivalent. The rutile structure is a simple example—others are described later. 

We have already noted examples of structures in which tetrahedra or octahedra share various numbers of vertices. Structures in which tetrahedra and octahedra share only vertices form an intermediate group ... 

FIG. 5.43. Chains formed from tetrahedra and octahedra sharing only vertices. 

Only the second one is possible and we should have n — p. If j = 3, then G is Octahedron. At every vertex of Octahedron, we have two ways to put 2-gons. Those possibilities are indexed by a pair of 3-gons sharing only an edge. Since we have 6 vertices, this makes 64 possibilities and, up to isomorphism, 7 possibilities. Clearly, every 3-gon should belong to at least one pair. This restricts us to three possibilities. The first possibility is shown below with the number associated to each vertex ... 

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