Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 40

We have seen how T₊, T_– and O sites occur between pairs of cp layers, Fig. 1.23, and how fcc and ccp structures are often equivalent, Fig. 1.20. We can now see where the T₊, T_– and O sites are located within an fcc unit cell that has anions, X, at corners and face centres (A, B, C, and D), Fig. 1.24. The octahedral sites are easiest to locate; they are at edge‐centre 1, 2, 3 and body‐centre 4 positions. If the unit cell has length a, the M–X distance for octahedral sites is a/2.

In order to see the T₊, T_– sites clearly, it is convenient to divide the unit cell into eight minicubes by bisecting each cell edge (dashed lines). These minicubes contain anions at only four of the eight corners; in the middle of each minicube is a tetrahedral site, either T₊ or T_–. There are two orientations for the minicubes, and therefore for the T sites, as shown in Fig. 1.25; relative to the parent unit cell, Fig. 1.24, the T_– tetrahedra have their apices pointing in the [] direction and the T₊ sites in the [111] direction.

At this point, we note that it is exceptionally useful to be able to regard a tetrahedral site as occupying the centre of a cube with alternate corners absent, Fig. 1.25(a) and (b). For such an arrangement, it is easy to calculate bond lengths and bond angles (c) and to demonstrate the various symmetry elements associated with a tetrahedron. Thus, for the minicube of edge dimension a/2,

Figure 1.24 Available cation sites, 1–12, in an fcc anion array.

Twofold rotation axes and fourfold inversion axes run parallel to minicube edges but pass through pairs of opposite faces; there are three of these in total (see also Fig. 1.7c); mirror planes occur parallel to the set of lattice planes with indices {110}, and there are six of these in total [as in Fig. 1.9(c)]; threefold axes run parallel to <111> directions, four in total, Fig. 1.9(a); further details are in Appendix C.