Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 46

These structures are considered together because they all have ccp/fcc anions and differ only in the positions of the cations. In Fig. 1.24 are shown the anions in a fcc unit cell with all possible O, T₊ and T_– sites for the cations. There is no rule as to which sites should be labelled T₊ and T_–; the choice is yours. The choice of origin is also arbitrary; for present purposes, it is more convenient to place the anions at the origin and also, therefore, at face centre positions. The different structures are generated as follows:

 rock salt: O occupied; T+ and T– empty

 zinc blende: T+ (or T–) occupied; O, T– (or T+) empty

 antifluorite: T+, T– occupied; O empty.

Unit cells are shown in Fig. 1.29, in oblique projection (a–c) and as projections on the ab face of the unit cell in (d–f). Each is described in more detail later.

Figure 1.29 Unit cell of (a, d) NaCl, (b, e) ZnS, sphalerite, and (c, f) Na2O. Closed circles are cations; open circles are anions.

Table 1.7 Two ways to describe the antifluorite structure

	Old cell	New cell
Anions	000, ½½0, ½0½, 0½½	¾¾¾, ¼¾¾, ¼¾¼, ¾¼¼
Cations	¼¼¼, ¼¼¾, ¼¾¼, ¾¼¼	000, 00½, 0½0, ½00
	¼¾¾, ¾¼¾, ¾¾¼, ¾¾¾	0½½, ½0½, ½½0, ½½½

A general rule regarding coordination numbers is that in any structure of formula A _x X _y , the coordination numbers of A and X must be in the ratio of y:x. In both rock salt and zinc blende, x = y and therefore, in each, anions and cations have the same coordination number.

In antifluorite, of formula A₂X, the coordination numbers of cation and anion must be in the ratio of 1:2. Since the cations occupy tetrahedral sites, the anion coordination number must be eight. In order to see this, it is convenient to redefine the origin of the unit cell to coincide with a cation rather than an anion. This is done by displacing the unit cell along a body diagonal by one‐quarter of the length of the diagonal. The cation at X in Fig. 1.29(c), with coordinates ¼¼¼, may be chosen as the new origin of the unit cell. The coordinates of all atoms in the new cell are given by subtracting ¼¼¼ from their coordinates in the old cell, as in Table 1.7.

In cases where negative coordinates occur as a result of this subtraction, e.g. –¼–¼–¼, the position lies outside the new unit cell and it is necessary to find an equivalent position within the unit cell. In this particular case, 1 is added to each coordinate, giving ¾¾¾. Addition of 1 to, say, the x coordinate is equivalent to moving to a similar position in the next unit cell in the x direction. The new unit cell of antifluorite with its origin at cation X, is shown in Fig. 1.30(a). It contains cations at corners, edge centres, face centres and body centre.

In order to see the anion coordination more clearly, the unit cell may be imagined as divided into eight minicubes (as in Fig. 1.24). Each minicube in Fig. 1.30(a) has cations at all eight corners and at the centre of each is an eight‐coordinate site. Anions occupy four of these eight minicubes such that parallel to the cell axes the eight‐coordinate sites are alternately occupied and empty. The eightfold coordination for one anion, A, is shown in Fig. 1.30(b).

In the antifluorite structure, the effect of changing the origin from an anion to a cation is to show the structure in a completely different light. This does not happen with the rock salt and zinc blende structures. In these, the cation and anion positions are interchangeable and it is immaterial whether the origin coincides with an anion or a cation.

So far, the NaCl, ZnS and Na₂O structures have been described in two ways: (a) as cp structures and (b) in terms of their unit cells. A third way is to regard them as built of space‐filling polyhedra. Each ion and its nearest neighbours may be represented by the appropriate polyhedron, e.g. in zinc blende a tetrahedron represents one Zn with four S neighbours (or vice versa). It is then necessary to consider how neighbouring polyhedra are linked in 3D. Let us now consider each of these structures in more detail.

Figure 1.30 Alternative view of the antifluorite structure.