Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 30

Many metallic, ionic, covalent and molecular crystal structures can be described using the concept of close packing. The guiding factor is that structures are usually arranged to have the maximum density. The principles involved can be understood by considering the most efficient way of packing equal‐sized spheres in three dimensions.

The most efficient way to pack spheres in two dimensions is shown in Fig. 1.16(a). Each sphere, e.g. A, is surrounded by, and is in contact with, six others, i.e. each sphere has six nearest neighbours and its coordination number, CN, is six. By regular repetition, infinite sheets called close packed layers form. The coordination number of six is the maximum possible for a planar arrangement of contacting, equal‐sized spheres. Lower coordination numbers are, of course, possible, as shown in Fig. 1.16(b), where each sphere has four nearest neighbours, but the layers are no longer close packed, cp. Note also that within a cp layer, three close packed directions occur. Thus, in Fig. 1.16(a) spheres are in contact in the directions xx′, yy′ and zz′ and sphere A belongs to each of these rows.

The most efficient way to pack spheres in three dimensions is to stack cp layers on top of each other. There are two simple ways to do this, resulting in hexagonal close packed and cubic close packed structures as follows.

The most efficient way for two cp layers A and B to be in contact is for each sphere of one layer to rest in a hollow between three spheres in the other layer, i.e. at P or R in Fig. 1.16(c) and (d). Two layers in such a position relative to each other are shown in Fig. 1.17. Atoms in the second layer may occupy either P or R positions, but not both together, nor a mixture of the two. Any B (dashed) sphere is therefore seated between three A (solid) spheres, and vice versa.

Figure 1.16 (a) A cp layer of equal‐sized spheres; (b) a non‐cp layer with coordination number 4; (c, d) alternative positions P and R for a second cp layer.

Figure 1.17 Two cp layers arranged in A and B positions. The B layer occupies the P positions of Fig. 1.16.

Addition of a third cp layer to the two shown in Fig. 1.17 can also be done in two ways, and herein lies the distinction between hexagonal and cubic close packing. In Fig. 1.17, suppose that the A layer lies underneath the B layer and we wish to place a third layer on top of B. There is a choice of positions, as there was for the second layer: the spheres can occupy either of the new sets of positions S or T but not both together nor a mixture of the two. If the third layer is placed at S, then it is directly over the A layer. As subsequent layers are added, the following sequence arises:

This is known as hexagonal close packing, hcp. If, however, the third layer is placed at T, then all three layers are staggered relative to each other and it is not until the fourth layer is positioned (at A) that the sequence is repeated. If the position of the third layer is called C, this gives (Fig. 1.18)

This sequence is known as cubic close packing (ccp). The two simplest stacking sequences are hcp and ccp and these are by far the most important in structural chemistry. Other more complex sequences with larger repeat units, e.g. ABCACB or ABAC, occur in a few materials; some of these larger repeat units are responsible for the phenomenon of polytypism.

In a 3D cp structure, each sphere is in contact with 12 others, and this is the maximum coordination number possible for contacting and equal‐sized spheres. [A common non‐cp structure is the body centred cube, e.g. in α‐Fe, with a coordination number of eight; see Fig. 1.11(e).] Six of these neighbours are coplanar with the central sphere, Fig. 1.16(a); from Fig. 1.17 and Fig. 1.18, the remaining six are in two groups of three spheres, one in the plane above and one in the plane below (Fig. 1.19); hcp and ccp differ in the relative orientations of these two groups of three neighbours.

Figure 1.18 Three close packed layers in ccp sequence.

Figure 1.19 Coordination number 12 of shaded sphere in (a) hcp and (b) ccp structures. The shaded sphere is in the B layer, the layer underneath is A, and the layer above is either (a) A or (b) C.

Many structures, not just of metals and alloys, but also ionic, covalent and molecular structures, can be described using close packing ideas. Sometimes the atoms that form the cp array are as closely packed as possible, but in other cases their arrangement is as in cp but the atoms are clearly not touching. Such structures are known as eutactic structures. Some guidelines as to whether it is appropriate to consider a structure in terms of a cp arrangement are given in Appendix D.