details of the bonding requirements or band structures of the metal. Structural details of elements not included in Table 1.3 are given in Appendix D.
1.15.2 Alloys
Alloys are intermetallic phases or solid solutions and, as is the case for pure metals, many have cp structures. For example, Cu and Au have ccp structures, both as pure elements and when mixed to form Cu–Au alloys. At high temperatures, a complete range of solid solutions between Cu and Au forms. In these the Cu and Au atoms are distributed statistically over the lattice points of the fcc unit cell and therefore the ccp layers contain a random mixture of Cu and Au atoms. On annealing the compositions AuCu and AuCu3 at lower temperatures, the Au and Cu atoms order themselves; ccp layers still occur but the arrangement within the layers is no longer statistical. Such order–disorder phenomena occur commonly in both metallic and ionic structures.
Table 1.3 Structures, unit cell dimensions, and metallic radii of some common metals
| ccp | hcp | bcc | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Metal | r/Å | a/Å | Metal | r/Å | a/Å | c/Å | Metal | r/Å | a/Å |
| Cu | 1.28 | 3.6147 | Be | 1.12 | 2.2856 | 3.5842 | Fe | 1.24 | 2.8664 |
| Ag | 1.45 | 4.0857 | Mg | 1.60 | 3.2094 | 5.2105 | Cr | 1.25 | 2.8846 |
| Au | 1.44 | 4.0783 | Zn | 1.37 | 2.6649 | 4.9468 | Mo | 1.36 | 3.1469 |
| Al | 1.43 | 4.0495 | Cd | 1.52 | 2.9788 | 5.6167 | W | 1.37 | 3.1650 |
| Ni | 1.24 | 3.5240 | Ti | 1.47 | 2.9506 | 4.6788 | Ta | 1.43 | 3.3026 |
| Pb | 1.75 | 4.9502 | Zr | 1.60 | 3.2312 | 5.1477 | Ba | 2.17 | 5.019 |
1.15.3 Ionic structures
The structures of materials such as NaCl, Al2O3, Na2O and ZnO, in which the anion is larger than the cation, are built of cp layers of anions with the cations placed in interstitial sites. Many structures are possible in which the variables are the anion stacking sequence, either hcp or ccp, and the number and type of interstitial sites occupied by cations. The cations are, however, often too large for the prescribed interstitial sites and the structure can accommodate them only by expanding the anion array. Consequently, the anion arrangement is the same as in cp, but the anions may not be in contact. O'Keeffe suggested the term eutactic for structures such as these. In the discussions that follow, use of the terms hcp and ccp for the anion arrays does not necessarily imply that the anions are in contact but rather that the structures are eutactic. A further complication, as we shall see later, is that the rigid sphere model is an oversimplification of reality since, in ionic structures, it can be difficult to specify ion sizes exactly.
It is useful at this stage to identify the criteria for an ionic structure to be regarded as close packed, since both cation and anion arrangements are involved. As with metal structures, the first criterion is that close packed anions exist with a coordination number, by other anions, of twelve. Six of these should be co‐planar, forming cp layers, as in Fig. 1.16(a). The other six form part of two cp layers to either side; the relative orientation of these two layers determines the overall stacking sequence, whether ccp or hcp.
Various departures from ideal cp structures occur. Thus, in some structures, such as solid solutions or doped materials, anion vacancies may be present, within the overall cp array. These vacancies are usually at random throughout the anion array, such as in the perovskite (La1−x Sr x )(Ga1−x Mg x )O3−x , but the ordering of oxygen vacancies may occur as in, for example, the beta alumina family of structures in which every fifth layer in a cp oxide sequence has ¾ of the oxygens missing, Fig. 8.23. In other cp structures, particularly those containing large cations such as K+, Sr2+ or La3+ in perovskites, mixed cp layers are present because these cations are large enough to be surrounded by 12 anion neighbours. Thus, in SrTiO3, Sr and O together form a ccp sequence in which the Sr and O positions are ordered rather than randomised, Fig. 1.41(e). And in some other structures, especially fluorite, the cations rather than the anions form a cp array. Finally, the structures may be eutactic in that the cp arrangement of one type of ion is clearly present, but the ions are pushed apart and therefore, the structures are not closest packed.
Within a cp anion array, interstitial sites for cations are restricted to either tetrahedral or octahedral, as discussed next. For the anion coordination number, however, there is no such limitation. Although the anion–anion coordination number (above) is twelve, this refers to the second coordination sphere around a particular anion whereas the anion–cation coordination number refers to the primary coordination sphere, with the smallest interatomic distances. Various examples of anion–cation coordination numbers and arrangements are given throughout this chapter, as well as examples of structures that are distorted away from an ideal cp arrangement. First, we consider the cases of the standard tetrahedral and octahedral coordinations.
1.15.3.1 Tetrahedral and octahedral sites
Two types of interstitial site, tetrahedral and octahedral, are present in cp structures, Fig.
