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.
1.11 Relationship Between Cubic Close Packed and Face Centred Cubic
The unit cell of a ccp arrangement is the familiar face centred cubic (fcc) unit cell, Fig. 1.11(c), with spheres at corner and face centre positions. The relation between ccp and fcc is not immediately obvious since the faces of the fcc unit cell do not correspond to cp layers. The cp layers are, instead, parallel to the {111} planes of the fcc unit cell. This is shown in Fig. 1.20 and Appendix B. The spheres labelled 2–7 in Fig. 1.20(a) form part of a cp layer, as revealed by removing a corner sphere 1 in (b) and comparing (b) with Fig. 1.16(a). The orientations of (a) and (b) in Fig. 1.20 are the same but the spheres in (b) are shown larger. A similar arrangement to that shown in (b) would be seen on removing any corner sphere in (a) and, therefore, in a ccp structure, cp layers occur in four orientations. These orientations are perpendicular to the body diagonals of the cube (the cube has eight corners but only four body diagonals and, hence, four different orientations of the cp layers). The cp layers in one orientation are seen edge‐on in (c) and in another orientation, perpendicular to the layers in (d); (c) is the same as (a) but rotated slightly; similarly, (d) is the same as (b) but also rotated slightly. In (d), all atoms of the unit cell, spanning four cp layers, ABCA, are shown in projection down a <111> direction.
1.12 Hexagonal Unit Cell and Close Packing
An hcp arrangement of spheres has a hexagonal unit cell (Fig. 1.21). The basal plane of the cell coincides with a cp layer of spheres (b). The unit cell contains two spheres, one at the origin (and hence at all corners) and one inside the cell at positions ⅔, ⅓, ½ [pink circle in (a) and (b)]. Note that although the two a axes of the basal plane are equal, we need to distinguish them by a 1 and a 2 for the purpose of describe atomic coordinates of the positions ⅔, ⅓, ½. The use of such fractional coordinates to represent positions of atoms inside a unit cell is discussed later.
cp layers occur in only one orientation in an hcp structure. These are parallel to the basal plane, as shown for one layer in Fig. 1.21(b). The two axes in the basal plane are of equal length; a = 2r, if the spheres of radius r touch; the angle Γ is 120° (Table 1.1).
Figure 1.20 Face centred cubic, fcc, unit cell of a ccp arrangement of spheres.
The symmetry of the hexagonal unit cell is deceptively simple. The basal plane in isolation has a sixfold rotation axis but the adjacent B layer along the c axis reduces this to threefold rotational symmetry, as shown in Fig. 1.21(c): note the crystallographic symbol for a threefold axis, which is a solid triangle.
The structure does, however, possess a 63 screw axis parallel to c and
