TOPICS
sp 3 Hybridization: a scheme for tetrahedral and related species
5.2 Packing of spheres
Many readers will be familiar with descriptions of metal lattices based upon the packing of spherical atoms, and in this section we provide a re´sume´ of common types of pack-ing, and introduce the terms unit cell and interstitial hole.
Cubic and hexagonal close-packing
Let us place a number of equal-sized spheres in a rectangular box, with the restriction that there must be a regular arrange-mentof spheres. Figure 5.1 shows the most efficient way in which to cover the floor of the box. Such an arrangement is close-packed, and spheres that are not on the edges of the assembly are in contact with six other spheres within the layer. A motif of hexagons is produced within the assem-bly. Figure 5.2a shows part of the same close-packed arrangement of spheres; hollows lie between the spheres and we can build a second layer of spheres upon the first by placing spheres in these hollows. However, if we arrange the spheres in the second layer so that close-packing is again achieved, it is possible to occupy only every other hollow.
This is shown on going from Figure 5.2a to 5.2b.
Now consider the hollows that are visible in layer B in Figure 5.2b. There are two distinct types of hollows. Of the four hollows between the grey spheres in layer B, one lies over a red sphere in layer A, and three lie over hollows in layer A. The consequence of this is that when a third layer of spheres is constructed, two different close-packed arrange-ments are possible as shown in Figures 5.2c and 5.2d. The arrangements shown can, of course, be extended sideways, and the sequences of layers can be repeated such that the
Chapter
5
TOPICS
& Packing of spheres
& Applications of the packing-of-spheres model
& Polymorphism
& Alloys and intermetallic compounds
& Band theory
& Semiconductors
& Sizes of ions
& Ionic lattices
& Lattice energy
& Born–Haber cycle
& Applications of lattice energies
& Defects in solid state lattices
fourth layer of spheres is equivalent to the first, and so on.
The two close-packed arrangements are distinguished in that one contains two repeating layers, ABABAB . . . , while the second contains three repeating layers, ABCABC . . . (Figures 5.2d and 5.2c respectively).
Close-packing of spheresresults in the most efficient use of the space available; 74% of the space is occupied by the spheres.
The ABABAB . . . and ABCABC . . . packing arrange-ments are called hexagonal close-packing (hcp) and cubic close-packing (ccp), respectively. In each structure, any given sphere is surrounded by (and touches) 12 other spheres and is said to have 12 nearest neighbours, to have a coordina-tion numberof 12, or to be 12-coordinate. Figure 5.3 shows representations of the ABABAB . . . and ABCABC . . . arrangements which illustrate how this coordination number arises; in these diagrams, ‘ball-and-stick’ representa-tions of the lattice are used to allow the connectivities to be seen. This type of representation is commonly used but does not implythat the spheres do not touch one another.
The unit cell: hexagonal and cubic close-packing
A unit cell is a fundamental concept in solid state chemistry, and is the smallest repeating unit of the structure which carries all the information necessary to construct unambigu-ouslyan infinite lattice.
The smallest repeating unit in a solid state lattice is a unit cell.
Fig. 5.2 (a) One layer (layer A) of close-packed spheres contains hollows that exhibit a regular pattern. (b) A second layer (layer B) of close-packed spheres can be formed by occupying every other hollow in layer A. In layer B, there are two types of hollow;
one lies over a sphere in layer A, and three lie over hollows in layer A. By stacking spheres over these different types of hollow, two different third layers of spheres can be produced. The blue spheres in diagram (c) form a new layer C; this gives an ABC sequence of layers. Diagram (d) shows that the second possible third layer replicates layer A; this gives an ABA sequence.
Fig. 5.1 Part of one layer of a close-packed arrangement of equal-sized spheres. It contains hexagonal motifs.
The unit cells in Figure 5.4 characterize cubic (ccp) and hexagonal close-packing (hcp). Whereas these respective descriptors are not obviously associated with the packing sequences shown in Figures 5.2 and 5.3, their origins are clear in the unit cell diagrams. Cubic close-packing is also called face-centred cubic (fcc) packing, and this name clearly reflects the nature of the unit cell shown in Figure 5.4a. The relationship between the ABABAB . . . sequence and the hcp unit cell is easily recognized; the latter consists of parts of three ABA layers. However, it is harder to see the ABCABC . . . sequence within the ccp unit cell since the close-packed layers are not parallel to the base of the unit cell but instead lie along the body-diagonal of the cube.
Interstitial holes: hexagonal and cubic close-packing
Close-packed structures contain octahedral and tetrahedral holes (or sites). Figure 5.5 shows representations of two
Fig. 5.3 In both the (a) ABA and (b) ABC close-packed arrangements, the coordination number of each atom is 12.
Fig. 5.4 Unit cells of (a) a cubic close-packed (face-centred cubic) lattice and (b) a hexagonal close-packed lattice.
Fig. 5.5 Two layers of close-packed atoms shown (a) with the spheres touching, and (b) with the sizes of the spheres reduced so that connectivity lines are visible. In (b), the tetrahedral and octahedral holes are indicated.
Chapter 5 . Packing of spheres 133
layers of close-packed spheres: Figure 5.5a is a ‘space-filling’
representation, while in Figure 5.5b, the sizes of the spheres have been reduced so that connectivity lines can be shown (a ‘ball-and-stick’ diagram). This illustrates that the spheres lie at the corners of either tetrahedra or octahedra; conversely, the spheres pack such that there are octahedral and tetra-hedral holes between them. There is one octatetra-hedral hole per sphere, and there are twice as many tetrahedral as octahedral holes in a close-packed array; the octahedral holes are larger than the tetrahedral sites. Whereas a tetrahedral hole can accommodate a sphere of radius 0.23 times that of the close-packed spheres, a sphere of radius 0.41 times that of the close-packed spheres fits into an octahedral hole.
Non-close-packing: simple cubic and body-centred cubic arrays
Spheres are not always packed as efficiently as in close-packed arrangements; ordered arrays can be constructed in which the space occupied by the spheres is less than the 74% found for a close-packed arrangement.
If spheres are placed so as to define a network of cubic frameworks, the unit cell is a simple cube (Figure 5.6a). In the extended lattice, each sphere has a coordination number of 6. The hole within each cubic unit is not large enough to accommodate a sphere equal in size to those in the array, but if the eight spheres in the cubic cell are pulled apart slightly, another sphere is able to fit inside the hole. The result is the body-centred cubic (bcc) arrangement (Figure 5.6b). The coordination number of each sphere in a bcc lattice is 8.