Close packing

Figure 20.32 shows a close-packed layer of identical spheres, one with maximum utilization of space. A close-packed three-dimensional structure is obtained by stacking such close-packed layers on top of one another. However, this stacking can be done in different ways, which result in close-packed polytypes, or structures that are identical in two dimensions (the close-packed layers) but differ in the third dimension. In all polytypes, the spheres of second close-packed layer lie in the depressions of the first layer (Fig. 20.33). The third layer may be added in either of two ways. In one, the spheres are placed so that they reproduce the first layer (Fig. 20.34a), to give an ABA pattern of layers. Alternatively, the spheres may be placed over the gaps in the first layer (Fig. 20.34b), so giving an ABC pattern. Two polytypes are formed if the two stacking patterns are repeated in the vertical direction. If the ABA pattern is repeated, to give the sequence of layers ABABAB..., the spheres are hexagonally close-packed

Fig. 20.32 The first layer of close-packed spheres used to build a three-dimensional close-packed structure.

Fig. 20.33 The second layer of close-packed spheres occupies the dips of the first layer. The two layers are the AB component of the close-packed structure.

Fig. 20.34 (a) The third layer of close-packed spheres might occupy the dips lying directly above the spheres in the first layer, resulting in an ABA structure, which corresponds to hexagonal close-packing. (b) Alternatively, the third layer might lie in the dips that are not above the spheres in the first layer, resulting in an ABC structure, which corresponds to cubic close-packing.

(hcp). Alternatively, if the ABC pattern is repeated, to give the sequence ABCABC ..., the spheres are cubic close-packed (ccp). We can see the origins of these names by referring to Fig. 20.35. The ccp structure gives rise to a face-centred unit cell, so may also be denoted cubic F (or fcc, for face-centred cubic).2 It is also possible to have random sequences of layers; however, the hcp and ccp polytypes are the most import ant. Table 20.2 lists some elements possessing these structures. The compactness of close-packed structures is indicated by their coordination number, the number of atoms immediately surrounding any selected atom, which is 12 in all cases. Another measure of their compactness is the packing fraction, the fraction of space occupied by the spheres, which is 0.740 (see the following Justification). That is, in a close-packed solid of identical hard spheres, only 26.0 per cent of the volume is empty space. The fact that many metals are close-packed accounts for their high densities.

Fig. 20.35 A fragment of the structure shown in Fig. 20.34 revealing the (a) hexagonal (b) cubic symmetry. The tints on the spheres are the same as for the layers in Fig. 20.34.

Fig. 20.36 The calculation of the packing fraction of an ccp unit cell.

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مواضيع ذات صلة

Energetics

Structure

Structure refinement

The phase problem

Bragg’s law

X-ray diffraction

Lattices and unit cells

Surface pressure

Membrane formation

Micelle formation

The electrical double layer

Classification and preparation