The close packing of spheres

Key points: The close packing of identical spheres can result in a variety of polytypes, of which hexagonal and cubic close-packed structures are the most common. Many metallic and ionic solids can be regarded as constructed from entities, such as atoms and ions, represented as hard spheres. If there is no directional covalent bonding, these spheres are free to pack together as closely as geometry allows and hence adopt a close-packed structure, a structure in which there is least unfilled space. The coordination number (CN) of a sphere ²Nearly all the structures in this text are available as rotatable, three-dimensional versions in the Online Resource Centre for this text.

in a close-packed arrangement (the ‘number of nearest neighbours’) is 12, the greatest number that geometry allows.3 When directional bonding is important, the resulting structures are no longer close-packed and the coordination number is less than 12. Consider first a single layer of identical spheres (Fig. 3.11). The greatest number of immediate neighbours is 6 and there is only one way of constructing this close-packed layer.4 A second close-packed layer of spheres is formed by placing spheres in the dips between the spheres of the first layer. (Note that only half the dips in the original layer are occupied, as there is insufficient space to place spheres into all the dips.) The third close packed layer can be laid in either of two ways and hence can give rise to either of two polytypes, or structures that are the same in two dimensions (in this case, in the planes) but different in the third. Later we shall see that many different polytypes can be formed, but those described here are two very important special cases. In one polytype, the spheres of the third layer lie directly above the spheres of the first. This ABAB...pattern of layers, where A denotes layers that have spheres directly above each other and likewise for B, gives a structure with a hexagonal unit cell and hence is said to be hexagonally close-packed (hcp, Figs 3.12a and 3.13). In the second polytype, the spheres of the third layer are placed above the gaps in the first layer. The second layer covers half the holes in the first layer and the third layer lies above the remaining holes. This arrangement results in an ABCABC...pattern, where C denotes a layer that has spheres not directly above spheres of the A or the B layer positions (but they will be directly above another C type layer). This pattern corresponds to a structure with a cubic unit cell and hence it is termed cubic close-packed (ccp, Figs 3.12b and 3.14). Because each ccp unit cell has a sphere at one corner and one at the centre of each face, a ccp unit cell is sometimes referred to as face-centred cubic (fcc).

Fig. 3.12 The formation of two close-packed polytypes. (a) The third layer reproduces the first to give an ABA structure. (b) The third layer lies above the gaps in the first layer, giving an ABC structure. The different colours identify the different layers of identical spheres.

The unoccupied space in a close-packed structure amounts to 26 per cent of the total volume (see Example 3.3). However, this unoccupied space is not empty in a real solid because electron density of an atom does not end as abruptly as the hard-sphere model suggests. The type and distribution of holes are important because many structures, including those of some alloys and many ionic compounds, can be regarded as formed from an expanded close packed arrangement in which additional atoms or ions occupy all or some of the holes.

³That this arrangement, where each sphere has 12 nearest-neighbours, is the highest possible density of packing spheres was conjectured by Johannes Kepler in 1611; the proof was found only in 1998. 4A good way of showing this yourself is to get a number of identical coins and push them together on a flat surface; the most efficient arrangement for covering the area is with six coins around each coin. This simple modelling approach can be extended to three dimensions by using any collection of identical spherical objects such as balls, oranges, or marbles.

Fig. 3.13 The hexagonal close-packed (hcp) unit cell of the ABAB... polytype. The colours of the spheres correspond to the layers in Fig. 3.12a.

Fig. 3.16 The structure of solid C₆₀ showing the packing of C₆₀ polyhedra on an fcc unit cell.

The ccp and hcp arrangements are the most efficient simple ways of filling space with identical spheres. They differ only in the stacking sequence of the close-packed layers and other, more complex, close-packed layer sequences may be formed by locating successive planes in different positions relative to their neighbours (Section 3.4). Any collection of identical atoms, such as those in the simple picture of an elemental metal, or of approximately spherical molecules, is likely to adopt one of these close-packed structures unless there are additional energetic reasons—specifically covalent interactions—for adopting an alternative arrangement. Indeed, many metals adopt such close-packed structures (Section 3.4), as do the solid forms of the noble gases (which are ccp). Almost spherical molecules, such as C₆₀, in the solid state also adopt the ccp arrangement (Fig. 3.16), and so do many small molecules that rotate around their centres and thus appear spherical, such as H2, F2, and one form of solid oxygen, O₂.

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مواضيع ذات صلة

Polytypism

Lattices and unit cells

The orbitals

Bonding and antibonding orbitals

Hybridization

Hypervalence

Promotion

Polyatomic molecules

Homonuclear diatomic molecules

The hydrogen molecule

The basic shapes

Resonance