KEY POINT The close packing of identical spheres can result in a vari- ety of polytypes, of which hexagonal and cubic close-packed structures are the most common.
Many metallic and ionic solids can be regarded as con- structed from 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.
Consider first a single layer of identical spheres (Figs 4.11 and 4.12a). The greatest number of immediate neighbours is six, and there is only one way of constructing this close- packed layer.2 Note that the environment of each sphere in
Zn2+
S2–
a b c
FIGURE 4.6 The cubic ZnS structure.
Cs+ Cl–
FIGURE 4.7 The cubic CsCl structure.
W
(a)
(b)
(0,1)
ẵ
FIGURE 4.8 (a) The structure of metallic tungsten and (b) its projection representation.
EXAMPLE 4.1 Identifying lattice types
Determine the translational symmetry present in the structure of cubic ZnS (Fig. 4.6) and identify the lattice type to which this structure belongs.
Answer We need to identify the displacements that, when applied to the entire cell, result in every atom arriving at an equivalent location (same atom type with the same coordination environment). In this case, the displacements (0,+ẵ,+ẵ), (+ẵ,+ẵ,0), and (+ẵ,0,+ẵ), where +ẵ in the x, y, or z coordinate represents a translation along the appropriate cell direction by a distance of a/2, b/2, or c/2 respectively, have this effect.
For example, starting at the labelled Zn2+ ion towards the near bottom left-hand corner of the unit cell (the origin), which is surrounded by four S2− ions at the corners of a tetrahedron, and applying the translation (+ẵ,0,+ẵ), we arrive at the Zn2+
ion towards the top front right-hand corner of the unit cell, which has the same tetrahedral coordination to sulfur. Identical translational symmetry exists for all the ions in the structure.
These translations correspond to those of the face-centred lattice, so the lattice type is F.
Self-test 4.1 Determine the lattice type of CsCl (Fig. 4.7).
2 A 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.
the layer is identical, with six others placed around it in a hexagonal pattern. A second close-packed layer of spheres is formed by placing spheres in the dips between the spheres of the first layer so that each sphere in this second layer touches three spheres in the layer below (Fig. 4.12b). (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 arrangement of spheres in this second layer is identical to that in the first, each with six nearest neighbours; the
pattern is just slightly displaced horizontally. The third close- packed layer can be laid in either of two ways (remember, only half the dips in the preceding layer can be occupied).
This gives 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 dif- ferent 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 and each sphere in the second layer gains three more neighbours in this layer above it. 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 (often written in the short- ened form hcp) (Figs 4.12c and 4.13). In the second poly- type, the spheres of the third layer are placed above the dips that were not occupied in the first layer. The second layer fits into half the dips in the first layer and the third layer lies above the remaining dips. This arrangement results in an ABCABC . . . pattern, where C denotes a layer that has spheres that are not directly above spheres of the A or the
S
Si
(0,1)
(0,1)
(0,1)
ẳ ẳ
ẵ S
Si
(0,1)
(0, ( ( (0 ( ( ( (0 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
( 1)
(0,1)
ẳ ẳ
ẵ
FIGURE 4.10 The structure of silicon sulfide (SiS2).
FIGURE 4.11 A close-packed layer of hard spheres.
EXAMPLE 4.2 Drawing a three-dimensional representation in projection Convert the face-centred cubic lattice shown in Fig. 4.5 into a
projection diagram.
Answer We need to identify the locations of the lattice points by viewing the cell from a position perpendicular to one of its faces. The faces of the cubic unit cell are square, so the projection diagram viewed from directly above the unit cell is a square. There is a lattice point at each corner of the unit cell, so the points at the corners of the square projection are labelled (0,1). There is a lattice point on each vertical face, which projects to points at fractional
coordinate ẵ on each edge of the projection square. There are lattice points on the lower and on the upper horizontal face of the unit cell, which project to two points at the centre of the square at 0 and 1, respectively, so we place a final point in the centre of a square and label it (0,1). The resulting projection is shown in Fig. 4.9.
Self-test 4.2 Convert the projection diagram of the unit cell of the SiS2 structure shown in Fig. 4.10 into a three-dimensional representation.
(0,1)
(0,1)
ẵ
ẵ
FIGURE 4.9 The projection representation of a face-centred cubic unit cell.
a close-packed arrangement (the ‘number of nearest neigh- bours’) is 12, formed from 6 touching spheres in the orig- inal close-packed layer and 3 from each layer above and below it. This is the greatest number that geometry in three dimensions allows.3 When directional bonding is important, the resulting structures are no longer close-packed and the coordination number is less than 12.
A A
B A
B A A
B
A (directly above A) A
B A
A BBBBBBBBBBBBBBBBBBBBBBB C A
B
C
(a) (b) (c) (d)
FIGURE 4.12 The formation of two close-packed polytypes. (a) A single close-packed layer, A. (b) The second close-packed layer, B, lies in dips above A. (c) The third layer reproduces the first to give an ABA arrangement structure (hcp). (d) The third layer lies above the gaps in the first layer, giving an ABC arrangement (ccp). The different colours identify the different layers of identical spheres.
3 That 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.
FIGURE 4.13 The hexagonal close-packed (hcp) unit cell of the ABAB . . . polytype. The colours of the spheres correspond to the layers in Fig. 4.12c.
A NOTE ON GOOD PRACTICE
The descriptions ccp and fcc are often used interchangeably, although strictly ccp refers only to a close-packed arrangement, whereas fcc refers to the lattice type of the common representation of ccp. Throughout this text the term ccp will be used to describe this close-packing arrangement. It will be drawn as the cubic unit cell, with the fcc lattice type, as this representation is easiest to visualize.
B layer positions (but 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 (short- ened to ccp) (Figs 4.12d and 4.14). 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). The coordination number (CN) of a sphere in
The occupied space in a close-packed structure amounts to 74 per cent of the total volume (see Example 4.3).
However, the remaining unoccupied space, 26 per cent, 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 these spaces between the close- packed spheres are known as ‘holes’. They 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.
4r
√8r
√8r
√8r
FIGURE 4.15 The dimensions involved in the calculation of the packing fraction in a close-packed arrangement of identical spheres of radius r.
The ccp and hcp arrangements are the most efficient sim- ple 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 4.4). 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, such as covalent bond- ing interactions, for adopting an alternative arrangement.
Indeed, many metals adopt such close-packed structures (Section 4.4), as do the solid forms of the noble gases (which are ccp). Almost-spherical molecules, such as fullerene, C60, in the solid state, also adopt the ccp arrangement (Fig. 4.16), and so do many small molecules that rotate around their centres in the solid state and thus appear spherical, such as H2, F2, and one form of solid oxygen, O2.