A crystal of an element or compound can be regarded as constructed from regularly repeating structural elements, which may be atoms, molecules, or ions. The ‘crystal lattice’
is the geometric pattern formed by the points that represent the positions of these repeating structural elements.
(a) Lattices and unit cells
KEY POINTS The lattice defines a network of identical points that has the translational symmetry of a structure. A unit cell is a subdivision of a crystal that when stacked together following translations reproduces the crystal.
A lattice is a three-dimensional, infinite array of points, the lattice points, each of which is surrounded in an iden- tical way by neighbouring points. The lattice defines the repeating nature of the crystal. The crystal structure itself is obtained by associating one or more identical structural units, such as atoms, ions, or molecules, with each lattice point. In many cases the structural unit may be centred on the lattice point, but that is not necessary.
A unit cell of a three-dimensional crystal is an imaginary parallel-sided region (a ‘parallelepiped’) from which the entire crystal can be built up by purely translational dis- placements;1 unit cells so generated fit perfectly together with no space excluded. Unit cells may be chosen in a variety of ways but it is generally preferable to choose the smallest cell that exhibits the greatest symmetry. Thus, in the two-dimensional pattern in Fig. 4.1, a variety of unit cells (a parallelogram in two dimensions) may be chosen, each of
1 A translation exists where it is possible to move an original figure or motif in a defined direction by a certain distance to produce an exact image. In this case a unit cell reproduces itself exactly by translation parallel to a unit cell edge by a distance equal to the unit cell parameter.
which repeats the contents of the box under translational dis- placements. Two possible choices of repeating unit are shown, but (b) would be preferred to (a) because it is smaller. The relationship between the lattice parameters in three dimen- sions as a result of the symmetry of the structure gives rise to the seven crystal systems (Table 4.1 and Fig. 4.2). All ordered structures adopted by compounds belong to one of these crystal systems; most of those described in this chapter,
(a) Possible unit cell (b) Preferred unit cell choice (c)Not a unit cell
FIGURE 4.1 A two-dimensional solid and two choices of a unit cell. The entire crystal is produced by translational displacements of either unit cell, but (b) is generally preferred to (a) because it is smaller.
which deals with simple compositions and stoichiometries, belong to the higher symmetry cubic and hexagonal sys- tems. The angles (α, β, γ) and lengths (a, b, c) used to define the size and shape of a unit cell, relative to an origin, are the unit cell parameters (the ‘lattice parameters’); the angle between a and b is denoted γ, that between b and c is α, and that between a and c is β; a general triclinic unit cell is illustrated in Fig. 4.2.
TABLE 4.1 The seven crystal systems
System Relationships between lattice parameters
Unit cell defined by Essential symmetries
Triclinic a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90° a b c α β γ None
Monoclinic a ≠ b ≠ c, α = γ = 90°, β ≠ 90° a b c β One two-fold rotation axis and/or a mirror plane Orthorhombic a ≠ b ≠ c, α = β = γ = 90° a b c Three perpendicular two-fold axes and/or mirror planes Rhombohedral a = b = c, α = β = γ ≠ 90° a α One three-fold rotation axis
Tetragonal a = b ≠ c, α = β = γ = 90° a c One four-fold rotation axis Hexagonal a = b ≠ c, α = β = 90°, γ = 120° a c One six-fold rotation axis
Cubic a = b = c, α = β = γ = 90° a Four three-fold rotation axes tetrahedrally arranged
a a
a a
c c
a a
b
β a
b α γ
β
α
a c
a a
α α Cubic
Triclinic Hexagonal
Tetragonal Orthorhombic
Monoclinic Rhombohedral
(trigonal)
120°
a
a a b
c c a
c b
FIGURE 4.2 The seven crystal systems.
A primitive unit cell (denoted by the symbol P) has just one lattice point in the unit cell (Fig. 4.3) and the trans- lational symmetry present is just that on the repeating unit cell. More complex lattice types are body-centred (I, from the German word innenzentriert, referring to the lat- tice point at the unit cell centre) and face-centred (F) with two and four lattice points in each unit cell, respectively, and additional translational symmetry beyond that of the unit cell (Figs 4.4 and 4.5). The additional translational
symmetry in the body-centred cubic (bcc) lattice, equivalent to the displacement (+ + +12, 12, 12) from the unit cell origin at (0,0,0), produces a lattice point at the unit cell centre;
note that the surroundings of each lattice point are identi- cal, consisting of eight other lattice points at the corners of a cube. Centred lattices are usually preferred to primitive (although it is always possible to use a primitive lattice for any structure), for with them the full structural symmetry of the cell is more apparent.
We use the following rules to work out the number of lattice points in a three-dimensional unit cell. The same pro- cess can be used to count the number of atoms, ions, or molecules that the unit cell contains (Section 4.9).
• A lattice point within the body of a cell belongs entirely to that cell and counts as 1.
• A lattice point on a face is shared by two cells and con- tributes ẵ to the cell.
• A lattice point on an edge is shared by four cells and hence contributes ẳ.
• A lattice point at a corner is shared by eight cells that share the corner, and so contributes 18.
Thus, for the face-centred cubic lattice depicted in Fig.
4.5 the total number of lattice points in the unit cell is (8× + × =18) (6 12) 4. For the body-centred cubic lat- tice depicted in Fig. 4.4, the number of lattice points is (1 1) (8× + × =18) 2.
(b) Fractional atomic coordinates and projections
KEY POINT Structures may be drawn in projection, with atom posi- tions denoted by fractional coordinates.
The position of an atom in a unit cell is normally described in terms of fractional coordinates, coordinates expressed as a fraction of the length of a side of the unit cell. Thus, the position of an atom, relative to an origin (0,0,0), located at xa parallel to a, yb parallel to b, and zc parallel to c is denoted (x,y,z), with 0 ≤ x, y, z ≤ 1. Three-dimensional repre- sentations of complex structures are often difficult to draw and to interpret in two dimensions. A clearer method of rep- resenting three-dimensional structures on a two-dimensional surface is to draw the structure in projection by viewing the unit cell down one direction, typically one of the axes of the unit cell. The positions of the atoms relative to the projec- tion plane are denoted by the fractional coordinate above the base plane and written next to the symbol defining the atom in the projection. If two atoms lie above each other, then both fractional coordinates are noted in parentheses.
For example, the structure of body-centred tungsten, shown in three dimensions in Fig. 4.8a, is represented in projection in Fig. 4.8b.
a b c
(+1,0,0) O
FIGURE 4.3 Lattice points describing the translational symmetry of a primitive cubic unit cell. The translational symmetry is just that of the unit cell; for example, the a lattice point at the origin, O, translates by (+1,0,0) to another corner of the unit cell.
a b c
O
(+ẵ,ẵ,ẵ)
FIGURE 4.4 Lattice points describing the translational symmetry of a body-centred cubic unit cell. The translational symmetry is that of the unit cell and (+ẵ,+ẵ,+ẵ), so a lattice point at the origin, O, translates to the body centre of the unit cell.
a b c
O
(+ẵ,ẵ,0) (+ẵ,0,ẵ) (0,+ẵ,ẵ)
FIGURE 4.5 Lattice points describing the translational symmetry of a face-centred cubic unit cell. The translational symmetry is that of the unit cell and (+ẵ,+ẵ,0), (+ẵ,0,+ẵ), and (0,+ẵ,+ẵ) so a lattice point at the origin, O, translates to points in the centres of each of the faces.