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 without rotation or reflection, reproduces the crystal.

A lattice is a three-dimensional, infinite array of points, the lattice points, each of which is surrounded in an identical way by neighbouring points, and which defines the basic repeating structure of the crystal. In some cases the structural unit may be centred on the lattice point, but that is not necessary. The crystal structure itself is obtained by associating one or more identical structural units (such as molecules or ions) with each lattice point. A unit cell of the crystal is an imaginary parallel-sided region (a ‘parallelepiped’) from which the entire crystal can be built up by purely translational displacements;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. 3.1, a variety of unit cells may be chosen, each of which repeats the contents of the box under translational displace ments. 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 dimensions as a result of the symmetry of the structure gives rise to the seven crystal systems (Table 3.1 and Fig. 3.2). All ordered structures adopted by compounds belong to one of these crystal systems; most of those described in this chapter, which deals with simple compositions and stoichiometries, belong to the higher symmetry cubic and hexagonal systems. The angles ( α,ß ,ý ) and lengths (a, b, c) used to define the size and shape of a unit cell are the unit cell parameters (the ‘lattice parameters’); the angle between a and b is denoted , that between b and c is , and that between a and c is ; see the triclinic unit cell in Fig. 3.2. A primitive unit cell (denoted by the symbol P) has just one lattice point in the unit cell (Fig. 3.3) and the translational symmetry present is just that on the repeating unit cell. More complex lattice types are body-centred (I, from the German word innenzentriet, referring to the lattice 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 1A 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.

¹A 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.

Fig. 3.2 The seven crystal systems.

the unit cell (Figs 3.4 and 3.5). The additional translational symmetry in the body-centred cubic (bcc) lattice, equivalent to the displacement (+1 2 ,+1 2 ,+1 2 ) 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 identical, consisting of eight other lattice points at the corners of a cube. Centred lattices are sometimes preferred to primitive (although it is always possible to use a primitive lattice for any structure) for with them the essential 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 process can be used to count the number of atoms, ions, or molecules that the unit cell contains (Section 3.9).

1. Alattice point in the body of, that is fully inside, a cell belongs entirely to that cell and counts as 1.

2. A lattice point on a face is shared by two cells and contributes 1 2 to the cell.

3. A lattice point on an edge is shared by four cells and hence contributes 1/4.

 4. A lattice point at a corner is shared by eight cells that share the corner, and so contributes 1/8.

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مواضيع ذات صلة

Polytypism

The close packing of spheres

The orbitals

Bonding and antibonding orbitals

Hybridization

Hypervalence

Promotion

Polyatomic molecules

Homonuclear diatomic molecules

The hydrogen molecule

The basic shapes

Resonance