To describe the atomic positions in Figure 1.1a we chose OXAY as one unit parallelogram. We could equally well have chosen OXTA. The choice of a particular unit parallelogram or unit cell is arbitrary, subject to the constraint that the unit cell tessellates, that is, it repeats periodically, in this case in two dimensions. Therefore, NQPM is not a permissible choice – although NQPM is a parallelogram, it cannot be repeated to produce the graphene structure because P and M have environments which differ from those at N and Q.
The corners of the unit cell OXAY in Figure 1.1a all possess identical surroundings. We could choose and mark on the diagram all points with surroundings identical to those at O, X, A and so on. Such points are N, Q, R, S and so on. The array of all such points with surroundings identical to those of a given point we call the mesh or net in two dimensions (a lattice in three dimensions). Each of the points is called a lattice point.
A formal definition of the lattice is as follows: A lattice is a set of points in space such that the surroundings of one point are identical with those of all the others. The type of symmetry described by the lattice is referred to as translational symmetry. The lattice of the graphene crystal structure drawn in Figure 1.1a is shown in Figure 1.1b. It consists of a set of points with identical surroundings. Just a set of points: no atoms are involved. Various primitive unit cells are marked in Figure 1.1c. A primitive unit cell is defined as a unit parallelogram which contains just one lattice point. The conventional unit cell for graphene corresponding to OXAY in Figure 1.1a is outlined in Figure 1.1c in heavy lines and the corresponding x‐ and y‐axes are marked.
In general, the conventional primitive unit cell of a two‐dimensional net which shows no obvious symmetry is taken with its sides as short and as nearly equal as possible, with γ, the angle between the x‐ and y‐axes, taken to be obtuse, if it is not equal to 90°. However, the symmetry of the pattern must always be taken into account. The net shown in Figure 1.1b is very symmetric and in this case we can take the sides to be equal, so that a = b and γ = 120°.
Comparisons of Figures 1.1a and b emphasize that the choice of the origin for the lattice is arbitrary. If we had chosen O′ in Figure 1.1a as the origin instead of O and then marked all corresponding points in Figure 1.1a, we would have obtained the identical lattice with only a change of origin. The lattice then represents an essential aspect of the translational symmetry of the crystal however we choose the origin.
In three dimensions the definition of the lattice is the same as in two dimensions. The unit cell is now the parallelepiped containing just one lattice point. The origin is taken at a corner of the unit cell. The sides of the unit parallelepiped are taken as the axes of the crystal, x, y, z, using a right‐handed notation. The angles α, β, γ between the axes are called the axial angles (see Figure 1.2). The smallest separations of the lattice points along the x‐, y‐ and z‐axes are denoted by a, b, c, respectively and called the lattice parameters.
Figure 1.2 Definition of the smallest separations a, b and c of the lattice points along the x‐, y‐ and z‐axes respectively, together with the angles α, β and γ between the axes for a lattice in three dimensions
Inspection of the drawing of the arrangement of the ions in a crystal of caesium chloride, CsCl, in Figure 3.17 shows that the lattice is an array of points such that a = b = c, α = β = γ = 90°, so that the unit cell is a cube. There is one caesium ion and one chlorine ion associated with each lattice point. If we take the origin at the centre of a caesium ion, then there is one caesium ion in the unit cell with coordinates (0, 0, 0) and one chlorine ion with coordinates
Figure 1.3 The numbers give the elevations of the centres of the atoms, along the z‐axis, taking the lattice parameter c as the unit of length
It is apparent from this drawing of the crystal structure of caesium chloride that the coordination number for each caesium ion and each chlorine ion is eight, each ion having eight of the other kind of ion as neighbours. The separation of these nearest neighbours, d, is easily seen to be given by:
(1.1)
since a = b = c.
The number of units of the formula CsCl per unit cell is clearly 1.
1.2 Lattice Planes and Directions
A rectangular mesh of a hypothetical two‐dimensional crystal with mesh parameters a and b of very different magnitude is shown in Figure 1.4. Note that the parallel mesh lines OB, O′B′, O″B″, and so on all form part of a set and that the spacing of all lines in the set is quite regular; this is similar for the set of lines parallel to AB: A′B′, A″B″ and so on. The spacing of each of these sets is determined only by a and b (and the angle between a and b, which in this example is 90°). Also, the angle between these two sets depends only on the ratio of a to b, where a