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 and b are the magnitudes of a and b, respectively. If external faces of the crystal formed parallel to O″B and to AB, the angle between these faces would be uniquely related to the ratio a : b. Furthermore, this angle would be independent of how large these faces were (see Figure 1.5). This was recognized by early crystallographers, who deduced the existence of the lattice structure of crystals from the observation of the constancy of angles between corresponding faces. This law of constancy of angle states: In all crystals of the same substance the angle between corresponding faces has a constant value.
Figure 1.4 A rectangular mesh of a hypothetical two‐dimensional crystal with mesh parameters a and b of very different magnitude
Figure 1.5 Demonstration of the law of constancy of angles between faces of crystals: the angle φ between the faces is independent of the size of the faces
The analogy between lines in a mesh and planes in a crystal lattice is very close. Crystal faces form parallel to lattice planes and important lattice planes contain a high density of lattice points. Lattice planes form an infinite regularly spaced set which collectively passes through all points of the lattice. The spacing of the members of the set is determined only by the lattice parameters and axial angles, and the angles between various lattice planes are determined only by the axial angles and the ratios of the lattice parameters to one another.
Prior to establishing a methodology for designating a set of lattice planes, it is expedient to consider how directions in a crystal are specified. A direction is simply a line in the crystal. Select any two points on the line, say P and P′. Choose one as the origin, say P (Figure 1.6). Write the vector r between the two points in terms of translations along the x‐, y‐ and z‐axes so that:
(1.2)
where a, b and c are vectors along the x‐, y‐ and z‐axes, respectively, and have magnitudes equal to the lattice parameters (Figure 1.6). The direction is then denoted as [uvw] – always cleared of fractions and reduced to its lowest terms. The triplet of numbers indicating a direction is always enclosed in square brackets. Some examples are given in
