Anthony R. West - Solid State Chemistry and its Applications

The general symbol for Miller indices is ( hkl ). It is not necessary to use commas to separate the three letters or numbers and the indices are enclosed in parentheses, (). The brackets {} are used to indicate sets of planes that are equivalent; for example, the sets (100), (010) and (001) are equivalent in cubic crystals and may be represented collectively as {100}. In the examples chosen so far, all of the Miller indices are either 0 or positive but, as we shall see later, it is also important to define lattice planes that intersect axes in their negative directions. In these cases the relevant index has a bar above the number; thus planes are referred to as ‘bar h , bar k, bar l ’.

Figure 1.15 Examples of Miller indices: (a) (101); (b) (100); (c) (200); (d) (h00); (e) indices of directions [210] and [].

To label lattice planes in hexagonal unit cells, a modification to the method is required. For hexagonal unit cells, with a = a ≠ c, there are three possible choices for the two a axes in the basal plane, a 1 , a 2and a 3, as shown in Fig. 1.14. Let us choose a 1and a 2to define the unit cell with the origin at O. Next, consider the set of planes shown, which are parallel to c ; assign Miller indices in the usual way but in this case, find the intercepts of plane 1 on all three a axes as well as the c axis; since these planes in Fig. 1.14 are parallel to c , plane 1 does not intersect the c axis. This leads to four indices for this set, representing the intercepts on a 1, a 2, a 3and c , i.e. . The third index is redundant since, writing the indices as ( hkil ), the condition

holds (i.e. –1 + 3 – 2 = 0). Sometimes, all four indices are specified; sometimes, only three are specified (i.e. in the usual way); sometimes, the third index may be represented as a dot (i.e. ).

Directions in crystals and lattices are labelled by first drawing a line that passes through the origin and parallel to the direction in question. Let the line pass through a point with general fractional coordinates x, y, z ; the line also passes through 2 x , 2 y , 2z; 3 x , 3 y , 3 z , etc. These coordinates, written in square brackets, [ x , y , z ], are the indices of the direction; x, y and z are arranged to be the set of smallest possible integers, by division or multiplication throughout by a common factor. Thus [½½0], [110], [330] all describe the same direction, but by convention [110] is used.

For cubic systems, an [ hkl ] direction is always perpendicular to the ( hkl ) plane of the same indices, but this is only sometimes true in non‐cubic systems. Sets of directions which, by symmetry, are equivalent, e.g. cubic [100], [010], are written using angle brackets, <100>. Some examples of directions and their indices are shown in Fig. 1.15(e). Note that the [210] direction is defined by taking the origin at the bottom‐left‐front corner of the unit cell and taking the fractional coordinates: 1, 0.5, 0, to define the direction. For the direction [ ], where the bar sign indicates a negative direction, a different origin is chosen, at the bottom‐right‐front corner of the unit cell and passing through the fractional coordinates: –1, 0.67, 1.

We have already defined the d ‐spacing of a set of planes as the perpendicular distance between any pair of adjacent planes in the set and it is this d value that appears in Bragg's law. For a cubic unit cell, the (100) planes simply have a d ‐spacing of a, the value of the cell edge, Fig. 1.15(b). For (200) in a cubic cell, d = a /2. For orthogonal crystals (i.e. α = β = γ = 90°), the d ‐spacing for any set of planes is given by

(1.1)

The equation simplifies for tetragonal crystals, in which a = b, and still further for cubic crystals with a = b = c:

(1.2)

As a check, for cubic (200): h = 2, k = l = 0; 1/ d 2= 4/ a 2; d = a /2.

Monoclinic and, especially, triclinic crystals have much more complicated d ‐spacing formulae because each angle that is not equal to 90° is an additional variable. The formulae for d ‐spacings and unit cell volumes of all crystal systems are given in Appendix A.

The unit cell, by definition, must contain at least one formula unit, whether it be an atom, ion pair, molecule, etc. In centred cells, the unit cell contains more than one formula unit and more than one lattice point. A simple relation exists between cell volume, the number of formula units in the cell, the formula weight (FW) and the bulk crystal density ( D ):

where N is Avogadro's number. If the unit cell, of volume V , contains Z formula units, then

Therefore,

(1.3)

V is usually expressed in Å 3and must be multiplied by 10 –24to convert V to cm –3and to give densities in units of g cm –3. Substituting for N, the equation reduces to

(1.4)

and, if V is in Å 3, the units of D are g cm −3. This simple equation has a number of uses, as shown by the following examples: