indices.
Figure 1.14 Miller indices for a hexagonal lattice.
In order to assign Miller indices to a set of planes, there are four stages:
1 From the array of lattice points, or the crystal structure, identify the unit cell, choose the origin and label the axes a, b, c and angles α (between b and c), β (between a and c) and γ (between a and b).
2 For a particular set of lattice planes, identify that plane which is adjacent to the one that passes through the origin.
3 Find the intersection of this plane on the three axes of the cell and write these intersections as fractions of the cell edges. The plane in question, Fig. 1.13(b), cuts the x axis at a/2, the y axis at b and the z axis at c/3; the fractional intersections are therefore ½, 1, ⅓.
4 Take reciprocals of these fractions and write the three numbers in parentheses; this gives (213). These three integers, (213), are the Miller indices of the plane and all other planes parallel to it and separated from adjacent planes by the same d‐spacing.
Some other examples are shown in Fig. 1.15. In (a), the shaded plane cuts x, y and z at 1a, ∞b and 1c, i.e. the plane is parallel to b. Taking reciprocals of 1, ∞ and 1 gives us (101) for the Miller indices. A Miller index of 0 means, therefore, that the plane is parallel to that axis. In Fig. 1.15(b), the planes of interest comprise opposite faces of the unit cell. We cannot determine directly the indices of plane 1 as it passes through the origin. Plane 2 has intercepts of 1a, ∞b and ∞ C and Miller indices of (100).
Figure 1.15(c) is similar to (b) but there are now twice as many planes as in (b). To find the Miller indices, consider plane 2, which is the one that is closest to the origin but without passing through it. Its intercepts are ½, ∞ and ∞ and the Miller indices are (200). A Miller index of 2, therefore indicates that the plane cuts the relevant axis at half the cell edge. This illustrates an important point. After taking reciprocals, do not divide through by the highest common factor. A common source of error is to regard, say, the (200) set of planes as those planes interleaved between the (100) planes, to give the sequence (100), (200), (100), (200), (100), …. The correct labelling is shown in Fig. 1.15(d). If extra planes are interleaved between adjacent (100) planes then all planes are labelled as (200).
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
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 2 and a 3, as shown in Fig. 1.14. Let us choose a 1 and a 2 to 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 3 and c, i.e.
holds (i.e. –1 + 3 – 2 = 0). Sometimes, all four indices are specified; sometimes, only three are specified (i.e.
1.6 Indices of Directions
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 2x, 2y, 2z; 3x, 3y, 3z, 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 [
1.7 d‐Spacing Formulae
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)
