Anthony Kelly - Crystallography and Crystal Defects

Figure 1.8The plane ( hkl ) in a crystal making intercepts of a/ h , b/ k and c/ l along the x ‐, y ‐ and z ‐axes, respectively

Thus, for example, the plane marked Y in Figure 1.9makes intercepts on the axes of infinity, 2 b and infinity, respectively. Taking the reciprocals of these intercepts gives h = 0, k = and l = 0. Clearing the fractions gives h = 0, k = 1 and l = 0. Hence, the set of lattice planes parallel to Y is designated (010). The triplet of numbers describing the Miller index is always enclosed in round brackets. Similarly, the plane marked P in Figure 1.9has intercepts 1 a , 2 b and c . Therefore, taking the reciprocals of these intercepts, h = 1, k = and l = 3. Clearing the fractions, we have (216) as the Miller indices. The indices of a number of other planes are shown in Figure 1.9. Negative values of the intercepts are indicated in the Miller index notation by a bar over the appropriate index (see the examples in Figure 1.9).

Figure 1.9Examples of various lattice planes in a crystal. The indices of the planes P and Y are discussed in the accompanying part of the text in Section 1.2

The reason for using Miller indices to index crystal planes is that they greatly simplify certain crystal calculations. Furthermore, with a reasonable choice of unit cell, small values of the indices ( hkl ) belong to widely spaced planes containing a large areal density of lattice points. Well‐developed crystals are usually bounded by such planes, so that it is found experimentally that prominent crystal faces have intercepts on the axes which when expressed as multiples of a , b and c have ratios to one another that are small rational numbers. 4

This law expresses the mathematical condition for a vector [ uvw ] to lie in a plane ( hkl ). This condition can be determined through elementary vector considerations. Consider the plane ( hkl ) in Figure 1.10with the normal to the plane .

Figure 1.10The plane ( hkl ) in a crystal making intercepts of a/ h , b/ k and c/ l along the x ‐, y ‐ and z ‐axes, respectively, together with the vector normal to ( hkl )

A general vector, r, lying in ( hkl ) can be expressed as a linear combination of any two vectors lying in this plane, such as and . That is:

(1.3)

for suitable λ and μ . Hence, expressing and in terms of a, band c, it follows that:

(1.4)

If we re‐express this as r= u a+ v b+ w c– a general vector [ uvw ] lying in ( hkl ) – it follows that:

(1.5)

and so:

(1.6)

which is the condition for a vector [ uvw ] to lie in the plane ( hkl ): Weiss zone law. 5It is evident from this derivation that it is valid for arbitrary orientations of the x ‐, y ‐ and z ‐axes with respect to one another.

Frequently, a number of important crystal lattice planes all lie in the same zone; that is, they intersect one another in parallel lines. For instance, in Figure 1.9the planes (100), and (110) are all parallel to the direction [001]. They would be said to lie in the zone [001], since [001] is a common direction lying in all of them. The normals to all of these planes are perpendicular to [001]. This is not an accident – the normals are constrained to be perpendicular to [001] by the Weiss zone law.

To see why, we can make use of elementary vector algebra relationships discussed in Appendix 1, Section A1.1. Consider the plane ( hkl ) shown in Figure 1.10. The vector normal to this plane, n, must be parallel to the cross product . Hence:

(1.7)

and so, after some straightforward mathematical manipulation, making use of the identities

it is apparent that nis parallel to the vector h a* + k b* + l c*. That is:

(1.8)