Crystal directions and planes

1.7 Crystal directions and planes

In a structurally disordered material, such as fully annealed silica glass, the value of a physical property is independent of the direction of measurement; the material is said to be isotropic. Conversely, in many single crystals, it is often observed that a structurally sensitive property, such as electrical conductivity, is strongly direction dependent because of variations in the periodicity and packing of atoms. Such crystals are anisotropic. We therefore need a precise method for specifying a direction, and equivalent directions, within a crystal. The general method for defining a given direction is

5 The notion that the striking external appearance of crystals indicates the existence of internal structural units with similar characteristics of shape and orientation was proposed by the French mineralogist Hauy in 1784. Some 130 years

elapsed before actual experimental proof was provided by the new technique of X-ray diffraction analysis. 6 Lattices are imaginary and limited in number; crystal structures are real and virtually unlimited in their variety.

Atoms and atomic arrangements 17 Orthorhombic

a a a a Triclinic

120 a Hexagonal

aa a b

Rhombohedral

Axial angles Cubic

System

Axes

a 1 ⫽ a 2 ⫽ a 3 All angles ⫽ 90⬚ Tetragonal

a 1 ⫽ a 2 ⫽ c All angles ⫽ 90⬚ Orthorhombic

All angles ⫽ 90⬚ Monoclinic

a⫽b⫽c

a⫽b⫽c

Two angles ⫽ 90⬚; 1 angle ⫽ 90⬚

Triclinic

a⫽b⫽c

All angles different; none equal 90⬚

Hexagonal a 1 ⫽ a 2 ⫽ a 3 ⫽ c Angles ⫽ 90⬚ and 120⬚ Rhombohedral

a 1 ⫽ a 2 ⫽ a 3 All angles equal, but not 90⬚ Figure 1.7 The seven systems of crystal symmetry (S = skew operation).

18 Physical Metallurgy and Advanced Materials

Figure 1.8 Indexing of directions (a) and planes (b) in cubic crystals. to construct a line through the origin parallel to the required direction and then to determine the

coordinates of a point on this line in terms of cell parameters (a, b, c). Hence, in Figure 1.8a, the direction − AB is obtained by noting the translatory movements needed to progress from the origin O → to point C, i.e. a = 1, b = 1, c = 1. These coordinate values are enclosed in square brackets to give

the direction indices [1 1 1]. In similar fashion, the direction −→ DE can be shown to be [ ¯ 1 2 ¯1 ¯1] with the bar sign indicating use of a negative axis. Directions which are crystallographically equivalent in a given crystal are represented by angular brackets. Thus, and comprises [1 0 0], [0 1 0], [0 0 1], [¯1 0 0], [0 ¯1 0] and [0 0 ¯1] directions. Directions are often represented in non-specific terms as [u v w] and

Physical events and transformations within crystals often take place on certain families of parallel equidistant planes. The orientation of these planes in three-dimensional space is of prime concern; their size and shape are of lesser consequence. (Similar ideas apply to the corresponding external facets of a single crystal.) In the Miller system for indexing planes, the intercepts of a representative

plane upon the three axes (x, y, z) are noted. 7 Intercepts are expressed relatively in terms of a, b, c. Planes parallel to an axis are said to intercept at infinity. Reciprocals of the three intercepts are taken and the indices enclosed by round brackets. Hence, in Figure 1.8b, the procedural steps for indexing the plane ABC are:

Intercepts

Reciprocals

Miller indices

The Miller indices for the planes DEFG and BCHI are (0 ¯1 0) and (1 1 0), respectively. Often it is necessary to ignore individual planar orientations and to specify all planes of a given crystallographic type, such as the planes parallel to the six faces of a cube. These planes constitute a crystal form and have the same atomic configurations; they are said to be equivalent and can be represented by a single group of indices enclosed in curly brackets, or braces. Thus, {1 0 0} represents a form of six planar

7 For mathematical reasons, it is advisable to carry out all indexing operations (translations for directions, intercepts for planes) in the strict sequence a, b, c.

Atoms and atomic arrangements 19

(a) (b) Figure 1.9 Prismatic, basal and pyramidal planes in hexagonal structures.

orientations, i.e. (1 0 0), (0 1 0), (0 0 1), (¯1 0 0), (0 ¯1 0) and (0 0 ¯1). Returning to the (1 1 1) plane ABC of Figure 1.8b, it is instructive to derive the other seven equivalent planes, centering on the

origin O, which comprise {1 1 1}. It will then be seen why materials belonging to the cubic system often crystallize in an octahedral form in which octahedral {1 1 1} planes are prominent.

It should be borne in mind that the general purpose of the Miller procedure is to define the orientation of a family of parallel equidistant planes; the selection of a convenient representative plane is a means to this end. For this reason, it is permissible to shift the origin provided that the relative disposition of a, b and c is maintained. Miller indices are commonly written in the symbolic form (h k l). Rationalization of indices, either to reduce them to smaller numbers with the same ratio or to eliminate fractions, is unnecessary. This often-recommended step discards information; after all, there is a real difference between the two families of planes (1 0 0) and (2 0 0).

As mentioned previously, it is sometimes convenient to choose a non-primitive cell. The hexagonal structure cell is an important illustrative example. For reasons which will be explained, it is also appropriate to use a four-axis Miller–Bravais notation (h k i l) for hexagonal crystals, instead of the

three-axis Miller notation (h k l). In this alternative method, three axes (a 1 ,a 2 ,a 3 ) are arranged at 120 ◦ to each other in a basal plane and the fourth axis (c) is perpendicular to this plane (Figure 1.9a). Hexagonal structures are often compared in terms of the axial ratio c/a. The indices are determined by taking intercepts upon the axes in strict sequence. Thus, the procedural steps for the plane ABCD, which is one of the six prismatic planes bounding the complete cell, are:

Miller–Bravais indices

20 Physical Metallurgy and Advanced Materials

Figure 1.10 Typical Miller–Bravais directions in (0 0 0 1) basal plane of hexagonal crystal. Comparison of these digits with those from other prismatic planes such as (1 0 ¯1 0), (0 1 ¯1 0)

and (1 ¯1 0 0) immediately reveals a similarity; that is, they are crystallographically equivalent and belong to the {1 0 1 0} form. The three-axis Miller method lacks this advantageous feature when applied to hexagonal structures. For geometrical reasons, it is essential to ensure that the plane indices comply with the condition (h + k) = −i. In addition to the prismatic planes, basal planes of (0 0 0 1) type and pyramidal planes of the (1 1 ¯2 1) type are also important features of hexagonal structures (Figure 1.9b).

The Miller–Bravais system also accommodates directions, producing indices of the form [u v t w]. The first three translations in the basal plane must be carefully adjusted so that the geometrical condition u + v = −t applies. This adjustment can be facilitated by subdividing the basal planes into triangles (Figure 1.10). As before, equivalence is immediately revealed; for instance, the close-packed directions in the basal plane have the indices [2 ¯1 ¯1 0], [1 1 ¯2 0], [¯1 2 ¯1 0], etc. and can be represented by