For hexagonal crystals, it is convenient to use a reference system with four axes (a1, a2, a3, and c), as shown in Figure 2.13, to specify crystallographic planes and

directions. Three axes (a1t a2, and a3) are coplanar and lie on the base of the hexagonal prism of the unit cell with a 120° angle between them. The fourth axis (c) is perpendicular to the base. Thus, a four-digit notation (hkil), known as Miller-Bravais indices, can be used for denoting planes, and [uvtw] for direc­tions in a hexagonal crystal. The use of the Miller-Bravais indices enables denoting crystallographically equivalent planes by the same set of indices. How­ever, the basic procedure for the Miller-Bravais indices are the same as that of the Miller indices. Let us take a look at the plane shown in Figure 2.13a. The intercepts created by the four axes are 1, 1, —1, and 1. Thus, the Miller-Bravais indices of the plane is (1/1 1/i —1/1 1/1) or (1011). Similarly, Miller-Bravais indices of the other planes shown in Figure 2.13b-d can be derived. Note that in all the four cases, the condition h + k = — i is satisfied as a1, a2, and a3 axes are coplanar vectors.

For the Miller-Bravais indices of a direction in the hexagonal crystal, the basic procedure again remains the same. However, it is bit different. For example, the direction along or parallel to the axis a1 can be resolved into components along a2 and a3, each component being —1, as shown in Figure 2.14. The first index can be obtained from the relation u + v = — t or u = —(t + v) = —(—1 — 1) = 2. The direction does not have any component in the c direction. Thus, the Miller-Bravais index of the direction is [2110].

One can use the alternative method of using points as per the cubic, but in this case we first start from defining a1, a2, and c axes as points with the coordinates:

a1 — (100), a2 — (010), and c =(001).

As per the cubic system, find the direction as a subtraction of tail from the head and call it [h’k’l’]. Find the four-axes notation [hkil] by using the following relations and clearing of fractions and reducing to lowest integers:

h = 1 (2h’ — k’), k = 3(2k’- h’), i = —1(h’ + k’), ‘ =

Thus, the Miller index of а1, for example, is easily found as (100)-(000) so that h’ = 1, k = 0, and l’ = 0 or h = 2/3, k = —1/3, i = -1/3, and l = 0 or Miller index of aj is [2110]. This is a very useful procedure in deriving the Miller indices for direc­tions in HCP crystals.

2.1.6