Miller-Bravais indices for hexagonal

MILLER-BRAVAIS INDICES FOR HEXAGONAL COORDINATE SYSTEMS... 

Determination of the Miller-Bravais Indices for a Plane within a Hexagonal Unit Cell... 

Determine the Miller-Bravais indices for the plane shown in the hexagonal unit cell. 

Since a subsidiary axis is assumed in addition to the and axes in the hexagonal system (and in the hexagonal expression of the trigonal system), the index is expressed by four indices for a general face (hktl). This is called the Miller-Bravais index. For example, in Fig. A.3.1(b) a face ACB cuts the + and + axes at 1 (= OA, OB), and the - axis at 0C = 0D = OB/2. Therefore this face is indexed as (1120). From geometry, h(l) + k(l) = i(2). 

There is also the three-digit system for directions in hexagonal crystals. For planar indices, it uses intercepts on the ai, a2, and c axes. The indices (HKL) are related to the Miller-Bravais indices (hkil) by... 

For hexagonal crystal planes, a slightly different indexing nomenclature is used relative to cubic crystals. To index a plane in the hexagonal system, four axes are used, called Miller-Bravais indices. In addition to both a and b axes, another axis... 

Since i is determined by h and k, it is sometimes replaced by a dot and the plane symbol writ n hk /). Sometimes even the dot is omitted. However, this usage defeats the purpose for which Miller-Bravais indices were devised, namely, to give similar indices to similar planes. For example, the side planes of the hexagonal prism in Fig. 2-11(b) are all similar and symmetrically located, and their relationship is clearly shown in their full Miller-Bravais symbols (lOTO), (OlTO), (TlOO), (To 10), (OTlO), (iTOO). On the other hand, the abbreviated symbols of these planes, (10-0), (01 -0), (Tl 0), (TO-O), (OT-0), (lT-0) do not immediately suggest this relationship. 

Here, uppercase U, V, and W indices are associated with the three-index scheme (instead of u, V, and w as previously), whereas lowercase u, v, f, and w correlate with the Miller-Bravais four-index system. For example, using these equations, the [010] direction becomes [1210]. Several directions have been drawn in the hexagonal unit cell of Figure 3.9. 

For crystals having hexagonal symmetry, it is desirable that equivalent planes have the same indices as with directions, this is accomplished by the Miller-Bravais system shown in Figure 3.8. This eonvention leads to the four-index (hkil) scheme, which is favored in most instances because it more clearly identifies the orientation of a plane in a hexagonal crystal. There is some redimdaney in that i is determined by the sum of h and k through... 

---