Miller-Bravais index system

The Miller-Bravais index system for identifying planes and directions in hexagonal crystals is similar to the Miller index system except that it uses four axes rather than three. The advantage of the four-index system is that the symmetry is more apparent. Three of the axes, ai, a2, and a3, he in the hexagonal (basal) plane at 120° to one another and the fourth or c-axis is perpendicular to then, as shown in Figure 3.1. 

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). 

However, there should be noted that, in terms of directions, many crystal-lographers prefer not to make the transformation within Miller-Bravais axes system, while working with the Miller system for directions and with Miller-Bravais only for the planes indexing of the hexagonal system. 

The Miller indices of planes in crystals with a hexagonal unit cell can be ambiguous. In order to eliminate this ambiguity, four indices, (hkil), are often used. These are called Miller-Bravais indices and are only used in the hexagonal system. The index i is given 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... 

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... 

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