Miller-Bravais Indices for Hexagonal Coordinate Systems

9 MILLER-BRAVAIS INDICES FOR HEXAGONAL COORDINATE SYSTEMS 

The Miller indices of faces equivalent by the symmetry operations of the crystal class are derived by application of equation (1.21) of paragraph 1.2,4. In the 

A complication arises from the action of a threefold axis in a hexagonal coordinate system. In this case, the indices of equivalent faces are not obtained simply by permutations and sign changes of (hkl). To circumvent this problem, we use four indices (hkil) defined with respect to the vectors ai,a2,a3 perpendicular to the threefold axis (Fig. 2.37), and c parallel to the threefold axis. For /i, fc 0, the length cut by the plane on a3 is negative, hence we obtain  

Because the threefold axis permutes the vectors a, a2 and a3, we obtain the following recipe to obtain the indices of equivalent faces generated by a threefold 

The same recipe cannot be applied for the indices luvw] of translations (zones, edges), A method which defines four indices [uvcow] has been published (L. Weber, Z, Kristallogr. 57,200-203,1922) however, it is not well known and little used. We would discourage the use of four indices for translations. We recall (Section 1,2,4) that hkl) and [uvw] do not transform in the same way, the first being covariant and the latter contravariant. Accordingly, a threefold axis produces the equivalent indices [uvw ], [v u — v)w], [(y — m)mw]. 

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