Indexing hexagonal crystal system example

A problem arises for crystals having hexagonal symmetry in that some eqnivalent crystallographic directions do not have the same set of indices. For example, the [111] direction is equivalent to [101] rather than to a direction with indices that are combinations of Is and —Is. This situation is addressed nsing a fonr-axis, or MUler-Bravais, coordinate system, which is shown in Figure 3.8. The three ], aj, and a axes are all contained within a single plane (called the basal plane) and are at 120° angles to one another. The Z axis is perpendicular to this basal plane. Directional indices, which are obtained as described earlier, are denoted by four indices, as [uvtw, by convention, the u, v, and t indices relate to vector coordinate differences referenced to the respective a, Oj, and a- axes in the basal plane the fonrth index pertains to the z axis. 

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