Crystallographic directions hexagonal crystals

For a crystal having a hexagonal symmetry, a set of four numbers, [uvtw], named the Miller-Bravais coordinate system (see Figure 1.5), is used to describe the crystallographic directions, where the first three numbers, that is, u, v, t, are projections along the axes at, a2, and a3, describing the basal plane of the hexagonal structure, and w is the projection in the z-direction [2,3],... 

Careful measurement of mineral specimens allowed crystals to be classified in terms of six crystal families, called anorthic, monoclinic, orthorhombic, tetragonal, hexagonal and isometric. This classification has been expanded slightly by crystallographers into seven crystal systems. The crystal systems are sets of reference axes, which have a direction as well as a magnitude, and hence are vectors1. The crystal families and classes are given in Table 1.1. 

Fortunately, but also evidently, many simplifications arise when considering the uniaxial indicatrix of crystals belonging to the trigonal, tetragonal, and hexagonal crystal systems. First, there is only one optic axis which, by convention, always lies along the z crystallographic axis hence, X, Y, and Z disappear, as z suffices to define this direction. The refractive index associated with this direction is called or n. The plane perpendicular to the optic axis is necessarily a circular section whose diameters all have the same refractive index denoted or n. Thus, and Hy disappear. Optically positive means... 

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. 

When plotting crystallographic directions for hexagonal crystals it is sometimes more convenient to modify the four-axis coordinate system shown in Figure 3.8 to that of Figure 3.10 here, a grid has been constructed on the basal plane that consists of sets of lines parallel to each of the a, a2, and a, axes. The intersections of two sets of parallel... 

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