Hexagonal crystal system direction indices

As with Miller indices, to specify directions in hexagonal crystals a four-index system, u v t w ] is sometimes used. The conversion of a three-index set to a four-index set is given by the following rules ... 

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... 

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. 

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... 

Refractive index data are very useful for the quantitation of isotropic (liquid and cubic liquid crystal) phases, and for the calibration of cell thickness and nonflatness. Hovever, the analysis of birefringent phases using refractive index data has been found to be unreliable (9). A problem arises from the fact that the orientation of such phases relative to the direction of the light path, as veil as the system variables, influence refractive indices. In order to use refractive index data for quantitation, a phase must spontaneously orient in a reproducible fashion. Such orientation does occur in the case of fluid lamellar phases (as in short chain polyoxyethylene nonionic systems (7)), but viscous lamellar phases, hexagonal phases, and crystal phases do not orient to a sufficient degree. 

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. 

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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