Crystallographic planes hexagonal crystals

Transmission electron microscopy ( ) analysis reveals that these materials crystallize as hexagonal planar particles with marked anisotropic shape,8,37 as shown in Figure 6. When appropriate preparation methods are used, plate-like crystals are obtained with small thickness of about 20-30 nm and an aspect ratio D/h=5-10. Selected area diffraction (SAD) patterns of incident beams perpendicular and parallel to the large hexagonal facet show that they correspond to the crystallographic planes perpendicular to the c axis. The anisotropic shape of the... 

Figure 1.37. Three possibilities to select the crystallographic basis in hexagonal and trigonal crystal systems and the family of (1120) crystallographic planes in the hexagonal crystal system. Indices are shown in the unit cell based on the vectors b and c. Three additional symmetrically related families of planes have indices (1120), (1210) and (2110) in the same basis and we leave their identification to the reader.

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... 

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... 

The orientations of planes for a crystal structure are represented in a similar manner. Again, the unit cell is the basis, with the three-axis coordinate system as represented in Figure 3.5. In all but the hexagonal crystal system, crystallographic planes are specified by three Miller indices as hkl). Any two planes parallel to each other are equivalent and have identical indices. The procedure used to determine the h, k, and I index numbers is as follows ... 

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],... 

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