Anthony Kelly - Crystallography and Crystal Defects

Figure 2.16A stereogram of a trigonal crystal of class m with a rhombohedral unit cell. When poles in the upper and lower hemispheres coincide in projection, the indices shown refer to the poles in the upper hemisphere

In plotting a stereogram of a trigonal crystal with a rhombohedral unit cell from a given value of the angle α , it is useful to note that the angle γ between any of the crystal axes and the unique triad axis is given by:

(2.8)

(see Problem 2.2).

For many purposes it is more convenient to use a hexagonal cell when dealing with trigonal crystals, irrespective of whether the lattice of the trigonal crystal is primitive hexagonal or rhombohedral. The shape of the cell chosen is the same as for hexagonal crystals (Figure 1.19j) and, since it is chosen without reference to the lattice, may or may not be primitive. The value c / a is characteristic of the substance.

A stereogram of a trigonal crystal of the point group m with planes indexed according to the Miller–Bravais scheme is shown in Figure 2.17. This is the same crystal as that in Figure 2.16( c / a = 1.02). When using the hexagonal cell, the x ‐, y ‐ and u ‐axes are chosen parallel to the diads in m .

Figure 2.17The same crystal as in Figure 2.16indexed using a hexagonal cell. When poles in the upper and lower hemispheres superpose in projection, the indices refer to poles in the upper hemisphere

The relationship between the indices in the two stereograms can be easily worked out using the zone addition rule and the Weiss zone law, Eq. (1.6), from the orientation relationship of the rhombohedral and hexagonal cells. In Figures 2.16and 2.17this is (0001) || (111) 6and (10 1) || (100). The plane (10 0) in Figure 2.17then has indices (2 ) in Figure 2.16. The indices (2 ) could be deduced by noting that the plane (2 ) contains the [111] direction (and so the pole of (2 ) must lie on the primitive if [111] is at the centre of the stereogram), is equally inclined to the y ‐ and z ‐axes, and lies in the zone containing (111) and (100). The plotting of a stereogram and the determination of axial ratios for a trigonal crystal referred to hexagonal axes then proceed as for the hexagonal system. 7

In the class m , special forms lie (i) normal to the triad: {0001}, (ii) parallel to the triad: { hki 0}, (iii) normal to mirror planes: { h 0 l }, and (iv) equally inclined to two diads { hh l }. The six faces in the form:{ h 0 l }, make a rhombohedron; {10 2}would be an example, consisting of the planes (10 2), ( 102), (0 12), ( 01 ), (1 0 ), and (01 ). This form is similar in appearance to {0 h l }, which is also a rhombohedron, rotated 60° with respect to the first one.

The relationship between { h 0 l } and {0 h l } (or, equivalently, {0 k l }) is shown in Figure 2.18. They are actually quite separate forms and each one is a special form. We therefore need to add {0 h l } to the list of special forms, in addition to { h 0 l }. It is apparent from Figure 2.16that when using the rhombohedral cell the two forms {10 1} and {01 1} have different indices, since the face above (00 ) in the projection in Figure 2.16would have indices (22 ).