Читать книгу Crystallography and Crystal Defects - Anthony Kelly - Страница 35

This crystal system is defined by the possession of a single triad axis. It is closely related to the hexagonal system. The possession of a single triad axis by a crystal does not, by itself, indicate whether the lattice considered as a set of points is truly hexagonal, or whether it is based on the staggered stacking of triequiangular nets. When the lattice is rhombohedral, a cell of the shape of Figure 1.19k can be used. The cell in Figure 1.19k is a rhombohedron and the angle α (< 120°) is characteristic of the substance. When the lattice of a trigonal crystal is hexagonal, it is not appropriate to use a rhombohedral unit cell.

The symmetry elements in the holosymmetric class m are shown in Figure 2.6 and the repetition of a single pole in accordance with this symmetry is also demonstrated in this figure. In m, three diad axes arise automatically from the presence of and the three mirrors lying parallel to . These diad axes, which intersect in the inverse triad axis, do not lie in the mirror planes. If the rhombohedral cell is used for such a crystal then the axes cannot be chosen parallel to prominent axes of symmetry.

A stereogram of a trigonal crystal indexed according to a rhombohedral unit cell is shown in Figure 2.16. The value of α is 98°. The x‐, y‐ and z‐axes are taken to lie in the mirror planes and the inverse triad is a body diagonal of the cell, therefore lying along the direction [111], which, from the geometry of the rhombohedral unit cell, is also parallel to the normal to the (111) plane. It is clear that the x‐, y‐ and z‐axes – that is, the directions [100], [010] and [001] – do not lie normal to the (100), (010) and (001) planes, respectively. However, these directions are easily located. For example, the z‐axis, [001], is the pole of the zone containing (10), (010), (100) and (10), shown as a great circle in Figure 2.16. Likewise, the y‐axis is the pole of the zone containing (10), (100), (001) and (01), and the x‐axis is the pole of the zone containing (01), (001), (010) and (01).

Figure 2.16 A 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 2.17 The same crystal as in Figure 2.16 indexed 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 2.16 and 2.17 this is (0001) || (111)⁶ and (101) || (100). The plane (100) in Figure 2.17 then 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: {hki0}, (iii) normal to mirror planes: {h0l}, and (iv) equally inclined to two diads {hhl}. The six faces in the form:{h0l}, make a rhombohedron; {102}would be an example, consisting of the planes (102), (102), (012), (01), (10), and (01). This form is similar in appearance to {0hl}, which is also a rhombohedron, rotated 60° with respect to the first one.

The relationship between {h0l} and {0hl} (or, equivalently, {0kl}) is shown in Figure 2.18. They are actually quite separate forms and each one is a special form. We therefore need to add {0hl} to the list of special forms, in addition to {h0l}. It is apparent from Figure 2.16 that when using the rhombohedral cell the two forms {101} and {011} have different indices, since the face above (00) in the projection in Figure 2.16 would have indices (22).

Figure 2.18 The relationship between the special forms {101} and {011} in the trigonal system

The other trigonal point groups are shown in Figure 2.6. In 3 and in there are neither mirrors nor diads. The class 3m has three mirrors intersecting in the triad axis; the x‐, y‐ and u‐axes of the hexagonal cell are taken to lie perpendicular to the mirror planes. The point group 32 contains diads normal to 3 (consistent with the geometry for permissible combinations of rotational symmetry operations in Table 1.2), which are taken as the crystallographic axes. If three mirror planes intersect in an inversion triad, as in m, then diads automatically arise normal to the mirror planes; the diads are chosen as the x‐, y‐ and u‐axes. Finally, we note that the class 3/m is not specified as a point group in the trigonal system: it is placed in the hexagonal system because it is equivalent to .