Crystallography and Crystal Defects. Anthony Kelly
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specified in both ways are given in Figure 2.14. In relating planes and zone axes using Eq. (1.6), it is usually best to work entirely in the three‐index notation for both planes and directions and to translate the three‐index notation for a direction into the four‐index system at the end of the calculation. It is also useful to note that the condition needed for a four‐index vector [UVTW] to lie in a four‐index plane (hkil) is:
(2.5)
Other point groups besides 6/mmm in the hexagonal system are shown in Figure 2.6. We note that (≡ 3/m) is placed in this system because of the use of rotoinversion axes to describe symmetry operations of the second sort; 6,
, 6/m and 6mm show no diad axes, just like their counterparts in the tetragonal system. The crystal axes for 6mm are usually chosen to be perpendicular to one set of mirrors (they then lie in the other set) and
m2 could be developed as
m (≡ 3/mm). The diads automatically arise and are chosen as crystallographic axes. Of course, 622 contains diads. It could be developed as 62, since the second set of diads arises automatically (see Table 1.2). The axes are chosen parallel to one set of diads. Only 6/m and 6/mmm are centrosymmetric in this system.
It is apparent from the stereogram in Figure 2.12b that stereograms with 0001 at the centre showing {hki0} poles are straightforward to plot. To plot more general poles on a stereogram with 0001 at the centre, it is apparent that the c/a ratio has to be used. Thus, for example, this has to be used to determine the angle between faces such as (0001) and (hhl), for example (11
1). It is convenient to choose a (hh
l) plane because such a plane is equally inclined to the x‐ and the y‐axes.
From Figure 2.15, the angle θ between the (0001) pole and the (hhl) pole is seen to be given by:
(2.6)
Figure 2.15 Geometry to determine the angle θ between the (0001) pole and the () pole
Similarly, the angle θ between (0001) and (h0l) is:
(2.7)
An example of a stereogram centred at (0001) with poles of the forms {111}, {10
1} and {12
1} indicated for a hexagonal cell is shown in Section 2.6 in connection with crystals of the trigonal system. The special forms in the various classes of the hexagonal system are listed in Table 2.1.
2.6 Trigonal System
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 (1
0), 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 (0
1), (001), (010) and (01
).
Figure 2.16 A stereogram