Crystallography and Crystal Defects. Anthony Kelly

Figure 1.17 Examples of possible allowed combinations of rotational symmetries in crystals. In (a) a tetrad, A, is perpendicular to two diads, B and C, at 45° to one another, while in (b) the tetrad, A, is 45° away from a diad, B, and 54.74° from a triad, C, with B and C 35.26° apart. In (a) other diad axes which must be present are also indicated, and in (b) other triad axes (but not other tetrad and diad axes) which must be present are also indicated

For case (ii), we can again choose n_A to be [001]. n_B can be chosen to be a vector such as:

      (1.38)

      Therefore, in Eq. (1.30):

      (1.39)

      and so:

      (1.40)

      i.e. γ = 120°, γ′ = 240°, and n_C is a unit vector parallel to [111], making an angle of 54.74° with the fourfold axis and 35.26° with the twofold axis. This arrangement is shown in Figure 1.17b, again with the original axes marked. It should be noted that the presence of the tetrad at A automatically requires the presence of the other triad axes (and of other diads, not shown), since the fourfold symmetry about A must be satisfied. The triad axes lie at 70.53° to one another.

      As a third example, suppose that the rotation about n_A is a hexad, so that α = 60° and α/2 = 30°, and suppose the rotation about n_B is a tetrad, so that β = 90° and β/2 = 45°. Under these circumstances, Eq. (1.32) becomes:

(1.41)

      Since n_A · n_B has to be less than 1, and cos γ ≥ 0, because from Table 1.1 permitted values of γ are 60°, 90°, 120° and 180°, it follows that there are no solutions for n_A and n_B in Eq. (1.41) for Statement (1.33) to be valid. Therefore, we have shown that a sixfold axis and a fourfold axis cannot be combined together in a crystal to produce a rotation equivalent to a single sixfold, fourfold, threefold or twofold axis.

Statement (1.33) and Eq. (1.35) can be studied to find the possible combinations of rotational axes in crystals. The resulting permissible combinations and the angles between the axes corresponding to these are listed in Table 1.2, following M.J. Buerger [7].

Table 1.2 Permissible combinations of rotation axes in crystals

      u is the angle between n_B and n_C, v is the angle between n_C and n_A, and w is the angle between n_A and n_B.

In deriving these possibilities from Eqs. (1.33) and (1.35), it is useful to note that cos⁻¹ = 54.74°, cos⁻¹ = 35.26°, and cos⁻¹(1/3) = 70.53°. The sets of related rotations shown in Table 1.2 can always