Crystallography and Crystal Defects. Anthony Kelly
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(1.26)
using the Einstein summation convention (Section A1.4).
An equivalent, more elegant, way is to use quaternion algebra. The rotation matrix described by Eq. (1.23) is homomorphic (i.e. exactly equivalent) to the unit quaternion:
(1.27)
satisfying:
(1.28)
In quaternion algebra, the equation equivalent to Eq. (1.24) is one in which two quaternions qB and qA are multiplied together. The multiplication law for two quaternions p and q is (Section A1.4):
(1.29)
The quaternion p · q is also a unit quaternion, from which the angle and the direction cosines of the axis of rotation of this unit quaternion can be extracted using Eq. (1.27). Therefore, if:
(1.30)
the angle γ′ satisfies the equation:
(1.31)
The term (n1A n1B + n2A n2B + n3A n3B) is simply the cosine of the angle between nA and nB, and therefore nA · nB, since nA and nB are of unit length. Defining γ = 2π − γ′, so that γ and γ′ are the same angle measured in opposite directions, and rearranging the equation, it is evident that:
(1.32)
Permitted values of γ are 60°, 90°, 120° and 180°, as shown in Table 1.1, and so possible values of cos γ/2 are:
(1.33)
To apply these results to crystals, let us assume that the rotation about nA is a tetrad, so that α = 90° and α/2 = 45°. Suppose the rotation about nB is a diad, so that β = 180° and β/2 = 90°. Then, in Eq. (1.32):
(1.34)
Since nA · nB has to be less than one for non‐trivial solutions of Eq. (1.34), the possible solutions of Eq. (1.34) are when nA and nB make an angle of (i) 90° or (ii) 45° with one another, corresponding to values of γ of 180° and 120°, respectively. From Eq. (1.29) the direction cosines n1C, n2C and n3C of the axis of rotation, nC, are given by the expressions:
(1.35)
because sin
(1.36)
and so:
(1.37)
That is, γ = 180°, γ′ = 180°, and nC is a unit vector parallel to [1
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 nA to be [001]. nB 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 nC 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,