Anthony Kelly - Crystallography and Crystal Defects

Combinations of successive rotations about different axes are always inextricably related in groups of three. This arises because a rotation about an axis of unit length, say n A, of an amount α followed by a rotation about another axis of unit length, say n B, of amount β can always be expressed as a single rotation about some third axis of unit length, n C, of amount γ ′. 11

In an orthonormal coordinate system (one in which the axes are of equal length and at 90° to one another) the rotation matrix Rdescribing a rotation of an amount θ (in radians) about an axis nwith direction cosines n 1, n 2and n 3takes the form (Section A1.4):

(1.23)

If we first apply a rotation of an amount α about an axis n A, described by a rotation matrix R A, after which we apply a rotation of an amount β about an axis n B, described by a rotation matrix R B, then in terms of matrix algebra the overall rotation is:

(1.24)

where R Cis the rotation matrix corresponding to the equivalent single rotation of an amount γ ′ about an axis n C. It is evident that one way of deriving γ ′ and the direction cosines n 1C, n 2Cand n 3Cis to work through the algebra suggested by Eq. (1.24)and to use the properties of the rotation matrix evident from Eq. (1.23):

(1.25)

(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 q Band q Aare multiplied together. The multiplication law for two quaternions pand qis (Section A1.4):

(1.29)

The quaternion p· qis 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 ( n 1A n 1B+ n 2A n 2B+ n 3A n 3B) is simply the cosine of the angle between n Aand n B, and therefore n A· n B, since n Aand n Bare 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 n Ais a tetrad, so that α = 90° and α /2 = 45°. Suppose the rotation about n Bis a diad, so that β = 180° and β /2 = 90°. Then, in Eq. (1.32):

(1.34)

Since n A· n Bhas to be less than one for non‐trivial solutions of Eq. (1.34), the possible solutions of Eq. (1.34) are when n Aand n Bmake 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 n 1C, n 2Cand n 3Cof the axis of rotation, n C, are given by the expressions:

(1.35)

because sin γ = sin γ ′. If we choose n Ato be the unit vector [001] in the reference orthonormal coordinate system then for case (i) we can choose n Bto be a vector such as [100]. Hence in Eq. (1.30):

(1.36)

and so:

(1.37)

That is, γ = 180°, γ ′ = 180°, and n Cis a unit vector parallel to [1 0]. This possibility is illustrated in Figure 1.17a, with axes A, B and C lettered according to these assumptions. The other diad axes marked in Figure 1.17a must automatically be present, since A is a tetrad.

Figure 1.17Examples 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