Читать книгу Crystallography and Crystal Defects - Anthony Kelly - Страница 22

As we have just shown in Section 1.5, the axes of n‐fold rotational symmetry which a crystal can possess are limited to values of n of 1, 2, 3, 4, or 6. These axes lie normal to a net. In principle, a crystal might conceivably be symmetric with respect to many intersecting n‐fold axes. However, it turns out that the possible angular relationships between axes are severely limited. To discover these we need a method to combine the possible rotations. One possible method is to use spherical trigonometric relationships, such as the approach adopted by Euler and developed by Buerger [7,8]. An equivalent approach is to make use of the homomorphism between unit quaternions and rotations, described and developed by Grimmer [9], Altmann [10], and Kuipers [11].

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 R describing a rotation of an amount θ (in radians) about an axis n with direction cosines n₁, n₂ and n₃ takes 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_C is 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_2C and n_3C is 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_B and q_A 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 (n_1A n_1B + n_2A n_2B + n_3A n_3B) is simply the cosine of the angle between n_A and n_B, and therefore n_A · n_B, since n_A and n_B 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 n_A is a tetrad, so that α = 90° and α/2 = 45°. Suppose the rotation about n_B is a diad, so that β = 180° and β/2 = 90°. Then, in Eq. (1.32):

(1.34)

Since n_A · n_B has to be less than one for non‐trivial solutions of Eq. (1.34), the possible solutions of Eq. (1.34) are when n_A and n_B 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 n_1C, n_2C and n_3C of the axis of rotation, n_C, are given by the expressions:

(1.35)

because sin γ = sin γ′. If we choose n_A to be the unit vector [001] in the reference orthonormal coordinate system then for case (i) we can choose n_B to 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_C is a unit vector parallel to [10]. 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 1.17 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

Axes	α	β	γ	u	v	w	System
A	B	C
2	2	2	180°	180°	180°	90°	90°	90°	Orthorhombic
2	2	3	180°	180°	120°	90°	90°	60°	Trigonal
2	2	4	180°	180°	90°	90°	90°	45°	Tetragonal
2	2	6	180°	180°	60°	90°	90°	30°	Hexagonal
2	3	3	180°	120°	120°	70.53°	54.74°	54.74°	Cubic
2	3	4	180°	120°	90°	54.74°	45°	35.26°	Cubic

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 be designated by three numbers, such as 222, 233, or 234, each number indicating the appropriate rotational axis.