Читать книгу Introduction to Solid State Physics for Materials Engineers - Emil Zolotoyabko - Страница 16

As was already mentioned, translational symmetry imposes tight constraints on possible kinds of rotation axes in crystals. This is one of the strongest and most important results in crystallography. In fact, let us consider the network of equivalent points in space, which are related to each other by translational symmetry (i.e. are produced from some origin by linear combinations of translation vectors, see Eq. (1.1)). Suppose that the rotation axis, which is characterized by the elementary rotation angle φ, crosses one of these points (say point A), perpendicularly to the plane of drawing (Figure 1.17). Point B is shifted from point A by translation vector a, the distance AB being equal to the translation length, a. Since point B belongs to the same network, the identical rotation axis is passing also through it. Let rotate point B by angle φ about the axis passing through point A. In such a way, we receive third equivalent point B′. Repeating this procedure for point A (i.e. rotating it by angle φ about the axis passing through point B), we obtain fourth equivalent point A′. Being equivalent, all four points belong to the same network. It means that the distance A′B′ should be an integer number of translations, a (since vector A′B′ is parallel to vector AB = a), i.e. A′B′ = pa, where p is an integer. On the other hand, by considering the geometry in Figure 1.17, one finds that A′B′ = AB + 2AB′ · sin (φ−90°). Using AB = AB′ = a yields pa = a + 2a· sin(φ−90°), and finally:

(1.A.1)

Restrictions imposed on possible types of rotation axes by Eq. (1.A.1), mathematically follow from two conditions: |cos φ | ≤ 1 and integer numbers of p. Possible values of cos φ and p, as well as the order of the axis, n = 360°/φ, are given in Table 1.2.

First numerical line in Table 1.2 contains trivial symmetry element, i.e. rotation axis 1 (n = 1, φ = 360°). It is trivial since each geometrical figure transforms into itself under rotation by 360°. According to Table 1.2, as non-trivial elements, there are only four different rotation axes in crystals, which are compatible with translational symmetry: 2 (n = 2, φ = 180°), 3 (n = 3, φ = 120°), 4 (n = 4, φ = 90°), and 6 (n = 6, φ = 60°).

As next important question, we ask: how many rotation axes of the same high order (3, 4, or 6) can simultaneously exist in a crystal? To address it, let take one n-fold axis, marked as OA in Figure 1.18, and add to it another axis of this kind (OB) inclined by angle δ. Then, we can rotate the axis OB around OA by the elementary rotation angle, φ = 360°/n, and produce the third axis, OD. The duplication procedure can be continued. However, all these axes are equivalent and, thus, the angles between them should be equal δ. The latter condition imposes strong restrictions on possible configurations of equivalent rotation axes in crystals and angles between them.

Table 1.2 Possible types of rotation axes permitted by translational symmetry.

p	cos φ	φ
−1	1	360°	1
0	½	60°	6
1	0	90°	4
2	−½	120°	3
3	−1	180°	2

Figure 1.18 Illustration of the simultaneous appearance of several high-order rotation axes in a crystal.

In fact, it follows from triangle DOB that DB = 2⋅DO⋅sin(δ/2). The plane of triangle DAB is perpendicular to the axis OA and hence the angle ∠ OAD = 90°. If so, DA = DO⋅sinδ and from triangle DAB, the segment DB = 2⋅DA⋅sin(φ/2) = 2⋅DO⋅sin(φ/2)⋅sin δ . Comparing two expressions for segment DB, given in this paragraph, yields , or

(1.A.2)

For a threefold rotation axis (axis 3, φ = 120°), Eq. (1.A.2) predicts that and, correspondingly, cosδ = −1/3. This angle, δ = 109.47°, is between the space cube diagonals, e.g. between the [111]- and []-diagonals. In other words, crystal symmetry permits the existence of single axis 3, as follows from translational symmetry, or the simultaneous existence of four such axes (no more, not less), arranged in 3D space as spatial diagonals of a cube. The arrangement of four threefold rotation axes is the leading combination in the cubic symmetry system, existing in all five its classes (point groups) (Table 1.1).

For a fourfold rotation axis (4, φ = 90°), Eq. (1.A.2) gives cos, i.e. δ = 90°. This is the angle between cube edges. In other words, crystal symmetry permits the existence of single axis 4, or three such axes, arranged as cube edges, i.e. with angles δ = 90° between them. This combination exists in classes 432 and of the cubic symmetry system (Table 1.1). In cubic class , we find three fourfold roto-inversion axes, which combine 90° rotation followed by inversion operation.

For a sixfold rotation axis (6, φ = 60°), Eq. (1.A.2) yields cos(δ/2) = 1, i.e. δ = 0. In other words, crystal symmetry allows the existence of single sixfold axis only in all classes of hexagonal symmetry (Table 1.1).