to Figure 1.14a, the left‐hand diagram shows the net of points consistent with a twofold axis of symmetry normal to the net and with no axis of higher symmetry. This net corresponds with the solutions N = −1 or 3, α = 0 or 180° in Table 1.1 and is based on a parallelogram. The lengths of two adjacent sides of the parallelogram (a, b in Figure 1.14a) and the value of the included angle γ can be chosen arbitrarily without removing the consistency with twofold symmetry. If a motif showing twofold symmetry normal to the net were associated with each lattice point, then twofold symmetry axes would be present at all the points shown in the right‐hand diagram of Figure 1.14a.
All regular nets of points are consistent with twofold symmetry axes normal to the net because such a net of points is necessarily centrosymmetric and in two dimensions there is no difference between a centre of symmetry and a twofold or diad axis. The net corresponding to N = 1, α = 90° in Table 1.1 is based upon a square, shown in the left‐hand diagram of Figure 1.14b. If a two‐dimensional crystal possesses fourfold symmetry, it must necessarily possess this net. In addition, the atomic motif associated with each of the lattice or net points must also possess fourfold rotational symmetry. Provided the motif fulfils this condition, there must be a fourfold axis at the centre of each of the basic squares of the net and twofold axes at the midpoints of the sides, as shown in the right‐hand diagram of Figure 1.14b. A two‐dimensional crystal possessing fourfold rotational symmetry cannot possess fewer symmetry operations than those shown in the right‐hand diagram of Figure 1.14b.10
The net consistent with both α = 60° and α = 120°, corresponding to the possession of hexad symmetry and triad symmetry, respectively, is the triequiangular net shown in Figure 1.14c. The primitive unit cell of this net has both sides equal and the included angle is necessarily 120°. It must be noted clearly that such a mesh of points is always consistent with sixfold symmetry. If the atomic motif associated with each lattice point is consistent with sixfold symmetry then diad and triad axes are automatically present, as shown in the central diagram of Figure 1.14c. A two‐dimensional crystal will possess threefold symmetry provided the atomic motif placed at each lattice point of the lattice shown in Figure 1.14c possesses threefold symmetry. The only symmetry operations necessarily present are then just the threefold axes arranged as indicated in the far right‐hand diagram of Figure 1.14c.
The simple rectangular net shown in Figure 1.14d has a primitive cell with a and b not necessarily equal. The angle between a and b is 90°. This net of points is consistent with the presence of diad axes at the intersection of the mirror planes, as is the mesh shown in Figure 1.14e. The simplest unit cell for the net in Figure 1.14e is a rhombus, indicated with the dotted lines. This has the two sides of the cell equal, and the angle between them, γ, can take any value. When dealing with a net based on a rhombus, it is, however, often convenient to choose as the unit cell a rectangle which contains an additional lattice point at its centre. This cell, outlined with full lines in the left‐hand diagram of Figure 1.14e, has the angle between a and b necessarily equal to 90°. Hence it contains an additional lattice point inside it, which is called a non‐primitive unit cell. The primitive unit cell is the dotted rhombus. The non‐primitive cell clearly has twice the area of the primitive one and contains twice as many lattice points. It is chosen because it is naturally related to the symmetry, and is called the centred rectangular cell. This feature of choosing a non‐primitive cell, because it is more naturally related to the symmetry operations, is one we shall meet often when dealing with the three‐dimensional space lattices. The arrangements of diad axes and mirror planes consistent with the rectangular net and with the centred rectangular net are shown in the right‐hand diagrams of Figures 1.14d and 1.14e, respectively.
1.6 Possible Combinations of Rotational Symmetries
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 nA, of an amount α followed by a rotation about another axis of unit length, say nB, of amount β can always be expressed as a single rotation about some third axis of unit length, nC, 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 nA, described by a rotation matrix RA, after which we apply a rotation of an amount β about an axis nB, described by a rotation matrix RB, then in terms of matrix algebra the overall rotation is:
(1.24)
where RC is the rotation matrix corresponding to the equivalent single rotation of an amount γ′ about an axis nC. It is evident that one way of deriving γ′ and the direction cosines n1C, n2C and n3C 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)
