1.4.4 Reflection Symmetry
The fourth symmetry is reflection symmetry. The operation is that of reflection in a mirror. Mirror symmetry relates, for example, our left and right hands. The dotted plane running normal to the paper, marked m in Figure 1.12, reflects object A to A′, and vice versa. A′ cannot be moved about in the plane of the paper and made to superpose on A. The dotted plane is a mirror plane. It should be noted that the operation of inversion through a centre of symmetry also produces a right‐handed object from a left‐handed one.
Figure 1.12 Reflection symmetry
1.5 Restrictions on Symmetry Operations
All crystals show translational symmetry.8 A given crystal may, or may not, possess other symmetry operations. Axes of rotational symmetry must be consistent with the translational symmetry of the lattice. A onefold rotation axis is obviously consistent. To prove that in addition only diads, triads, tetrads and hexads can occur in a crystal, we consider just a two‐dimensional lattice or net.
Let A, A′, A″, … in Figure 1.13 be lattice points of the mesh and let us choose the direction AA′A″ so that the lattice translation vector t of the mesh in this direction is the shortest lattice translation vector of the net. Suppose an axis of n‐fold rotational symmetry runs normal to the net at A. Then the point A′ must be repeated at B by an anticlockwise rotation through an angle α = A′AB = 2π/n. Also, since A′ is a lattice point exactly similar to A, there must be an n‐fold axis of rotational symmetry passing normal to the paper through A′. This repeats A at B′ through a clockwise rotation, as shown in Figure 1.13. That these two rotations are in opposite senses does not matter – they are both a consequence of the n‐fold axis of rotational symmetry under consideration.
Figure 1.13 Diagram to help determine which rotation axes are consistent with translational symmetry
B and B′ define a lattice row parallel to AA′. Therefore, the separation of B and B′ by Eq. (1.12) must be an integral number times t. Call this integer N. From Figure 1.13 the separation of B and B′ is (t − 2t cos α). Therefore, the possible values of α are restricted to those satisfying the equation:
or:
(1.22)
where N is an integer. Since −1 ≤ cos α ≤ 1 the only possible solutions are shown in Table 1.1. These correspond to onefold, sixfold, fourfold, threefold, and twofold axes of rotational symmetry. No other axis of rotational symmetry is consistent with the translational symmetry of a lattice and hence other axes do not occur in crystals.9
Table 1.1 Solutions of Eq. (1.22)
| N | −1 | 0 | 1 | 2 | 3 |
| cos α | 1 |
|
0 |
− |
−1 |
| α | 0° | 60° | 90° | 120° | 180° |
Corresponding to the various allowed values of α derived from Eq. (1.22), three two‐dimensional lattices, also known as nets or meshes, are defined. These are shown as the first three diagrams on the left‐hand side of Figure 1.14. It should be noted that the hexad axis and the triad axis both require the same triequiangular mesh, the unit cell of which is a 120° rhombus (see Figure 1.14c).
Figure 1.14 The five symmetrical plane lattices or nets. Rotational symmetry axes normal to the paper are indicated by the following symbols: ♦ = diad; ▴ = triad; ▪ = tetrad;
In the same way that the possession of rotational symmetry axes perpendicular to the net places restriction on the net, restrictions are placed upon the net by the possession of a mirror plane: consideration