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 of this identifies the two additional nets shown in Figures 1.14d and e. To see this, let A and A′ be two lattice points of a net and let the vector t joining them be a lattice translation vector defining one edge of the unit cell. A mirror plane can be placed normal to the lattice row AA′, as in Figure 1.15a, or as in Figure 1.15b. It cannot be placed arbitrarily anywhere in between A and A′. It must either lie midway between A and A′, as in Figure 1.15a, or pass through a lattice point, as in Figure 1.15b. Since AA′ determines a row of lattice points, a net can be built up consistent with mirror symmetry by placing a row identical to AA′ parallel with AA′, but displaced from it. There are just two possible arrangements, which are both shown in Figure 1.16, with the original lattice vector t indicated and all of the mirror planes consistent with the arrangement of the lattice points marked on the two diagrams. Hence, the spatial arrangements shown in Figure 1.16 give rise to the nets shown in Figures 1.14d and e.
Figure 1.15 Restrictions placed on two‐dimensional lattices through the imposition of mirror planes. These can be placed normal to a lattice row AA′ only as in (a) or (b)
Figure 1.16 The two possible arrangements of nets consistent with
