Читать книгу Crystallography and Crystal Defects - Anthony Kelly - Страница 21

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; = hexad. Nets in (d) and (e) are both consistent with mirror symmetry, with the mirrors indicated by thick lines

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 mirror symmetry

Returning 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.