Anthony Kelly - Crystallography and Crystal Defects

For case (ii), we can again choose n Ato be [001]. n Bcan be chosen to be a vector such as:

(1.38)

Therefore, in Eq. (1.30):

(1.39)

and so:

(1.40)

i.e. γ = 120°, γ ′ = 240°, and n Cis a unit vector parallel to [111], making an angle of 54.74° with the fourfold axis and 35.26° with the twofold axis. This arrangement is shown in Figure 1.17b, again with the original axes marked. It should be noted that the presence of the tetrad at A automatically requires the presence of the other triad axes (and of other diads, not shown), since the fourfold symmetry about A must be satisfied. The triad axes lie at 70.53° to one another.

As a third example, suppose that the rotation about n Ais a hexad, so that α = 60° and α /2 = 30°, and suppose the rotation about n Bis a tetrad, so that β = 90° and β /2 = 45°. Under these circumstances, Eq. (1.32) becomes:

(1.41)

Since n A· n Bhas to be less than 1, and cos γ ≥ 0, because from Table 1.1permitted values of γ are 60°, 90°, 120° and 180°, it follows that there are no solutions for n Aand n Bin Eq. (1.41)for Statement (1.33)to be valid. Therefore, we have shown that a sixfold axis and a fourfold axis cannot be combined together in a crystal to produce a rotation equivalent to a single sixfold, fourfold, threefold or twofold axis.

Statement (1.33) and Eq. (1.35)can be studied to find the possible combinations of rotational axes in crystals. The resulting permissible combinations and the angles between the axes corresponding to these are listed in Table 1.2, following M.J. Buerger [7].

Table 1.2Permissible combinations of rotation axes in crystals

u is the angle between n Band n C, v is the angle between n Cand n A, and w is the angle between n Aand n B.

In deriving these possibilities from Eqs. (1.33) and (1.35), it is useful to note that cos −1 = 54.74°, cos −1 = 35.26°, and cos −1(1/3) = 70.53°. The sets of related rotations shown in Table 1.2can always be designated by three numbers, such as 222, 233, or 234, each number indicating the appropriate rotational axis.

The permissible combinations of rotation axes, listed in Table 1.2, are each identified with a crystal system in the far right‐hand column of that table. A crystal system contains all those crystals that possess certain axes of rotational symmetry. In any crystal there is a necessary connection between the possession of an axis of rotational symmetry and the geometry of the lattice of that crystal. We shall explore this in the next section, and we have seen some simple examples in two dimensions in Section 1.5. Because of this connection between the rotational symmetry of the crystal and its lattice, a certain convenient conventional cell can always be chosen in each crystal system. These systems are listed in Table 1.3, in which the name of the system is given, along with the rotational symmetry operation or operations which define the system and the conventional unit cell, which can always be chosen. This cell is in many cases non‐primitive; that is, it contains more than one lattice point. The symbol ≠ means ‘not necessarily equal to’. The general formula for the volume, V , of the unit cell of a crystal with cell dimensions a , b , c , α , β and γ is:

(1.42)

(see Problem 1.17).

Table 1.3The crystal systems

System	Symmetry	Conventional cell
Triclinic	No axes of symmetry	a ≠ b ≠ c ; α ≠ β ≠ γ
Monoclinic	A single diad	a ≠ b ≠ c ; α = γ = 90° < β
Orthorhombic	Three mutually perpendicular diads	a ≠ b ≠ c ; α = β = γ = 90°
Trigonal	A single triad
Tetragonal	A single tetrad	a = b ≠ c ; α = β = γ = 90°
Hexagonal	One hexad	a = b ≠ c ; α = β = 90°, γ = 120°
Cubic	Four triads	a = b = c ; α = β = γ = 90°

aRhombohedral unit cell.

bThis is also the conventional cell of the hexagonal system.

Particular note should be made of the trigonal crystal system in Table 1.3. Here, there are two possible conventional unit cells, one rhombohedral and one the same as the hexagonal crystal system. This is because some trigonal crystals have a crystal structure based on a rhombohedral lattice, while others have a crystal structure based on the primitive hexagonal lattice [12].