Crystallography and Crystal Defects. Anthony Kelly
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As a third example, suppose that the rotation about nA is a hexad, so that α = 60° and α/2 = 30°, and suppose the rotation about nB is a tetrad, so that β = 90° and β/2 = 45°. Under these circumstances, Eq. (1.32) becomes:
(1.41)
Since nA · nB has to be less than 1, and cos
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 1.2 Permissible combinations of rotation axes in crystals
Axes | α | β | γ | u | v | w | System | ||
A | B | C | |||||||
2 | 2 | 2 | 180° | 180° | 180° | 90° | 90° | 90° | Orthorhombic |
2 | 2 | 3 | 180° | 180° | 120° | 90° | 90° | 60° | Trigonal |
2 | 2 | 4 | 180° | 180° | 90° | 90° | 90° | 45° | Tetragonal |
2 | 2 | 6 | 180° | 180° | 60° | 90° | 90° | 30° | Hexagonal |
2 | 3 | 3 | 180° | 120° | 120° | 70.53° | 54.74° | 54.74° | Cubic |
2 | 3 | 4 | 180° | 120° | 90° | 54.74° | 45° | 35.26° | Cubic |
u is the angle between nB and nC, v is the angle between nC and nA, and w is the angle between nA and nB.
In deriving these possibilities from Eqs. (1.33) and (1.35), it is useful to note that cos−1
1.7 Crystal Systems
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