Читать книгу Crystallography and Crystal Defects - Anthony Kelly - Страница 23

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 1.3 The 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°

^a Rhombohedral unit cell.

^b This 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].