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 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].
1.8 Space Lattices (Bravais12 Lattices)
All of the symmetry operations in a crystal must be mutually consistent. There are no fivefold axes of rotational symmetry because such axes are not consistent with the translational symmetry of the lattice. In Section 1.6 we derived the possible combinations of pure rotational symmetry operations that can pass through a point. These combinations are classified into different crystal systems and we will now investigate the types of space lattice (that is, the regular arrangement of points in three dimensions as defined in Section 1.1) that are consistent with the various combinations of rotation axes. We shall find, as we did for the two‐dimensional lattice (or net) consistent with mirror symmetry (Section 1.5), that more than one arrangement of points is consistent with a given set of rotational symmetry operations. However, the number of essentially different arrangements of points is limited to 14. These are the 14 Bravais, or space, lattices. Our derivation is by no means a rigorous one because we do not show that our solutions are unique. This derivation of the Bravais lattices is introduced to provide a background for a clear understanding of the properties of imperfections studied in Part II of this book. The lattice is the most important symmetry concept for the discussion of dislocations and martensitic transformations.
We start with the planar lattices or nets illustrated in Figure 1.14. To build up a space lattice we stack these nets regularly above one another to form an infinite set of parallel sheets of spacing z. All of the sheets are in identical orientation with respect to an axis of rotation normal to their plane, so that corresponding lattice vectors t1 and t2 in the nets are always parallel. The stacking envisaged is shown in Figure 1.18. The vector t3 joining lattice points in adjacent nets is held constant from net to net. The triplet of vectors t1, t2, t3 defines a unit cell of the Bravais lattice.
Figure 1.18 Stacking of nets to build up a space lattice. The triplet of vectors t1, t2, t3 defines a unit cell of the Bravais lattice
We now consider the net based on a parallelogram (Figure 1.14a). If we stack nets of this form so that the points of intersection of twofold axes in successive nets do not lie vertically above one another then we destroy the twofold symmetry axes normal to the nets. We have a lattice of points showing no rotational symmetry. The unit cell is an arbitrary parallelepiped with edges a, b, c, no two of which are necessarily equal, and where the angles of the unit cell α, β, γ can take any value; the cell is shown in Figure 1.19a. By choosing a, b, c appropriately we can always ensure that the cell is primitive. Although this lattice contains no axis of rotational symmetry, the set of lattice points is of course necessarily centrosymmetric.