Translational Symmetry
II. Crystal Systems 
The trigonal and hexagonal unitcell information in the table below is reference material only.
Seven Crystal Systems
Most crystals usually have some type of rotational symmetry combined with the pure translational symmetry of the lattice. We now consider the constraints that rotational symmetry imposes upon lattices.
The figure below represents a twodimensional section of the
abplane of a lattice, red balls indicating equivalent lattice points.
With no rotational symmetry constraints imposed upon the lattice, the angle
γ may take any arbitrary value.
Suppose a twofold rotation axis
is combined with the lattice along the baxis direction as shown below:
Combining a lattice with the different orders of rotation symmetry leads to the seven possible crystal systems. The lattices of the seven crystal systems consist of an infinite array of identical points each with an identical environment. The point symmetry of this environment is known as the syngony of the lattice. The symbols of the syngonies will be explained later. The seven possible syngonies characterise the seven crystal systems:
Crystal System  Characteristic Symmetry 
Syngony  UnitCell Parameters  Independent Parameters 

Triclinic  1× 1fold  1  a ≠ b ≠ c; α ≠ β ≠ γ  6 
Monoclinic  1× 2fold  2/m  a ≠ b ≠ c; α = γ = 90°; β ≠ 90°  4 
Orthorhombic  3× 2fold  mmm  a ≠ b ≠ c; α = β = γ = 90°  3 
Tetragonal  1× 4fold  4/mmm  a = b ≠ c; α = β = γ = 90°  2 
Trigonal (see note)  1× 3fold  6/mmm (P) 3m (R) 
a = b ≠ c; α = β = 90°; γ = 120°  2 
Hexagonal  1× 6fold  6/mmm  a = b ≠ c; α = β = 90°; γ = 120°  2 
Cubic  4× 3fold  m3m  a = b = c; α = β = γ = 90°  1 
Note: In the trigonal system, the syngony of the lattice may have a higher symmetry than the characteristic symmetry of the crystal system depending on whether the lattice is primitive (P) or rhombohedrally centred (R).
The rhombohedral system (R) is a subgroup of the trigonal system in which either a triplycentred hexagonal cell may be used (as in the table) or, alternatively, a primitive rhombohedral cell may be used with a = b = c, α = β = γ ≠ 90°. The number of cell parameters is two irrespective of the choice of unit cell.
© Copyright 19952006.
Birkbeck College, University of London.

Author(s):
Jeremy Karl Cockcroft Huub Driessen David Moss Andreas Wostrack 