15.9: Crystal Systems

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The rotational symmetry of a crystal places constraints on the shape of the conventional unit cell we choose to describe the structure. On this basis we divide all structures into one of 7 crystal systems. For example, for crystals with 4 fold symmetry it will always be possible to choose a unit cell that has a square base with a = b and γ = 90°:

4-fold symmetry example

There are 14 unique combinations of the 7 crystal systems with the possible types of primitive and non-primitive lattices. These are referred to as the 14 Bravais lattices.

Crystal systems, lattices and symmetry elements

Crystal System	Defining Symmetry	Unit Cell Geometry
Triclinic	Translational Only	a≠b≠c; α≠β≠γ
Monoclinic	A diad axis (parallel to [010])	a≠b≠c; α=γ=90°; β>90°
Orthorhombic	3 diads (each should be parallel to each axis)	a≠b≠c; α=β=γ=90°
Trigonal For more information click here	1 triad (parallel to [001])	a=b≠c; α=β=90°; γ=120° ( or a=b=c; 120° > α=β=γ ≠ 90°)
Hexagonal	1 hexad (parallel to [001])	a=b≠c; α=β=90°; γ=120°
Tetragonal	One tetrad (parallel to the [001] vector)	a=b≠c; α=β=γ=90°
Cubic	4 triads (all parallel to <111> directions)	a=b=c; α=β=γ=90°

Bravais Lattice Structures

And you can use the Wolfram Demonstration Project Viewer to look at the Bravais lattices