The Labyrinth of Symmetry in crystallography
Table of Contents
The symmetry analysis is done first for simple Bravais lattice and then for the crystal structures. Bravais lattice is the simplest one, as it contains similar points in a space in different patterns. In crystal structure contains atoms, ions or molecules at each points of the bravais lattice. Hence,
Crystal structure = Bravais lattice + atoms/ions/molecules/basis
1. Bravais Lattice, Space group, Point Groups, Crystal symmetry
- From the group theory point of view the set of operations which takes a Bravias lattice into itself is called a symmetry group. A symmetry group can contains three types of operations: (i) translational, (ii) point symmetry operations (rotation, inversion, reflection), (iii) both translational and point symmetry operations.
- Point symmetry groups are the set of operations which transforms the lattice into itself after series of symmetry operations, while keeping a point fixed.
- Bravais lattice and space groups are same, the first is the terminology from lattice point of view and the second terminology is from group theory point of view. There are 14 space groups.
- Crystal symmetry and Point groups are the same, first is the terminology from lattice point of view and the second terminology is from group theory point of view. There are 7 space groups.
2. List of point group which is in a space group
- The Wikipedia article: Crystal Structure
- A bravais lattice/space group will always contain elements from the crystal symmetry/point groups. As the point groups are less than the space group, we can write a point group and enumerate all the space groups inside which contains that point group.
- For example the cubic point group is contained in the FCC, BCC and simple cubic Bravais lattice.
| Point Groups/ | Properties of | Space Groups/ | List of Bravais Lattice | Properties of | Comments |
|---|---|---|---|---|---|
| Crystal Systems | Point group | Bravais Lattice | space group | ||
| Cubic | \(a=b=c\) | 3 | Primitive | ||
| body-centered | |||||
| face-centered | |||||
| Tetragonal | \(a=b\neq c\) | 2 | FCC, primitive | ||
| Orthorhombic | \(a \neq b\neq c\) | 4 | primitive | ||
| base centered | |||||
| body centered | |||||
| face centered | |||||
| Monoclinic | \(a \neq b\neq c\) | ||||
| \(\angle ab = \alpha =\pi/2\) | 2 | primitive | |||
| \(\beta bc = \beta =\pi/2\) | base centered | ||||
| \(\gamma ac = \gamma \neq \pi/2\) | |||||
| Triclinic | \(a \neq b\neq c\) | ||||
| \(\angle ab = \alpha \neq\pi/2\) | 1 | primitive | |||
| \(\angle bc = \beta \neq\pi/2\) | |||||
| \(\angle ac = \gamma \neq\pi/2\) | |||||
| Trigonal | \(a = b = c\) | 1 | primitive | ||
| \(\alpha=\beta=\gamma \neq \pi/2\) | |||||
| Hexagonal | base is hexagon | 1 | primitive |
3. Crystallographic point group and space group
- The Bravais lattice with arbitrary basis gives rise to the crystal structure. In the previous cases we analyse the system where the basis of the bravais lattice is symmetric, i.e. a sphere. However in this case the basis of the bravais lattice is arbitrary.
- There are 32 point groups and 230 space groups.
- The five symmetry operations are:
- Rotation by \(2\pi/n\) around some axis
- Rotation-reflection
- Rotation-inversion
- Reflection
- Inversion