# Symmetry, Crystal Systems and Bravais Lattices

## The Role of Symmetry

During this course we will focus on discussing crystals with a discrete translational symmetry, i.e. crystals which are formed by the combination of a Bravais lattice and a corresponding basis.

Despite this restriction there are still many different lattices left satisfying the condition. However, there are some lattices types that occur particularly often in nature. Or some - actually distinct - lattices share certain properties. These aspects arise from the symmetry of the lattices.

Furthermore, when describing spatial arrangements of objects this is done by using a cordinate system. The choice of the coordinate system (e.g. cartesian, cylindrical, spherical, ...) usually depends on the symmetry as well. Therefore it is helpful to classify crystal structures according to its symmetry.

### Point Symmetry

Besides the translational symmetry mentioned above we will now also make use of point symmetries, i.e. the group of symmetry operations that leaves at least one point unchanged. These comprise rotations, reflections, inversions or any combinations of these.^{[1]}
They are the basis of the classification of cystals: One can for example count the number of axes of rotation and their respective multiplicities in order to compare different crystals in regard of their symmetry.

However, I do not want to go into further details of symmetry and group theory at this point because it is rather abstract and not essential for the following. If you are interested in this you may look for a good book on crystallography.

## The Seven Crystal Systems

*First notice:* The intention of the following listing is to give you an overview rather than making you feel required to learn them by heart!

In a first step one divides the Bravais lattices into 7 crystal systems which are defined by the lengths $a$, $b$, $c$ and angles $\alpha$, $\beta$, $\gamma$ between the primitive translation vectors. The resulting crystal systems are listed and visualised below.^{[2]}^{[3]}^{[4]}

Crystal System | Lengths | Angles |
---|---|---|

cubic | $a = b = c$ | $\alpha = \beta = \gamma = 90^\circ$ |

trigonal | $a = b = c$ | $\alpha = \beta = \gamma \lt 120^\circ$, $\neq 90^\circ$ |

hexagonal | $a = b \neq c$ | $\alpha = \beta = 90^\circ$, $\gamma = 120^\circ$ |

tetragonal | $a = b \neq c$ | $\alpha = \beta = \gamma = 90^\circ$ |

orthorhombic | $a \neq b \neq c$ | $\alpha = \beta = \gamma = 90^\circ$ |

monoclinic | $a \neq b \neq c$ | $\alpha = \beta = 90^\circ \neq \gamma$ |

triclinic | $a \neq b \neq c$ | $\alpha \neq \beta \neq \gamma$ |

## The 14 Bravais Lattices

So one classifies different lattices according to the shape of the parallelepiped spanned by its primitive translation vectors.

However, this is not yet the best solution for a classification with respect to symmetry. Consider for example the unit cells (a) and (b) presented before: While cell (a) is the actual unit cell spanned by the primitive translation vectors, it does not show the symmetry of the lattice properly whereas cell (b) clearly shows the two axes of rotation.

So sometimes it makes sense not to use a primitive unit cell but one which fits better to the symmetry of the problem. This idea leads to the 14 Bravais Lattices which are depicted below ordered by the crystal systems:

### Cubic

### Tetragonal

### Orthorhombic

### Hexagonal

### Monoclinic

### Trigonal and Triclinic

## References

[1] | Festkörperphysik De Gruyter 2014 (p. 8f.) |

[2] | Festkörperphysik De Gruyter 2014 (ch. 1.1.2.2.) |

[3] | Oxford Solid State Basics Oxford 2013 (p. 122) |

[4] | Festkörperphysik De Gruyter 2014 (p. 58) |