# nLab crystal

Contents

### Context

#### Solid state physics

This entry is about the notion of “crystal” in solid state physics. For the notion in algebraic geometry see at crystal (algebraic geometry).

# Contents

## Idea

In solid state physics a crystal is a bound state of atomic nuclei and electrons (much as atoms and molecules are, too, but) in which the nuclei occupy a discrete “regular lattice” which is invariant – as a discrete subset of Euclidean space and as far as the crystal extends – under a discrete subgroup of the Euclidean group of translations, reflections and rotations.

These subgroups are also known as crystallographic groups and are used to classify crytalline structures.

In addition to these spatial symmetries, the dynamics of the electrons in a crystalline material may also have internal symmetries and/or CPT-symmetries (like time-reversal symmetry) which together with the crystallographic symmetry affect the physical properties of the crystalline material, for instance whether it behaves like a conductor/semi-conductor/insulator with respect to electric currents or whether it admits topological phases of matter which are “protected” by these symmetries.

## References

Historical articles identifying the point-lattice nature of crystals:

• Auguste Bravais, Mémoire sur les Systèmes Formés par les Points Distribués Régulièrement sur un Plan ou dans L’espace, J. Ecole Polytech. 19 (1850) 1 $[$ark:12148/bpt6k96124j$]$

• William Barlow, Probable Nature of the Internal Symmetry of Crystals, Nature 29 (1883) 186–188 $[$doi:10.1038/029186a0$]$

Textbook accounts:

With focus on the mathematics of crystallographic groups:

• Harold Hilton, Mathematical crystallography and the theory of groups of movements, Oxford Clarendon Press (1903) $[$web$]$

• Peter Engel, Geometric Crystallography – An Axiomatic Introduction to Crystallography, D. Reidel Publishing (1986) $[$doi:10.1007/978-94-009-4760-3$]$