nLab crystal


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



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.


Historical articles identifying the point-lattice nature of crystals:

Textbook accounts:

With focus on the mathematics of crystallographic groups:

See also:

Last revised on September 10, 2022 at 11:47:07. See the history of this page for a list of all contributions to it.