nLab Bravais lattice

Contents

Context

Representation theory

Solid state physics

Contents

Idea

A Bravais lattice (terminology originating in the study of crystals in solid state physics) is the equivalence class of a crystal lattice with maximal point symmetry.

Mathematically, this the equivalence class of the corresponding crystallographic group ΛG pt dO(d)\Lambda \rtimes G_{pt} \;\subset\; \mathbb{R}^d \rtimes \mathrm{O}(d) (for G ptStab O(d)(Λ)G_{pt} \coloneqq Stab_{O(d)}(\Lambda) the full stabilizer group) under conjugation by ambient linear transformations ϕ\phi \,\in\, GL ( d ) GL(d) :

ΛG ptϕ(Λ)(ϕG ptϕ 1). \Lambda \rtimes G_{pt} \;\;\sim\;\; \phi(\Lambda) \rtimes \big( \phi \circ G_{pt} \circ \phi^{-1} \big) \,.

(eg. Schwarzenberger 1972, p. 325)

Variant definitions exist, for instance it is of interest to retain handedness and divide out only by SL ( d ) SL(d) .

All crystal lattices with non-maximal point symmetry may be understood as unions of affine transformations of Bravais lattices.

References

The concept goes back to:

  • 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]]

A good review is in:

Further discussion:

Lecture notes and textbook accounts:

See also:

Last revised on June 14, 2022 at 07:11:51. See the history of this page for a list of all contributions to it.