nLab Brillouin torus

Contents

Contents

Idea

Given a crystalline material, its Brillouin torus is the space of distinguishable momenta/wave vectors of its electronic excitations. Due to the periodicity of the crystal, also these momenta/wave vectors are (dually) periodically identified, which makes this space an n-torus.

Mathematically, the discrete space underlying a crystal is a subset of a Euclidean space SE n nS \;\subset\;E^n \simeq \mathbb{R}^n (the set of atomic sites) which is preserved (as a subset) by the action of a crystallographic group ΛGIso(E)\Lambda \rtimes G \,\subset\, Iso(E), where Λ\Lambda is a full lattice (and GG the corresponding point group). Then its Brillouin torus is the Pontrjagin dual group of this lattice (eg. Freed-Moore 13):

𝕋 Br n=Hom Grp(Λ,S 1). \mathbb{T}^n_{Br} \,=\, Hom_{Grp}\big(\Lambda, \, S^1 \big) \,.

References

Textbooks on solid state physics traditionally speak of the “reciprocal lattice” (e.g. Kittel 1953, p. 27) which is the dual lattice Hom Grp(Λ,)Hom_{Grp}\big(\Lambda, \, \mathbb{Z} \big), e.g.:

The resulting Brillouin torus Hom Grp(Λ,)/Hom Grp(Λ,)Hom_{Grp}\big(\Lambda, \, \mathbb{R} \big)/Hom_{Grp}\big(\Lambda, \, \mathbb{Z} \big) is often left implicit. It is made explicit in:

Last revised on June 14, 2022 at 06:41:15. See the history of this page for a list of all contributions to it.