Brillouin zone

Zones de Brillouin (Fr). Brillouin Zonen (Ge). Zonas de Brillouin (Sp). Zone di Brillouin (It). ブリュアンゾーン (Ja)

Definition

A Brillouin Zone is a particular choice of the unit cell of the reciprocal lattice. It is defined as the Wigner-Seitz cell (also called Dirichlet or Voronoi Domain) of the reciprocal lattice. It is constructed as the set of points enclosed by the Bragg planes, the planes perpendicular to a connection line from the origin to each lattice point and passing through the midpoint. Alternatively, it is the set of points closer to the origin than to any other reciprocal lattice point. The whole reciprocal space may be covered without overlap with copies of such a Brillouin Zone.

For high-symmetry lattices one introduces sometimes the notion of n-th Brillouin Zone. This is the set of points one reaches with a straight line from the origin and passing through n-1 Bragg Planes. In this terminology, the Brillouin Zone defined above is the first Brillouin Zone. The n-th Brillouin Zone is a shell around lower Brillouin Zones and their shape becomes fast rather complicated.

Vectors in the Brillouin Zone or on its boundary characterize states in a system with lattice periodicity, e.g. phonons or electrons. States are non-equivalent if they belong to different vectors in a unit cell of the reciprocal lattice. This is not necessarily the Brillouin Zone. Especially, for low-symmetry systems Brillouin Zones are sometimes difficult to visualize, and another choice of unit cell may be useful, e.g. the parallelepiped spanned b the basis vectors.

See also

Section 2.2 of International Tables of Crystallography, Volume D

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