
Introduction
In order to define the Brillouin zone we need to define first the reciprocal lattice. The set of all wave vectors K that yield plane waves with the periodicity of a given Bravais lattice is known as its reciprocal lattice. Analytically, K belongs to the reciprocal lattice of a Bravais lattice of points R, provided that the relation
holds for any r, and all R in Bravais lattice. Factoring out
we can characterize the reciprocal lattice as the set of wave vectors K satisfying
for all R in the Bravais lattice.
The reciprocal lattice is itself a Bravais lattice
and its primitive vectors can be generated from the vectors of the direct
lattice. Let a_{1},
a_{2},
a_{3}_{ }
be a set of primitive vectors, then the reciprocal lattice can be generated
by the three primitive vectors:
Using the relations between direct and reciprocal lattice it can be
shown that the reciprocal lattice of sc is sc (at k
space), the reciprocal of bcc is fcc, and reciprocal of fcc
is bcc.
The first Brillouin zone is defined to be the WignerSeitz
primitive cell of the reciprocal lattice, or it could be defined as the
set of points in k space that can be reached from the origin without
crossing any Bragg plane. The second Brillouin zone is the set of points
that can be reached from the first zone by crossing only one Bragg plane.
The
Alternatively, the nth Brillouin zone can
be defined as the set of points that can be reached from the origin by
crossing
The first three Brillouin zones for some lattices
are illustrated below (they were constructed by the program BrillouinZone):
















Programs
We have two program packages to generate Brillouin zones:

