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 a₁, a₂, a₃  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 Wigner-Seitz 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 (n + 1)th Brillouin zone is the set of points not in the (n - 1)th zone that can be reached from the nth zone by crossing n - 1 Bragg planes.
    Alternatively, the nth Brillouin zone can be defined as the set of points that can be reached from the origin by crossing n - 1 Bragg planes, but no fewer.
    The first three Brillouin zones for some lattices are illustrated below (they were constructed by the program BrillouinZone):
 

	First zone	Second zone	Third zone
sc
fcc
bcc

Programs

    We have two program packages to generate Brillouin zones:

• zone - by Moshe Levi
• BrillouinZone - by Nir Shvalb