International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. D, ch. 2.2, p. 298
Section 2.2.7. The vector and the Brillouin zone^{a}Institut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165TC, A1060 Vienna, Austria 
The vector plays a fundamental role in the electronic structure of a solid. In the above, several interpretations have been given for the vector that
Starting with one of the 14 Bravais lattices, one can define the reciprocal lattice [according to (2.2.2.4)] by the Wigner–Seitz construction as discussed in Section 2.2.2.2. The advantage of using the BZ instead of the parallelepiped spanned by the three unit vectors is its symmetry. Let us take a simple example first, namely an element (say copper) that crystallizes in the facecentredcubic (f.c.c.) structure. With (2.2.2.4) we easily find that the reciprocal lattice is bodycentredcubic (bcc) and the corresponding BZ is shown in Fig. 2.2.7.1. In this case, f.c.c. Cu has symmetry with 48 symmetry operations (point group). The energy eigenvalues within a star of (i.e. ) are the same, and therefore it is sufficient to calculate one member in the star. Consequently, it is enough to consider the irreducible wedge of the BZ (called the IBZ). In the present example, this corresponds to 1/48th of the BZ shown in Fig. 2.2.7.1. To count the number of states in the BZ, one counts each point in the IBZ with a proper weight to represent the star of this vector.

The Brillouin zone (BZ) and the irreducible wedge of the BZ for the f.c.c. direct lattice. After the corresponding figure from the Bilbao Crystallographic Server (http://www.cryst.ehu.es/ ). The IBZ for any space group can be obtained by using the option KVEC and specifying the space group (in this case No. 225). 
The BZ is purely constructed from the reciprocal lattice and thus only follows from the translational symmetry (of the 14 Bravais lattices). However, the energy bands , with lying within the first BZ, possess a symmetry associated with one of the 230 space groups. Therefore one can not simply use the geometrical symmetry of the BZ to find its irreducible wedge, although this is tempting. Since the effort of computing energy eigenvalues increases with the number of points, one wishes to restrict such calculations to the basic domain, but the latter can only be found by considering the space group of the corresponding crystal (including the basis with all atomic positions).
One possible procedure for finding the IBZ is the following. First a uniform grid in reciprocal space is generated by dividing the three unitcell vectors by an integer number of times. This is easy to do in the parallelepiped, spanned by the three unitcell vectors, and yields a (moreorless) uniform grid of points. Now one must go through the complete grid of points and extract a list of nonequivalent points by applying to each point in the grid the pointgroup operations. If a point is found that is already in the list, its weight is increased by 1, otherwise it is added to the list. This procedure can easily be programmed and is often used when integrations are needed. The disadvantage of this scheme is that the generated points in the IBZ are not necessarily in a connected region of the BZ, since one member of the star of is chosen arbitrarily, namely the first that is found by going through the complete list.