Section 2.2.7.2. The Brillouin zone (BZ)

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 face-centred-cubic (f.c.c.) structure. With (2.2.2.4) we easily find that the reciprocal lattice is body-centred-cubic (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.

Figure 2.2.7.1 | top | pdf | 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).