For convenience define a new space which can be related to the real crystal lattice space by the relations between basis vectors:
b1 = [a2a3]/v0 b2 = [a3a1]/v0 b3 = [a2a1]/v0
These new basis vectors define a new lattice, called the reciprocal lattice. The volume v0 of the unit cell in real space is a1a2a3 cm3, and the volume v00 of the unit cell in this reciprocal space or k-space is b1b2b3 cm-3.
The periodic lattice field creates energy gaps in the bands of allowed quantum states as explained in the introduction. In k-space these occur at critical values of k = ± p/a. In 3 dimensions an allowed energy band contains k-states for every direction through the crystal, and the lattice structure fixes the critical k values at ± p/a and the associated energy gaps, differently in the different crystallographic directions of this motion. The map of this variation in a k-space diagram is the Brillouin Zone of the structure
To find in this representation the Bragg reflection conditions equivalent to k = ± p/a in one dimension we construct in the reciprocal lattice a Wigner-Seitz cell by drawing connecting lines from the origin to nearby lattice sites and then bisecting each such line by a plane. The Wigner-Seitz cell is the volume inside all these planes. The boundary of this cell defines the condition for Bragg reflection , as shown. And because of this the Wigner-Seitz cell in reciprocal space is the Brillouin zone of the crystal represented by this particular reciprocal lattice. The number of quantum (wave) states in the Brillouin zone is equal to the number of unit cells in the crystal. There are of course other reflecting planes in a crystal of closer spacing and corresponding to these there are other bisecting planes and higher zones. The nth Brillouin zone is the zone bounded by the set of points that can be reached from the origin by crossing n-1 bisecting planes. The higher zones are seen to be fragmented pieces separated from each other by the lower zones inside them. Each higher zone translates into the first zone. This is a general result and means that each zone occupies the same volume of k-space. It further follows that, not just the first zone, but every zone contains N quantum states for a crystal of N unit cells, each of which can of course hold 2 electrons of opposite spin.
Wigner-Seitz cells are also used to ennumerate 'vibrational states' of the crystal lattice.
Prepared by Laura Malcolm -- last update : March 27, 1997