Brillouin Zone

Any Bravais lattice will become reciprocal lattice after Fourier transformation. In physics, reciprocal lattice is often used to describe the periodic of the momentum space (also called "k-space"). For any kind of Bravais lattice, if the lattice vectors \tilde{a}_1, \tilde{a}_2 , and \tilde{a}_3 are fixed, the corresponding reciprocal lattice vectors are defined as

The first Brillouin zone is a primitive cell in the reciprocal space, and be the set of points in k-space that can be reached from origin without crossing any Bragg plane. The irreducible Brillouin zone be the first Brillouin zone reduced by all of the symmetries in the point group of the lattice. All the corner of the Brillouin zone be a linear transformation by the reciprocal lattice vectors :

The importance of the Brillouin zone is derived from the Bloch wave description of waves in a periodic medium where the solutions can be fully characterized by their behavior in a single Brillouin zone.

Since we transferred the lattice vectors \tilde{a}_1, \tilde{a}_2 , and \tilde{a}_3 into the lattice translation vectors a_1, a_2, and a_3 in FAME, the associated Brillouin zone need to be adjusted. In the following, we provide original Brillouin zone of various lattice and their transformation relation. It is worth nothing that the band structure can be computed by solving the eigenvalue problems associated with the k's along the segments connecting any two corner points of the associated irreducible Brillouin zone.