Brillouinzone

Brillouin zone is the primitive cell of the reciprocal lattice of a crystal. It is defined as the Wigner–Seitz cell around the origin in reciprocal space and represents the set of wavevectors k that are physically equivalent under the lattice translations. The first Brillouin zone consists of all k points closer to the origin than to any other reciprocal lattice point; higher-order zones are defined similarly relative to other reciprocal points.

Construction and geometry are based on planes that are perpendicular to reciprocal lattice vectors and bisect the lines from the origin to neighboring reciprocal points. This yields a polyhedral region in k-space whose boundaries are the Bragg planes, reflecting conditions for constructive interference of waves scattered by the lattice.

Shapes of the first Brillouin zone depend on the crystal lattice. The simple cubic lattice has a cubic first Brillouin zone. For a body-centered cubic lattice the zone is a truncated octahedron, while for a face-centered cubic lattice the zone is a rhombic dodecahedron. These shapes are the Wigner–Seitz cells of the corresponding reciprocal lattices.

Significance in solid-state physics is substantial. The Brillouin zone provides a natural domain for labeling electron states via Bloch wavevectors, and is central to the study of electronic band structure. Energy bands are often plotted along high-symmetry paths within the first zone, and band gaps frequently occur at zone boundaries due to Bragg reflection. The concept also underpins interpretations of diffraction patterns and the relationship between real-space lattices and reciprocal space in various dimensions.