Brillouin zone

The first Brillouin zone is a fundamental cell in reciprocal space. For a one-dimensional lattice of spacing $a$, one conventional choice is

The first Brillouin zone is $|k_x|<K,|k_y|<L$. The second consists of four disconnected lobes. Its right lobe has vertices $(K,-L),(2K,-L+K^2/L),(2K,L-K^2/L),(K,L)$; its upper lobe has vertices $(-K,L),(K,L),(0,L+K^2/L)$. Reflection in the coordinate axes gives the other two. The bisectors to $(2K,2L)$ and $(4K,0)$ truncate the lobes. Their combined area is $4KL$, equal to the first zone's area. Counting the Bragg planes crossed from the origin proves the construction.

The Gamma point is zero crystal wavevector, modulo reciprocal-lattice vectors. A direct band gap at this point allows a low-energy vertical optical transition with negligible photon momentum. Crystal symmetry constrains the degeneracies and symmetry labels of the band-edge states.

A reduced Brillouin zone is the reciprocal primitive cell for an enlarged real-space unit cell. Doubling a one-dimensional unit cell halves the momentum interval but adds a second internal sublattice degree of freedom. Band folding therefore preserves the number of normal modes.

The extended-zone region reached from the first zone by crossing one Bragg plane from the origin, excluding boundaries. For a one-dimensional lattice with spacing $a$, its two components are $\pi /a<|k|<2\pi /a$. It is not simply a translated first-zone cell.

The Wigner-Seitz cell of a reciprocal lattice centered at the origin. It contains wavevectors at least as close to zero as to any other reciprocal lattice point. Boundary sharing conventions do not change its area or volume.

Putting $M=m$ in the diatomic-chain dispersion relation with equal springs gives ${\omega }_{-}=2\sqrt{K/m}|\sin (kd/2)|$ and ${\omega }_{+}=2\sqrt{K/m}|\cos (kd/2)|$ in the reduced zone $|k|\le \pi /(2d)$. The branches touch at its edge. If the sites are actually identical, the true primitive unit cell has length $d$; the apparent optical phonon branch is the outer part of the monatomic-chain dispersion relation folded by an unnecessarily doubled cell. A gap does not appear merely from redrawing the zone.

A Bragg point is a crystal momentum at which two plane waves differing by a reciprocal lattice vector are degenerate. A periodic potential mixes the waves there, producing an avoided crossing and opening a band gap.

The reciprocal lattice of a two-dimensional triangular Bravais lattice is triangular, so its Wigner-Seitz cell is a regular hexagon. Opposite boundary points are identified modulo reciprocal-lattice translations, and the six corners fall into two inequivalent classes of three corners each, conventionally denoted $K$ and $K^{\prime }$.