Brillouin Zone

Physics: The Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice, defined by reciprocal vectors bᵢ with aᵢ·bⱼ = 2πδᵢⱼ. Square lattice: BZ is a square of side 2π/a with symmetry points Γ(0,0), X(π/a,0), M(π/a,π/a). Hexagonal lattice: hexagonal BZ with special K=(2π/3a, 2π/3√3a) and K' points where graphene Dirac cones live. FCC real lattice has BCC reciprocal lattice (truncated octahedron BZ). Zone folding in superlattices of period na creates mini-BZ boundaries at k = ±π/na, opening mini-gaps and creating flat bands.