reciprocallattice

Reciprocal lattice is a mathematical construct used in crystallography and solid-state physics to describe the periodicity of a crystal in reciprocal space. For a real-space Bravais lattice defined by primitive vectors a1, a2, a3, the reciprocal lattice is defined by vectors b1, b2, b3 that satisfy bi · aj = 2π δij. Equivalently, b1 = 2π (a2 × a3) / V, b2 = 2π (a3 × a1) / V, b3 = 2π (a1 × a2) / V, where V = a1 · (a2 × a3) is the cell volume. Any reciprocal-lattice vector is G = h b1 + k b2 + l b3 with integers h, k, l.

The magnitude |G| is related to the interplanar spacing d_hkl of the corresponding real-space planes by |G| = 2π / d_hkl. The reciprocal lattice provides a natural framework for diffraction: constructive interference occurs when the scattering vector equals a reciprocal-lattice vector, satisfying the Laue condition. In diffraction experiments, the observed reflections correspond to reciprocal lattice points, with intensities governed by the structure factor F(G), which depends on the arrangement of atoms within the unit cell.

Geometrically, the first Brillouin zone is the Wigner–Seitz cell of the reciprocal lattice and plays a central role in electronic band theory. The reciprocal lattice is the Fourier transform of the direct lattice and, despite the presence of a basis, preserves lattice periodicity in reciprocal space. Real-space and reciprocal-space descriptions are complementary tools for analyzing crystal structure, diffraction patterns, and electronic structure.