A vector describing the propagation direction and wavelength of a wave in a crystal.The wave vector is a mathematical description of a wave that encodes both its direction of propagation and its spatial frequency, which is inversely related to wavelength. In a crystal, the wave vector is essential for describing how waves — whether they are electron waves, phonons (lattice vibrations), or photons — move through the periodic arrangement of atoms. It is a fundamental concept in solid-state physics, as it allows scientists to classify the allowed energy states of electrons in a material and to understand phenomena like heat conduction, electrical conductivity, and the interaction of waves with crystal structures.
The concept of the wave vector was developed in the early 20th century alongside the birth of quantum mechanics and solid-state physics, with key contributions from Felix Bloch and Léon Brillouin in the 1920s and 1930s.
The wave vector, often denoted as k, points in the direction the wave travels, and its magnitude equals 2π divided by the wavelength. In a crystal, the periodic potential of the atoms modifies the relationship between the wave vector and energy, leading to the formation of energy bands and gaps. When a wave scatters off the crystal lattice, the change in its wave vector is governed by the crystal's reciprocal lattice, which is a key concept in X-ray diffraction and neutron scattering. The wave vector also determines how heat is carried by phonons — the quantized vibrations of the lattice — and how electrons respond to electric and magnetic fields.
The wave vector is used in the design of semiconductors, in X-ray crystallography to determine molecular structures, and in the analysis of neutron scattering experiments to study magnetic and lattice properties of materials.
In a crystal, the wave vector is only defined up to a multiple of the reciprocal lattice vector, which is why physicists often work in the 'first Brillouin zone' — a concept that is central to understanding electron behavior in solids.
| \omega | variable | angular frequency — angular frequency radian per second |
| \mathbf{k} | variable | wave vector — wave vector radian per metre |
| J | variable | joint density of states — density of states per joule per cubic metre |
| E | variable | photon energy — energy joule |
| E_c | variable | conduction band energy — energy joule |
| E_v | variable | valence band energy — energy joule |
| \mathbf{k} | variable | crystal momentum — wave vector per metre |
| \int_{\text{BZ}} d^3 | operator | integration over the first Brillouin zone — integral over brillouin zone |
| \delta | operator | selects states satisfying energy conservation — dirac delta function |