kristallmomentum

Kristallmomentum, or crystal momentum, is a quantum number that characterizes the motion of electrons in a crystalline solid. It arises from the translational symmetry of a periodic lattice and is closely tied to Bloch states. In a periodic potential, electronic eigenfunctions can be written as ψ_{n,k}(r) = e^{i k·r} u_{n,k}(r), where u_{n,k}(r) has the periodicity of the lattice and k is the wavevector in the reciprocal lattice. The crystal momentum is defined as p_crystal = ħ k and is conserved in the absence of processes that break the lattice periodicity, modulo a reciprocal lattice vector G; physical momentum is not generally conserved due to lattice forces.

Because k is confined to the first Brillouin zone, the energy spectrum forms bands ε_n(k). The group velocity of an electron in band n is v_n,k = (1/ħ) ∇_k ε_n(k). Under an external force F, semiclassical dynamics give ħ dk/dt = -e [E + v_n,k × B], where E and B are external electric and magnetic fields; the velocity is determined by the band structure. Crystal momentum changes by reciprocal lattice vectors under scattering processes that involve lattice momentum transfer, such as Umklapp processes.

Limitations: crystal momentum is not a true mechanical momentum; in finite crystals, at surfaces, with disorder, or with phonons, translational symmetry is broken and crystal momentum is not strictly conserved. Nevertheless it provides a robust framework for understanding electron dynamics in solids.

Significance: crystal momentum is central to band theory and semiconductor physics, underpinning semiclassical transport, Bloch oscillations, and many topological phenomena through the geometry of Bloch bands.