kristallmomenta

Kristallmomentum, or crystal momentum, is a quantum-mechanical concept used to describe the momentum-like character of electrons in a crystalline solid. In a periodic lattice potential, Bloch's theorem states that electron eigenstates can be written as psi_{n,k}(r) = e^{i k·r} u_{n,k}(r), where u_{n,k}(r) has the lattice periodicity. The quantity ħk serves as the crystal momentum and acts as a label for the Bloch states in the n-th energy band.

Crystal momentum is conserved with respect to the lattice’s translational symmetry and is defined only modulo a reciprocal lattice vector G; states with k and k+G describe the same physical state. It is not the actual physical momentum of the electron, but a quasi-momentum that governs the electron’s behavior inside the periodic potential.

Under external forces, the crystal momentum evolves according to ħ dk/dt = F_ext, where F_ext is the applied force (for example, in an electric field E, F_ext = -eE). The group velocity of a Bloch electron is given by v_n(k) = (1/ħ) ∇_k E_n(k), linking crystal momentum to the band structure E_n(k).

Crystal momentum is central to understanding electronic structure, transport, and optical properties of solids. It underpins phenomena such as Bloch oscillations and Brillouin-zone dynamics. In modern solid-state physics, Berry phase and related topological properties are defined as integrals over crystal momentum space, highlighting the fundamental role of k in characterizing electronic states.