Kristallpotential

Kristallpotential, commonly referred to as crystal potential in solid-state physics, denotes the periodic potential energy landscape experienced by charged particles in a crystalline solid due to the orderly arrangement of ions and electron density. It represents the electrostatic and exchange-correlation potential produced by the lattice and is periodic with the lattice: U(r+R) = U(r) for all lattice vectors R.

Mathematically, the crystal potential can be written as U(r) = sum_R V(r − R), where V is the potential associated with a single lattice site. In reciprocal space it has Fourier components U_G corresponding to the reciprocal lattice vectors G. The periodic potential leads to Bloch's theorem: electron wavefunctions take the form ψ_k(r) = e^{i k·r} u_k(r), with u_k(r) periodic in the lattice.

The crystal potential underpins the formation of electronic energy bands and band gaps. Models such as the nearly-free electron model treat the potential as a weak perturbation to free electrons, while the tight-binding model uses localized atomic-like orbitals. More sophisticated descriptions employ pseudopotentials or ab initio methods to account for core electrons and many-body effects.

Because it governs the motion of electrons and holes, the crystal potential determines a material's electrical, optical, and transport properties. It is central to semiconductor physics and to calculations of band structure, density of states, and effective masses used to describe charge carrier dynamics.