Wigner Crystal

Electron localization into a crystalline lattice at low density (high rₛ)

rₛ ≡ (3/4πn)^(1/3)/aB
Crystal: rₛ ≳ 100 (2D ≳ 37)
Γ = e²/(rₛ·kT) = coupling
Wigner Crystal: At sufficiently low electron density (high rₛ = r₀/aB), Coulomb repulsion energy dominates kinetic energy, and electrons spontaneously localize into a periodic triangular lattice. The dimensionless coupling parameter: Γ = (potential energy) / (kinetic energy) = e²/(kT · rₛaB) ~ rₛ/T Wigner (1934) predicted crystallization at rₛ ≳ 100 (3D); in 2D the threshold is rₛ ≳ 37. This simulation uses a 2D Ewald-summed Coulomb potential with Langevin dynamics. The pair correlation function g(r) reveals short-range vs long-range order.