Langmuir wave

A longitudinal electron plasma oscillation whose long-wavelength frequency approaches the electron plasma frequency when ions can be treated as stationary. Its dispersion relation and Landau damping depend on the electron phase-space distribution function. Compact velocity support can permit exactly undamped branches outside the support.

The long-wavelength root of the degenerate-plasma dielectric response. Expanding its logarithm at large $s=\omega /(k{v}_F)$ gives $ϵ=1-{\omega }_p^2/{\omega }^2-3{\omega }_p^2{k}^2{v}_F^2/(5{\omega }^4)+\cdots$. The occupied Fermi sphere has $⟨v_{\parallel }^2⟩=v_F^2/5$, accounting for the dispersive coefficient. The static electron response instead defines the Thomas-Fermi screening length $v_F/(\sqrt{3}{\omega }_p)$.