ThomasFermiScreening

Thomas-Fermi screening refers to the screening of electric fields by electrons in a degenerate electron gas within the Thomas-Fermi model, a semiclassical approach introduced by Llewellyn Thomas and Enrico Fermi in the late 1920s. It provides a self-consistent description of how a static external potential is shielded by the rearrangement of electrons in atoms and metals.

In this framework the electron density n(r) is treated as a local function of the electrostatic potential φ(r). For a uniform system at zero temperature, n(r) is related to the local chemical potential μ − e φ(r) by n(r) = (1/3π^2) (2m/ħ^2)^{3/2} [μ − e φ(r)]^{3/2} for μ > e φ(r), and n = 0 otherwise. The electrostatic potential satisfies Poisson’s equation ∇^2 φ = −ρ/ε0 with ρ = e[n(r) − N+(r)], where N+ is the positive background charge (ions). Linearizing around a neutral background yields a screened potential of the form φ(r) ∝ e^{−k_TF r}/r, where k_TF is the Thomas-Fermi wavevector and the inverse length 1/k_TF defines the screening length.

The resulting Thomas-Fermi screening length is typically on the order of tenths of an angstrom to a few angstroms in metals, reflecting short-range screening. The TF model captures the qualitative tendency of electronic systems to screen external perturbations but has limitations: it ignores quantum oscillations (Friedel oscillations), exchange and correlation effects, and detailed band structure; it provides only a rough description of finite-temperature effects.

Extensions of the idea include finite-temperature Thomas-Fermi theory and more sophisticated approaches such as density functional theory and the Lindhard dielectric function, which incorporate quantum mechanics more fully.