A charged impurity dropped into a metal's free-electron gas doesn't feel a bare 1/r Coulomb field for long — the mobile electrons redistribute around it, piling up (for a positive charge) or thinning out (for a negative one) until the net field decays exponentially instead of algebraically. This is Thomas-Fermi screening, the T = 0, degenerate-gas cousin of classical Debye screening. This 2D view looks straight down on the impurity's plane, so the exponential falloff of the cloud is read directly as a shrinking disc radius.
k_F = (3π² n)^(1/3) Fermi wavevector
E_F = ħ²k_F² / (2mₑ) Fermi energy
g(E_F) = mₑk_F / (π²ħ²) density of states at E_F
k_TF² = e² g(E_F) / ε₀ Thomas-Fermi wavevector
φ(r) = (Ze / 4πε₀r) · e^(−r/λ_TF) screened potential, λ_TF = 1/k_TF
- Carrier density n — sets how many conduction electrons per unit volume respond; higher n means a larger Fermi energy and a much shorter screening length (real metals sit around n ≈ 5–25 ×10²⁸ m⁻³, giving λ_TF ≈ 0.4–1 Å).
- Impurity charge Z — sign and size of the point charge at the centre; the electron cloud (blue) is pulled in for Z > 0 and pushed out for Z < 0.
- Toggle bare Coulomb — overlays the unscreened 1/r potential on the graph so you can see how sharply the exponential cutoff differs from it.
Real-world relevance: Thomas-Fermi screening is why metals are opaque and why doped semiconductors, electrolytes and even nuclear matter all shield internal charges over a finite, density-dependent range rather than an infinite one.