Thomas-Fermi Screening in a Metal

Electron Gas

Carrier density n 8.5 ×10²⁸ m⁻³

Impurity charge Z +1 e

Cloud settling speed 1.0×

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Fermi wavevector k_F

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Fermi energy E_F

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Screening length λ_TF

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DOS at E_F

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How it works

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.

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.

Impurity charge Screening electron cloud

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