Friedel Oscillations: Screened Charge Density Around an Impurity

Electron Gas

Fermi wavevector k_F 1.20

Impurity charge Z 1.0

Probe radius r 3.00

Live Readouts

δn(r) at probe

0.000

Oscillation λ = π/k_F

2.62

T-F screening length

1.30

Model

Friedel

How it works

Drop a bare point charge Ze into a degenerate electron gas (a metal). The conduction electrons rearrange to screen it — but because the electron gas has a sharp Fermi surface at wavevector k_F, linear-response (Lindhard) theory shows the induced density does not decay smoothly. It oscillates and decays algebraically:

δn(r) ≈ -Z · k_F · cos(2 k_F r) / (2π² r²)   (Friedel oscillation, 3D)

The period of the ripples is set purely by the Fermi surface: λ = π / k_F — it is a direct real-space fingerprint of the Fermi wavevector, independent of the impurity strength. The crude Thomas–Fermi approximation instead assumes a smooth, structureless response and predicts a monotonic Yukawa-like decay:

δn(r) ≈ -Z · k_TF² · e^(−k_TF r) / (4π r)     with k_TF ≈ 1.11·√k_F

k_F slider — sets the electron density (n = k_F³/3π²); higher k_F means a tighter oscillation period and shorter Thomas–Fermi screening length.
Z slider — impurity charge; scales the amplitude of the response linearly (this is linear-response theory).
r probe — moves the highlighted shell; the readout shows the exact signed density perturbation there under whichever model is active.
Friedel / Thomas–Fermi buttons — switch which formula drives the 3D cloud, so you can see the ringing pattern collapse into TF's smooth exponential decay.

Friedel oscillations are directly observable: they are what STM tips image as concentric ripples around a single adsorbed atom on a metal surface, and they mediate the oscillatory RKKY exchange coupling between magnetic impurities in a metal.

Excess electron density (screening) Depleted density Probe shell

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