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 kF, 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
This 2D view renders the same radially-symmetric δn(r) two ways at once: the left square is a flat cross-section slice through the impurity's plane, colored pixel-by-pixel from r = √(x²+y²) using the exact formula above (a 2D density-field map, not a camera view of a 3D cloud); the right strip is the same δn(r) drawn as a signed curve against radius, with the probe marked on both.
- kF 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 ring/marker; the readout shows the exact signed density perturbation there under whichever model is active.
- Friedel / Thomas–Fermi buttons — switch which formula drives the field and curve, 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.