Wiedemann-Franz Law: Electron Gas Transport in a Metal

Sample

Metal preset

Temperature T 300 K

Impurity / defect level 10%

Applied electric field E 40%

Live Transport Readouts

Electrical σ

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S/m

Thermal κ

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W/(m·K)

Lorenz ratio L / L₀ (Wiedemann–Franz)

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L₀ = 2.44×10⁻⁸ WΩK⁻²

Scattering time τ

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Mean free path ℓ

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How it works

Conduction electrons in a metal are treated as a free Fermi gas that scatters off lattice vibrations (phonons) and impurities, exactly as in the Drude–Sommerfeld model:

σ = n e² τ / mₑ            (electrical conductivity)
κ = (π²/3)(k_B/e)² σ T    (Wiedemann–Franz law)
L₀ = κ/(σT) = (π²/3)(k_B/e)² ≈ 2.44×10⁻⁸ WΩK⁻²

The same electrons that carry charge also carry heat, so the two conductivities are locked together by the universal Lorenz number L₀ — true for any metal, independent of n, τ or crystal structure, as long as scattering is elastic.

Total scattering rate 1/τ combines two channels via Matthiessen's rule: 1/τ = 1/τ_impurity + 1/τ_phonon(T). Elastic impurity scattering never spoils L/L₀ = 1, but inelastic small-angle electron-phonon scattering (dominant around T ≈ θ_D/5, where θ_D is the Debye temperature) is far more effective at relaxing the electric current than the heat current, so the Lorenz ratio dips measurably below 1 in that intermediate range and recovers at both very low and very high T — a real, textbook deviation from the "ideal" Wiedemann–Franz law.

Metal preset — sets the real electron density n and Debye temperature θ_D for Cu, Ag, Al or Fe.
Temperature — drives phonon-scattering resistivity ρ_ph(T) and the Lorenz-ratio dip.
Impurity level — adds elastic residual resistivity; more impurities suppress the dip and pin L/L₀ closer to 1, since elastic scattering never violates Wiedemann–Franz.
Electric field — visual drift bias on the electron gas; higher conductivity produces a faster net drift for the same field.

Every particle in the 3D view undergoes real thermal jitter plus a Monte-Carlo scattering event with probability dt/τ per frame, so the animation speed of collisions directly tracks the computed relaxation time.

Conduction electrons Ion cores (lattice)

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