Engineering & Science
Bloch-Grüneisen Metal Resistivity: T^5 Cryogenic vs Linear High-T Law
Model pure metal electrical resistivity ρ(T) across the Thirring high-temperature expansion ρ_0 + ρ_Θ*(T/Θ_R)*(1 - Θ_R^2/(18*T^2)) and Bloch’s cryogenic T^5 acoustic-phonon law.
Why pure copper resistivity scales linearly with T at 300 K—but collapses as T^5 at 20 K!
Above half the resistive Debye temperature (T >= 0.5 * Θ_R), all acoustic phonon modes are thermally populated and electron-phonon scattering grows linearly with temperature according to the Bloch-Grüneisen Thirring expansion: ρ(T) = ρ_0 + ρ_Θ * (T / Θ_R) * (1 - Θ_R^2 / (18 * T^2)). Deep in the cryogenic regime (T << Θ_R), both phonon population (~T^3) AND small-angle scattering inefficiency (~T^2) combine into Felix Bloch’s famous T^5 law: ρ_low(T) = ρ_0 + 497.6 * ρ_Θ * (T / Θ_R)^5!
For oxygen-free high-conductivity copper (ρ_0 = 0.0020 μΩ·cm, Θ_R = 333 K, ρ_Θ = 1.88 μΩ·cm): at T = 300 K, high-T Bloch-Grüneisen predicts ρ(300 K) = 1.5798 μΩ·cm (climbing to 3.3314 μΩ·cm at 600 K), whereas cooling to T = 20 K slashes phonon scattering to 0.000731 μΩ·cm so total resistivity settles at 0.002731 μΩ·cm!
| Copper Temperature Regime (ΘR = 333 K, ρΘ = 1.88 μΩ·cm) | Scaling Law Active | Predicted Resistivity ρ(T) | Scattering Physics |
|---|---|---|---|
| Cryogenic Magnet Coil (T = 20 K) | ρ0 + 497.6 · ρΘ · (T/ΘR)^5 | 0.002731 μΩ·cm | Small-Angle T^5 Phonon Freeze-Out |
| Ambient Interconnect (T = 300 K) | Thirring Linear-T Expansion | 1.5798 μΩ·cm | Full Brillouin-Zone Phonon Bath |
| Power Busbar Hot Spot (T = 600 K) | Thirring Linear-T Expansion | 3.3314 μΩ·cm | Asymptotic Linear Slope ρΘ / ΘR |
High-Temperature Bloch-Grüneisen Resistivity (T >= 0.5 * Θ_R, μΩ·cm)
Evaluates the quantum-corrected Thirring expansion ρ_0 + ρ_Θ * (T / Θ_R) * (1 - Θ_R^2 / (18 * T^2)) for ambient and elevated temperatures.
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0, B5 < 0.5 * B3), "Use Low-T T^5 Formula or Check Inputs", ROUND(B2 + B4 * (B5 / B3) * (1 - (B3 ^ 2) / (18 * (B5 ^ 2))), 4))Cryogenic Bloch T^5 Resistivity Limit (T <= 0.15 * Θ_R, μΩ·cm)
Evaluates ρ_0 + 497.6 * ρ_Θ * (T / Θ_R)^5 using 4 * 124.4 = 497.6 from the ζ(5) Riemann integral.
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid BG Inputs", ROUND(B2 + 497.6 * B4 * ((B5 / B3) ^ 5), 6))Try the example
Use the original values or change them to test your understanding.