Engineering & Science
Matthiessen’s Rule & Cryogenic Residual Resistivity Ratio (RRR)
Separate impurity/defect resistivity rho_0 from thermal phonon scattering rho_ph(T) via Augustus Matthiessen’s Rule rho_tot = rho_0 + rho_ph and grade cryogenic copper/niobium purity via RRR = rho(300 K) / rho(4.2 K).
Why CERN accelerator cavities & MRI magnets specify RRR >= 300 instead of "99.99% pure"!
At room temperature (300 K), electron-phonon lattice vibrations dominate copper’s electrical resistivity (~16.78 nOhm*m), masking trace parts-per-million oxygen, sulfur, and dislocation defects! Cool that same metal into liquid helium (4.2 K), and thermal phonons freeze out completely—leaving ONLY static impurity and grain-boundary scattering rho_0 via Augustus Matthiessen’s (1864) additive scattering law: rho(T) = rho_0 + rho_ph(T)!
Cryogenic and superconducting engineers therefore grade high-purity Oxygen-Free High-Conductivity (OFHC) copper stabilizer and SRF niobium cavities by the dimensionless Residual Resistivity Ratio RRR = rho(300 K) / rho(4.2 K) = (rho_0 + rho_ph,300) / rho_0! High-purity RRR-300 OFHC copper with rho_0 = 0.056 nOhm*m and phonon resistivity rho_ph(300 K) = 16.78 nOhm*m has rho(300 K) = 16.836 nOhm*m and RRR = 16.836 / 0.056 = 300.6, whereas commercial ETP wire with rho_0 = 0.350 nOhm*m drops to RRR = 48.9!
| Copper Cryogenic Grade (rho_ph,300K = 16.78 nOhm*m) | 4.2 K Residual rho_0 (nOhm*m) | 300 K Total rho (nOhm*m) | Residual Resistivity Ratio RRR |
|---|---|---|---|
| Superconducting Magnet Stabilizer (OFHC) | 0.056 nOhm*m | 16.836 nOhm*m | 300.6 (High Cryogenic Conductivity) |
| Commercial Electrolytic Tough Pitch (ETP) | 0.350 nOhm*m | 17.130 nOhm*m | 48.9 |
| Cold-Worked Brass Alloy | 42.000 nOhm*m | 58.780 nOhm*m | 1.4 (Impurity Dominated) |
Total Electrical Resistivity by Matthiessen’s Rule rho_tot = rho_0 + rho_ph(T) (nOhm*m)
Sums temperature-independent defect/impurity resistivity B2 and lattice phonon resistivity B3.
=IF(OR(B2 <= 0, B3 < 0), "Invalid Resistivity Inputs", ROUND(B2 + B3, 3))Cryogenic Residual Resistivity Ratio RRR = (rho_0 + rho_ph,300K) / rho_0
Divides 300 K total resistivity by 4.2 K residual defect resistivity rho_0 (assuming negligible 4.2 K phonon scattering).
=IF(OR(B2 <= 0, B3 < 0), "Invalid Resistivity Inputs", ROUND((B2 + B3) / B2, 1))Try the example
Use the original values or change them to test your understanding.