Matthiessen’s rule at cryogenic temperatures For a conductor with total resistivity ρ = ρi + aT (ρi: residual resistivity; a: constant; T: absolute temperature), which statement best describes the low-temperature limit?

Introduction / Context:
Matthiessen’s rule decomposes metal resistivity into a temperature-independent part from defects/impurities and a temperature-dependent phonon (lattice vibration) part. Understanding the limiting behavior guides low-temperature measurements and purity assessments of metals and cryogenic wiring.

Given Data / Assumptions:

• Total resistivity: ρ = ρi + aT.
• ρi is residual (defect) resistivity; aT is phonon contribution.
• “Very low temperatures” means T approaches 0 K but is not exactly zero.

Concept / Approach:

As T decreases, phonon scattering diminishes rapidly, so the temperature-dependent term aT shrinks toward zero. The resistivity then approaches the residual limit ρi set by impurities and defects. At sufficiently low T, aT is much smaller than ρi, though strictly aT = 0 only at T = 0 K (idealized limit).

Step-by-Step Solution:

Verification / Alternative check:

Residual resistivity ratio (RRR = ρ(300 K)/ρ(4.2 K)) is large for pure metals, evidencing that low-T resistivity is dominated by ρi, with aT ≪ ρi near liquid helium temperatures.

Why Other Options Are Wrong:

A claims aT = 0, which holds only at exactly 0 K. B and C contradict low-T behavior; E states exact equality without basis.

Common Pitfalls:

Assuming the phonon term vanishes at any small T; confusing trend (“much less than”) with exact equality at 0 K.

Final Answer:

aT < ρi