Past exam of the physics A course of the University of Cambridge / 2018 / Paper 2 / Section C / C11 / d / Solution

In a normal metal, the electrical resistivity can often be approximated using Matthiessen's rule,Static impurities, defects and grain boundaries scatter conduction Electrons even as $T\to 0$, giving the residual resistivity. Thermal lattice phonons add a temperature-dependent contribution through electron-phonon scattering. In a clean three-dimensional acoustic model, small-angle electron-phonon transport at low temperature gives ${\rho }_{\mathrm{phonon}}\propto T^5$ at low $T$; once thermal phonon populations are approximately classical, the resistivity is approximately linear in $T$. An electron-electron contribution proportional to $T^2$ can be relevant in some low-temperature ranges.

Carbon in mild steel, together with associated disorder, strain and microstructural changes, adds scattering and generally raises the residual resistivity relative to purer iron. Thus a dilute alloy may have a similar broad phonon-dependent rise but a larger baseline. The sketch is a generic phonon-plus-disorder model, not measured iron data. Iron's magnetic scattering and phase changes, and real steel's phases and processing history, can add structure; the simple sum is approximate rather than an exact material law. The expected explanation is greater Electron scattering from the added carbon and resulting disorder.