Temperature Dependence of Resistance

For most conductors, over a reasonably small temperature range, the change in resistance with temperature is found to follow a simple linear law:

where $R_0$ is the resistance at some reference temperature $T_0$ (commonly $0^\circ\text{C}$ or room temperature), $R_T$ is the resistance at temperature $T$, and $\alpha$ is called the temperature coefficient of resistance of the material, with SI unit $\text{K}^{-1}$ or, equivalently, ${}^\circ\text{C}^{-1}$.

Metals. For pure metals, $\alpha$ is small and POSITIVE (typically around $0.004\ {}^\circ\text{C}^{-1}$), so resistance rises fairly linearly with temperature over an ordinary working range. This behaviour follows directly from the microscopic picture of Section 3.3: as temperature rises, the lattice ions vibrate more energetically, so free electrons collide with them more often, reducing the relaxation time $\tau$; since $\rho = m/(ne^2\tau)$, a smaller $\tau$ directly means a LARGER resistivity, and hence a larger resistance. The free-electron density $n$ itself stays essentially constant with temperature in a metal, so this fall in $\tau$ is the whole story.

Alloys such as nichrome, manganin and constantan are specially formulated to have a very small $\alpha$, so their resistance stays almost constant despite large temperature changes -- exactly the property wanted for a heating element (nichrome, used in electric heaters and toasters) or a standard resistance coil (manganin, used in resistance boxes) whose value must not drift as current flows through it and warms it up. …