Metals — Question 10

Question 10

Independent scattering channels add as inverse lifetimes

A metal has independent impurity and phonon relaxation times $\tau_i=6.00\times10^{-14}\,\mathrm s$ and $\tau_p=3.00\times10^{-14}\,\mathrm s$. Survival probabilities multiply: $P(t)=e^{-t/\tau_i}e^{-t/\tau_p}$.

1. Derive the effective relaxation time and the fraction of scattering events due to impurities.

2. If phonon scattering is suppressed while impurities remain, find the conductivity increase and explain why resistivities add.

Question 10 - Solution

Independent scattering channels add as inverse lifetimes

(a) Method and calculation

$$P(t)=e^{-t(1/\tau_i+1/\tau_p)}\Rightarrow\boxed{\tau=\frac{\tau_i\tau_p}{\tau_i+\tau_p}=2.00\times10^{-14}\,\mathrm s}.$$$$\boxed{f_i=(1/\tau_i)/(1/\tau_i+1/\tau_p)=1/3}.$$

(b) Interpretation and further calculation

$$\sigma\propto\tau\Rightarrow\boxed{\sigma_{\mathrm{cold}}/\sigma=\tau_i/\tau=3}.$$$$\boxed{\rho=\frac{m}{ne^2}\left(\frac1{\tau_i}+\frac1{\tau_p}\right)=\rho_i+\rho_p}.$$

Physical check

Adding rates assumes independent mechanisms with the same simple transport weighting. The result need not hold for strongly anisotropic or correlated scattering.