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Is there any intuitive explanation of why resistivity of metal goes as $T^5$ at low temperature?
The Debye theory gives that the phonon distribution goes as $n(\omega)\sim T $ at higher temperature and $n(\omega)\sim T^3$ at low temperature? What is responsible for this extra $T^2$ suppression?
Also, how do I go about rigorously deriving the $T^5$ dependence from scattering theory?
Following this set of course notes (p. 94-95): the number of phonons scales as $n(\omega)\sim T^3$, but for low temperatures one phonon scattering is not going to knock an electron far along the Fermi surface. So to really randomize the electron direction one needs enough scatterings to make the sum of angular changes of order unity. The scattering angle scales roughly like $\approx \omega^2/2k_F^2 c_{sound}^2 \propto T^2$, so one needs $1/T^2$ extra scatterings. So the total scaling becomes $T^5$... except as the text (and daydreamer's comment) points out, the scaling has often a lower exponent because of the shape of the Fermi surface.
