# Reconciling two views of Matthiessen's rule for electron mobility

My understanding of Matthiessen's rule is that when electron conduction in a solid is inhibited by more than one mechanism, that the overall resistivity $$\rho$$ is formed as if these resistance source were added "in series," $$\rho = \sum_i \rho_i$$, where $$i$$ enumerates the distinct resistive phenomena. This is easy for me to conceptualize by associating each resistivity with a scattering rate $$\rho_i\propto\nu_i$$. It seems reasonable then that the total resistivity is proportional to a total scattering rate $$\rho\propto\nu$$ formed from a weighted sum of the individual collision rates.

Now let's say I want to abstractly represent a solid as a circuit element, where each scattering mechanism is a resistor that contributes to the overall resistance of the solid: one for scattering off of ions, one for scattering off phonons, etc. The argument above says I should lay these resistors in series.

If we didn't have that knowledge, though, one might hypothesize that for a homogeneous solid, the electron current is split up equally amongst the different scattering mechanisms. This would lead to a circuit representation with the different scattering mechanisms' resistances added in parallel. Is there a way to deduce that this is wrong without appealing to kinetic theory ($$\rho_i\propto\nu_i$$)?

First, if we take the Matthiessen's rule formula, $$\frac 1 {\mu}=\frac 1 {\mu_{impurities}}+\frac 1 {\mu_{lattice}}$$ and take into account that the resistivity is inversely proportional to the mobility, we'll get $$\rho=\rho_{impurities}+\rho_{lattice}$$, which implies a series connection of resistors representing various scattering mechanisms.