Mathematically, how does the electron mean free path or electron-lattice mean free path depend on temperature?
Is there any formula linking the two?
Check this link: http://hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/menfre.html. You can find formula there.
In gas phase the general expression of the mean free path of an electron with fixed velocity is v/f where v is the velocity of the particle and f is the collision frequency
f=Nc*|v-vc|*s, where Nc is the density of target particles vc their velocity and s the cross section. If you have a mixture, you should consider a mean cross section weighted on the gas composition.
In the case of electron collisions,
the only dependence on the gas temperature, considering a system at constant density, is contained in vc. Therefore the mean value of |v-vc| (consider velocities as vectors) over the maxwell distribution, will give you the dependence on the gas temperature.
However vc is much smaller than v, therefore |v-vc| ~ v giving for the mean free path 1/Nc*s.
However the cross section can depend on the electron velocity, therefore a successive mean over the electron distribution should be calculated, giving the dependence on the electron temperature, that depend on the gas species.
In solid phase there are different contributions. One due to the atoms in the lattice, which depend again on the density. This contribution in independent on the temperature if you do not consider thermal expansion. The second contribution is due to collisions with phonon which number is a function of the temperature.
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