Doubts regarding Ohm's resistance from impurity scattering It is told that electrons can scatter on impurities in crystals, which leads to resistance and Ohm's law. My problem is that even though the potential is not exactly periodic, the wave packet can move in such a random potential elastically. Resistance would indicate energy loss, hence, electrons should transfer energy to impurities, but this is not present since the random potential is rigid (I mean conservative). The story is different for electron-phonon scattering, where the phonon field gets excited by electron-phonon interaction, and the atomic lattice takes up energy, which is just the Joule heat. So, does this mean that impurity scattering can not be the source of electric resistance?
 A: Impurity scattering definitely matters. How much and in what context depends on the material and the device.
In a metal the interaction with phonons in the lattice dominates, but if you cool down to freeze out the phonons there will be residual resistance that depends on impurity scattering as well as other effects.
In semiconductors, the conductivity is controlled by the doping, but the mobility is also controlled by the doping concentration, or the amount of charges impurities in the lattice. You can also have electron electron scattering too.  If you look at the resistance/conductivity  or mobility as a function of temperature you will see the two competing effects.
So to think about resistance $$J=\sigma E$$ and one way to think about the conductivity $\sigma$ is to think about the velocity distribution of the electrons and treat the electrons as a gas. When a field is applied the net change in velocity is the drift velocity. When we do this we usually think about a mean time between collisions, or using Boltzmann equation a relaxation time approximation.
For the case of the charged impurity, we can use a  Coulomb potential and see how the electron interacts(Rutherford Scattering) and to your point this is an elastic interaction.  Conwell in a 1950 paper.
But because you have a change in direction of the electron the resultant direction of the electron has changed after the scattering event. This changes the velocity distribution.
If the electron was moving in the direction of the applied field, it would have less velocity in that direction after it scatters. Also with increased impurity the mean time between collision events can decrease.
Then you use the modified velocity distribution and solve the Boltzmann transport equation.
Also I guess the impurity atom is in the lattice so even though it is probably very small there is some  recoil and transfer to the lattice, but the change in angle with the scattering effect dominates the mobility and thus the resistance.
