A Wignerfunction approach to phonon scattering

We consider the motion of a single electron under phonon scattering caused by a crystal lattice. Starting from the Frohlich Hamiltonian in the second quantization formalism we derive a kinetic transport model by using the Wig ner transformation. Under the assumption of small electron-phonon interacti on we derive asymptotically the operator representing electron-phonon scatt ering in the Wigner equation. We then consider some scaling limits and fina lly we give the connection of our result to the well known Barker-Ferry equ ation.