Semi-classical limits in a crystal with a Coulombian self-consistent potential: Effective mass theorems

We study the semi-classical limit of the dynamics of electrons in a crystal under the influence of a self-consistent Coulombian potential. To do this, we introduce two small parameters:, the typical length of the crystal, and h, the macroscopic Planck constant. When andh tend to zero, under special conditions, the electrons distribution limit follows a Vlasov equation where the electrons mass is replaced by an effective mass.