  We study a semi-classical Schr{\"o}dinger equation which describes the
dynamics of an electron in a crystal in the presence of impurities. It is
well-known that under suitable assumptions on the initial data, the wave
function can be approximated in the semi-classical limit by the solution of a
simpler equation, the effective mass equation. Using Floquet-Bloch
decomposition and with a non-degeneracy condition on the critical points of the
Bloch bands, as it is classical in this subject, we establish effective mass
equations for more general initial data. Then, when the critical points are
degenerated (which may occur in dimension strictly larger than one), we prove
that a similar analysis can be performed, leading to a new type of effective
mass equations which are operator-valued and of Heisenberg form. Our analysis
relies on Wigner measure theory and, more precisely, to its applications to the
analysis of dispersion effects.
