Semiclassical quantization of electrons in magnetic fields: the   generalized Peierls substitution
  A generalized Peierls substitution which takes into account a Berry phase term must be considered for the semiclassical treatment of electrons in a magnetic field. This substitution turns out to be an essential element for the correct determination of the semiclassical equations of motion as well as for the semiclassical Bohr-Sommerfeld quantization condition for energy levels. A general expression for the cross-sectional area is derived and used as an illustration for the calculation of the energy levels of Bloch and Dirac electrons. 
Pierre Gosselin (IF), Hocine Boumrar, Herve Mohrbach (FCN, LPMC - EA   3468)