Bloch Electrons in a Magnetic Field

A formalism recently developed by the author is used to simplify Kohn's derivation of an effective Hamiltonian. The new derivation is valid for arbitrary symmetry and with the inclusion of spin-orbit effects and for bands with degeneracies. The actual form is obtained to second order in the magnetic field. It is explicitly shown that the procedure is asymptotically convergent. This Hamiltonian is then used to obtain the susceptibility in a convenient form. The nature of the actual energy levels is discussed with particular attention to the broadening, gaps in the continuous region, and magnetic breakdown.