Compton scattering in a uniform, constant magnetic field

The literature review indicated the lack of a general Compton cross section for magnetic fields. An unrestricted expression for the magnetic Compton effect would be of value in laboratory work with megagauss fields where, although the magnetic fields may be weak in comparison to the quantum mechanical characteristic field B0, the electron's energy quantum number n can be large enough to make its energy extremely relativistic. It would also be of value in astrophysics work with the magnetic fields of order B0 inferred for the surface of neutron stars. A general expression for the Compton cross section for a uniform, constant magnetic field was derived based on the Dirac equation and second order perturbation theory, the electrons represented by single particle wave functions and the photons by plane waves. The solution was carried out as far as practical analytically and was programmed for numerical solution by computer; the results were displayed graphically. The solution possesses the usual energy conservation for the system of the electron plus the photon; in addition, the component along the magnetic field of the linear momentum of the system is conserved in the magnetic Compton effect. There is no general conservation of momentum. These conclusions are from the analytical portion of the work. The numerical portion reveals the existence of the Larmor peak, that is, a resonance in the total cross section when the frequency of the incoming photon is near the electron's Larmor frequency. At the peak, one photon polarization mode exhibits a severe dip for outgoing photons moving perpendicular to the magnetic field