ABSTRACT
In this final chapter, a number of important concepts are introduced and discussed in depth. We begin with the classical theory of Compton scattering, where the Lorentz force equation is used to describe the covariant dynamics of an electron in a plane wave and where radiation reaction effects are neglected. The classical differential Compton scattering cross-section is obtained, and it closely matches its QED equivalent in the limit where the incident photon energy, as observed in the rest frame of the electron, is much smaller than the rest energy of the electron. In this regime, recoil is negligible, and the classical description proves useful, as the coherent field of a laser, for example, can be treated as a classical, continuous electromagnetic field. Using the differential Compton scattering cross-section, the influence of the phase space of an ensemble of electrons upon the scattered radiation can then be studied in detail. In the case of a relativistic electron beam, the transverse emittance and the axial energy spread can be modeled analytically using Gaussian distributions, and incoherent summations can be performed, leading to analytical expressions for the scattered radiation.
