Retardation effects in nonrelativistic two-photon electron bremsstrahlung in the Coulomb field

We investigate the deviations from the dipole approximation for the two-photon bremsstrahlung cross section sigma(4), corresponding to the observation of the emitted photons (but not the scattered electron) in coincidence, which has been the subject of recent experiments. Our analysis is for the Coulomb field, and it is done in the Born approximation. We find that retardation effects are important, even in the lower energy range (8-12 keV) of the Hippler experiments, but not in the particular configuration for which experiments were performed. Using the simple analytic expression that we derive, including retardation, in the nonrelativistic Born approximation, we show that, in the incident electron energy range 10-50 keV, the differences between the relativistic Born (Smirnov's approach) and nonrelativistic dipole approximation results are mainly retardation effects.