Relativistic regimes in which Compton scattering doubly differential cross sections obtained from impulse approximation are accurate due to cancelation of errors.

There is no simple parameter that can be used to predict when impulse approximation (IA) can yield accurate Compton scattering doubly differential cross sections (DDCS) in relativistic regimes. When Z is low, a small value of the parameter \textless p\textgreater /q (where \textless p\textgreater is the average initial electron momentum and q is the momentum transfer) suffices. For small Z the photon electron kinematic contribution described in relativistic S-matrix (SM) theory reduces to an expression, X$^{rel}$, which is present in the relativistic impulse approximation (RIA) formula for Compton DDCS. When Z is high, the S-Matrix photon electron kinematics no longer reduces to X$^{rel}$, and this along with the error characterized by the magnitude of \textless p\textgreater /q contribute to the RIA error $\Delta $. We demonstrate and illustrate in the form of contour plots that there are regimes of incident photon energy $\omega_{i}$ and scattering angle $\theta$ in which the two types of errors at least partially cancel. Our calculations show that when $\theta$ is about 65$^{\circ}$ for Uranium K-shell scattering, $\Delta $ is less than 1{\%} over an $\omega_{i}$ range of 300 to 900 keV.