Triply differential cross section for Compton scattering

Abstract

The triply differential cross section for Compton scattering from atomic electrons is obtained numerically in a full relativistic second-order $S$-matrix calculation based on the independent particle approximation. We compare our results with the results of more approximate approaches. Special attention is paid to the validity of the impulse approximation (IA), which has often been used for calculating the doubly differential cross section even when the photon momentum transfer K is similar to the average momentum ${p}_{\mathrm{av}}$ of the bound electron, which is ionized (and IA is found to be fairly accurate even in such circumstances). We here show that, on the contrary, IA calculations of the (less averaged) triply differential cross section are quite inaccurate for $|\mathbf{K}|\ensuremath{\sim}{p}_{\mathrm{av}},$ even near the peak in the triply differential cross section (where the free kinematics for scattering from an initial free electron at rest are satisfied and where IA should work the best). We conclude that electron momentum distribution determination through the Compton profile, using the doubly differential cross section, is more accurate at lower energies than direct determination through the measurement of the triply differential cross section at the same energy. In addition, viewing the total cross section for double ionization in Compton scattering as another observable less averaged than the doubly differential cross section in single ionization, we estimate that IA predictions of the total cross section for double ionization in Compton scattering from Helium are adequate above about 50 keV.