Prediction of Compton doubly and tripply differential cross sections and Compton profiles at high energy from modified nonrelativistic theories: Effect of treating the momentum of the ejected electron relativistically

With increasing atomic number and incident photon energy ($\omega_1$), the nonrelativistic (nr) matrix element based on the interaction Hamiltonian [$H_{int}=(e^2A^2/2) - e(p\cdot A)$] becomes insufficient for the accurate prediction of Compton cross sections, even in the Compton peak region, except when v/c (v=velocity of the ejected electron) is small, in which case it remains valid. Under such circumstances one could use the more exact relativistic S-matrix (SM) theory. However we find that for doubly and triply differential cross sections in the vicinity of the Compton peak, an $A^2$ matrix element based on Schr\"odinger wavefunctions works even for $v/c \rightarrow 1$, if the momentum of the ejected electron is treated relativistically. However an entirely nr treatment (including for momentum) of the Compton profiles (CP), as a function of $p_z$ ($p_z$ = z component of the incident electron energy), unlike for the cross sections, is in surprisingly good agreement with relativistic SM (with relativistic $p_z$) results even when $v/c \rightarrow 1$, due to partial cancellation of relativistic factors in CP and $p_z$.