Why criteria for impulse approximation in Compton scattering fail in relativistic regimes

The assumption behind impulse approximation (IA) for Compton scattering is that the momentum transfer q is much greater than the average $

$ of the initial bound state momentum distribution p. Comparing with S-matrix results, we find that at relativistic incident photon energies ($\omega_i$) and for high Z elements, one requires information beyond $

/q$ to predict the accuracy of relativistic IA (RIA) diferential cross sections. The IA expression is proportional to the product of a kinematic factor $X^{nr}$ and the symmetrical Compton profile J, where $X^{nr}=1+cos^2\theta$ ($\theta $ is the photon scattering angle). In the RIA case, $X^{nr}$, independent of p, is replaced by $X^{rel}(\omega ,\theta ,p)$ in the integrand which determines J. At nr energies there is virtually no RIA error in the position of the Compton peak maximum ($\omega_f^{pk}$) in the scattered photon energy ($\omega_f$), while RIA error in the peak magnitude can be characterized by $

/q$. This is because at low $\omega_i$, the kinematic effects described by S-matrix (also RIA) expressions behave like $X^{nr}$, while in relativistic regimes (high $\omega_i$ and Z), kinematic factors treated accurately by S-matrix but not RIA expressions become significant and do not factor out.