Characterization of the relativistic contributions to Compton doubly differential cross sections

Illustrations and a detailed analysis will be provoded on how relativistic factors enter into the predictions of Compton doubly differential cross sections (DDCS), using the impulse approximation (IA) theory as a model. Within IA theory it was found that one can analyze the relativistic contributions to DDCS in terms of two components, a kinematic factor $K(p_z)$ and the Compton profile $J(p_z,\rho_{rel} )$, both functions of $p_z$, the z component of the ejected electron. $J(p_z,\rho_{rel} )$ is also a function of the relativistic charge density $\rho_{rel} $. It will be shown by taking the nonrelativistic limit of $p_{min}$ (the relativistic version of $p_z$), which accounts for most of the relativistic effects on DDCS, that the relativistic shift of the Compton peak in DDCS to higher energy, determined by $J(p_z,\rho_{rel})$, as well as most of the relativistic increase in the peak amplitude determined by $K^{rel}(p_{min})$, is due to the only term in $p_{min}$ that differs from $p_z$, that is $\omega_1 \omega_2$ in the former and ${\omega_1^2 + \omega_2^2 \over 2}$ in the latter, the two terms becoming equal as $\omega_2 \rightarrow \omega_1$, ($\omega_1$ and $\omega_2$ are the incident and scattered photon energies). However the true nonrelativistic limit of $K(p_{min})$ is not obtained unless $\omega_2(1-cos \theta )$ is neglected, which is valid if $\omega_2 << mc^2$, also if $\theta $ is small even at high energies. Finally the much smaller relativistic contribution due to $\rho_{rel} $, which decreases the Compton peak height with increasing nuclear charge Z will be examined.