Correction of a relativistic impulse approximation expression used to obtain Compton profiles from photon scattering doubly differential cross sections

Abstract

A Compton profile (CP) can provide useful information about the electron populations and distributions in atoms, molecules, and ions to assess many physical properties of matter. However, a CP cannot be measured directly, but must be obtained from scattering data. The CPs discussed in this study are derived from photon-atom doubly differential cross sections (DDCS) via the following expression which is derived from an impulse approximation (IA) theory given by DDCS = KJ, where K represents a kinematic factor and J represents the CP. A relativistic version of this expression (i.e., RKJ)—an approximation of the full relativistic IA expression—is used for relativistic regimes; however, it does not yield accurate results for the inner and middle shells of moderate to heavy atoms. In this study, expressions from nonrelativistic (NR) hydrogen-like wavefunctions with a relativistic QED kinematic factor K rel and relativistic electron energy were derived to correct the RKJ expression for the K, L, M, and N atomic subshells. This derivation made it possible for relativistic contributions and screening effects to largely cancel, for any regime of energy angle and Z. Thus, the RKJ error which can be greater than 30% is reduced to within few percent over 99% of the electron momentum distribution range of any subshell CP when compared to published tabulated theoretical values. Two simple versions of the relativistic QED kinematic component of the corrected RKJ expressions were obtained and tested: one valid at high photon energies, and the other at small scattering angles. RKJ corrections were applied to the extraction of CP from the full spectrum K-N shell DDCS, which resulted in much improved accuracy for the K-shell. Good agreement was observed with tabulated beyond K-shell CPs around the tail regions, but systematic differences were found to occur at the maxima. The details of this phenomenon are illustrated and discussed in this article.