Low-momentum-transfer nonrelativistic limit of the relativistic impulse approximation expression for Compton-scattering doubly differential cross sections and characterization of their relativistic contributions

Abstract

The nonrelativistic (nr) impulse approximation (NRIA) expression for Compton-scattering doubly differential cross sections (DDCS) for inelastic photon scattering is recovered from the corresponding relativistic expression (RIA) of Ribberfors [Phys. Rev. B 12, 2067(1975] in the limit of low momentum transfer (q → 0), valid even at relativistic incident photon energies ω 1 > m provided that the average initial momentum of the ejected electron p i is not too high, that is, p i < m. This corresponds to a binding energy E b < 10 keV. This q → 0 nr limit is simultaneous with the approach of the scattering angle θ to 0 • (θ → 0 • ) around the Compton peak maximum. This explains the observation that it is possible to obtain an accurate Compton peak (CP) even when ω 1 > m using nr expressions when θ is small. For example, a 1% accuracy can be obtained when ω 1 = 1 MeV if θ < 20 • . However as ω 1 increases into the MeV range, the maximum θ at which an accurate Compton peak can be obtained from nr expressions approaches closer to zero, because the θ at which the relativistic shift of CP to higher energy is greatest, which starts at 180 • when ω 1 < 300 keV, begins to decrease, approaching zero even though the θ at which the relativistic increase in the CP magnitude remains greatest around θ = 180 • . The relativistic contribution to the prediction of Compton doubly differential cross sections (DDCS) is characterized in simple terms using Ribberfors further approximation to his full RIA expression. This factorable form is given by DDCS = KJ , where K is the kinematic factor and J the Compton profile. This form makes it possible to account for the relativistic shift of CP to higher energy and the increase in the CP magnitude as being due to the dependence of J (p min , ρ rel ) (where p min is the relativistic version of the z component of the momentum of the initial electron and ρ rel is the relativistic charge density) and K(p min ) on p min . This characterization approach was used as a guide for making the nr QED S-matrix expression for the Compton peak kinematically relativistic. Such modified nr expressions can be more readily applied to large systems than the fully relativistic version. L. A. LAJOHNPHYSICAL REVIEW A 81, 043404 (2010) taking the nr limit of the second-order perturbation relativistic QED S-matrix element to obtain expressions for DDCS and triply differential cross sections (TDCS), free of spurious singularities but having a separable A 2 and p · A form (not separable in the fully relativistic S-matrix expression). He then introduced relativistic kinematics, resulting in expressions that can be used to obtain reasonably accurate K-shell DDCS at all θ if ω 1 < 300 keV. At higher energies this expression only works at smaller scattering angles. This was illustrated by their example calculation in which ω 1 = 661 keV and θ 60 • .In NRIA theory, DDCS are proportional to the Compton profile, given bywhere p is the initial electron momentum. ρ nr (p) is the Fourier transform of the nr charge density ρ nr (r)...

Relativistic impulse approximation in Compton scattering

“…where E(p z ) = m 2 c 4 + p 2 z c 2 and q is the modulus of the momentum transfer vector q ≡ k i − k f . Furthermore, previous study revealed that p z and E(p z ) are exactly the energy and momentum minimum of the initial state electrons activated in Compton scattering [27], namely…”

“…Recently, LaJohn compared various treatments of RIA formulation in a similar manner, and achieved the nonrelativistic limit of RIA for low-momentum-transfer cases [27]. However, his work is limited to hydrogen-like systems.…”

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

“…NR expressions for DDCS can provide fairly accurate predictions for the DDCS of light atoms (see [18,19]) if the incident photon energy is ω i = m or θ → 0° [20]; otherwise, relativistic factors must be included ([6, 7, 21-24] for examples). If there is a need to obtain a CP from DDCS in a relativistic regime, the most straightforward approach for Compton scattering is RKJ approximation, which is fairly accurate overall ω i values; if the binding energy (E b ) is low, it becomes less accurate with increasing E b .…”

A low energy bound atomic electron Compton scattering model for Geant4

“…Alternative models to Ribberfors' DDCS have been developed and even expanded to the Triply Differential Compton scattering Cross-Section (TDCS), including the dependence of the ejection direction of the Compton electron [16][17][18][19][20][21]. These models have primarily explored the numerical calculation of K-shell Compton scattering cross-sections for numerous materials over an energy range of a few keV [16,17,20] to 1 MeV [20].…”

“…These models have primarily explored the numerical calculation of K-shell Compton scattering cross-sections for numerous materials over an energy range of a few keV [16,17,20] to 1 MeV [20]. Of the wide range of different approaches in both the RIA and Independent Particle Approximation (IPA), one of the most promising is that described in the seminal paper from Kaliman et al [17].…”