Retardation, multipole, and relativistic kinematics effects for x- andγ-ray Compton scattering fromK-shell electrons

Abstract

The aim of this paper is to obtain the right expression of the fully nonrelativistic $K$-shell Compton scattering amplitudes and cross sections. Using the Coulomb Green's function method, we prove that by adequate analytical calculation of the nonrelativistic limit of the second order $S$-matrix element, it is possible to get Compton amplitudes which have no spurious singularities, as the old nonrelativistic result with retardation [M. Gavrila, Phys. Rev. A 6, 1348 (1972)] presents. A pole-free formula for Compton triply differential cross section, giving the right dependence on the angles of both the final electron and photon is obtained. This was possible by introducing relativistic kinematics terms and noticing that some of them cancel the retardation terms that generate the spurious poles. Consequently, the right nonrelativistic limit of the Compton amplitudes should be considered only after the cancellation of these terms. In this way the contributions corresponding to ${\mathbf{A}}^{2}$ and $\mathbf{p}∙\mathbf{A}$ operators present in the nonrelativistic Hamiltonian are correctly revealed. We show that good predictions are obtained for the whole Compton spectrum and any photon scattering angles for incoming photon energies ${\ensuremath{\omega}}_{1}$ up to $\ensuremath{\alpha}Zm$, if relativistic kinematics terms are included. The doubly differential cross section for Compton scattering of unpolarized photons from $K$-shell electrons is obtained by numerical integration over the angles of the final photon. Comparing our exact nonrelativistic formulae predictions for Compton scattering on intermediate and high atomic number targets, with experimental results and fully relativistic numerical evaluation of Bergstrom et al. [Phys. Rev. A 48, 1134 (1993)], a good agreement within 10% is found for the whole spectrum and any scattering angle for incoming photon energy below 200 keV. This shows that spin effects are small for unexpected large incoming photon energies and high $Z$ targets. Also, in the case of $s$ bound states, the nonrelativistic wave function approximates fairly well the Dirac spinor even for large energies if momentum transfer is not too high. A second numerical integration of the doubly differential cross section over scattered photon energies was performed, giving the singly differential cross section. A good agreement with fully relativistic data is found for incoming photon energies up to 300 keV and any atomic target, if relativistic kinematics effects are taken into account. The same is true in the case of Compton scattering on lead at 661 keV for scattering angles less than 60\ifmmode^\circ\else\textdegree\fi{}.