Multipole decomposition of the elastic atomic photon scattering form factor and the anomalous scattering factor

Abstract

The elastic atomic photon scattering form factor amplitude Af0, obtained from the high energy and low momentum transfer (K) limit of the exact (within the independent particle approximation (IPA)) Rayleigh relativistic second order S-matrix scattering amplitude Asm is decomposed into an infinite sum of its electric and magnetic field multipoles. The energy-dependent part of Af0 (the scattering angle-dependent part of Af0 is incorrect due to the low K approximation), combined with the angle-dependent part of Asm, yields a form factor amplitude Af (different versions of Af and Af0 will be described in the text) that can be used to decompose the anomalous scattering factor Aasf into its multipoles. Af0 which initially contains coherent and incoherent factors is reduced to a single sum over the multipoles of only coherent terms by assuming that the charge density is spherically symmetric. At photon energies up to about ten times the K-shell binding energy it takes up to quadrupole Aasf for light atoms of Z ≤ 20 and up to the fourth multipole for heavy atoms such as uranium for an accurate Aasf (within IPA). The multipole contributions were found to become somewhat larger as the scattering angle becomes larger (greater than 120°); however there were partial cancellations of such effects.