Eur. Phys. J. B (2017) 90: 246
https://doi.org/10.1140/epjb/e2017-80266-5

Regular Article

Gauge invariance and relativistic effects in X-ray absorption and scattering by solids

¹ Sorbonne Universités, UPMC Univ Paris 06, UMR CNRS 7590, Muséum National d’Histoire Naturelle, IRD UMR 206, Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie, 4 place Jussieu, 75005 Paris, France
² Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, BP 48, 91192 Gif-sur-Yvette Cedex, France

^a e-mail: christian.brouder@impmc.upmc.fr

Received: 8 May 2017
Received in final form: 13 October 2017
Published online: 11 December 2017

Abstract

There is an incompatibility between gauge invariance and the semi-classical time-dependent perturbation theory commonly used to calculate light absorption and scattering cross-sections. There is an additional incompatibility between perturbation theory and the description of the electron dynamics by a semi-relativistic Hamiltonian. In this paper, the gauge-dependence problem of exact perturbation theory is described, the proposed solutions are reviewed and it is concluded that none of them seems fully satisfactory. The problem is finally solved by using the fully relativistic absorption and scattering cross-sections given by quantum electrodynamics. Then, a new general Foldy-Wouthuysen transformation is presented. It is applied to the many-body case to obtain correct semi-relativistic transition operators. This transformation considerably simplifies the calculation of relativistic corrections. In the process, a new light-matter interaction term emerges, called the spin-position interaction, that contributes significantly to the magnetic X-ray circular dichroism of transition metals. We compare our result with the ones obtained by using several semi-relativistic time-dependent Hamiltonians. In the case of absorption, the final formula agrees with the result obtained from one of them. However, the correct scattering cross-section is not given by any of the semi-relativistic Hamiltonians.