Gauge invariance and relativistic effects in photon absorption and scattering by matter
Résumé
Several semi-relativistic Hamiltonians have been proposed to describe the interaction of light with magnetic matter. In the calculation of x-ray magnetic circular dichroism, these different Hamiltonians lead to different results. Gauge invariance is supposed to help choosing the proper Hamiltonian, but we show that the semi-classical formulation of the absorption cross-section is not gauge invariant in a commonly accepted sense. To avoid the problem of choosing the proper semi-relativistic Hamiltonian, we start from the fully relativistic absorption and scattering cross-sections given by quantum electrodynamics and we derive a semi-relativistic expansion of the cross-sections by formulating a new many-body Foldy-Wouthuysen transformation of the wavefunctions in the transition matrix elements. 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. In the case of absorption, the final formula agrees with the result obtained from one of the semi-relativistic Hamiltonians. However, the correct scattering cross-section is not given by any of the semi-relativistic Hamiltonians.
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