Chapter 4
X-ray Dichroisms in Spherical Tensor
and Green’s Function Formalism
Hebatalla Elnaggar, Pieter Glatzel, Marius Retegan, Christian Brouder,
and Amélie Juhin
Abstract In this book chapter, our goal is to provide experimentalists and theoreticians with an accessible approach to the measurement or calculation of X-ray
dichroisms in X-ray absorption spectroscopy (XAS). We start by presenting the key
ideas of different calculation methods such as density functional theory (DFT) and
ligand-field multiplet (LFM) theory and discuss the pros and cons for each approach.
The second part of the chapter is dedicated to the expansion of the XAS cross section
using spherical tensors for electric dipole and quadrupole transitions. This expansion
enables to identify a set of linearly independent spectra that represent the smallest
number of measurements (or calculations) to be performed on a sample, in order
to extract all spectroscopic information. Examples of the different dichroic effects
which can be expected depending on the type of transitions and on the symmetry of
the system are then given.
4.1 Introduction
4.1.1 The X-ray Absorption Cross Section
The X-ray absorption cross section is obtained by dividing the transition rate by the
flux of photons and summing over all possible final states. It is given in (4.1) where
ω is the photon energy, and α the fine structure constant. I (E I ) and F(E F ) are the
initial and final state wave functions (energies), and T is the transition operator,
H. Elnaggar
Debye Institute for Nanomaterials Science, Utrecht University, 3584 CA Utrecht,
The Netherlands
P. Glatzel · M. Retegan
European Synchrotron Radiation Facility, 71 Rue des Martyrs, 38000 Grenoble, France
Ch. Brouder · A. Juhin (B)
Institut de Minéralogie, Physique des Matériaux et Cosmochimie, CNRS-Sorbonne Université,
4 Place Jussieu, 75252 Paris Cedex 05, France
e-mail: amelie.juhin@sorbonne-universite.fr
© The Author(s) 2021
H. Bulou et al. (eds.), Magnetism and Accelerator-Based Light Sources,
Springer Proceedings in Physics 262,
https://doi.org/10.1007/978-3-030-64623-3_4
83
X-ray Dichroisms in Spherical Tensor
and Green’s Function Formalism
Hebatalla Elnaggar, Pieter Glatzel, Marius Retegan, Christian Brouder,
and Amélie Juhin
Abstract In this book chapter, our goal is to provide experimentalists and theoreticians with an accessible approach to the measurement or calculation of X-ray
dichroisms in X-ray absorption spectroscopy (XAS). We start by presenting the key
ideas of different calculation methods such as density functional theory (DFT) and
ligand-field multiplet (LFM) theory and discuss the pros and cons for each approach.
The second part of the chapter is dedicated to the expansion of the XAS cross section
using spherical tensors for electric dipole and quadrupole transitions. This expansion
enables to identify a set of linearly independent spectra that represent the smallest
number of measurements (or calculations) to be performed on a sample, in order
to extract all spectroscopic information. Examples of the different dichroic effects
which can be expected depending on the type of transitions and on the symmetry of
the system are then given.
4.1 Introduction
4.1.1 The X-ray Absorption Cross Section
The X-ray absorption cross section is obtained by dividing the transition rate by the
flux of photons and summing over all possible final states. It is given in (4.1) where
ω is the photon energy, and α the fine structure constant. I (E I ) and F(E F ) are the
initial and final state wave functions (energies), and T is the transition operator,
H. Elnaggar
Debye Institute for Nanomaterials Science, Utrecht University, 3584 CA Utrecht,
The Netherlands
P. Glatzel · M. Retegan
European Synchrotron Radiation Facility, 71 Rue des Martyrs, 38000 Grenoble, France
Ch. Brouder · A. Juhin (B)
Institut de Minéralogie, Physique des Matériaux et Cosmochimie, CNRS-Sorbonne Université,
4 Place Jussieu, 75252 Paris Cedex 05, France
e-mail: amelie.juhin@sorbonne-universite.fr
© The Author(s) 2021
H. Bulou et al. (eds.), Magnetism and Accelerator-Based Light Sources,
Springer Proceedings in Physics 262,
https://doi.org/10.1007/978-3-030-64623-3_4
83