Multiple Scattering Theory for Spectroscopies pp 345-350 | Cite as

# Local Geometry by XANES and RXS

## Abstract

A brief overview of the capability of x-ray absorption near edge structure (XANES) and resonant x-ray scattering (RXS) in the hard x-ray region to determine the local geometrical structure around the photoabsorbing atom is presented.

## 22.1 Introduction. Geometry Beyond the Radial Distribution Function

X-ray absorption spectroscopy (XAS) is the appropriate technique to investigate the neighbourhood of a photoabsorber atom embedded in a medium [1]. The x-ray absorption process is a transition between two quantum states: from an initial state with an x-ray and a core electron to a final state with no x-ray, a core-hole and a photoelectron. The inner shell electron is well described by atomic wave functions and the photoelectron final states wave function for an isolated atom is well described by a free spherical wave outgoing from the atom. The final states outgoing wave becomes from the superposition of the free outgoing wave and the scattered waves by the atoms surrounding the absorber one. This interference between the outgoing wave and the scattered ones produces minima and maxima in space. Such a simplified picture describes the multiple scattering approximation to the XAS spectrum [2, 3, 4, 5], where the cross section can be factorized as: \(\sigma = \sigma _0 (1+ \sum \; \chi _n)\). Here \(\sigma _0\) represents the cross section for the isolated atom (the free outgoing wave function) and \(\sum \; \chi _n\) comes from the processes in which the photoelectron is dispersed \(n-1\) times by the neighbour atoms before interfering with the outgoing wave at the origin. These multiple scattering events carry information on the topological order around the absorber, i.e. interatomic distances and angles between bonds. Such multiple scattering signals are always present in the XAS spectrum but their contribution rapidly decreases with increasing the energy (or wave vector \(\mathbf {q}\)).

The information on the local geometry contained in XAS comes from multiple scattering processes of the outgoing wave. Joined to this, direct information on the geometry can be obtained from the tensor character of the absorption cross section [6]. Therefore in anisotropic materials the spectrum is dependent on the relative orientation of the electric field polarization of the x-ray beam to the sample orientation. In the EXAFS region interatomic distances on the direction of the electric field vector can be obtained [1]. On the other hand the energy of the absorption edge highly depends on the polarized direction. This strong difference at the edge allows to detect by RXS the scattered intensity for photon energies at the absorption edge in forbidden reflections [7]. Within this contribution, we will shortly review some experimental probes of these facts. i.e. local geometry determined by (i) multiple scattering description (ii) polarized absorption spectra and (iii) ordering of distortions by resonant x-ray scattering.

## 22.2 Local Geometry in XANES Spectrum. Multiple Scattering

*K*-edge absorption spectra of a water solution of potassium permanganate where MnO\(_4^-\) ion is tetrahedrally coordinated and a water solution of Mn\(_{2}^+\) where this ion is octahedrally coordinated [5]. These two spectra after rescaling the energy and renormalizing the amplitude shows the same sinusoidal behaviour in the energy region sensitive only to the pair correlation function (EXAFS region). A strong difference is observed in the XANES region due to the different geometrical arrangement of the ligand atoms around the metal ion in the two complexes.

As illustration, Fig. 22.1 shows the expansion in the different MS contributions of the [Mn(H\(_2\)O)\(_6\)]\(^{2+}\) octahedral cluster compared with the experimental spectrum. As we can see the main line A for octahedral clusters is determined by a full multiple scattering resonance where all the multiple scattering contributions are in phase [5, 8]. Multiple scattering theory is nowadays the usual method to analyze the XANES spectra. There is a lot of codes to calculate the XANES spectra as: mcms, gnxas, mxan, fdmnes, feff, etc. The reader is addressed to other chapters of this book to know the recent improvements of the different codes. Theoretical fitting of the XANES spectra can be performed [9] and it is generalized the implementation of the MS contributions in the analysis of the extended part of the XAS spectrum (EXAFS) [10].

## 22.3 Anisotropy in the Local Geometry from Polarized XANES Spectra

As it is shown, the local geometry can be retrieved from XANES spectra through the theoretical simulation using MS framework. But independently of this fact, the absorption coefficient is a symmetric tensor of rank 2 in the dipole approximation related to the electric field vector of the incident x-ray beam (\(\mathbf {e}_{\mathbf {q}}\)). In principle, six different components are necessary to completely describe it. Depending on the local symmetry the number of components is reduced in such a way that for octahedral or tetrahedral symmetries the tensor is diagonal and the three components are equal, i.e. it behaves as a scalar. For tetragonal symmetry and taken the reference frame as the tetragonal axis and the two directions perpendicular to it, the tensor is also diagonal with two different components, along the tetragonal axis and perpendicular to it [6]. It is obvious that XAS spectra on oriented samples will provide direct information on the local symmetry. In the case of the EXAFS region where the pair distribution function is obtained, the observed path contribution is factorized by the scalar product of \(\mathbf {e}_{\mathbf {q}}\) and the bond direction [1]. For example, the EXAFS contribution to the surrounding atoms located perpendicular to \(\mathbf {e}_{\mathbf {q}}\) is exactly zero. In the XANES spectra differences come mainly from the energy position of the absorption edge, an anisotropic shift is observed. As a matter of illustration, La\(_{1-x}\)Sr\(_{1+x}\)MnO\(_4\) compounds present a tetragonal distorted local structure around the Mn atom. The experiments were performed with the polarization of the incident beam parallel and perpendicular to the *c*- axis. XANES spectra for the three single crystals show an anisotropic splitting being larger the for \(x=0\) sample and the smaller for the \(x=0.5\) one [11]. The magnitude of this splitting correlates with the larger Mn-O interatomic distance along the *c*-axis for the \(x=0\) composition. Therefore, the anisotropic splitting measures the degree of tetragonal distortion of the MnO\(_6\) octahedron. Needless to say that from the polarized EXAFS spectra, the in-plane Mn-O interatomic distances are determined when \(\mathbf {e}_{\mathbf {q}}\) is perpendicular to the *c*-axis whereas the out of plane Mn-O distances are obtained for \(\mathbf {e}_{\mathbf {q}}\) parallel to *c*.

## 22.4 Ordering of the Local Distortions by Resonant X-Ray Scattering (RXS)

RXS combines absorption and diffraction as they have in common the x-ray atomic scattering factor (ASF), *f*, which is usually written as: \(f = f_0 + f' + If"\) [1]. It contains an energy independent part, \(f_0\), corresponding to the classical Thomson scattering and two energy-dependent terms, \(f'\) and *f*", also known as the atomic anomalous scattering factor. RXS occurs when the x-ray energy is tuned near the absorption edge of an atom in the crystal. We recall here the intimate relationship between the atomic anomalous scattering factor (ASF) and the absorption coefficient, i.e. the imaginary part of the ASF is proportional to the absorption coefficient, whereas the real part is related to the imaginary part through the Kramers–Kronig transformation. Therefore, the ASF has the same tensorial character as the absorption coefficient. The polarization dependence of ASF is in the origin of the observation of RXS intensity in forbidden reflections. These reflections that are forbidden by symmetry due to the scalar character of the Thomson scattering. They are forbidden by symmetry elements with translation components (screw axes and glide planes) and appear due to the presence of local anisotropy (sometimes assigned to orbital ordering (OO)). As we have seen in the previous section, the anisotropy is mainly reflected in the anisotropic shift of the polarized XANES spectra (and the ASF) and therefore, the enhancement of the scattered intensity just appear close to the absorption threshold.

*Pbnm*and \((\text {odd},0,0)\), \((0,\text {odd},0)\) and \((0,0,\text {odd})\) reflections are forbidden. This is due to the fact that their structure factor comes from the difference between equivalent atoms in the unit cell. The difference in the ASF among equivalent Mn atoms is originated by the different orientation of the local tetragonal distortion, as it is depicted in Fig. 22.2. Therefore, scattered reflected intensity will be observed at energies close to the absorption edge due to the anisotropic shift between the two polarizations parallel and perpendicular to the local tetragonal axis. As it is shown in Fig. 22.2 maximum of the scattered intensity occurs just at the white line. The energy dependence of the scattered intensity of the (030) and (003) forbidden reflections together with and the calculated ones using the mxan code [13] are shown in Fig. 22.2.

## 22.5 Summary

Within this short paper we pretend to briefly illustrate the capability of the hard x-ray spectroscopies to determine the local geometry around a specific atom. There is a lot of points that are not discussed, such as DAFS, DANES, or high energy resolution XAS spectroscopies. The reader can be addressed to the large bibliography on these subjects, for instance the book by J.A. van Bokhoven and C. Lamberti [14]. Regarding the multiple scattering theory and its recent developments they can refer to the other papers contained in this book.

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