Chapter 2.16. X-ray linear dichroism: dependence of XAFS on the orientation of the sample with respect to the incoming radiation

An important intuitive concept in XANES analysis is the link between specific spectral features and small well defined structural units. Modelling the angular dependence of the spectra plays a prominent role in this respect. Analysis of the V K-edge spectrum of V₂O₅ serves as an illustrative example (Šipr et al., 1999): by comparing experimental spectra with spectra calculated for simple structural units, it was found that the dominant sharp peak at the onset of the spectrum for is only formed if the nearest neighbourhood of the V atom is asymmetric and a very short V—O bond parallel to the direction is present at the same time (Fig. 1).

Figure 1 Simulating essential features of polarized V K-edge spectra of V2O5 employing simple structural units, as performed by Šipr et al. (1999).

The calculations of Šipr et al. (1999) relied on a real-space formalism. However, similar information can also be obtained when band-structure (reciprocal-space) methods are used. Cabaret et al. (2013) interpreted peaks in the polarized Ca K edge of CaC₆ in terms of hybridization between orbitals associated with the photoabsorbing Ca atom and orbitals associated with neighbouring atoms. This information was then used as guidance for the analysis of spectra of Li–Ca intercalated graphite.

The additional information carried by the angular dependence of the spectra can be beneficial for investigating the structure of complicated bulk systems such as minerals or clays because in such a case there are many structural parameters to be fitted and any extra condition constraining them is welcome. The structure optimization proceeds along the same lines as for unpolarized spectra, but there are more data to fit. One example is the structural refinement of a fine-grained layer silicate: Garfield nontronite (Manceau et al., 1998). Here, angle-dependent extended X-ray absorption fine-structure (EXAFS) spectra measured at the Fe K edge were simulated by ab initio modelling to obtain information about the flattening angle of the Fe(O,OH)₆ octahedra, about the cation distribution in the octahedral sheet and about the differentiation between di-octahedral and tri-octahedral structures. Modelling of angle-dependent polarized EXAFS was also crucial for studying the local environment of zinc sorbed in phyllomanganate birnesite (Manceau et al., 2002).

When X-ray absorption spectroscopy is used for structural studies the primary interest lies in the EXAFS region, because extracting the structural information from EXAFS is much more straightforward than from XANES. However, there are situations in which relying on EXAFS is not convenient and one has to turn to XANES. This happens, for example, for diluted systems where extracting EXAFS with a good signal-to-noise ratio is very difficult, while the more intensive XANES signal can be acquired relatively easily. Another situation where it may be useful to engage XANES is when one is interested in bond angles (apart from bond lengths).

One approach to obtain information about the local structure around the photoabsorber, which proved to be especially useful for biological systems, is best-fitting the experimental spectrum with XANES calculations for model structures. If angle-dependent spectra are available, there are more data to fit and hence the fitting procedure is more reliable. Application of this procedure to polarized Fe K-edge XANES of the iron protein carbonmonoxy myoglobin (MbCO) and of its photoproduct Mb*CO is demonstrated in Fig. 2. Distances as well as bond angles around the Fe atom were determined in this way (Della Longa et al., 2001).

Figure 2 Polarized Fe K-edge XANES of MbCO and Mb*CO. Experimental data are plotted as circles and best-fitted data are plotted as solid lines. Reprinted with permission from Della Longa et al. (2001). Copyright (2001) by the American Physical Society.

An additional increase in accuracy can be achieved if more techniques are combined. Arcovito et al. (2007) employed best-fitting of polarized XANES with X-ray diffraction to refine structural models of cyanomet sperm whale myoglobin.

Cooperation between measurement and modelling is needed to obtain information about the structure around doped impurities. This was demonstrated in the study of cobalt-doped ZnO by Ney et al. (2008). The comparison of experimental and theoretical Co K-edge XLD provides convincing proof that the Co atoms are located in zinc-substitutional sites (Fig. 3).

Figure 3 Measured and calculated XANES (a) and XLD (b) spectra at the Co K edge of cobalt-doped ZnO. A zinc-substitutional cobalt position was assumed. Reprinted with permission from Ney et al. (2008). Copyright (2008) by the American Physical Society.

The intensity of the pre-peak at the K edge of an orientationally averaged (powder) spectrum has commonly been used to distinguish between tetrahedral and octahedral coordination of transition-metal impurities. Namely, for tetrahedral coordination the hybridization of the semi-localized 3d atomic states of the impurity with the ligand states leads to states with a p symmetry with respect to the photoabsorber, giving rise to a sharp intense peak formed by dipole transitions. For an octahedral coordination this is not possible and the pre-peak can thus only be formed by less probable quadrupole transitions (Yamamoto, 2008). More detailed information, for example about possible distortions from the cubic symmetry, can be obtained by resorting to angle-dependent spectra. As an example one can mention a study of chromium impurities in an MgAl₂O₄ spinel, where it was found that while the pre-edge region of the orientationally averaged Cr K spectrum is not significantly affected by a trigonal distortion, the dichroic signal is quite sensitive to it (Juhin et al., 2008).

The dependence of the X-ray absorption spectrum on the direction of the polarization vector is especially strong at surfaces where the symmetry is substantially broken. This can be used to determine adsorption sites and bond lengths for adatoms and molecules on surfaces. Vvedensky et al. (1987) simulated polarized O K-edge XANES for c(2×2)O on Cu(100) assuming various adsorption sites and bond lengths to find that the O overlayer occupies hollow sites 0.7 Å above the copper surface (Fig. 4).

Figure 4 Comparison of O K-edge XANES of c(2×2)O on Cu(100) measured for two directions of the polarization vector with calculated spectra obtained within full multiple-scattering (solid lines) and single-scattering (open circles) frameworks for the hollow, bridge and atop adsorption sites. Reprinted with permission from Vvedensky et al. (1987). Copyright (1987) by the American Physical Society.

A similar approach was employed by Nordlund et al. (2004) to determine the surface structure of a thin ice film grown on Pt(111). Here, the measured polarized O K-edge XANES was modelled by superposing the theoretical spectra of oriented molecular clusters to conclude that crystalline ice is terminated with a large abundance of isotropically distributed free OH groups and a distorted subsurface.

Dipole transitions dominate over quadrupole transitions over practically the whole XAFS range, so the quadrupole transitions can usually be ignored. However, sometimes quadrupole transitions are important; for example, in the pre-edge region of spectra of transition-metal atoms at the K edge. This is because the K-edge quadrupole transitions are to the d states, the density of which is much higher in the respective energy region than the density of the p states (to which the dipole transitions occur). Deciding whether a spectral feature is of dipole or quadrupole nature is difficult; among other reasons, this is because theoretical calculations are especially challenging close to the edge, where effects due to the core hole, electron correlations and the nonspherical part of the potential can be significant. The ultimate answer may come from analysis of the angular dependence of the spectra, because depending on the setup this dependence can sometimes be quite different for dipole and quadrupole transitions.

This approach was first used for the analysis of the pre-peak in the Cu K-edge spectrum of the (creat)₂CuCl₄ complex (Hahn et al., 1982); see the left panel of Fig. 5 for the full angle-dependent spectrum. The symmetry of the CuCl₄ complex can approximately be described by point group D_4h, meaning that the dipole transitions should not exhibit any dependence on the azimuthal angle φ. Nevertheless, the experiment shows that the height of the pre-peak oscillates with a fourfold periodicity (right panel of Fig. 5), consistent with the angular dependence of the quadrupole transition for the respective symmetry (Brouder, 1990),

Figure 5 Left: Cu K-edge spectrum of (creat)2CuCl4 with highlighted pre-peak area. Right: variation of the height of the pre-peak when the polarization vector is rotated around the crystal c axis. Reprinted with permission from Hahn et al. (1982). Copyright (1982) by Elsevier.

As mentioned in Section 2, quadrupole transitions can induce angular dependence in spectra of cubic systems where the dipolar transitions are fully isotropic. A comprehensive analysis of the angular dependence of the Fe K-edge spectrum of cubic pyrite (FeS₂) was performed by Cabaret et al. (2001) for the whole XANES range. They demonstrated that the spectrum is indeed isotropic except for the pre-edge region, where the spectrum clearly contains a quadrupole component, in agreement with theory. Another instructive example is the Cr K-edge spectrum of an MgAl₂O₄:Cr³⁺ spinel. By combining the analysis of the experimental angular dependence of the pre-peak with ab initio and model multiplet calculations, Juhin et al. (2008) presented evidence for its quadrupole character. They also derived some general implications for studying the local geometry around the photoabsorbing atom by means of XLD.

A robust way to identify quadrupole transitions even for low-symmetry systems is to analyse spectra recorded with orientations of the sample and the X-ray beam such that the cross sections for dipole transitions are identical while the cross sections for quadrupole transitions differ. This approach was employed by Bocharov et al. (2001), who demonstrated in this way that the pre-peak in the Cu K-edge spectrum of monoclinic CuO has a quadrupole origin.

The extra information provided by polarized XAS becomes convenient if an accurate comparison between theory and experiment is required. This happens, for example, when one tries to identify many-body effects in the XANES. Initially this approach was adopted when studying molecular complexes, because accurate configuration–interaction calculations could be performed for these systems. By comparing the experimental angular dependence of the spectra with quantum-chemical calculations, many-body shake-down features were identified in the Cu K-edge spectra of a planar CuCl₄ complex (Kosugi et al., 1984) and a linear CuCl₂ complex (Yokoyama et al., 1986).

Another example in which comparison between experimental and theoretical angle-dependent spectra was used to demonstrate the presence of many-body effects is the study of the Cu K edge of CuO (Calandra et al., 2012). Here, the incorporation of many-body effects into theoretical spectra was achieved by first performing an ab initio calculation of a single-particle spectrum and then convoluting it with an experimental core-level photoemission spectrum to account for shake-up features. Comparisons of experimental angle-dependent spectra with calculations that either do or do not account for shake-up effects are shown in Fig. 6.

Figure 6 Identifying shake-up contributions to the Cu K-edge XANES of CuO by comparing calculated angle-dependent spectra with experimental spectra. Reprinted with permission from Calandra et al. (2012). Copyright (2012) by the American Physical Society.

The next level of sophistication is not just to show that many-body effects are present in the spectra but to employ spectroscopy to study the nature of these effects in greater detail. Juhin et al. (2010) analysed the polarized Co K-edge XANES of LiCoO₂ to find that the screening of the Co 1s core hole is angle-dependent. In their approach, they combined the GGA+U description of the correlations among the d electrons with supercell calculations for a system with one 1s electron removed to account for core-hole effects.

Finally, let us mention an example in which investigating the angular dependence of XANES together with the angular dependence of the X-ray magnetic circular dichroism (XMCD) helps to understand the role of many-body effects not just in the spectra but also in the physical mechanism that determines the distance between the adatom and substrate. This was achieved by Sessi et al. (2014) when studying angle-dependent spectra for iron, cobalt, nickel and copper adatoms on graphene and graphite. By comparing experiments with model Hamiltonian calculations, they were able to assess the role of dispersive (van der Waals) forces in the physisorption or chemisorption of 3d adatoms.