InternationalX-ray absorption spectroscopy and related techniquesTables for Crystallography Volume I Edited by C. T. Chantler, F. Boscherini and B. Bunker © International Union of Crystallography 2022 |
International Tables for Crystallography (2022). Vol. I. Early view chapter
https://doi.org/10.1107/S1574870720007570 ## Diffraction anomalous fine structure: basic formalism
The aim of this chapter is to give a basic formalism for diffraction anomalous fine-structure spectroscopy (DAFS) in combination with multi-wavelength anomalous diffraction for the study of the structural properties of materials. The experimental aspects of DAFS are reported together with a few examples of its application. Keywords: multi-wavelength anomalous diffraction; diffraction anomalous fine structure. |

Similarly to X-ray absorption fine-structure (XAFS) spectroscopy, diffraction anomalous fine-structure (DAFS) spectroscopy provides information about the empty electronic orbitals, oxidation state and local atomic environment of resonant (anomalous) atoms selected by the diffraction condition. It belongs to the family of resonant elastic X-ray scattering methods (Galéra *et al.*, 2012). The appearance of fine structures in the diffracted intensity spectra close to absorption edges was first reported in the mid-1950s by Y. Cauchois (Cauchois, 1956). In the late 1980s and early 1990s, Arčon and coworkers and Stragier and coworkers clearly described the similarity of XAFS and DAFS oscillations and demonstrated the interest of DAFS to the XAFS community (Arčon *et al.*, 1987; Stragier *et al.*, 1992). Soon afterwards, Pickering and coworkers measured DAFS spectra of powder samples (Pickering *et al.*, 1993). The reader can find extensive information about DAFS spectrocopy in several review articles (Sorensen *et al.*, 1994; Hodeau *et al.*, 2001; Favre-Nicolin *et al.*, 2012; Renevier & Proietti, 2013).

The atomic scattering factor is writtenwhere **Q** is the scattering vector, *Q* is its amplitude, *E* is the photon energy, *f*^{0} is the Thomson atomic scattering factor and *f*′ and *f*′′ are the real and imaginary parts of the anomalous scattering factor, respectively. The anomalous scattering factor depends on the valence and local environment of the resonant atoms and on the polarization direction of the incoming and scattered X-ray beams when the point-group symmetry of the Wykcoff position is noncubic. The anisotropy of the anomalous scattering must be considered when analysing resonant diffraction data (Joly, 2001); usually, *f*′ and *f*′′ are retrieved from polarized XAFS spectra measured prior to performing multi-wavelength anomalous diffraction (MAD) experiments.

The scattered intensity from the crystal structure can be written by separating out the resonant scattering contributionwhere *F*_{T}, the phase of which is φ_{T}, denotes the nonresonant partial structure factor. *F*_{A} corresponds to the Thomson scattering of all resonant atoms *A* only. *F*_{T} = *F*_{A} + *F*_{N}, where *F*_{N} is the partial structure factor corresponding to all nonresonant atoms.

The classical MAD equation (Hendrickson, 1991), giving the relationship between the intensity and the photon energy, can then be written aswhere φ_{N} and φ_{A} are the phases of the partial structure factors *F*_{N} and *F*_{A}, respectively.

In equation (3), , and are experimental data, so that three unknowns remain: |*F*_{N}|, |*F*_{A}| and φ_{NA} = φ_{N} − φ_{A}. These can be determined if the scattered intensity is measured for at least three optimally chosen energies around the absorption edge. MAD is a powerful tool for determining the atomic composition at crystallographic sites. For instance, in the case of a semiconductor alloy the unit-cell structure is usually known and therefore so is φ_{NA}, so that a measurement at two energies can be sufficient to determine the composition inside the sample region selected by the diffraction condition (Magalhães-Paniago *et al.*, 2002). In the case of semiconductor nanostructures this is only true as long as *isostrain* regions within the nano-objects can be distinguished in the diffraction pattern (Kegel *et al.*, 2000), *i.e.* in practice when the objects are large enough and present large strain gradients (see Fig. 1). For smaller or buried objects, no *a priori* values can be assumed for φ_{NA} and at least three energies must be used (Favre-Nicolin, 2011).

In the extended region above the edge, the atomic scattering factor of resonant atom *j* can be split into *smooth* and *oscillatory* parts: = + and = + , where + is the anomalous scattering of bare neutral atoms and is the contribution of the resonant scattering to . The smooth atomic scattering factor is *f*_{0A}(**Q**, *E*) = .

The real and imaginary components of the oscillatory fine structure are related by the Kramers–Kronig transform (or the optical theorem) and the imaginary component is related to the EXAFS χ_{j} by χ_{j}(*E*) = (Stragier *et al.*, 1992; Cross *et al.*, 1998).

One can write the structure factor as to first order (Proietti *et al.*, 1999), where *F*_{j}, the phase of which is φ_{j}, is the partial structure factor of atom *j*. The intensity can then be written aswhere *F*_{0} = |*F*_{0}|exp(*i*φ_{0}), *I*_{0} = |*F*_{0}|^{2} and χ_{Q} is the first-order extended DAFS (Favre-Nicolin *et al.*, 2012; Renevier & Proietti, 2013),where is a normalization factor. By fitting equation (2) to the experimental DAFS spectrum with and , one can determine and φ_{TA} = φ_{T} − φ_{A}, which are used to calculate the scale factor *S*_{D} and the phase difference φ_{0A} = φ_{0} − φ_{A}. It is worth noting that there is no need for a crystallographic model to obtain an experimental χ_{Q}.The crystallographic weights of resonant atom *j* to χ_{Q} (equation 6) are written aswhere φ_{0j} = φ_{0} − φ_{j} represents the orthogonal projection of and on the vector *F*_{0} in the complex plane: . One can calculate them provided that the crystallographic structure is known, or possibly determine them if the individual and are known by fitting equation (6) to the experimental χ_{Q}. Note that is to be compared with the extended XAFS oscillations . The fundamental difference are the weights that give EDAFS site/spatial selectivity. As an example, we calculated EDAFS weights for a GeSi uncapped island (dome) grown on Si(001) (Katcho *et al.*, 2011), the Ge content being uniform and equal to 0.6 (Fig. 1). The uncapped dome relaxes in the growth direction (*z* axis), *i.e.* the in-plane lattice parameter increases with *z*. The isostrain regions, selected by the diffraction condition along the *z* direction, are clearly evident as a function of *h*. The maps give a direct and quantitative estimation of the spatial resolution in the *z* direction.

Using the EXAFS path formalism (Stragier *et al.*, 1992; Cross *et al.*, 1998; Proietti *et al.*, 1999), the *complex extended fine structure* that depends on the local atomic environment of a resonant atom *j* isThe sum runs over all scattering paths γ of the virtual photoelectron, *k* = (1/ℏ)[2*m*_{e}(*E* − *E*_{0})]^{1/2} is the photoelectron wavenumber, ℏ is the Planck constant, *m*_{e} is the electron mass, *E*_{0} is the edge energy, *R*_{γj} is the effective length of path γ*j* and φ_{γj}(*k*) is the net scattering photoelectron phase shift.

Equations (6) and (11) lead to the following expression for χ_{Q}(*k*):When the virtual photoelectron probes the same local atomic environments for all resonant atoms *j* in a region selected by the diffraction condition (see Fig. 1), one can set *A*_{γj} = *A*_{γ}, φ_{γj} = φ_{γ} and average on photoelectron scattering paths γ*j* over all *j* sites. Then, equation (6) can be written in a simple form, where 〈*R*〉_{γ} is the average effective length of path γ and σ_{γ} is the bond-length disorder (static and dynamic Debye–Waller factors). Note that the same kind of averaging is performed in the case of EXAFS, but here a weighting of the γ*j* paths is performed by *w*_{j} factors, which express the diffraction condition and allow one to select an isostrain region of the sample, as shown in Fig. 1. In equation (13), the χ_{Q} expression is very similar to the EXAFS expression, with the only difference being the crystallographic phase Δψ = φ_{0}(*k*) − φ_{A} − (π/2) in the sine argument.

A diffraction anomalous fine-structure measurement is a multi-wavelength anomalous diffraction measurement enriched by a continuous energy scan across resonance edges and over a wide energy range, including the extended fine-structure region (extended DAFS) above the resonance. It allows access, in the same experiment, to long-range crystallographic properties, such as for instance site-dependent atomic population, alloy composition and crystal polarity, and to the very local atomic environment of the resonant atoms selected by diffraction, mostly the atomic population and bond lengths of the nearest and next-nearest neighbours. DAFS can give access to information that is out of the reach of XAFS, such as as for instance local structural properties of heterogeneous materials (Maurizio *et al.*, 2020) and nanostructures embedded in heterostructures (Létoublon *et al.*, 2004) or free-standing on a bulk substrate (Katcho *et al.*, 2011). Indeed, due to the reduced, if not negligible, self-absorption and fluorescence effects, it is well suited to the study of nanostructures and thin films (Lamberti & Agostini, 2013). Also, it makes it possible to analyse atomic ordering in alloys (Meyer *et al.*, 1999; Ersen *et al.*, 2003) and determine the Warren–Cowley short-range order parameters in alloys (Belloeil *et al.*, 2020), which define the presence of short-range ordering, clustering or random atomic distribution, using a single experiment.

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