Chapter 2.21. Diffraction anomalous fine structure: basic formalism

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, f⁰ 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 Wyckoff 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).

Figure 1 Spatial selectivity of a diffraction experiment shown for a simulated GeSi uncapped dome grown on Si(001) (Katcho et al., 2011). The two-dimensional maps represent the diffraction weights for different Q(h, 0, 0) vectors with h = 3.97, 3.96, 3.95 and 3.93, close to the Si 400 reflection. Decreasing h corresponds to increasing the height z of the scattered region in the Bragg condition. The weights are summed along the [100] direction and plotted in the yz plane.

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)[2m_e(E − E₀)]^1/2 is the photoelectron wavenumber, ℏ is the Planck constant, m_e is the electron mass, E₀ is the edge energy, R_γj is the effective length of path γj 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 γj 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 γj 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 = φ₀(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.