International
Tables for
Crystallography
Volume I
X-ray absorption spectroscopy and related techniques
Edited by C. T. Chantler, F. Boscherini and B. Bunker

International Tables for Crystallography (2022). Vol. I. Early view chapter
https://doi.org/10.1107/S1574870720007570

Diffraction anomalous fine structure: basic formalism

Hubert Reneviera and Maria Grazia Proiettib*

aLMGP, Université Grenoble Alpes, Grenoble INP, Grenoble, France, and bDepartamento de Física de la Materia Condensada, Universidad de Zaragoza, INMA, CSIC–UNIZAR, Zaragoza, Spain
Correspondence e-mail:  proietti@unizar.es

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.

1. Introduction

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[link]). 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[link]). 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[link]; Stragier et al., 1992[link]). Soon afterwards, Pickering and coworkers measured DAFS spectra of powder samples (Pickering et al., 1993[link]). The reader can find extensive information about DAFS spectrocopy in several review articles (Sorensen et al., 1994[link]; Hodeau et al., 2001[link]; Favre-Nicolin et al., 2012[link]; Renevier & Proietti, 2013[link]).

2. Multi-wavelength anomalous diffraction

The atomic scattering factor is written[f({\bf Q},E) = f^{0}(Q)+f'(E)+if''(E), \eqno (1)]where 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 Wykcoff position is noncubic. The anisotropy of the anomalous scattering must be considered when analysing resonant diffraction data (Joly, 2001[link]); 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 contribution[I({\bf Q},E)\propto \left|F_{T}+F_{A}{{f''_{A}+if''_{A}} \over {f^{0}_{A}}}\right|^{2}, \eqno (2)]where FT, the phase of which is φT, denotes the nonresonant partial structure factor. FA corresponds to the Thomson scattering of all resonant atoms A only. FT = FA + FN, where FN is the partial structure factor corresponding to all non­resonant atoms.

The classical MAD equation (Hendrickson, 1991[link]), giving the relationship between the intensity and the photon energy, can then be written as[\eqalignno {I({\bf Q},E)& \propto|F_{N}|^{2} [(f_{A}^{0}+f'_{A})^{2}+f_{A}^{\prime\prime2}] \left|{{F_{A}} \over {f^{0}_{A}}}\right|^{2} + {{2|F_{N}F_{A}|} \over {f_{A}^{0}}}\cr &\ \quad  {\times}\ [(f_{A}^{0}+f'_{A})\cos(\varphi_{N}-\varphi_{A})+f''_{A}\sin(\varphi_{N}-\varphi_{A})], & (3)}]where φN and φA are the phases of the partial structure factors FN and FA, respectively.

In equation (3)[link], [f^{0}_{A}], [f'_{A}] and [f''_{A}] are experimental data, so that three unknowns remain: |FN|, |FA| and φNA = φNφA. 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 φNA, 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[link]). 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[link]), i.e. in practice when the objects are large enough and present large strain gradients (see Fig. 1[link]). For smaller or buried objects, no a priori values can be assumed for φNA and at least three energies must be used (Favre-Nicolin, 2011[link]).

[Figure 1]

Figure 1

Spatial selectivity of a diffraction experiment shown for a simulated GeSi uncapped dome grown on Si(001) (Katcho et al., 2011[link]). The two-dimensional maps represent the diffraction weights [w'_{j={\rm Ge}}] 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.

3. Diffraction anomalous fine structure

3.1. Extended DAFS (EDAFS)

In the extended region above the edge, the atomic scattering factor of resonant atom j can be split into smooth and oscillatory parts: [f'_{j}] = [f'_{0A}] + [\Delta f''_{0A}\chi '_{j}] and [f''_{j}] = [f''_{0A}] + [\Delta f''_{0A}\chi ''_{j}], where [f'_{0A}] + [if''_{0A}] is the anomalous scattering of bare neutral atoms and [\Delta f''_{0A}] is the contribution of the resonant scattering to [f''_{0A}]. The smooth atomic scattering factor is f0A(Q, E) = [f_{A}^{0}+f_{0A}^{\prime}+if_{0A}^{\prime\prime}].

The real and imaginary components of the oscillatory fine structure [\tilde{\chi} = \chi_{j}^{\prime}+i\chi_{j}^{\prime\prime}] are related by the Kramers–Kronig transform (or the optical theorem) and the imaginary component [\chi ''_{j}] is related to the EXAFS χj by χj(E) = [{\rm Im}\tilde{\chi}({\bf Q} = 0,E)] (Stragier et al., 1992[link]; Cross et al., 1998[link]).

One can write the structure factor as [F({\bf Q},E) = F_{0}({\bf Q},E) +{{\Delta f_{0A}^{\prime\prime}(E)} \over {f_{A}^{0}(Q)}}\textstyle \sum \limits_{j}F_{j}({\bf Q})[\chi_{j}^{\prime}(E)+i\chi_{j}^{\prime\prime}(E)] \eqno(4)]to first order (Proietti et al., 1999[link]), where Fj, the phase of which is φj, is the partial structure factor of atom j. The intensity can then be written as[I({\bf Q},E) = FF^{*}\approx I_{0}+2{{\Delta f_{0A}^{\prime\prime}|F_{A}F_{0}|} \over {f_{A}^{0}}}\chi_{\bf Q}, \eqno (5)]where F0 = |F0|exp(iφ0), I0 = |F0|2 and χQ is the first-order extended DAFS (Favre-Nicolin et al., 2012[link]; Renevier & Proietti, 2013[link]),[\chi _{\bf Q} = S_{D}\left({{I-I_{0}} \over {I_{0}}}\right) = \cos(\varphi_{0A})\textstyle \sum \limits_{j=1}^{N_{A}} w_{j}^{\prime}\chi_{j}^{\prime} + \sin(\varphi_{0A})\textstyle \sum \limits_{j=1}^{N_{A}} w_{j}^{\prime\prime}\chi_{j}^{\prime\prime}, \eqno (6)]where [S_{D} = {{f_{A}^{0}|F_{0}|} \over {2\Delta f_{0A}^{\prime\prime}|F_{A}|}}]is a normalization factor. By fitting equation (2)[link] to the experimental DAFS spectrum with [f'_{0A}] and [f''_{0A}], one can determine [\beta = |F_{A}|/f_{A}^{0}|F_{T}|] and φTA = φTφA, which are used to calculate the scale factor SD and the phase difference φ0A = φ0φA. It is worth noting that there is no need for a crystallographic model to obtain an experimental χQ.[S_{D} = {{|F_{0}|} \over {|F_{T}|}}\left({{1} \over {2\beta\Delta f_{0A}^{\prime\prime}}}\right), \eqno (7)][\tan(\varphi_{0A}) = {{\sin(\varphi_{TA})+\beta f_{0A}^{\prime\prime}} \over {\cos(\varphi_{TA})+\beta f_{0A}^{\prime}}}, \eqno (8)][{{|F_{0}|} \over {|F_{T}|}} = \left\{[\cos(\varphi_{TA})+\beta f_{0A}^{\prime}]^{2}+[\sin(\varphi _{TA})+\beta f_{0A}^{\prime\prime}]^{2} \right\}^{1/2}. \eqno (9)]The crystallographic weights of resonant atom j to χQ (equation 6[link]) are written as[w_{j}^{\prime} = {{|F_{j}|\cos(\varphi_{0j})} \over {|F_{A}|\cos(\varphi_{0A})}}, \quad w_{j}^{\prime\prime} = {{|F_{j}|\sin(\varphi_{0j})} \over {|F_{A}|\sin(\varphi_{0A})}}, \eqno (10)]where φ0j = φ0φj represents the orthogonal projection of [\chi '_{j}] and [\chi ''_{j}] on the vector F0 in the complex plane: [\textstyle\sum_{j=1}^{N_{A}} w_{j}^{\prime} = \sum_{j=1}^{N_{A}} w_{j}^{\prime\prime} = 1]. One can calculate them provided that the crystallographic structure is known, or possibly determine them if the individual [\chi '_{j}] and [\chi ''_{j}] are known by fitting equation (6)[link] to the experimental χQ. Note that [\textstyle \sum_{j=1}^{N_{A}}w_{j}^{\prime\prime}\chi_{j}^{\prime\prime}] is to be compared with the extended XAFS oscillations [\chi_{\rm EXAFS} = \textstyle \sum _{j=1}^{N_{A}}\chi_{j}^{\prime\prime}]. The fundamental difference are the weights [w''_{j}] 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[link]), the Ge content being uniform and equal to 0.6 (Fig. 1[link]). 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.

4. EDAFS path formalism

Using the EXAFS path formalism (Stragier et al., 1992[link]; Cross et al., 1998[link]; Proietti et al., 1999[link]), the complex extended fine structure that depends on the local atomic environment of a resonant atom j is[\chi_{j}^{\prime}+i\chi_{j}^{\prime\prime} = -\textstyle \sum\limits_{\gamma=1}^{\Gamma_{j}} A_{\gamma j} (k)\exp\{-i[2kR_{\gamma j}+\varphi_{\gamma j}(k)]\}. \eqno (11)]The sum runs over all scattering paths γ of the virtual photoelectron, k = (1/ℏ)[2me(EE0)]1/2 is the photoelectron wavenumber, ℏ is the Planck constant, me is the electron mass, E0 is the edge energy, Rγj is the effective length of path γj and φγj(k) is the net scattering photoelectron phase shift.

Equations (6)[link] and (11)[link] lead to the following expression for χQ(k):[\eqalignno {\chi_{\bf Q}(k) &= {\textstyle \sum \limits_{j=1}^{N_{A}}} {{|F_{j}|} \over {|F_{A}|}} {\textstyle \sum \limits_{\gamma=1}^{\Gamma_{j}}} A_{\gamma j}(k) \cr &\ \quad { \times}\ \sin\left [2k R_{\gamma j}+\varphi_{\gamma j}(k)+\varphi_{0}(k)-\varphi_{j}-{{\pi} \over {2}}\right]. &(12)}]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[link]), one can set Aγj = Aγ, φγj = φγ and average on photoelectron scattering paths γj over all j sites. Then, equation (6)[link] can be written in a simple form, [\eqalignno {\chi_{\bf Q}(k) & = {\textstyle \sum \limits_{\gamma}} A_{\gamma}(k) \exp(-2k^{2}\sigma_{\gamma}^{2}) \cr &\ \quad {\times}\ \sin \left [ 2k\langle R\rangle_{\gamma}+\varphi_{\gamma}(k)+\varphi_{0}(k)-\varphi_{A}-{{\pi} \over {2}} \right], &(13)}]where 〈Rγ is the average effective length of path γ and σγ 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 wj factors, which express the diffraction condition and allow one to select an isostrain region of the sample, as shown in Fig. 1[link]. In equation (13)[link], the χQ expression is very similar to the EXAFS expression, with the only difference being the crystallographic phase Δψ = φ0(k) − φA − (π/2) in the sine argument.

5. Conclusion

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[link]) and nanostructures embedded in heterostructures (Létoublon et al., 2004[link]) or free-standing on a bulk substrate (Katcho et al., 2011[link]). 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[link]). Also, it makes it possible to analyse atomic ordering in alloys (Meyer et al., 1999[link]; Ersen et al., 2003[link]) and determine the Warren–Cowley short-range order parameters in alloys (Belloeil et al., 2020[link]), which define the presence of short-range ordering, clustering or random atomic distribution, using a single experiment.

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