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/S1574870722001562 ## Multiple-scattering theory of X-ray absorption spectroscopy as a structural tool
The physical meaning and the potentialities of X-ray absorption spectroscopy (XAS) from core states are illustrated in the framework of multiple-scattering theory. By exploiting the mathematical and physical relationships between photoelectron diffraction (PED) and XAS, the energy modulation of the absorption coefficient above the absorption edge is interpreted as a three-dimensional holographic image of the atomic positions of the sample projected onto the energy axis. The theoretical conditions under which this interpretation is possible are specified, making PED and XAS a powerful tool for structural investigations in surface and material science. Keywords: multiple-scattering theory; absorption spectroscopy; structural techniques. |

The process of photoemission from an atom embedded in an environment by an incoming photon of sufficient energy ℏω to excite a core state is a quantum process that promotes a core electron into a continuum state with a kinetic energy given by the Einstein relation. The excited electron can either be detected (photoelectron diffraction; PED) or integrated over all emission angles to inclusively count the total number of core holes to calculate the absorption coefficient (X-ray absorption spectroscopy; XAS).

Ignoring the complications arising from electronic correlations, and therefore also the energy losses of the excited electron, one may describe the PED process in simple terms as a process in which the external photon generates an electron in the state of a spherical wave which, after traversing the emitter atom (the source), is scattered elastically (*i.e.* conserving the energy) by the atoms of the environment (the object) before reaching the detector located at a macroscopic distance from the sample. In particular, it can occur that the photoelectron passes directly from the emitter to the detector. At each event along a scattering path the photoelectron amplitude is modified by the scattering power of the atomic potential involved and multiplied by a phase. According to the prescriptions of quantum mechanics, all of the amplitudes of all the scattering paths with a common final state are to be summed together before taking their square modulus to obtain their intensity at the detector.

It is clear that the interference patterns thus measured by the detector depend on the actual positions of the atoms in the sample and represent a three-dimensional hologram of the sample object (*i.e.* the atomic positions in the system under study) in momentum and energy space, associated with the three degrees of freedom of the two polar and azimuthal emission angles and the final kinetic energy of the photoelectron (Fig. 1), as pointed out by Barton (1988). For an in-depth discussion on this aspect from an experimental point of view, see Fadley *et al.* (1994).

The analogy with holography is more than a verbal similarity. In optical holography a reference light wave is split into two beams by a splitter mirror. The main beam is sent directly to the detector, whereas the less intense secondary beam (usually 10% of the main beam) impinges on the object before arriving at the detector (usually a photographic plate). The interference patterns thus measured are then illuminated to reconstruct a virtual image of the object in real space.

The similarity to PED is striking, with three main differences. The first one is that in PED we are dealing with an electronic wave instead of a light wave to construct the hologram (although originally Gabor thought of electrons for the holographic process). The second difference is related to the fact that the incident wave (the source) is external to the object, while in PED it is generated internally in the object. Also, the intensity of the `primary beam' in PED (*i.e.* the electronic wave that passes directly from the photoemitter to the detector) cannot be controlled by the experimentalist but depends on the features of the photoemission process. The third difference concerns the reconstruction of the object, which in PED must be carried out with the help of a computer. Apart from these differences, the two techniques are based on the same quantum process of interference between scattered waves.

If we do not observe the photoemitted electrons and integrate over all emission angles at a fixed electron kinetic energy, we have a measure of the number of core holes created and therefore of the number of photoemission events. This number is proportional to the absorption coefficient at the chosen energy, as borne out by the mathematical derivation given below in the framework of multiple-scattering theory (MST; Mustre de Leon, 1989).

The integration process eliminates the physical detector located outside the sample and replaces it by the atomic emitter (since we count the number of holes), which in this way becomes both the source and the detector of the photoelectrons. This feature translates into the fact that only closed scattering paths are observed in absorption, as opposed to both closed and open paths that contribute to the interference patterns in PED. The observed modulations in the absorption coefficients as a function of the photoelectron kinetic energy therefore represent the remnants of these patterns after averaging over the emission angles. They are due to interference of the outgoing photoelectron wave with the returning wave due to the double role (source and detector) played by the emitter and can be used for structural analysis.

Throughout this chapter, we will use atomic units for lengths and Rydberg units for energies.

In the independent particle approach the initial and final states of the system are Slater determinants so that, ignoring for the moment the presence of the core hole in the final state, the photoemission cross section for the ejection of a photoelectron along the direction is given in the dipole approximation by where α = 1/137 is the fine-structure constant, ℏω is the incoming photon energy with polarization vector **ɛ**, |φ_{c}(**r**)〉 is the initial core-level wavefunction and is the time reversal of the continuum scattering state describing the excited photoelectron with kinetic energy *E* (*k* = *E*^{1/2}) given by the Einstein relation *E* = ℏω − *I*_{c}, where *I*_{c} is the core ionization potential referred to the vacuum level. The time reversal is necessary to impose the boundary condition that no electron exists in a continuum state in the remote past. The details of the derivation are given in Sébilleau *et al.* (2006).

The continuum state satisfies the Schrödinger equation (SE) with positive energy with an outgoing wave boundary condition and normalization to one state per Rydberg, where the symbol → indicates asymptotic behaviour. In equation (2) *V*_{eff}(**r**) represents an effective optical potential, in general complex, coming from the reduction of the many-body problem to an effective one-particle problem. We defer discussion of the complex case to a later section and assume a real potential for the moment. Details of the reduction process are provided in Sébilleau *et al.* (2006).

By introducing the free Green's function (GF)we transform the SE with the above boundary conditions into the equivalent Lippmann–Schwinger equation (LSE), Due to the linearity of this equation and the expansion of a plane wave in partial spherical waves, we can write so that obeys the same equation (5) but in response to an exciting spherical wave *J*_{L}(**r**; *k*) of angular momentum *L*, Remembering that where **r**_{<} and are the lesser and greater of **r** and **r**′, respectively, taking **r** at distances much greater than the effective range of the potential, assumed to decay more rapidly than 1/**r**, we find asymptotically where we have defined as The quantity is known as the **T**-matrix of the potential and measures the scattering response to an incident wave of angular momentum *L* into one with angular momentum *L*′. We therefore recover the same asymptotic behaviour as in equation (3) if one multiplies all terms of equation (10) by *A*_{L}(**k**) and takes into account the behaviour of at great distances. In the above, we have used for short and , where *j*_{l}, *h*_{l} denote spherical Bessel and Hankel functions of order *l*, respectively, and *L* stands for *l*, *m*. Throughout the paper we will use real spherical harmonics.

At its most basic, multiple-scattering theory (MST) is a technique for solving a linear partial differential equation over a region of space with certain boundary conditions. It is implemented by dividing the space into non-overlapping domains Ω_{j} (cells), solving the differential equation separately in each of the cells and then assembling the partial solutions together into a global solution that is continuous and smooth across the whole region and satisfies the given boundary conditions. This task is accomplished by introducing a partition of the potential that follows the same partition of the space, so that , where *v*_{j}(**r**) coincides with *V*_{eff}(**r**) within the cell Ω_{j} and is zero outside. We then introduce local scattering solutions obeying the same equation (8) but with the potential *v*_{j}(**r**). These solutions are local in that they are defined only in the reference cell Ω_{j} whose centre is located at **R**_{j}, so that **r**_{j} = **r** − **R**_{j}. In terms of these we can also define the scattering power of the various cells Ω_{j}. Hatada *et al.* (2010) provide the method for solving this local problem numerically.

It is now reasonable, and can be shown rigorously, that the global solution inside cell Ω_{j} can be written as a superposition of local solutions in the form Then, after the integral over all space is partitioned over the cells Ω_{j}, equation (5) transforms the differential problem into an algebraic condition for the coefficients , where originates from the inhomogeneous term in equation (6) and the partial wave propagators come from the two-centre decomposition of the free GF,

Equation (13) has a simple physical interpretation. It shows that the local amplitude at site *j* and angular momentum *L* is obtained by summing the inhomogeneous term to the contributions coming from all of the other sites *i* after scattering from the corresponding cell through the potential **T**-matrix active there and free propagation from *i* to *j* via the spherical wave propagator . In a sense, it transforms the point-to-point propagation of equation (5) into a cell-to-cell propagation by replacing the point potential scattering with the corresponding cell **T**-matrix.

While in equation (12) the expansion coefficients are dimensionless quantities, in MST it is expedient to work with scattering amplitudes defined as This means introducing new local basis functions given by such that On the basis of equation (13), the new MS equation are easily seen to be whereby the solution for the amplitude at site *n*, is given in terms of the scattering path operator **τ**, which is the inverse of the MS matrix (**T**^{−1} − **G**), As usual, we have introduced matrices labelled by the site and angular momentum indices. The matrix **τ** shows the typical separation, peculiar to MST, between the atomic dynamics represented by the cell **T**-matrix and the geometrical arrangement of the atoms descibed by the spherical wave propagator matrix **G**.

The series expansion is possible when ρ(**T****G**), the spectral radius (maximum eigenvalue) of **T****G**, is less than 1. In the case of real potentials the scattering amplitudes satisfy the relation which is a kind of generalized optical theorem. This relation is very important, since it establishes the connection between the photoemission and photoabsorption cross section (Hatada *et al.*, 2010).

On the basis of equations (17) and (1), we are now in a position to calculate the PED cross section. Due to the localization of the initial core state at site *c*, we have taking into account spin degeneracy in nonmagnetic systems, which we assume to be dealing with, and introducing the atomic transition-matrix element Remembering equation (19), we obtain the PED cross section by taking the product of the amplitude for creating a photoelectron in a state of spherical wave *L* selected according to the dipole selection rule from an initial core state *L*_{c}, multiplied by the amplitude of propagation from site *c* to any site *j*, starting with angular momentum *L* and ending with angular momentum *L*′ after any number of scattering events, multiplied by the phase difference exp(*i***k** · **R**_{j}) between the initial and final sites *c* and *j*, multiplied by the spherical wave amplitude for escaping towards the detector. All of these amplitudes are summed together and squared to obtain the intensity of the photoelectron current at the detector. The inversion of of the MS matrix (**T**^{−1} − **G**) by series expansion, when possible, allows us to decompose the full propagation from the emitter to any site in the sample in terms of scattering paths of various orders, beginning from order zero, which represents direct propagation of the photoelectron from the emitter to the detector. All of this supports the holographic interpretation of PED given in Section 1.

By integrating the PED cross section over all emission angles and exploiting equation (21), we obtainwhich can be shown to be equal to the absorption cross section (Sébilleau *et al.*, 2006). In fact, from the spectral representation of the full GF of the system, the solution of the equation can be written aswhere *n* runs over all bound and continuum states of the SE. Therefore, where the θ function assures that transitions are possible only to unoccupied states above the Fermi energy *E*_{F} of the system. Note, *en passant*, that features due to transitions to empty bound states below the vacuum level (no photoelectron in the vacuum) can appear in an absorption spectrum (see Fig. 8 in Chantler & Creagh, 2022). Despite their relevance to the understanding of the electronic structure of the system under study, we do not discuss them in the present treatment, which is mainly focused on the use of XAS for structural analysis. Fig. 2 shows a typical 1*s* absorption spectrum picturing the XANES and EXAFS regions. Quadrupole 1*s* → *nd* transitions are also shown.

One of the advantages of MST is that one can write an explicit form of the full GF in terms of the scattering path operator as where is the irregular solution of the SE inside cell Ω_{i} matching smoothly to *J*_{L}(**r**; *k*) on the bounding sphere of the cell. Notice that the singular part of the GF is real if the potential is real, so that its imaginary part is zero. By inserting this form of the GF into equation (27) one recovers equation (24).

According to this equation, the absorption cross section is the imaginary part of the product of the amplitudes for creating a photoelectron multiplied by the amplitude for its propagation from the emitter site back to the same site multiplied by the amplitude for destroying it. Therefore, the emitter acts as the source and detector of the photoelectron and only closed paths contribute to the absorption features.

The connection between integrated PED and XAS via the full GF makes an alternative point of view possible. By taking the imaginary part of both terms in equation (26) at **r** = **r**′, we see that is the total density of states at energy *E*. Using the form in equation (28) for the GF, we obtain showing that, due to the localization of the core hole, the polarization-averaged XAS is proportional to the *L* projected density of states onto the site of the photoabsorber for the final *L* selected by the dipole selection rule, since averaging over the incoming photon polarizations retains only diagonal terms in *L* in equation (24). This is not only true for the near-edge region of an absorption spectrum (XANES), as widely recognized in the literature (Ankudinov *et al.*, 1998), but also for the high-energy extended region (EXAFS), which is usually thought of as a scattering pattern rather than in terms of density of states.

The MS theory of PED and XAS presented in Section 2, although an aid to physical intuition, is only an approximation to the physical reality and cannot be used for accurate structural analysis. It provides oscillation of the absorption coefficient with amplitudes that are too large and maxima and minima that are a few eV off the corresponding features in the experimental spectrum, even if one uses a realistic (real) effective potential.

Corrections due to many-body effects must be considered. In fact, the excitation of a core electron leaves behind a core hole and an (*N* − 1)-electron system which, due to electron correlations, can remain either in its ground state (completely relaxed around the core hole, usually carrying most of the weight) or in an excited state with a probability depending on the system under consideration. Each of the ways that the initial total energy is partitioned between the photoelectron and the remaining (*N* − 1)-electron system is called a channel. The excitation energy appears as an energy loss for the photoelectron and is due to various inelastic processes, whether intrinsic (shake up, shake off) or extrinsic (plasmon excitations, scattering by electrons of the system *etc.*).

Due to the presence of inelastic channels, the integrated PED spectrum is no longer equal to the total absorption cross section, because the latter is an all-inclusive measurement that counts all processes, elastic and inelastic, whereas in PED one measures a particular channel selected by the final kinetic energy of the detected photoelectron.

According to multichannel scattering theory, in the description of one particular channel it is possible to eliminate all of the others and describe their effect by a complex effective optical potential, which is usually nonlocal and energy-dependent.

In an absorption experiment all of the photoemission channels with the same total energy instead interfere quantummechanically. Due to the fact that their various spectral features peak at different energies and add up with no fixed phase relations, the resulting effect on the absorption coefficient tends to smear out the strong diffraction features due to the single-particle density of states of the completely relaxed (primary) channel, sometimes shifting local maxima and minima. For a dynamical model describing this process, taking into account intrinsic and extrinsic escitations and their interference, see Campbell *et al.* (2002).

For our purposes here, suffice it to say that one can approximately describe this smearing effect by a complex, energy-dependent, local potential *V*_{eff}(**r**; *E*). We refer to Natoli *et al.* (2003) and Rehr & Albers (2000) for the physical meaning of this potential and its characterization. Its presence gives a finite mean free path to the excited photoelectron, given approximately by , where indicates a path average (see Section 3.3 of Sébilleau *et al.*, 2006). In this respect the core-hole width can be considered to be an added complex potential, so that λ_{t}(*E*) = *E*^{1/2}/. Due to this finite mean free path, PED and XAS become local techniques in the sense that they carry structural information only on atoms located within a sphere of radius λ_{t} for XAS and of radius 2λ_{t} for PED centred on the photoabsorber, in keeping with the different kind of paths (closed and open) observed in the two spectroscopies. This locality is also effective in periodic systems. In the following, we will concentrate mainly on absorption.

In order to take inelastic losses in the absorption process into account one needs to remember that in scattering theory with a complex potential the optical theorem relates the imaginary part of the forward-scattering amplitude to the sum of the elastic plus inelastic total cross sections (see Chapter 7 of Canto & Hussein, 2012). For this purpose, we solve equation (25) with the complex potential *V*_{eff}(**r**; *E*) and rewrite the full GF (equation 28) as where is now the irregular solution of the SE inside cell Ω_{i} matching smoothly to on the bounding sphere of the cell (Hatada *et al.*, 2010). Since τ and *T* are scattering amplitudes, it can be shown that equation (27) can also be used with a complex potential to account for both elastic and inelastic processes (Sébilleau *et al.*, 2006). Moreover, the presence of a core hole in the final state coupled to the reduction to an effective one-particle problem introduces an overlap factor *S*_{0}(ω) between the Slater determinants of the initial and final states, which is usually approximated by a constant of the order of 0.8–0.9 (for a dynamical model, see Campbell *et al.*, 2002).

When the potential is complex, the form (30) of the GF is to be preferred, since it makes explicit the separation between the atomic absorption and the scattering from the environment, due to the fact that the first term becomes zero in the absence of a neighbourhood to the photoemitter. Therefore, so that the absorption coefficient is composed of a background atomic absorption (usually smooth) and a superimposed structural signal due to the environment of the photoabsorber. In the following, we will restrict ourselves to analysis of the polarization averaged XAS.

Structural analysis in XAS rests on the interpretation of the energy modulations of the scattering path operator **τ** and the possibility of resolving it in terms of MS paths. As anticipated, the series expansion of **τ** is absolutely convergent when ρ(**T****G**) < 1, otherwise it is only conditionally (or asymptotically) convergent (Sébilleau & Natoli, 2009; Rehr & Albers, 2000). Therefore, with due precautions, one can almost always analyze the energy modulations in terms of MS paths. From equations (30) and (20) we find, taking into account that , using for short *C* = −8παℏω|*S*_{0}(ω)|^{2}.

The functional form of the general term of the series is given in Natoli & Benfatto (1986), together with exact expressions for the terms *n* = 2 (single scattering) and *n* = 3 (double scattering).

In general, the number of relevant paths depends on the size of ρ(**T****G**). If the spectral radius is near 1, many terms contribute to the series, whereas for ρ(**T****G**) near zero only the lowest terms with *n* = 2 contribute. Since ≃ , the size of ρ(**T****G**) mainly depends on the magnitude of , where δ_{l}(*E*) is the phase shift of the scattered wave and sinδ_{l}(*E*) measures the electron–atom interaction strength at energy *E*. According to potential scattering, δ_{l}(*E*) begins around *n*_{l}π at *E* = 0, where *n*_{l} is the number of bound states of orbital symmetry *l*, and tends to zero for *E* → ∞ (Canto & Hussein, 2012). Therefore, in an absorption spectrum, beginning from *E* = 0, usually (but not always) we find an energy region where an infinite (or very large) number of paths of high order contribute [the full MS (FMS) region, called XANES (X-ray absorption near-edge structure)], followed by a region where few low-order paths are present [intermediate MS (IMS) region]; a third much wider region then follows where only single-scattering paths are present (*n* = 2). This latter is called the EXAFS (extended X-ray absorption fine structure) region (Benfatto *et al.*, 1986). There are exceptions to this ordering that occur when the relevant phase shifts begin exactly at *n*_{l}π at *E* = 0. This happens, for example, for noble gases. In fact for all noble gases except helium the outermost shell always contains eight electrons in the shells *ns*, *np* (*n* = 2, 5). Therefore, the atomic potential just has bound *l* = 0, 1 states, so that the corresponding phase shifts δ_{0,1} start exactly at a multiple of π. As a consequence ρ(**T****G**) ≃ 0, so that one can analyze the XANES region only in terms of single-scattering paths, as in solid argon. Another example is copper, which has a 3*d* resonance at the Fermi level so that the phase shifts δ_{2} of the states ɛ*d* are very near π and δ_{0,1} is very small (Sébilleau & Natoli, 2009).

The analysis in MS paths, although useful to obtain structural information, is not very efficient and versatile. One can, however, reorder the MS series in such a way as to obtain a faster convergence rate and to relate each order *n* to an *n*-body distribution function. This method is the basis of the *GNXAS* analysis package, and we refer to Filipponi & Di Cicco (1995) and Filipponi *et al.* (1995) for an in-depth discussion. For a comprehensive review of alternative methods for structural analysis based on MST (in particular the path-by-path method), the bearing of many-body effects on the structural signal and a discussion of the convergence of the MS series taking into account the photoelectron mean free path and thermal vibrations, see Rehr & Albers (2000) and references therein.

When the Coulomb interaction of the excited photoelectron not only produces an average effect that can be described by adding a complex part to the static atomic exchange–correlation potential, but also generates quantum effects, in that it substantially changes the final density of states, one is obliged to take the dynamical reaction of the atomic system to the impinging electron into account. This is possible in the framework of multi-channel MST. This approach goes beyond the atomic multiplet approach in crystal field theory in that it takes into account both the local atomic electron correlations and the extended character of the electron wavefunction (band effect) in an *ab initio* way. It is expected that this approach will be required in the low-energy region of the absorption spectrum of strongly correlated systems, especially at the *L* edges, while the high-energy region is almost unaffected due to the fact that the interchannel matrix elements go to zero here and one reaches the sudden approximation limit in the photoemission spectra (Krüger & Natoli, 2004; Krüger, 2010; Sébilleau *et al.*, 2006).

The reduction of the many-body core excitation problem to an effective one-particle process in view of structural analysis is still an open problem that has not yet been completely solved. Questions such as the incomplete charge relaxation around the core hole, the concomitant excitonic effect in narrow band systems, the construction of a reliable yet practically usable optical potential other than the local density Hedin–Lundqvist (HL) exchange (Hedin & Lundqvist, 1971) and correlation potential, a good representation of the atomic background including multi-excitation processes, a good description of the core excitation in light-element systems and the low-energy interference effect between intrinsic and extrinsic excitations are all aspects of the absorption process that should be addressed for its improved use as a structural tool, apart from their intrinsic importance in fundamental physics.

Despite the presence of these unsolved aspects of the absorption problem, the approximate theory presented in Section 3 can be expected to work satisfactorily in all cases where the primary (elastic) channel in photoemission carries most of the weight (80–90%) with respect to the inelastic channels, so that many-body corrections are expected to be small. In this case the complex HL potential, a kind of local density approximation to the self-energy of the excited photoelectron describing only extrinsic losses due to local plasmon excitations, is sufficient for a realistic representation of the EXAFS region, although this is not the case in the XANES region of the spectrum. In this latter region, even in case of small interchannel coupling, subtle effects of quantum interference between intrinsic and extrinsic excitations and their cancellation may play a role in modifying the absorption spectrum (Campbell *et al.*, 2002). Indeed, the description of XANES is less satisfactory than that achieved in the EXAFS region and probably depends more heavily on the unsolved problems hinted at above. In the EXAFS region a multiple-scattering theory with HL complex potential and a constant reduction factor of the order of 0.8–0.9 instead seems to work rather well. For a deeper discussion of these points, we refer to Rehr & Albers (2000) and references therein.

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