International
Tables for Crystallography Volume I Xray absorption spectroscopy and related techniques Edited by C. T. Chantler, F. Boscherini and B. Bunker © International Union of Crystallography 2024 
International Tables for Crystallography (2024). Vol. I. ch. 5.5, pp. 653658
https://doi.org/10.1107/S1574870722005535 Shakeup and shakeoff processes^{a}Jožef Stefan Institute, Jamova 39, SI1000 Ljubljana, Slovenia, and ^{b}Faculty for Mathematics and Physics, University of Ljubljana, SI1000 Ljubljana, Slovenia Because of correlations, more than one atomic electron can change its orbital in a single photon absorption. Due to these multielectron photoexcitations (MPEs), the atomic photoabsorption cross sections exhibit characteristic tiny spectral structures above the corresponding absorption edges. Since the atomic cross section forms a background to the structural Xray absorption finestructure (XAFS) signal, MPEs are a source of uncertainty in highprecision structural analysis. Resonant inelastic Xray scattering (RIXS) can be used to isolate the MPE signal from the dominant singleelectron photoionization component that contains the structural information. In this review, MPE spectral types are sorted according to the underlying physical processes and the corresponding modelling approaches are briefly mentioned. The main experimental requirements for MPE studies with RIXS spectroscopy are presented, together with a few recent examples of RIXS MPE studies on noble gases. Keywords: multielectron photoexcitation; resonant inelastic Xray scattering. 
Ideally, structural information due to multiple photoelectron scattering from neighbouring atoms would modulate a smooth singleelectron photoionization cross section of an isolated atom. Indeed, an asymptotic decrease in the atomic photoabsorption background in the region far above the corresponding innershell edge may be described by a smooth power law. However, close to the edge this smooth dependence is modified by multielectron photoexcitations (MPEs) exhibiting weak spectral features. This MPE signal cannot be fully smoothed out and leads to systematic errors, limiting the precision of structural analysis (Filipponi, 1995; Kodre et al., 1997). MPEs are also of great fundamental interest in atomic physics, providing information on manybody excitation dynamics. Since the atomic MPE spectral features in ordered systems overlap with the structural signal, they have been most extensively studied for noble gases (Deslattes et al., 1983; Deutsch & Hart, 1986a,b; Deutsch et al., 1992; Schaphorst et al., 1993; Arčon et al., 1995). Subsequently, MPE studies have been extended to several metallic vapours (Filipponi et al., 1993; Padežnik Gomilšek, Kodre, Arčon & Prešeren, 1999; Padežnik Gomilšek et al., 2003, 2011; Kodre et al., 2002) and to some partly disordered solid systems (Filipponi et al., 1988; Padežnik Gomilšek, Kodre, Arčon, LoireauLozac'h et al., 1999) where the structural XAFS oscillations are primarily governed by a single frequency from the first nearest neighbour coordination shell.
Generally, the MPE contributions are classified into three groups, each with a different characteristic spectral shape in the absorption spectrum, and they are distinguished by different finalstate configurations. In the first case the excitation of an innershell electron to an unoccupied atomic bound state is accompanied by the excitation of another electron from a higher occupied atomic orbital, leading to a neutral atom final configuration. These double excitations exhibit sharp resonant peaks in the absorption spectrum. The second group are shakeup transitions leading to singly ionized atoms. Here, the removal of an innershell electron is accompanied by the excitation of another more loosely bound electron to a higher unoccupied atomic orbital. This MPE channel produces tiny sharp edges in the measured absorption spectrum. In the case of the shakeoff channel the second electron is also excited to the continuum, producing doubly ionized atoms. The onset of the shakeoff MPE contribution exhibits only a slow gradual rise without any sharp spectral features. These three classes of MPE contributions are presented schematically in Fig. 1.

Schematic presentation of resonant doubleexcitation (left), shakeup (middle) and shakeoff (right) multielectron transitions and their characteristic spectral shapes. 
In addition to a characteristic spectral shape, different MPE contributions are also distinguished by the threshold energies. For a given vacancy configuration [nln′l′] resonant double excitations will be situated at energies closest to the corresponding [nl] absorption edge, the shakeup edge will follow at slightly higher energies and finally the onset of the shakeoff contribution will appear. The relative intensities of MPE contributions depend on the atomic shell [n′l′] of the electron accompanying [nl] innershell photoionization. While discrete resonant peaks and shakeup edges dominate in the case of loosely bound outershell electrons, shakeoff prevails for deeper innershell ionization. The strongest MPE contributions would typically involve outershell electrons with relatively low binding energies. Consequently, the most pronounced MPE spectral features, which are easily observed in the measured absorption spectrum, contain significant contributions from discrete resonant and shakeup excitations. Because of low threshold energies, they are situated just above the corresponding innershell edge. In this nearedge region the single photoionization cross section typically exhibits a steeper slope compared with the extrapolation of the asymptotic trend reached far above the threshold, mainly on account of the core relaxation and postcollision interaction in corehole Auger decay (Tulkki & Åberg, 1985). This obscures the separation of MPE signal from the total photoabsorption cross section. For more tightly bound innershell electrons the MPE signal is shifted higher above the corresponding innershell edge, where the extrapolation of the single photoionization cross section is already quite accurate. However, in this case the shakeoff channel prevails, leaving us without any sharp spectral signature, which is again very difficult to isolate. In any case, absorption spectroscopy only provides the total cross section, which cannot be reliably separated into the relatively weak MPE signal and the dominant singleelectron photoionization component.
Apart from Xray photoabsorption spectra, MPEs are also manifested by satellite spectral lines in highenergy resolution Xray emission, Auger or photoelectron spectra emitted due to the relaxation of multiply excited states (Deslattes et al., 1983; Armen et al., 1985; Heiser et al., 1994). In the case of high experimental energy resolution the satellite lines can be resolved from the diagram lines corresponding to the decay of singly ionized atoms. When these emission techniques are coupled to selective photoexcitation using synchrotron radiation, a clean MPE spectrum can be obtained by monitoring the intensity of the satellite line while scanning the incoming photon energy across the corresponding MPE threshold. The main experimental problem hindering such studies is related to the low MPE cross sections in the nearthreshold region. This is additionally hampered by the relatively low efficiency of the high energy resolution spectrometers that are required to resolve the satellite lines. Resonant inelastic Xray scattering (RIXS), which explores the radiative emission channel, has so far been applied to MPE studies of solid materials (Deutsch et al., 1996; Diamant et al., 2003; Huotari et al., 2008). In order to avoid structural solidstate effects, these studies are restricted to core electrons being shaken off or knocked out (in the case of double 1s ionization). To extend these studies to valence electrons and inspect the relevance of different MPE mechanisms, gasphase experiments are required. Compared with electron emission techniques, bulk penetrating photonin/photonout RIXS spectroscopy in the tender and hard Xray energy ranges can successfully be performed with a thinwindow highpressure static gas cell and is therefore particularly well suited for MPE studies on gaseous atomic targets.
Successful implementation of RIXS spectroscopy requires the detection of emitted photons with an energy resolution comparable to the corehole lifetime broadening. Such high resolving power is provided by wavelengthdispersive Xray (WDX) crystal spectrometers. Quite large targettocrystal and crystaltodetector distances are typically needed to provide the required resolving power. In the tender Xray energy range (2–5 keV) a full invacuum enclosure is necessary in order to suppress photon absorption in the relatively long path to the detector. This places some constructional constraints on the Xray spectrometer, which typically makes use of a single cylindrically bent crystal analyzer in different focusing geometries. A vertical focusing perpendicular to the diffraction plane is applied in von Hamos geometry, which is combined with the positionsensitive detection of diffracted Xrays to provide a dispersive mode of operation (Hoszowska et al., 1996; Fig. 2). In the case of a focused incident photon beam, Johann geometry using pointtopoint focusing within the diffraction plane is most commonly applied (Journel et al., 2009). The application of Johanntype spectrometers is limited to the backscattering angles because of the geometrical cot^{2}(θ_{Bragg}) contribution to the energy broadening. In order to eliminate this problem, Johansson geometry can be applied (Kavčič et al., 2012). This is particularly important for tender Xray spectrometers, where due to a limited selection of appropriate analyzer crystals one is forced to use a wider Bragg angular range. In the hard Xray range the absorption is lower and the spectrometers typically operate under atmospheric pressure and/or use a helium bag to suppress absorption. Most commonly, spherical Johanntype focusing analyzers are employed working close to the backscattering condition. In order to enlarge the solid angle and enhance the collection efficiency, multiple crystals are commonly used in sophisticated hard Xray emission spectrometers, reaching a solid angle of a few tenths of a percent while keeping the resolving power E/ΔE within or close to a value of 10^{4} (Sokaras et al., 2013; Fig. 2).
Typically, an atomic multiconfiguration Hartree (Dirac)–Fock (MCH/DF) approach is needed to generate MPE wavefunctions (Dyall et al., 1989) because in the resonant case two weakly bound electrons exhibit strong correlation. The MCH/DF wavefunctions are used to calculate the resonant MPE energies and energy positions of shakeup/shakeoff thresholds. The MPE absorption cross sections and Xray emission probabilities are calculated from dipole matrix elements of separately optimized (and thus nonorthogonal) wavefunctions describing the initial and final atomic state, possibly with one electron in the continuum (Zatsarinny, 1996). The RATIP computer code package offers the possibility of performing MPE calculations in the relativistic framework, extending the range to heavier atoms and deeper atomic shells (Fritzsche, 2012). The calculation of MPE matrix elements is often simplified assuming that the ejected photoelectron leaves the atomic volume very fast, so that the escape time is small compared with the shakeup energy of the second electron divided by ℏ. In this socalled sudden approximation the corresponding change of the Hamiltonian is taken to be instantaneous, which leads to factorization of the MPE probability into absorption and relaxation parts, with the latter describing the transition probability for the second electron upon a sudden change of the potential. The absorption part is described by the absolute square of a singleelectron dipole matrix element, and the relaxation is described by the monopole term given by the squared overlap between the initial atomic and the relaxed ionic orbital of the second electron. However, in some situations, especially when two electrons are ejected with a small total energy (just above the shakeoff threshold) and/or the shakeoff energy is large (deeper shells), deviations from the sudden approximation are observed. This is taken into account by the Thomas formula (Thomas, 1984), which has the effect of postponing saturation of the sudden shakeoff probability to photon excitation energies up to several hundreds of electronvolts above the corresponding threshold, depending on the shaken atomic shell. At low excess energies a competitive mechanism for the secondary electron ejection is a knockout (KO) collision. Such KO contributions can be estimated by half of the electronimpact cross section for ionization of the corresponding ion, which is usually calculated in the frame of the binary encounter Bethe model (Santos et al., 2003). For the RIXS process, a detailed shape of the cross section is given by the Kramers–Heisenberg equation, which describes the probability of each isolated absorption–emission path as the absolute square of the product of the absorption and emission amplitudes, weighted by an inverse sum of the intermediatestate energy detuning and its lifetime (Žitnik et al., 2007). In the case of spectrally overlapping intermediate states with paths leading to the same finalstate interference occurs and the path amplitudes must be summed prior to squaring.
The Ar K edge photoabsorption spectrum in the tender Xray range exhibits a pronounced structure starting about 20 eV above the 1s ionization threshold at 3.206 keV, which is assigned to [1s3p] twoelectron excitations (Deslattes et al., 1983; Deutsch & Kizler, 1992). The KM Xray emission spectrum exhibits a satellite line (Fig. 3), which originates from the radiative relaxation of the 1s3p doubly photoexcited state, as confirmed by configuration–interaction (CI) calculations (Dyall & Grant, 1984). In order to obtain a clean 1s3p MPE spectrum, a RIXS experiment combining both absorption and emission was performed (Kavčič et al., 2009). The recorded 2D RIXS spectral plane corresponding to the evolution of the KMM^{2} satellite line in the 1s3p nearthreshold region is shown in the inset in Fig. 3.
In the measured RIXS map the signal of the twoelectron process is well resolved from the single 1s photoionization signal. In addition, different twoelectron processes are clearly recognized in the map. The first two contributions exhibit a linear Raman–Stokes dispersion, which is a signature of discrete resonant excitations. This is followed by a sharp shakeup edge, after which the dependence on the excitation energy vanishes, which is a clear indication of the excitation into continuum.
The isolated 1s3p doublephotoexcitation spectrum is given by integral counts of the satellite line versus the excitation energy (Fig. 4). The spectrum exhibits all three types of MPE contributions. In modelling, three different types of functional components are typically used to reconstruct separate MPE features. Lorentzian peaks describe resonant contributions, a cumulative Lorentzian distribution (arctan edge) models the shakeup channels and the slowly opening shakeoff channel is modelled by an appropriate exponential saturation profile at the crossover from the adiabatic to the sudden regime (Thomas, 1984).
In a similar way, RIXS spectroscopy has also been used in the hard Xray range to study MPE above the Kr K absorption edge at 14.328 keV (Kavčič et al., 2014). Besides the use of an inair Johanntype spectrometer, the main difference is the significantly larger corehole broadening. The KNN^{2} satellite line, corresponding to [1s4p] double photoexcitations starting about 15 eV above the K absorption edge, is therefore hidden and only produces an asymmetry on the highenergy side of the KN diagram line. In this case, a clean reference KN diagram spectrum recorded at an excitation energy below the [1s4p] threshold is needed to extract the satellite line contribution at each photon excitation energy and to build the final Kr [1s4p] doublephotoexcitation spectrum.
In conventional absorption spectroscopy the threshold energy is most commonly used to identify MPE channels. Comparison of theoretical data with experiments is usually limited to the calculation of energy levels only, except for a few examples where a full quantitative theoretical model was used for comparison with the absolute experimental cross sections (Saha, 1990; Schaphorst et al., 1993). In the RIXS approach the uncertainty in the estimation of the 1s singleionization cross section in the nearthreshold region is removed, yielding an isolated MPE spectrum, which can be critically compared with theory. The inset in Fig. 4 presents a comparison of the resonant part of the Ar 1s3p MPE spectrum with the theoretical spectrum calculated within the CI model using separate optimization of the ground state and the corehole atomic orbitals.
However, one needs to be aware that in general the projection of the RIXS signal onto the incoming photon energy axis actually corresponds to the differential RIXS cross section recorded at a particular angle relative to the incident polarization axis, and therefore is not necessarily proportional to the true absorption spectrum, unless taken at the magic angle of 54.7°. The absolute values might be modified because of the possible angular dependence of the fluorescence emitted by intermediate states with well defined total angular momentum. Moreover, the shape might be altered by a change in the fluorescence decay branching ratio due to an additional hole in the valence shell. As shown recently for the case of 1s2p double excitation in argon (Žitnik et al., 2014), the corresponding fluorescence decay rate of the singlet corehole state [1s2p]^{1}P is almost two times larger than the decay rate of the triplet state [1s2p]^{3}P, generating significant differences between the integrated RIXS signal and the true absorption signal. Finally, the RIXS spectrum might also be modulated by the interference of the corresponding absorption–emission paths as observed previously for isolated atoms (Žitnik et al., 2015) and also for molecules (Kavčič et al., 2010).
XAFS analysis is among the most widely applicable tools for structural analysis. A very high level of precision has been reached through constant improvements in experimental accuracy and the development of theoretical tools and extraction algorithms. In order to take full advantage of these developments, the same level of accuracy is also required for the atomic background to the structural signal. RIXS spectroscopy has the capability to experimentally isolate the MPE spectral features representing the main source of uncertainty in the determination of the atomic background. So far, RIXS MPE studies in the tender and hard Xray ranges have successfully been performed on noble gases. With dedicated Xray emission endstations currently available at various synchrotron facilities, similar RIXS studies can be expanded to other more complex systems.
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