Chapter 5.5. Shake-up and shake-off processes

Successful implementation of RIXS spectroscopy requires the detection of emitted photons with an energy resolution comparable to the core-hole lifetime broadening. Such high resolving power is provided by wavelength-dispersive X-ray (WDX) crystal spectrometers. Quite large target-to-crystal and crystal-to-detector distances are typically needed to provide the required resolving power. In the tender X-ray energy range (2–5 keV) a full in-vacuum enclosure is necessary in order to suppress photon absorption in the relatively long path to the detector. This places some constructional constraints on the X-ray 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 position-sensitive detection of diffracted X-rays 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 point-to-point focusing within the diffraction plane is most commonly applied (Journel et al., 2009). The application of Johann-type spectrometers is limited to the backscattering angles because of the geometrical cot²(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 X-ray spectrometers, where due to a limited selection of appropriate analyzer crystals one is forced to use a wider Bragg angular range. In the hard X-ray 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 Johann-type 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 X-ray 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 10⁴ (Sokaras et al., 2013; Fig. 2).

Figure 2 Schematic representation of the focusing geometry used in a von Hamos single-cylindrical crystal spectrometer (left; Hoszowska et al., 1996) and in a Johann-type multiple spherical crystal hard X-ray emission spectrometer (right; Sokaras et al., 2013).

Typically, an atomic multi-configuration 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 shake-up/shake-off thresholds. The MPE absorption cross sections and X-ray 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 shake-up energy of the second electron divided by ℏ. In this so-called 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 single-electron 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 shake-off threshold) and/or the shake-off 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 shake-off 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 electron-impact 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 intermediate-state energy detuning and its lifetime (Žitnik et al., 2007). In the case of spectrally overlapping intermediate states with paths leading to the same final-state interference occurs and the path amplitudes must be summed prior to squaring.

The Ar K edge photoabsorption spectrum in the tender X-ray range exhibits a pronounced structure starting about 20 eV above the 1s ionization threshold at 3.206 keV, which is assigned to [1s3p] two-electron excitations (Deslattes et al., 1983; Deutsch & Kizler, 1992). The KM X-ray 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 KM-M² satellite line in the 1s3p near-threshold region is shown in the inset in Fig. 3.

Figure 3 Ar K photoabsorption spectrum (black line) and KM emission spectrum at an excitation energy below (red line) and above (blue line) the [1s3p] double-excitation threshold. Double-electron excitations exhibit pronounced spectral features above an excitation energy of 3225 eV. The contour plot in the inset presents a full scan of the satellite line across the 1s3p near-threshold region.

In the measured RIXS map the signal of the two-electron process is well resolved from the single 1s photoionization signal. In addition, different two-electron 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 shake-up edge, after which the dependence on the excitation energy vanishes, which is a clear indication of the excitation into continuum.

The isolated 1s3p double-photoexcitation 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 shake-up channels and the slowly opening shake-off channel is modelled by an appropriate exponential saturation profile at the crossover from the adiabatic to the sudden regime (Thomas, 1984).

Figure 4 Isolated Ar 1s3p photoabsorption spectrum exhibiting [1s3p;3s]nln′l′ resonant double excitations (red line), [1s3p]4p;5p shake-up edges (black dashed lines) and the [1s3p] shake-off contribution (blue dashed line). The inset represents only the resonant [1s3p]nln′l′ part of the spectrum compared with the results of multiconfiguration Dirac–Fock calculation (Kavčič et al., 2009).

In a similar way, RIXS spectroscopy has also been used in the hard X-ray range to study MPE above the Kr K absorption edge at 14.328 keV (Kavčič et al., 2014). Besides the use of an in-air Johann-type spectrometer, the main difference is the significantly larger core-hole broadening. The KN-N² 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 high-energy 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] double-photoexcitation 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 single-ionization cross section in the near-threshold 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 core-hole 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 core-hole state [1s2p]¹P is almost two times larger than the decay rate of the triplet state [1s2p]³P, 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).