Relativistic and resonant effects in the ionization of heavy atoms by ultra-intense hard X-rays

An accurate description of the interaction of intense hard X-ray pulses with heavy atoms, which is crucial for many applications of free-electron lasers, represents a hitherto unresolved challenge for theory because of the enormous number of electronic configurations and relativistic effects, which need to be taken into account. Here we report results on multiple ionization of xenon atoms by ultra-intense (about 1019 W/cm2) femtosecond X-ray pulses at photon energies from 5.5 to 8.3 keV and present a theoretical model capable of reproducing the experimental data in the entire energy range. Our analysis shows that the interplay of resonant and relativistic effects results in strongly structured charge state distributions, which reflect resonant positions of relativistically shifted electronic levels of highly charged ions created during the X-ray pulse. The theoretical approach described here provides a basis for accurate modeling of radiation damage in hard X-ray imaging experiments on targets with high-Z constituents.

T he availability of ultra-intense and short-pulsed free-electron laser radiation in the hard X-ray domain has opened up numerous avenues for investigating the structure and dynamics of matter with atomic resolution [1][2][3][4] . For hard X-rays, the interaction of light with a polyatomic system is often dominated by the presence of heavy (high-Z) elements since their absorption cross-section is typically two to three orders of magnitude larger than for light atoms such as carbon, nitrogen, and oxygen 5 . Heavy atoms, which occur naturally or can be embedded artificially in many molecules of biological relevance, play a particularly important role for imaging applications because their strong contribution to the total scattering signal is commonly used for phasing of the X-ray diffraction patterns 6 . Moreover, it was suggested that anomalous-diffraction effects caused by heavy atoms can provide a clean solution to the crystallographic phase problem even at very high X-ray intensities 7,8 .
When a single X-ray photon is absorbed by a heavy atom, an electron is typically emitted from one of the strongly bound inner shells. The electron vacancy is filled by an electron from the higher-lying shells, and the energy difference of this transition is dissipated radiatively (fluorescence) or via emission of yet another electron (Auger decay). This often triggers the emission of further electrons via an Auger cascade 9 . In an intense X-ray free-electron laser (XFEL) pulse, several X-ray photons can be absorbed by a single atom, leading to the creation of extremely high charge states [10][11][12] . Early studies have shown that multiphoton ionization of light elements by an XFEL pulse can be described theoretically by considering a sequence of single-photon transitions to the continuum 10,13 . For heavy atoms, it was observed that resonant excitations of inner-shell electrons into densely-spaced Rydberg states and unoccupied valence orbitals can dramatically enhance the ionization in the soft X-ray regime via a process dubbed resonance-enabled or resonance-enhanced X-ray multiple ionization (REXMI) 11,12 . The enhancement has been qualitatively reproduced by theoretical models taking into account boundbound resonant transitions in the intermediate ionic states [14][15][16] . However, a quantitative description of the resonant enhancement, which is essential, for example, for the proposed high-intensity phasing schemes 7,8 , requires a relativistic treatment 17 . This is extremely challenging since the combination of deep inner-shell ionization, relativistic effects, and resonant excitations creates an enormous computational complexity, especially at high X-ray intensities, where more than 20 high-energy photons can be absorbed by a single atom.
Here, we present experimental data on the multiple ionization of Xe atoms by ultra-intense XFEL pulses at several hard X-ray wavelengths and, by comparison with theory, demonstrate that relativistic and resonant effects indeed play a crucial role in this regime. They significantly enhance ionization and give rise to surprisingly structured charge state distributions (CSDs), which reflect resonance positions of relativistically shifted electronic levels of highly charged ions reached during the X-ray pulse as a consequence of the X-ray-driven ionization dynamics. The good agreement between experimental and theoretical results demonstrates that in spite of the complexity of the problem, our current ionization model can fully explain the experimental findings for all X-ray beam parameters.

Results
Experimental charge state distributions. Our experiments (see Methods) were carried out at the hard-X-ray nanometer-focus end-station of the Linac Coherent Light Source (LCLS) with peak fluences up to 5.1 mJ/μm 2 , which is two orders of magnitude higher than in previous measurements 11,12,18,19 . A dilute target of Xe atoms was exposed to the X-ray pulses, and the resulting atomic CSDs were measured using ion time-of-flight (ToF) mass spectrometry. Figure 1 depicts the ion ToF spectrum of Xe atoms ionized with hard X-ray FEL pulses at different photon energies (all of which were chosen to be above the L-shell ionization thresholds of neutral Xe) but with the same nominal pulse energy. While the maximum observed charge state increases steadily as the photon energy increases, the yield for specific charge states varies nonmonotonically as a function of photon energy. The CSDs for 5.5, 6.5, and 8.3 keV photon energy are shown as black squares in Figs. 2a and 3a, b, respectively. The pulse energies and corresponding peak fluence values are given in Table 1 in the Methods section. At all three photon energies, the most abundant charge state is Xe 8+ , which is characteristic for single-photon ionization of the Xe 2s and 2p levels followed by an Auger cascade 9 . This strong contribution from single-photon ionization stems from the broad low-fluence areas in the wings of the X-ray focus profile. Charge states of Xe 12+ and beyond result from multiphoton ionization, and their yield mostly decreases towards higher charge states. At 5.5 keV, three broad peaks around Xe 27+ , Xe 31+ , and Xe 37+ stand out from this steady decrease, as seen in Fig. 2a.
Relativistic and resonant effects. In order to interpret the observed CSDs and to elucidate the origin of the peak structure, we performed ab initio ionization dynamics calculations based on a rate-equation approach using the XATOM toolkit [20][21][22] , which was recently extended to include resonant transitions and relativistic effects 17 . The former is essential to reproduce the resonant enhancement, while the latter is particularly relevant for heavy- The arrows indicate the n-th xenon charge state, 132 Xe n+ atom ionization, where the spin-orbit splitting of the Xe 2p level, for example, which is about 300 eV, opens up new decay channels such as Coster-Kronig transitions. The coupled rate equations, whose total number exceeds 10 68 in the present study, are solved via a Monte Carlo on-the-fly approach 17,18 (see Methods). In Fig. 2a, the theoretical results with and without including resonance and relativistic effects (see color coding in the legend) are presented in comparison with the experimental CSD (black line and symbols). For a quantitative comparison, the theoretical data are averaged over the fluence distribution in the focal volume (see Methods). The overall behavior and the number and position of the peaks in the experimental CSD are only reproduced by the calculations when both resonant transitions and relativistic effects are included (red). Without resonant transitions (green and blue), the ion yield rapidly drops after the charge state whose 2p binding energy surpasses the 5.5-keV photon energy. The non-relativistic, resonant calculation (gray) shows more than three peaks, and the peak positions are not matched with the experimental observation.
The resonant enhancements occur when the binding energy of the L-shell orbitals in the highly charged ions exceeds the photon energy, such that further direct ionization from the L-shell is no longer possible. However, an L-shell electron can undergo a bound-bound transition into one of the unoccupied levels, followed by Auger decay, typically leaving the excited electron as a spectator (see Supplementary Figure 2 and Supplementary  Table 1). This resonant Auger process may happen multiple times and drive a REXMI mechanism 11,12 . As can be seen from comparison with theory, the resonantly enhanced multiphoton ionization drastically extends the highest achievable charge state at 5.5 keV photon energy up to Xe 42+ and, moreover, leads to three pronounced peaks around Xe 27+ , Xe 31+ , and Xe 37+ . Both of these effects were not observed in the earlier experiment on Xe at 5.5 keV 18 because of much lower fluence values in that experiment.
To elucidate the underlying physics of the peak structure, we calculated the total photoabsorption cross-sections for the electronic ground state of each charge state, shown in Fig. 2b. For the resonant calculations, the cross-sections were convolved with a 1% bandwidth, corresponding to the approximate photon bandwidth of the experiment. The corresponding orbital binding energies are plotted in Fig. 2c for the relativistic case. The red, blue, and pale blue lines are offset from the 2s, 2p 1/2 , and 2p 3 Figure 3) reveals that for the relativistic case, the maxima of the cross-sections are at Xe 27+ due to 2s 1/2 → 4p 1/2 /4p 3/2 transitions, at Xe 30+ due to 2p 1/2 → 4d 3/2 transitions, and at Xe 36+ due to 2p 3/2 → 4d 5/2 transitions, which are highlighted by dashed lines extending over all figure panels. As a consequence of the increased cross-section, the CSD also exhibits three maxima centered at the charge states directly following these resonance positions, except for the first maximum around Xe 26+ , where several transitions from the L-shell to n > 4 also contribute at slightly lower charge states. The three distinct peaks can only be accurately reproduced by the calculations if relativistic effects such as spin-orbit splitting are included, and their experimental observation is thus a direct probe of the relativistically shifted electronic levels of highly charged ions. The non-relativistic, resonant calculation (gray) produces four peaks (mainly corresponding to 2s → 5p, 2p → 5d, 2s → 4p, and 2p → 4d transitions) instead of three peaks and, thus, does not reproduce the experimental data. We note that in the low-charge-state region below Xe 13+ , which is dominated by single-photon ionization, all four calculations show deviations from the experimental CSD as a result of neglecting higher-order manyelectron corrections. Such effects are expected to lose relevance for higher charge states, where fewer and fewer electrons remain bound. Figure 3 shows the experimental CSDs for X-ray energies of (a) 6.5 keV and (b) 8.3 keV, along with similar calculations as described above. As for 5.5 keV, photoionization predominantly proceeds via ionization of the 2s and 2p orbitals, so that the CSDs at all three photon energies look very similar up to Xe 22 + . At 6.5 keV, the 2p 3/2 binding energy surpasses the photon energy at Xe 35+ (non-relativistic) and Xe 34+ (relativistic), and the crosssection shows a peak structure at subsequent charge states due to resonant transitions. Again, the peak structure in the experimental data is well reproduced only by the relativistic, resonant calculation. At 8.3 keV, direct L-shell ionization only closes at Xe 47+ , and the photon energy is sufficiently far above the 2p binding energy for a wide range of charge states such that there is almost no difference between the relativistic and the nonrelativistic cross-sections. As a result, the theoretical CSDs from all four calculations at 8.3 keV are very similar to each other.

Discussion
Based on the good agreement of the CSDs obtained from experiment and theory including relativistic and resonance effects, we are able to quantitatively describe and predict the degree of ionization for heavy atoms after the interaction with intense hard X-rays as a function of fluence. Figure 4a presents the fluence-dependent CSDs at 5.5 keV without focal volume averaging (the peak-fluence dependence of the experimental charge state yields is given in Supplementary Figure 4 and discussed in the Supplementary Discussion). The calculations show that the three characteristic peaks emerge at different fluence regimes, which contribute to the experimentally observed CSD due to volume averaging. Further insight is provided by Fig. 4b, which shows the mean charge calculated from the final ion yields as a function of fluence at three different photon energies. The final mean charge is an important atomic benchmark in the formation of warm or hot dense matter and in the electronic radiation damage for molecular imaging. As can be seen in Fig. 4b, the final mean charge increases as the fluence increases, but it reaches saturation at different fluences for different photon energies. At 5.5 keV (red curve), saturation starts early (around 0.5 mJ/μm 2 ) when direct L-shell photoionization is no longer available. Further charging-up occurs via REXMI as well as via valence-shell photoionization and multiple-core-hole state formation, and it fully saturates when no further resonances can occur and all valence electrons are depleted, which corresponds to a charge state of Xe 44+ . On the other hand, direct L-shell photoionization at 8.3 keV closes at Xe 47+ , and the 8.3-keV curve (green) therefore saturates around 5 mJ/μm 2 , eventually approaching Xe 50+ . Interestingly, the plot shows crossings of the curves in the fluence region between 1-5 mJ/μm 2 . At lower fluences, lower photon energies induce a higher final charge because of higher single-photon ionization cross-sections. In the high fluence regime (>5 mJ/μm 2 ), however, the opposite trend is observed, and higher photon energies lead to a higher final mean charge. This counterintuitive behavior demonstrates that one cannot predict the degree of ionization solely based on the crosssection of the neutral atom, as often done in an approximate fashion, and that ab initio calculations are needed to quantify the degree of ionization.
In summary, we report experimental and theoretical results on the multiphoton inner-shell ionization of a high-Z atom, xenon, at experimental parameters relevant for molecular structure determination via femtosecond coherent diffractive imaging and for other high-fluence, hard X-ray experiments. The peak photon fluence is two orders of magnitude higher than in previously reported experiments, which leads to the observation of resonance-enhanced X-ray multiple ionization in the hard X-ray regime. The enhancement is highest for photon energies several hundred electron-volts above the L-shell ionization threshold. It becomes smaller at higher photon energies as changes in the electronic structure narrow down the range of charge states with resonant transitions and move the resonances to higher, lessaccessible charge states. The good agreement of the experimentally observed CSDs with our calculations, in particular for three pronounced peaks at high charge states at 5.5 keV photon energy, underscores the importance of resonant and relativistic effects on The following parameters were used for the calculations in Figs. 2 and 3. The pulse energy and the number of shots are taken from experiment. The peak fluence, the width ratio, and the fluence ratio are fitted from the calibration, and the beamline transmission is calculated with these parameters, assuming the same focus size of 0.35 μm × 0.3 μm (FWHM). No calibration data are available for the photon energies of 7 and 7.5 keV used in Fig. 1 the hard X-ray multiphoton ionization of heavy atoms. The strong resonant enhancement for certain charge states allows to map the electronic structure of extremely highly charged ions. Furthermore, our results demonstrate the interplay of intensity and photon-energy dependence for X-ray ionization of heavy atoms and allow to predict, quantify, and possibly minimize the radiation damage, for example in high-fluence, hard X-ray imaging experiments.

Methods
Experiment. The experiment was performed at the Coherent X-ray Imaging (CXI) instrument 24,25 of the LCLS free-electron laser facility at SLAC, operated at 120-Hz repetition rate and with photon energies of 5.5, 6.5, 7, 7.5, and 8.3 keV. The photon energy bandwidth was estimated to be 1% (full-width at half-maximum, FWHM). The nominal pulse duration is 30 fs (FWHM). Using the 0.1 micron Kirkpatrick-Baez mirror system, the LCLS X-ray beam was focused in the center of an ion timeof-flight mass spectrometer, which is described in ref. 5 A 1 mm times 15 mm wide slit aligned perpendicular to the X-ray beam direction in the first spectrometer electrode was used to limit the spectrometer acceptance in order to reduce the ion current on the detector and to exclude ions generated out of focus. The CXI vacuum chamber was backfilled with xenon gas at a pressure of 2.3 × 10 −6 torr. The xenon ions were generated by the pulsed X-ray beam and extracted by a 83 mm long constant and homogeneous electric field of 22.8 V cm −1 . The ions passed a 315 mm long drift tube and were detected by a time-and positionsensitive 80-mm microchannel plate (MCP) detector equipped with a delay-line anode.
For each FEL shot, the MCP traces were recorded with an Acqiris DC282 digitizer. Each ion hit was identified in the post-analysis using a constant-fraction discriminator implemented in the software suite CASS 26,27 . The discriminator parameters were optimized for multi-hit detection, and the threshold was adjusted as a function of the time-of-flight in order to account for the velocity-dependent MCP detection efficiency. Signatures of seven xenon isotopes were observed. At increasing charge states, i.e., shorter flight times, the peaks of different isotopes of neighboring charge states start to overlap, and a deconvolution based on the natural isotopic abundance was performed to determine the ion yield for each charge state. The deconvolution for the 5.5-keV spectrum is illustrated in Supplementary Figure 1.
The shot-by-shot X-ray pulse energies were measured by gas detectors located upstream of the beamline optics. To extract the experimental CSDs shown in Figs. 2 and 3, only those shots with pulse energies within the range specified in Table 1 were selected in the analysis. To derive the actual X-ray fluence distribution (i.e., the energy per unit area at each position within the focal volume) in the interaction zone, which needs to take into account transmission losses in the beamline and the focal profile in the interaction region, calibration measurements on argon atoms were performed under the same experimental conditions. Focus parametrization. The XATOM toolkit [20][21][22] calculates the CSD of atoms ionized by an XFEL beam for given values of the photon fluence. For a comparison to experimental CSDs, the theoretical results must be averaged over the fluence distribution in the focal volume. In this work, the fluence distribution is described by a double Gaussian 28 in the two dimensions perpendicular to the beam propagation direction. The fluence values in the third dimension along the propagation axis are assumed to be constant since the Rayleigh length is larger than or comparable to the 1 mm entrance slit width of the spectrometer. The second Gaussian is introduced to model the low-fluence regions of the CXI nanofocus beam profile 29 . The double Gaussian profile consists of a linear combination of two independent Gaussians, which can be parameterized by three parameters: the peak fluence, the fluence ratio, and the width ratio 30 . The calibration of the focal volume was conducted using the XCALIB toolkit 30 . It optimizes the parameter set by minimizing the squared, nonlinear difference between theoretical and experimental argon CSDs, which were measured for calibration purposes. Calibrating the focal spot through CSDs of light elements is advantageous because their atomic properties are well known, and the CSD directly reflects the experimental conditions at the focal spot. To calibrate the high fluence regime, only highly charged argon ions were included in the calibration procedure (+11 to +18 for 5.5 keV, +10 to +18 for 6.5 keV, and +9 to +17 for 8.3 keV). Assuming a constant 0.35 μm × 0.3 μm focus size (FWHM) for all photon energies, XCALIB found the best fit to the experimental argon CSD with the peak fluence, width ratio, and fluence ratio listed in Table 1 for each photon energy. The beamline transmission is calculated with these fitted parameters 30 . The calibration reveals a decrease in beamline transmission for decreasing photon energy, which is expected due to an increase in the size of the unfocused X-ray beam at low photon energies, leading to an overfilling of the focusing mirrors. However, since no independent measurement of the focus size at each photon energy was performed, we note that the apparent decrease in beamline transmission is, most likely, also partly due to an increase in focus size. Our calibration cannot disentangle both effects since only the fluence distribution can be derived from the calibration.
Modeling. The multiple ionization dynamics of Xe atoms under the irradiation of ultra-intense X-rays are calculated using the XATOM toolkit [20][21][22] , which was recently extended to include relativistic effects and resonant excitations 17 . The relativistic energy corrections are calculated within first-order perturbation theory, and the relativistic electronic configurations are considered with spin-orbit splittings. In the course of the sequential multiphoton absorption process, all but the electrons in the K-shell of Xe can be ionized for the photon energies considered in this study. The q-hole electronic structure of Xe q+ is calculated on the basis of the Hartree-Fock-Slater method including the Latter tail correction in order to improve the asymptotic behavior of the effective potential and, thus, the calculated orbital eigenvalues (see ref. 21 for further details). Atomic data (photo-ionization cross-sections, Auger and Coster-Kronig rates, and fluorescence rates) are calculated in the lowest order perturbation theory. The total number of electronic configurations formed by removing 0, 1, or more electrons from Xe subshells (n ≥ 2) is 23,532,201 in the non-relativistic case without resonances. This number, which is equivalent to the number of coupled rate equations to be solved, increases to 5,023,265,625 when spin-orbit splittings are considered 17 . The number explodes when including resonant excitations in addition to the spin-orbit splittings. The maximum values of the principal quantum number and the orbital angular momentum quantum number employed in our calculations are n max = 30 and l max = 7 to obtain the converged total photoabsorption cross-sections shown in Fig. 2b. The total number of electronic configurations formed by removing 0, 1 or more electrons from initially occupied subshells of Xe q+ and placing them into unoccupied (n, l)-subshells is estimated to be~2.6 × 10 68 , according to the number partitioning expression used in ref. 16 To handle such a gigantic number of the rate equations, we apply the Monte Carlo on-the-fly approach, where only the probable pathways of ionization dynamics are followed and the atomic data are calculated only when the electronic configuration is visited in the pathway 17,18 . Thus, this approach imposes no limitation on the size of the configurational space. Our current model does not include higher order many body processes such as shakeoff and double Auger decay. Collisional processes such as dielectric recombination, impact ionization, and charge exchange are neglected because the mean free path of electrons and ions exceeded the sub-micron focus size by several orders of magnitude.

Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.