Electron population dynamics in resonant non-linear x-ray absorption in nickel at a free-electron laser

Abstract

Free-electron lasers provide bright, ultrashort, and monochromatic x-ray pulses, enabling novel spectroscopic measurements not only with femtosecond temporal resolution: The high fluence of their x-ray pulses can also easily enter the regime of the non-linear x-ray–matter interaction. Entering this regime necessitates a rigorous analysis and reliable prediction of the relevant non-linear processes for future experiment designs. Here, we show non-linear changes in the ${\mathrm{L}}_3$-edge absorption of metallic nickel thin films, measured with fluences up to 60 J/cm². We present a simple but predictive rate model that quantitatively describes spectral changes based on the evolution of electronic populations within the pulse duration. Despite its simplicity, the model reaches good agreement with experimental results over more than three orders of magnitude in fluence, while providing a straightforward understanding of the interplay of physical processes driving the non-linear changes. Our findings provide important insights for the design and evaluation of future high-fluence free-electron laser experiments and contribute to the understanding of non-linear electron dynamics in x-ray absorption processes in solids at the femtosecond timescale.

I.

INTRODUCTION

The modern understanding of complex materials relies on suitable approximations to the unabridged quantum mechanical description of the full, correlated many-body problem. To assess the predictive power of theoretical models and the selected approximations, detailed experimental studies of systems driven out of equilibrium are particularly insightful.

X-ray free-electron lasers (FELs) with their high-power (about 10¹⁰ photons/pulse after monochromatization) and ultrashort pulses (tens of fs) are uniquely suited for probing the non-equilibrium dynamics of matter in all states from solid state to plasma.^1,2

While schemes with separate pump and probe pulses can be used to observe dynamics slower than the pulse durations, transmission measurements of individual intense pulses can be used to infer the x-ray interaction and material dynamics within the pulse duration. With increasing pulse energy, the interaction with an exposed solid becomes highly non-linear, as the material state changes during the interaction. In non-linear absorption measurements, a single intense x-ray pulse acts as both a pump and a probe at the same time, and its total absorption depends on the changes it induces in the electronic system.^3–9 The highly excited solid evolves toward a state of warm dense matter (WDM) where individual electronic excitations reach up to hundreds of eV and excitation energies average out to many eV per atom, while the nuclear lattice still resembles that of the cold solid during the femtosecond pulse.^10–19

In this paper, we present new fluence-dependent x-ray absorption spectra recorded with monochromatic x rays on metallic nickel thin films around the nickel 2  ${\mathrm{p}}_{3/2}$ ( ${\mathrm{L}}_3$) edge, revealing changes in the valence electron system around the Fermi level, driven by FEL excitation densities spanning several orders of magnitude from the linear regime up to 60 J/cm² (corresponding to 2  $\times {10}^{15}$ W/cm²).

The electronic processes that ensue after the absorption of core-resonant photons trigger a complex series of dynamical processes of photon-absorption, electron excitation, and subsequent electronic scattering. Such non-equilibrium processes with a large amount of correlated particles involved and spanning orders of magnitude in internal energy are challenging to treat in ab initio simulations.^20–24 Rather than striving for a direct time-dependent solution to the quantum-mechanical many-body problem, our analysis, therefore, explores how much of the observed effects can be explained purely by the evolution of electronic populations within the ground-state density of states (DOS) and scaling the known ground-state probabilities of absorption, decay, and scattering processes with the current populations of the participating states. Therefore, we develop a rate equation model to provide an intuitive understanding of the electronic processes. Our model successfully describes the largest part of the non-linear changes in the spectra, affirming the dominant role of electron redistribution. However, especially in close vicinity of the resonance, our measurements deviate from the predictions of the rate model and indicate the need for more sophisticated theories. Nevertheless, our straightforward picture of an intense core-resonant x-ray pulse interaction with the valence system of a 3d metal lays a solid knowledge-based foundation for the planning and interpretation of non-linear x-ray spectroscopy experiments at FELs.

This paper is structured as follows: Sec. II describes the experimental setup used to acquire non-linear XANES spectra of the L₃-edge of metallic nickel films. In Sec. III, we introduce the working principle of the rate model used to interpret the experimental data. In Sec. IV, we show the experimental data in direct comparison to the simulation results and continue to interpret our findings in Sec. V. We conclude in Sec. VI.

II.

EXPERIMENT

X-ray absorption spectra of the nickel 2 ${\mathrm{p}}_{3/2}$ ( ${\mathrm{L}}_3$) edge were recorded at the Spectroscopy and Coherent Scattering Instrument (SCS) of the European XFEL.²⁵

The XAS (x-ray absorption spectroscopy) spectra were measured by continuously scanning the SASE3 monochromator²⁶ (synchronized with the undulator gap) back and forth many times in the range 846–856 eV. The photon bandwidth was about 420 meV full-width half-maximum (FWHM) and the FEL pulse duration on the sample was about 30 fs FWHM. The polarization was linear horizontal. The overall beam intensity was controlled using a gas attenuator filled with nitrogen and monitored using an x-ray gas-monitor (XGM) downstream of the monochromator.^27,28 Figure 1 illustrates the experimental concept. Panels (a) and (b) display the DOS (integrated over spin-up and spin-down states) next to the resulting XAS spectrum (including non-resonant background); panel (a) shows the situation of a low-fluence measurement, panel (b) shows the situation where sufficient photons are used to alter the DOS occupation and hence the measured XAS spectrum. Panel (c) shows the geometry of zone-plate, sample, and detector.

FIG. 1.

(a) Absorption at low fluences. The electronic system remains mostly in the ground state. The left side shows the density of states, red occupied and blue unoccupied, while the right side displays the resulting spectrum. (b) Absorption at high fluence. Later parts of the x-ray pulse probe a hot electronic system and see less unoccupied valence states at the resonant energy (bleaching). Unoccupied states and spectrum are shown in yellow. (c) Setup for non-linear XAS. The split-beam-normalization scheme uses a special zone plate,²⁹ which generates two adjacent beam foci for transmission through the sample and a reference membrane before the beams impinge on the detector.

For x-ray absorption measurements at FELs based on self-amplified spontaneous emission (SASE), beam-splitting schemes can deliver optimal normalization of SASE-fluctuations.^30–32 Here, we utilize such a scheme using a focusing and beam-splitting zone plate which also creates the required tight focusing to achieve extreme fluences. The zone plate combines an off-axis Fresnel structure for focusing and a line grating for beam-splitting in a single optical element.²⁹ It thus produces two μm-sized, identical foci in the sample plane, 1.9 mm apart, originating from the first-order diffraction of the zone plate, as well as the positive and negative first orders of the line grating. The sample has a square support of 25 mm size, containing square Si₃N₄ membrane windows (orange in Fig. 1) of (0.5 × 0.5) mm² size and 200 nm thickness with a distance of 2 mm between adjacent windows. Every second pair of rows (blue in Fig. 1) was additionally coated with a 20  $\text{nm}$ thick sample layer of polycrystalline metallic Ni by sputter deposition, on top of a 2  $\text{nm}$ bonding layer of Ta; a 2  $\text{nm}$ Pt capping layer was applied to prevent oxidation during sample-handling.

The sample frame was positioned such that one zone plate focus impinged on a nickel-coated membrane, while the other hit a bare silicon-nitride membrane. Thus, the difference in transmission of both beams can be dominantly attributed to absorption in the nickel film.

The detector was a fast readout-speed charge-coupled device (FastCCD) with a high dynamic range, enabling 10 Hz readout and increasing the fluence range available to the experiment.^33–35 Due to an unstable detector temperature, significant retroactive calibration of the detector was necessary (see Subsection 2 of Appendix B). To prevent detector saturation during measurements with the unattenuated beam, an additional aluminum filter of about 13  $\mu \mathrm{m}$ thickness was used between sample and detector.

During these high-intensity measurements, sample and reference films were locally damaged by intense individual FEL shots. Thus, the FEL was operated in single-shot mode at 10 Hz repetition rate, and the sample was scanned through the beam continuously at 0.5  $\text{mm}\hspace{0.17em}{\mathrm{s}}^{-1}$, resulting in ten shots per membrane window.

The shot craters in the reference membranes were later analyzed with scanning electron microscopy (SEM) to determine the effective focal size at specific photon energies. The resulting spot sizes were used to calibrate ray-tracing calculations which delivered the photon-energy-dependent spot size, ranging from 0.4  $\mu {\mathrm{m}}^2$ to about 3  $\mu {\mathrm{m}}^2$ (see Subsection 4 of Appendix B for details on the spot size determination).

III.

MODELING

Various approaches have been proposed to describe the interplay between photon absorption and the electronic structure evolution during the absorption of an FEL pulse. Ab initio methods such as Monte Carlo calculations, which explicitly calculate a large number of individual particles' interaction pathways,³⁶ and time-dependent density functional theory (TD-DFT), which sets out to solve the full quantum-mechanical many-body problem in terms of the electron density,³⁷ generally scale poorly with particle number. In contrast, rate models provide a simpler yet useful tool by describing the interplay between photon absorption and electronic system using non-quantized volume-average quantities and rates directly on a macroscopic scale.^5,38–41

Away from material resonances, rate models have been successfully used to describe fluence-dependent x-ray absorption in three-level systems, representing the ground, core-excited, and intermediate valence-excited states.^5,38 When probing the valence bands around material resonances, however, the evolution of the electronic system requires explicit modeling of the energy-resolved valence state populations.⁴² Tracking the full non-thermal population history proved crucial for accurately describing the non-linear absorption changes near and around the Fermi level. We assume that on the modeled femtosecond timescale, the DOS does not change significantly, which is motivated by the slower lattice reaction. This approach though cannot capture subtle changes in electron correlations.⁴³ Furthermore, since our measurements are done with linear polarization and are therefore not sensitive to the magnetization, we do not consider the exchange-split DOS but rather integrate over minority and majority electrons.

We describe the propagation of x-ray photons through the sample as well as the dynamics of electron populations within the sample using a set of ordinary differential equations. These are assembled from terms that each describe the rate of a specific physical process. The rate of each process is based on a tabulated or measured ground-state parameter, such as the Auger-lifetime or the absorption cross section, scaled with the appropriate fractional occupations at the simulated time.⁴⁴ The relevant process rates are compiled into differentials of electronic populations and photon density in space and time and implemented in a finite-element simulation to derive the electron population history and ultimately the x-ray transmission of a three-dimensional sample. Only the time constants for the valence band thermalization and the scattering cascades of free electrons are treated as free parameters and adapted to fit the experimental data.

The model considers an idealized three-dimensional sample traversed by an x-ray pulse with Gaussian shape in space and time. We make key approximations to reduce computational effort, such as neglecting any movement of electrons within the sample [consider the inelastic mean free path of relevant photoelectrons of about 1.3 nm (Ref. 45)] and describing photon propagation exclusively in the forward direction. The temporal evolution is solved using the fourth-order Runge–Kutta method with adaptive time-stepping. The propagation of photons in space is calculated as if it happened instantaneously in between the time-steps using the explicit fourth-order Runge–Kutta method.

In order to account for the two-dimensional Gaussian transversal intensity profile of the FEL spot, we first calculate the transmission of the sample for transversally uniform illumination for different fluences. Since we omit transversal coupling, the response to the Gaussian beam profile can then be reconstructed by appropriate radial integration over many values obtained for constant illumination. With these simplifications, the overall computational complexity is drastically reduced, as we simplify a problem with partial differentials in four dimensions into two separable one-dimensional initial value problems, one for photon propagation in space and one for the evolution of electronic populations in time.

The model describes the interaction between three types of population densities of electrons as well as incident photons via six distinct physical processes, listed in Tables I and II, respectively. Figure 2 schematically illustrates their relationships. The electron populations R_C and R_V describe the total number of electrons bound in the core and valence system, respectively, for an average single atom in the sample. Their values are limited by the number of available states, M_C and M_V. In the presented nickel L₃-edge spectra, the ground-state populations are R_C = 4, representing the 2 ${\mathrm{p}}_{3/2}$-electrons and R_V = 10, representing electrons from the 3d and 4s states. We describe the electronic population of the valence system in an energy-resolved manner, splitting it up into a discrete number of densities ρ_j, where j represents the index along the valence energy axis. The number of available states m_j for each energy bin in the valence system is derived from the calculated ground-state DOS^46,47 up to 30 eV above the Fermi level E_F. Beyond this value up to 800 eV above the Fermi level, the DOS of a free electron gas is used.²² All electrons with even higher energies, such as photo-electrons created via non-resonant absorption and Auger-electrons from the decay of core-holes, are described in a separate pool of electrons R_free without energy resolution, although the total energy of electrons in this pool is tracked by the parameter E_free.

TABLE I.

Electron and photon numbers in the rate model.

RC	Number of 2 ${\mathrm{p}}_{3/2}$ ( ${\mathrm{L}}_3$) core electrons, populating the available core states MC
RV	Number of valence electrons, populating available valence states MV; subdivided by energy bins with ρj and mj, which describe the energy-resolved DOS and its population
R free	Number of free electrons, created by Auger decay or photoemission before scattering with valence electrons
$N_i^{\mathrm{\text{phot}}}$	Density of x-ray photons incident per area at photon energy Ei in one simulation time step

TABLE II.

Process rates in the simulation: The index j enumerates the specific energy level E_j within the valence system; i enumerates the incident photon energy E_i.

$P_j^{\mathrm{\text{res}}}$	Rate of resonant interaction, i.e., absorption and stimulated emission between core level and bound valence states
$P_{i,j}^{\text{non}-\text{res}}$	Rate of non-resonant absorption from bound valence states to the unbound free electrons
$P_j^{\text{Auger}}$	Rate of Auger decay from the valence system to the core level
$P^{\text{scatt}}$	Rate of electron scattering of unbound electrons returning to the valence system
$P_j^{\text{red}}$	Redistribution of valence electrons due to cascading $P^{\text{scatt}}$ electrons
$P_j^{\text{therm}}$	Rate of thermalization of the valence system toward a Fermi–Dirac distribution

FIG. 2.

Photon, electron, and energy densities and their interactions. A photon density $N_i^{\mathrm{\text{phot}}}$ drives resonant interactions between the core electrons R_C and specific valence electrons ρ_j. It also drives non-resonant excitations from the entire valence electron system $R_V=\sum _j{\rho }_j$ to free electrons R_free, which have a total energy of E_free. Auger–Meitner decays transfer electrons from the valence system to both core states and free electrons; scattering cascades transfer electrons and energy from the free states to the valence system; thermalization drives the valence system toward a thermalized Fermi–Dirac distribution. $M_C,M_V$, and m_j represent the number of available states and are pictured as bars to represent the energy bins of the numerical calculation.

The processes of resonant absorption and stimulated emission are summarized in the resonant interaction process $P_j^{\mathrm{\text{res}}}$. This is possible because both processes describe a transfer of electrons between the 2 ${\mathrm{p}}_{3/2}$ core level and the valence system following the same interaction cross section, just with inverted sign. The non-resonant absorption process $P_{i,j}^{\text{non}-\text{res}}$ describes photon absorption at all other electronic levels. The non-resonant absorption length of the ground state is derived from the measured pre-edge absorption. Because of the relatively small contribution of the intermediate 3p and 3s electrons, the model ascribes all non-resonant absorption events to valence electrons, transferring them to the pool of high-energy electrons R_free. Sequential two-photon absorption (TPA) processes are implicitly treated by the model as a resonant absorption event followed by a non-resonant absorption event. This description does not account for the coherence and resonant enhancement of the TPA process. However, the fluences used here do not exceed 1.5  $\times {10}^{31}\text{photons}{\text{cm}}^{-2}\hspace{0.17em}{\mathrm{s}}^{-1}$, where according to the scaling law proposed by Szlachetko et al.,⁴⁸ the relative contribution of the direct TPA process should be on the order of 1%. Thus, the inaccuracy caused by not treating this process explicitly should be very small. Apart from photoelectrons from the valence system, the Auger–Meitner decay process $P_j^{\text{Auger}}$ likewise contributes to the free electron pool. These free electrons proceed to scatter with the valence electron system. The total rate of scattering $P^{\text{scatt}}$ is modeled by a simple lifetime parameter ${\tau }_{\text{scatt}}$, which characterizes how quickly free electrons re-join the valence system. The redistribution process $P_j^{\text{red}}$ distributes the kinetic energy of each scattering free electron among the valence system. Avoiding the complexity of explicitly calculating the multiple collisions involved in these electron scattering cascades, the algorithm instead approximates the effect of such a cascade: The energy of the scattering electron is spent to elevate an evenly distributed fraction of all valence electrons from occupied states to locally available unoccupied states (as indicated in purple in Fig. 2). Note that this process is not independent but represents an immediate consequence of the free electron scattering process; the scattering time ${\tau }_{\text{scatt}}$, thus, characterizes both $P^{\text{scatt}}$ and $P_j^{\text{red}}$ together. Finally, electronic thermalization is modeled with a bulk timescale ${\tau }_{\text{th}}$ (essentially quantifying electron–electron scattering) that moves the non-thermal valence electron distribution toward a target Fermi–Dirac distribution that corresponds to the momentary internal energy and population of the valence system.

The full mathematical description of the process terms is given in Subsection 1 of Appendix A, while the choice and derivation of input parameters is detailed in Subsection 3 of Appendix A.

A.

Differentials

From these process terms we assemble the time-differentials of the populations of electrons and photons listed in Table I. The movement of electrons between states is represented by process terms of electronic transitions appearing symmetrically in these differentials with positive and negative signs, thus ensuring the conservation of particle number. For example, the term for Auger decay appears twice with a negative sign in the valence electron differential, and once each with a positive sign in the differential for core- and free electrons.

The number of photons is reduced or increased by resonant interaction and reduced by non-resonant absorption. The model allows for an arbitrary number of incident photon energies E_i, each of which must be resonant to a specific bin of the valence energy system E_j. The temporal intensity profile of all incident photons is modeled as Gaussian. For the presented calculations, only a single resonant photon energy was used, representing measurements with monochromatic x rays. Incident photons are the only source of energy flow into the system, and all energy eventually emerges as the thermal energy of the valence system.

The valence system interacts via all modeled processes. The resonant absorption rate $P_j^{\mathrm{\text{res}}}$ changes the valence electron densities ρ_j at all incident photon energies E_i that are resonant. Via non-resonant absorption, incident photons can be absorbed by electrons from all valence states ρ_j. Auger processes depopulate the valence system. The thermalization drives electrons toward the Fermi–Dirac distribution based on the current internal energy and population of the valence band, without changing the total valence occupation. Electron scattering P^scatt causes electrons from the free electron pool to re-join the valence system in a random unoccupied state h_j, as well as redistributes electrons inside the valence system in an electron cascade triggered by the process, which is described via the $P_j^{\text{red}}$ term.

The population of core electrons is reduced (or increased, depending on the sign of $P_{i,j}^{\mathrm{\text{res}}}$) by resonant transitions of all incident photon energies E_i to states at all resonant energies ρ_j, and is increased by Auger decay from electrons of all energies j in the valence system. Note that the radiative emission channel is neglected in our model as it is designed for soft x-ray energies where Auger emission accounts for most core-hole decays (here specifically, 99.1% of the nickel L₃ core-hole decays^49,50). In another concession to the specific experiment simulated here, we further neglect fast electrons leaving the sample, since the electron mean free path of about 1.3 nm (Ref. 45) is much shorter than the sample thickness of 20 nm. While the model is generally suited to implement a loss process for free electrons, the total number of electrons in the system being strictly constant over time is a valuable indicator for the self-consistency of the calculation.

Free electrons are generated by non-resonant absorption from all incident photon energies E_i as well as Auger-decays from all energies in the valence band. The population is reduced by the free electron scattering rate $P^{\text{scatt}}$.

See Subsection 2 of Appendix A for the differential equations.

IV.

RESULTS

Figure 3 shows the measured spectra for the nickel ${\mathrm{L}}_3$-edge next to simulated spectra for increasing x-ray fluence over more than three orders of magnitude, from 0.03 to 60 J/cm². Each measured point represents an average of several FEL shots, sorted by x-ray fluence and photon energy. The varying statistical uncertainty is a result of the pulse intensity fluctuations of monochromatized SASE radiation⁵¹ in combination with photon energy-dependent spot sizes (see Subsection 3 of Appendix B for details on the shot classification).

FIG. 3.

Fluence-dependent Ni ${\mathrm{L}}_3$-edge spectra, measured (top) and simulated (bottom). The fluence of events contributing to each spectrum is given in the legend in terms of mean and standard deviation. Dashed simulated spectra do not have a corresponding measurement. The regions of interest from which absorbance changes shown in panels (b), (d), and (e) of Fig. 4 were quantified are shaded and labeled (I) (II), and (III), respectively. The error bars are shown for the measured spectra and represent the 95% confidence intervals for each bin of 102 meV width; solid lines of the measured spectra are smoothed using a Savitzky–Golay filter using windows of 21 bins and fourth-order polynomials. The experimental spectra are vertically offset by 100 mOD.

We observe four main fluence-dependent effects, which we quantify and compare to the simulated results in Fig. 4: (a) a redshift of the absorption edge of up to 0.9 ± 0.1 eV in the rising flank; (b) an increase in the pre-edge absorbance, as the rising edge of the absorption peak shifts and broadens; (c) a reduced peak absorbance; and (d) and (e), a reduced post-edge absorbance. The integration regions from which the effects (b), (d), and (e) are derived, are highlighted in Fig. 3 as (I), (II), and (III), respectively. Each is 0.6 eV wide. The shift of the absorption edge is quantified by the photon energy at which the absorbance reaches half of the peak value; its uncertainty is propagated from the statistical uncertainty of the absorption peak measurement.

FIG. 4.

Comparison of spectral effects between simulation (blue lines) and experiment (orange lines with error bars). The shift of the absorption edge in panel (a) represents the photon energy at which the half-maximum of the absorption peak is reached. The absorbance changes in panels (b), (d) and (e) are integrated from the gray shaded regions in Fig. 3, while panel (c) shows the global maximum of the spectrum.

As apparent from Fig. 4, the rise in absorption at the pre-edge (region I), the drop in post-edge absorption (region III), as well as the shift of the rising edge show good agreement within the measurement uncertainties between simulation and experimental data. The deviations observed in the absorption level of the resonance peak and just beyond will be discussed later.

We emphasize that this level of agreement with the experimental data are achieved across more than three orders of magnitude in fluence, based on a rather simplified description of well-known physical processes in combination with experimental or tabulated ground-state properties such as density, electronic configuration, and ground-state spectrum. Only the valence thermalization time ${\tau }_{\text{th}}$ and electron scattering time τ_scatt were varied to achieve the best match to the experimental results. We obtain a value of ${\tau }_{\text{th}}=6$ fs, which compares well to recent estimates for excitations on this energy scale.^40,41,52,53 The scattering time constant ${\tau }_{\text{scatt}}=1.5$ fs produces the best agreement with experimental data. This value appears reasonable as it summarizes a cascade of many individual electron scattering events, which we would expect to occur roughly every 100 as.⁴⁵

V.

DISCUSSION

Before further interpreting the non-linear effects shown in Fig. 4, let us first consider the example of a local valence band population history as shown in Fig. 5. The example is drawn from the uppermost 4 Å thick voxel of the simulated sample, excited with a Gaussian pulse profile centered around t = 0 with 30 fs FWHM duration and 30 J/cm² fluence. As such, the example is selected from the upper range of extreme excitations in this simulation to showcase the effects clearly. While panel (a) of Fig. 4 shows the calculated DOS as used by the simulation and published in Refs. 46 and 47, the colormap in (b) shows the occupation of these states over time. It is apparent that the occupation function mostly resembles a Fermi–Dirac distribution evolving from cold to hot. However, the states at E_j = E_i (highlighted by the blue ellipse), show greater population as they are directly populated by the resonant absorption process. We also show the effective electron temperature T and chemical potential μ, which are calculated from the internal energy and population of the valence system at every time step. Panel (c) shows the number of electrons per atom in the valence band below and above the Fermi level (blue solid and dashed curves, respectively) as well as the average number of core holes and the number of free electrons over time. One general observation is that for the given 30 fs pulse duration, the number of simultaneously existent core holes remains very small, even for high fluences. This has two reasons: On the one hand, the natural lifetime of the core-holes of 1.4 fs is small compared to the pulse duration.⁵⁰ On the other hand, the monochromatic excitation near the material resonance implies that the photons couple the core-level to a narrow selection of localized valence states.⁵⁴ In this case, the number of resonant valence states is small in comparison to the number of core electrons. Since the core-level and resonant valence states operate like a two-level system in which absorption and stimulated emission compete, the resonant absorption process saturates due to occupied valence states long before the core level is significantly depleted. This bleaching of valence holes is amplified over the pulse duration by an increasingly heated Fermi–Dirac distribution, which also increases the occupation of states above the Fermi level. Since both core-holes and free electrons decay so quickly, a majority of the absorbed energy is quickly translated into a broadening of the valence electron distribution. By the end of the pulse in this example, more than half of the 3d valence electrons are excited to valence states above the Fermi level, while the highest instantaneous number of core holes was only about one per 100 atoms, as shown in Fig. 5(c).

FIG. 5.

Evolution of electronic populations (simulation) in a single voxel at the sample surface for a pulse of 858.3 eV, with a pulse energy of 30  ${\mathrm{J}/\text{cm}}^2$. Panel (a) shows the total DOS used as an input for the simulation. Panel (b) shows the energy-resolved occupation (between 0 and 1) of the valence band over time, relative to the Fermi energy, and shares the corresponding axes with panels (a) and (c). The population (in electrons/atom/eV) is the product of the DOS and the occupation. The thermalized valence occupation lags a few femtoseconds behind the current chemical potential μ; the temperature T of the valence system rises rapidly, ultimately reaching up to 25 eV. The bleaching of valence states (highlighted with a blue dotted ellipse) is visible as a high non-thermal population at the resonant photon energy around 7 eV above the Fermi level. Panel (c) shows the number of core holes and free electrons over time, as well as the number of electrons in the valence system below and above the Fermi energy.

With these general observations about the evolution of the electronic system within the pulse duration in mind, we can now proceed to interpret the mechanisms responsible for the non-linear features in the spectra. Above the absorption edge, the decrease in absorption with increased fluence [see Fig. 4(d)] can be understood as a depletion of valence states available to the resonant core-to-valence transition. Similarly, below the absorption edge, the increase in absorption [see Fig. 4(b)] can be attributed to valence holes below the Fermi-level becoming available due to the thermalization process, as soon as the tail of the Fermi–Dirac distribution reaches the probed energy. The shift of the absorption edge [see Fig. 4(a)] can be explained by a non-linear combination of the two effects above. Consider that below the absorption edge at the beginning of the pulse, the sample only interacts with the x rays via the comparatively weak process of non-resonant absorption. However, once the sample is sufficiently heated that valence holes become available, additional resonant absorption begins to occur and accelerates further electronic heating—and in turn additional pre-edge absorption. Since the onset of this exponential process occurs earlier near the absorption edge, it contributes significantly to the observed spectral redshift. Another cause of the observed edge shift is the shift of the chemical potential μ, which strongly depends on the exact shape of the DOS and is shown in Fig. 5(b) as a green line. Initially, μ increases with absorbed fluence, as thermally excited electrons from the 3d states must spread out in energy to the lower DOS above the Fermi level. With rising electronic temperature, the high DOS of the 3d states becomes less relevant and the chemical potential drops again as expected in regular metals. A similar evolution of the chemical potential and electronic temperature was predicted for optically excited nickel by previous experiments and calculations.^14,55–57 It is remarkable that the experimentally observed redshift of 0.9 ± 0.1 eV can be reproduced by the rate model based on this very simple mechanism. However, this mechanism applies specifically to non-linear absorption using monochromatic x rays. Qualitatively similar redshifts have been observed in nickel after excitation with optical lasers even at up to three orders of magnitude lower excitation fluence.^43,58,59 These redshifts have recently been linked to modifications of the band structure due to the interplay of electronic correlations and optically induced demagnetization.⁴³ While such subtle, spin dependent effects may also occur in our high-fluence study, they are evidently overshadowed by the electron population dynamics. This aspect of an initially non-thermal electron distribution evolving toward a Fermi–Dirac distribution was also observed as a critical aspect of the optically excited spectra.^43,60 The timescale of electron thermalization was estimated to just over 100 fs, which is about 20 times slower than our estimate of 6 fs. This apparent discrepancy results from a scaling of the thermalization time with excitation density. Such a scaling is supported both by theory⁵² and recent pump–probe studies at the nickel M_2,3 edge⁵³ where the electron thermalization time decreased from 34 to 13 fs with rising optical pump fluence from 8 to 62 mJ/cm². This implies that the value of 6 fs found in our study represents an average time constant for the excitation densities in our experiments.

A significant deviation between model and experiment can be observed at the resonance peak itself, where the simulated electron dynamics lead us to expect a much stronger saturation effect than observed experimentally [Fig. 4(c)]. This saturation is reduced as the peak position redshifts and more thermally vacated d-states become resonant, leading the model to predict a slight recovery of absorbance at the highest fluences. Overestimating the saturation effect may be related to a fluence-dependent decrease in the excited state lifetime due to stimulated emission as well as increased carrier mobility around the Fermi edge, both leading to an energetic broadening of the resonant core–valence transition. Such a broadening would increase the number of resonant valence states and thus delay saturation especially at the edge, but is not considered in our model. While it may be expected that a purely population-based model cannot fully represent resonance effects at the resonance peak itself, the lack of any significant saturation around 852 eV [Fig. 4(d)] is more surprising. Both disagreements point to additional physical effects and call for more sophisticated models.

We speculatively propose three mechanisms which could contribute to these discrepancies: First, the transition matrix elements could get modified at higher excitation densities, especially around the resonance, while we model the absorption only based on the ground-state spectrum. Second, an energy dependence of the electron–electron scattering cross section could allow for particularly fast scattering of electrons with certain energies, counteracting the saturation. Third, a collective, correlated response of the electronic system (local field effects) could modify the DOS or the transitions even on the fast timescale of the FEL pulse duration.^43,61

A more detailed discussion of the model can be found in the Appendixes: In Subsection 4 of Appendix A, we show how a variation or elimination of specific processes leads to different predictions for the spectra, and in Subsection 5 of Appendix A, we discuss the limitations of the rate model and its suitability for future extension.

VI.

CONCLUSION

To summarize, we have analyzed fluence-dependent near-edge x-ray absorption spectra of the nickel 2 ${\mathrm{p}}_{3/2}$ core level up to x-ray fluences of 60 J/cm². We have developed a rate-equation model based on differential equations that describes the excitation and decay processes connecting populations of core and valence electronic states. Process rates are quantified by scaling known ground-state properties with evolving electron populations.

The model enables an understanding of the electronic population history under strong x-ray fluences and characterization of the resulting non-linear absorption near a core resonance. It successfully predicts the observed increase in absorption before and its decrease beyond the resonance, as well as the fluence-dependent redshift of the absorption peak over three orders of magnitude. However, the bleaching of the absorption peak is overestimated by the population-based model and will require more sophisticated models to accurately quantify. Here, the population dynamics rate model also provides a valuable point of reference for more advanced theoretical frameworks.

Providing the fundamental fingerprints of how strong x-ray fluences alter the electronic system and thus the absorption spectra, our straightforward picture of intense core-resonant x-ray pulse interaction can inform the design and interpretation of future FEL experiments. On the one hand, our model can guide the decision up to which point to maximize fluence for good statistics while keeping the absorption process linear, and to recognize the principal spectral fingerprints emerging at the onset of non-linear absorption due to electron dynamics within the pulse. On the other hand, an understanding of the population dynamics within high-fluence pulses, and in particular, an awareness of the dominant influence of electronic scattering processes, is crucial for emerging techniques that aim to utilize x-ray wave-mixing processes, such as stimulated core hole emission, in solids.^40,62–71