Transient absorption of warm dense matter created by an X-ray free-electron laser

The study investigates how intense femtosecond X-ray free-electron laser (XFEL) pulses drive solids into warm dense matter (WDM) and alter their electronic structure, focusing on transient X-ray absorption near-edge structure (XANES) at the Cu L2,3 edges. WDM lies between condensed matter and weakly coupled plasma regimes and remains challenging to characterize, particularly under non-equilibrium ultrafast excitation. Prior diagnostics have relied largely on X-ray emission spectroscopy, while time-resolved XANES of optically heated WDM has provided picosecond–subpicosecond insights. The authors leverage the broad spectral bandwidth of self-amplified spontaneous emission XFEL pulses to perform single-pulse, spectrally resolved XANES that simultaneously creates and probes WDM in copper, testing for nonlinear absorption phenomena—reverse saturable absorption (RSA) and saturable absorption (SA)—and benchmarking non-equilibrium electronic-structure models. They hypothesize that increasing XFEL intensity first enhances pre-edge absorption via transient 3d depletion (RSA) and then reduces absorption as transitions shift out of resonance due to further ionization and band shifts (SA).

- Nonlinear X-ray absorption phenomena have been observed in various materials, including SA and RSA, typically via spectrally integrated measurements (e.g., Nagler et al. 2009; Yoneda et al. 2014; Rackstraw et al. 2015; Hoffmann et al. 2022; Cho et al. 2017). Transitions between RSA and SA are well known in optics and 2D materials with significant applications in pulse shaping and mode locking. - XANES has been a powerful probe of electronic and atomic structure in optically/IR-heated WDM with ps–sub-ps resolution (e.g., Dorchies & Recoules 2016; Dorchies et al. 2011; Cho et al. 2011, 2016; Mahieu et al. 2018; Jourdain et al. 2021; Lee et al. 2021), revealing pre-edge features due to 3d vacancies and sensitivity to structural order (e.g., van Hove singularity signatures). - Theoretical frameworks include kinetic Boltzmann descriptions for ionization and non-equilibrium electron distributions in XFEL-irradiated solids (Ziaja et al. 2006, 2016, 2021) and finite-temperature DFT and finite-T Green’s function approaches for high-temperature XANES (Tan et al. 2021, 2023; Kas et al. 2022). A direct ab initio non-equilibrium XANES theory is lacking, necessitating hybrid approaches. Prior predictions (Cho et al.) suggested RSA→SA transitions but required higher XFEL intensity than previously available.

Experimental: - Facility/instrument: Spectroscopy and Coherent Scattering (SCS) instrument at European XFEL. - Samples: 100-nm and 500-nm thick copper foils on a 15-µm-thick Ni mesh; spectra with mesh hits were identified and removed. No oxidation signatures were found at low fluence. - XFEL parameters: 15 fs FWHM pulse duration; central photon energies near Cu L3 (≈932 eV) or L2 (≈952 eV) with ~0.7% bandwidth and ~40% spectral content above the edge; beam focused to 4 µm FWHM using Kirkpatrick–Báez mirrors; pulse energies up to 2 mJ; achieved on-target intensities up to ~7×10^16 W cm−2 (from main-text figures) with measurement accuracy estimated ~25% (Methods). - Detection: Rowland-geometry grating spectrometer (1,200 lines/mm, 5 m radius), 40 µm entrance slit, coupled to a CCD; energy calibrated by beamline monochromator; resolving power E/ΔE ≈ 2,400; transmitted beam attenuated by Al filters to avoid detector saturation. - Data acquisition: For each condition, ~2,000 reference spectra (no sample) and 200–1,000 sample spectra; averaging required due to SASE spiky spectra; absorption coefficient computed as μ = ln(Iref/Icu)/d; pulse transmission T computed from integrated spectral intensities Sref and Scu; spectrally resolved transmission evaluated over selected energy windows; pre-edge shift extracted from zero-crossings of the derivative after Savitzky–Golay smoothing with energy position accuracy ±0.11–0.22 eV. Theoretical modeling: - Kinetic Boltzmann solver models non-equilibrium evolution of charge-state populations and free-electron energy distribution under a 15 fs Gaussian XFEL pulse, including photoionization, Auger decay, electron-impact ionization (Lotz cross sections), three-body recombination, and e–e scattering (Fokker–Planck). Atomic process data from XATOM (Hartree–Fock–Slater). Pauli blocking neglected (classical electron assumption). Cu 2p level treated as degenerate at 939.11 eV with a constant −6.41 eV shift to align with the neutral Cu L3 edge at 932.7 eV. The pulse was approximated as monochromatic (~1 eV above L3) for kinetics. - Output of Boltzmann model: time- and intensity-dependent relative abundances of Cu configurations (dominantly varying 3d occupations; 3p-vacancy configurations at highest intensities) and transient free-electron distributions ρ(E). - Finite-temperature XANES: Real-space multiple-scattering calculations with FEFF10 to compute L3-edge absorption for relevant configurations. The local environment is approximated by atoms at the crystal lattice; an effective electronic temperature Teff is introduced to reproduce the average 3d occupation from Boltzmann output, allowing self-consistent charge-density relaxation. Absorption cross sections σA(ω, Teff) are computed per configuration and averaged over configurations weighted by Boltzmann abundances and over space-time with pulse intensity weighting to yield total σ(ω, x, y, z, t). A comparison approach (neglecting environmental changes with intensity) qualitatively reproduces trends but fails to capture the observed low-intensity redshift. - Analysis focuses on the 100-nm film at the L3 edge, with complementary L2 measurements and 500-nm results in the Supplementary Information.

- Spectrally resolved RSA to SA transition: - At the Cu L3 edge, pulse transmission versus intensity shows RSA (decreasing transmission) up to a minimum at I_tr ≈ 1×10^15 W cm−2, followed by SA (increasing transmission) at higher intensities. At the L2 edge, the transition shifts to higher intensity, I_tr ≈ 1×10^16 W cm−2. - Pre-edge feature and intensity dependence: - For I ≥ 1×10^14 W cm−2, a strong pre-edge absorption peak appears below L3 due to 2p3/2 → 3d transitions enabled by transient 3d vacancies. At I = 1×10^15 W cm−2, the pre-edge amplitude reaches ~2.5× the cold L3 edge amplitude. - With increasing intensity, the pre-edge peak initially redshifts (enhanced 3d depletion reduces screening and shifts the 3d band; concurrent 2p level shifts partly cancel net shifts), then at higher intensities blueshifts and broadens as higher charge states dominate and overall screening is further reduced. - Edge/spectral evolution at highest intensities: - At the highest intensities, substantial ionization and collisional processes broaden and shift features so that the L3 edge becomes unresolved; overall absorption decreases in the probed window and a moderate monotonic increase of absorption with photon energy is observed across the window. - Van Hove singularity signature: - A ~1 eV-wide peak at 936.7 eV, attributed to the zero-temperature van Hove singularity (fcc Cu), persists up to ~1×10^16 W cm−2, indicating no ion-motion-induced structural change within the 15 fs pulse. It is suppressed and disappears by ~7×10^16 W cm−2 due to strong electronic-state redistribution and broadening. - Spectrally resolved transmission trends (regions below/on/above L2,3): - Transmission minima occur at progressively higher intensities for higher photon-energy regions, consistent with absorption channels shifting with charge state and band evolution. Below the L3 pre-edge (region 1), transmission remains ~constant (close to cold value, ~87% for 100 nm) at low intensities; in the L2 pre-edge region (region 4), transmission drops sharply between 5×10^15 and 1×10^16 W cm−2, consistent with the opening of 2p3/2 → 3d channels in Cu2+, Cu3+, Cu4+. - Electron thermalization dynamics (from Boltzmann model): - Low intensity (5×10^12 W cm−2): high-energy photo/Auger peaks persist; below −30 eV the distribution grows and approaches Maxwell–Boltzmann (MB), but the system remains non-equilibrated at 20 fs (MB fit T ≈ 2.2 eV captures average kinetic energy only). - High intensity (≈1×10^17 W cm−2): rapid e–e scattering redistributes energy; by ~10 fs the electron distribution thermalizes to an MB distribution with kinetic temperature ≈90 eV at 20 fs. - Theory–experiment agreement: - The hybrid Boltzmann+FEFF10 model reproduces the qualitative RSA→SA transition, pre-edge growth and red/blue shifts with intensity, and the diminishing van Hove feature at high intensities.

The results demonstrate that intense, ultrafast XFEL pulses can both generate and diagnose WDM through transient XANES, revealing nonlinear absorption behavior. At low-to-intermediate intensities, transient 3d depletion increases the population of 2p−1 3d configurations, producing a strong pre-edge and RSA. The pre-edge redshift arises from reduced electronic screening and a lowered 3d-band energy relative to the chemical potential, partially offset by concomitant shifts of bound 2p levels. As intensity increases further, higher Cu charge states (with fewer 3d electrons) exhibit blueshifted and broadened excitations; absorption strength migrates to higher energies and eventually outside the measurement window, closing resonant channels and yielding SA. The shift of the RSA→SA transition to higher intensity at L2 reflects additional 2p3/2 → 3d channels of higher charge states falling within the L3 spectral region, which continue to contribute absorption even as lower-charge-state channels saturate. These findings validate the use of spectrally resolved XANES to benchmark non-equilibrium models of electronic structure in WDM and point to opportunities for X-ray pulse shaping (via controlled RSA/SA) at soft X-ray energies. The observed persistence and eventual disappearance of the van Hove singularity constrain structural dynamics on 15 fs timescales, implying negligible ion motion during the pulse at ≤10^16 W cm−2 and strong electronic-state redistribution at higher intensities. Discrepancies between theory and experiment (e.g., transition intensities, spectral shifts, high-energy cross sections) highlight the need for improved treatment of non-equilibrium electron distributions, solid-state multiplet effects, and intensity characterization.

This work reports the first spectrally resolved observation of a transition from reverse saturable absorption to saturable absorption in the X-ray regime at the Cu L2,3 edges, achieved by simultaneously creating and probing warm dense copper with 15 fs XFEL pulses. The study maps intensity-dependent XANES evolution, including a strong pre-edge due to transient 3d depletion, red-to-blue spectral shifts with increasing ionization, and the suppression of a van Hove singularity at the highest intensities. A hybrid kinetic Boltzmann plus finite-temperature FEFF10 framework qualitatively reproduces key trends, enabling benchmarking of non-equilibrium electronic-structure models for WDM and suggesting routes to X-ray pulse shaping. Future directions include implementing self-referencing detection to improve signal-to-noise, extending to two-color X-ray pump–probe XANES for time-resolved tracking of WDM formation and relaxation, and exploiting forthcoming attosecond XFEL capabilities for attosecond transient XANES to study matter under extreme conditions.

- Intensity calibration: The experimentally characterized beam intensity can differ by approximately 50% (as noted in the main text), complicating precise comparison with simulations; Methods estimate ~25% accuracy based on diagnostics. - Model approximations: The Boltzmann approach restricts to predominant excitation/relaxation paths, treats electrons classically (no Pauli blocking), and assumes a monochromatic pulse for kinetics; atomic levels are adjusted by a constant shift to match the neutral L3 edge. - Non-equilibrium electronic structure: FEFF10 calculations employ an effective electronic temperature to mimic non-equilibrium 3d occupations; direct ab initio non-equilibrium XANES theory is not available. Use of equilibrium-like electron distributions may misestimate cross sections, especially at higher energies. - Spectral broadening and multiplet effects: Possible overestimation of broadening and limited treatment of solid-state multiplet splitting may affect predicted shifts and line shapes. - Discrepancies: The calculated RSA→SA transition occurs at higher intensities than measured; the low-to-intermediate-intensity redshift is underpredicted; the high-intensity blueshift appears at higher simulated intensities; high-energy absorption reduction in theory is not observed experimentally. - Scope: Main-text analysis emphasizes the 100-nm film at L3; contributions from atoms with deeper-shell holes are neglected except at the highest intensities (≥3×10^16 W cm−2).