




\chapter{RIXS for Dynamically Compressed Systems}







This final chapter examines the use of RIXS to study laser-compressed materials, representing a key application of this methodology. This objective exploits RIXS's ability to capture a material's electronic structure in just a single snapshot, highlighting its potential for time-resolved studies, particularly of irreversible physical processes like laser compression. This capability holds significant promise as a diagnostic tool during capsule implosion in inertial confinement fusion research.

On the other hand, when investigating irreversible physical processes, the number of shots that can be recorded is often limited. This constraint poses a challenge for RIXS, given its relatively low cross-section, which can introduce significant noise into the averaged RIXS spectra and, consequently, into the extracted electronic structures.

This chapter analyses the RIXS spectra of compressed FeO, recorded during an experiment conducted in July 2021 at the Matter in Extreme Conditions (MEC) instrument of the LCLS facility~\cite{nagler2015matter}. The analysis begins with a review of the experimental setup, with a focus on the diagnostics used to assess the pressure conditions and crystalline structure of the compressed targets. Following this, the electronic structures reconstructed from the RIXS spectra are examined, comparing results from ambient conditions to those under compression. The aim is to retrieve the experimental evidence of the bandgap closure predicted by theoretical models.

As will be shown, while the data reveal trends consistent with bandgap closure, the poor SNR, which causes a low resolution in the extracted $\rho_{\rm eff}$, limits the precision of the results. Although these preliminary findings are not yet suitable for publication, they demonstrate the potential of this technique for such investigations, especially considering the ongoing technological advancements that promise to improve the SNR of RIXS spectra.






\section{Experimental Setup}






\begin{figure}
     \centering
     \includegraphics[width=\textwidth]{Figures/lcls setup.pdf}
     \caption[Setup of the LCLS experiment]{(a) A CAD model of the LCLS experimental setup, including laser beam paths and the main diagnostics. For clarity, the optics used to deflect the laser beams are not shown. The optical laser is split into two arms, further amplified, and recombined on the target at 20$^{\circ}$ and 50$^{\circ}$ angles relative to the sample normal. The target holder was specifically designed to ensure both the laser arms could illuminate the target during each shot. (b) Structure of the targets employed for the compressed shots.}
     \label{lcls setup}
 \end{figure}

In Figure \ref{lcls setup}a, the experimental setup of the LCLS experiment is shown, highlighting the key beam paths, including the XFEL, the optical laser used for compression, and the VISAR laser, which is discussed in more detail in the next section. The optical laser is a Nd:glass laser system operating at 527 nm, which can be moulded to the required pulse length and shape using an electro-optical modulator, with durations ranging from 2 ns up to 200 ns~\cite{nagler2015matter}. For this experiment, a simple flat-top pulse shape with a duration of approximately 10 ns was employed, focused to a 250 $\mu$m spot size using focal-length singlet lenses and incident at an angle of 35$^{\circ}$. Synchronisation with the XFEL, achieving time precision on the order of 10 fs, is possible using the beamline instruments outlined in~\cite{minitti2015optical}. The laser energy was varied between 40 and 60 J to generate different pressure conditions. 

For the compressed shots, the XFEL was tuned to a central energy of 7070 eV, just below the Fe absorption edge, and operated in seeded mode. To minimise heating of the target, the XFEL beam was defocused to approximately 60 $\mu$m\footnote{Excessively decreasing the XFEL beam size can degrade measurement resolution due to source broadening.} and was used solely to probe the compressed target via the RIXS mechanism. It is important to note that the XFEL at LCLS includes a third harmonic component of the fundamental photon frequency, with an intensity approximately 1\% of the fundamental. These higher-energy photons, exceeding the absorption edge, induce K$\alpha$ photodecays that can obscure the RIXS signal. To mitigate this issue, the central XFEL energy must be set sufficiently below the absorption edge, effectively shifting the RIXS spectrum away from the K$\alpha$ lines and preventing spectral overlap.

The RIXS photons were detected in reflection geometry using the same diagnostic setup as in the EuXFEL experiment but with a different von H\'amos spectrometer, which, along with the flat crystal spectrometer, was designed and built in Oxford. As in the EuXFEL experiment, this von H\'amos spectrometer features a RoC of 80 mm and a thickness of 40 $\mu$m, and it was paired with an ePix100 detector. The resolution of the spectrometers in this experiment was determined by comparing the measured widths of the K$\alpha$ lines with their tabulated values. The resolution was found to be 2.2 eV for the flat crystal spectrometer and 6.1 for the von H\'amos spectrometer. For analysing the compressed data, where few shots are available, exclusively the von H\'amos measurements will be used, as the flat crystal spectrometer lacks sufficient SNR for this limited dataset.

In addition, x-ray diffraction patterns were recorded in transmission geometry for each shot using four CSPAD detectors. These diffraction patterns were used to determine the crystalline structure of the compressed samples under various pressure conditions, including phase transitions and melting points along the Hugoniot curve. Correlating changes in the ionic structure with those in the density of states is crucial for understanding the interplay between electron and ion dynamics in iron oxides. A dedicated section will discuss the XRD analysis in more detail.

A schematic cross-section of the target is shown in Figure \ref{lcls setup}b. The targets consisted of multiple layers: a 52 $\mu$m plastic ablation layer designed to smooth irradiation non-uniformities and steepen the variable ramp compression into a shock before impinging on the target, a 8 $\mu$m FeO sample layer\footnote{This chapter focuses on FeO as it represents an interesting case and serves as a proxy for Fe and Fe$_2$O$_3$, which were also studied during this experiment.}, and a 20 $\mu$m sapphire window layer. This final layer is used for the pressure calculations, together with the VISAR system, employing the impedance matching technique. This layer was coated with aluminium to provide a reflective surface for VISAR measurements. Another aluminium flash coating (200 nm thick) was applied to the front face of all samples to minimise preheating from x-ray radiation emitted by the ablated plasma. Note that, due to the strict resonance conditions required for RIXS to occur, none of these other layers would produce RIXS photons that could affect the signal from the sample.

To perform the XRD analysis, the FeO samples were grown in a Fe (111) \textit{fibre texture}, meaning that the polycristallites comprising the sample are oriented with their normal closely aligned to the Fe (111) direction, though they remain randomly distributed in terms of their angular orientation around this axis. This fiber texture effectively rotates the reciprocal lattice of the FeO crystal around the Fe (111) axis, creating circumferences around this axis for each reciprocal lattice point (some of which may coincide).

As long as the x-ray beam is not aligned too closely to the fiber axis (in our experiment, a 35$^{\circ}$ angle of incidence was used), some of these circumferences intersect the \textit{Ewald sphere}~\cite{heighway2020shock}, resulting in bright spots along the \textit{Debye-Scherrer} rings. During compression, as the lattice undergoes deformation, these diffraction spots generally move, shedding light on active slip systems and how the target relieves shear stress under uniaxial compression~\cite{wehrenberg2017situ}. If the sample were composed of randomly oriented crystallites (as in a powder), the entire Debye-Scherrer rings would be illuminated due to diffraction interference, making it impossible to analyse the movement of individual diffraction spots. On the other hand, using a single crystal would make observing diffraction spots extremely challenging due to the unlikely intersections between the reciprocal lattice points and the Ewald sphere.








\subsection{VISAR Diagnostic}


% ``interface velocity'' 

\begin{figure}[t]
     \centering
     \includegraphics[width=0.6\textwidth]{Figures/VISAR.pdf}
     \caption[Scheme of the VISAR system with an example of camera image]{Scheme of the VISAR system with an example of a streak camera image from an experimental shot. In this image, a region of interest is displayed as a black rectangle together with the trajectory of a single fringe (solid black curve).}
     \label{visar}
 \end{figure}

As outlined in the Introduction, the pressure of the compressed system can be determined using Hugoniot's equations, provided the shock velocity and particle velocity behind the shock are known. These two quantities are measured using the Velocity Interferometer System for Any Reflector (VISAR), whose functioning now discussed.

The primary purpose of VISAR is to determine when the shock wave emerges from the rear surface of the sample and the velocity at which this interface begins to move. By measuring the time it takes for the shock to traverse the target and knowing its thickness, the shock velocity, $U_S$, can be calculated. At the same time, the interface velocity (also referred to as the free-surface velocity when the back of the sample is in direct contact with the vacuum) can be related to the particle velocity behind the shock. When the shock reaches the rear surface, a rarefaction wave propagates back into the target, resulting in a pressure release. By analysing this relaxation, the interface velocity, $U_I$, can be connected to the particle velocity, $U_P$, using the following relation~\cite{walsh1955equation}\footnote{In many cases, the approximate relation $U_I = 2U_P$ is used.}:

\begin{equation}
U_I = U_P + \int_{V(p_i)}^{V(p_f)} \biggl(\frac{dp_s}{dV}\biggr)^{1/2} \ dV \ ,
\label{particle velocity}
\end{equation} 

\noindent where $p_i$ is the shock pressure, $p_s$ is the pressure along the release path, $p_f$ is the released pressure, and $V$ is the sample volume. Then the others physical quantities can be found through the Rankine-Hugoniot equations (see Eqs.~\ref{rankine-hugoniot}). In practise, via many measurements over the years, there now exists a series of equation-of-state tables, making necessary only the measurement of $U_P$.

The scheme of the VISAR used in this experiment is shown in Fig. \ref{visar}, alongside an example of an experimental VISAR image. Various implementations and versions of the VISAR system exist, each tailored to different physical contexts~\cite{dolan2006foundations}. For this experiment, a line VISAR was employed, enabling the measurement of velocity profiles of a sample interface along one dimension (1D VISAR), so that shock planarity can be investigated. 

The VISAR system operates by directing a laser beam onto a moving reflective surface. The reflected beam is then split into two paths of different optical lengths, which are recombined on a streak camera to produce an interference pattern. This configuration enables the two beams to sample the position of the reflective surface at two different time points, separated by the optical delay between the paths. The optical delay is introduced using a combination of the refractive delay from an etalon filter and, in some cases, an additional translational offset of a mirror (M2 in the figure) perpendicular to its plane. The streak camera records the central slit region of the field of view and sweeps this signal over time across the output detector. Changes in the interference pattern over time provide information about the velocity history of the central strip of the reflective surface.

When the reflective surface is stationary, the interference pattern consists of fringes parallel to the time axis (see the upper part of Fig. \ref{visar}). This background pattern is achieved by slightly tilting one of the mirrors so that rays from the same beam travel slightly different distances before reaching the detector. The spatial frequency of this fringe pattern is determined by the mirror tilt and is typically adjusted to produce 20–30 fringes across the output image (as shown in Fig. \ref{visar}). A shift in the fringe pattern corresponds to the motion of the reflective surface. Therefore, to determine the velocity profile, the magnitude of the fringe shift for each spatial point must be analysed, as shown in Fig. \ref{visar}, where the trajectory of a single fringe over time is marked with a solid black line.

From this curve $U_{fr}(t)$, the interface velocity at that position can be calculated with the following transformation:

\begin{equation}
U_I(t) =  \frac{\lambda_L}{2\tau(1+\delta)} U_{fr}(t) = {\rm VPF}_0 \ U_{fr}(t) \ ,
\label{VPF}
\end{equation} 

\noindent where $\lambda_L$ is the wavelength of the laser light reflected from the surface, $\tau$ is the temporal delay between the two arms of the interferometer, and $\delta$ is a correction factor accounting for
the chromatic dispersion in the etalon. The term VPF$_0$ is the so-called velocity per fringe. A derivation of this formula, along with a more in-depth analysis of this measurement procedure, can be found in~\cite{hashim2016visar, barker1972laser}.

In many experiments, including ours, a transparent window is placed to the rear surface of the sample, allowing the shock wave to propagate into the window rather than breaking out into a vacuum. As described in Section \ref{Laser-Compressed Systems}, such windows are often necessary to sustain pressure in the material of interest over a longer duration, preserve interfacial reflectivity, or to better characterise a reflecting shock. When observing the sample through a transparent window, a correction factor must be applied to Eq. \ref{VPF} to account for the change in the refractive index as the laser beam passes through the window:

\begin{equation}
{\rm VPF} = {\rm VPF}_0/a \ ,
\label{VPF corrected}
\end{equation} 

\noindent where $a$ is the constant term in the expression of the window's refractive index $n$, which is assumed to scale linearly with the window's density ($n = a + b\rho$). When a window is present, to establish the relationship between the measured velocities and the thermodynamic quantities, an impedance matching analysis is required (see Section \ref{Laser-Compressed Systems}).

From the example VISAR image in Fig. \ref{visar}, one can determine the shock breakout time—the moment the shock reaches the back surface of the sample—by identifying the first shift in the fringe pattern. This breakout time is critical not only for calculating the shock velocity but also for determining the required delay between the XFEL and the optical laser to ensure that the XFEL probes the sample under uniform compression. Additionally, the image reveals the slight deviations in shock planarity and the region of the sample illuminated by the optical laser, which provides valuable information for aligning the optical laser with the XFEL beam.

In this experiment, pressure conditions were determined using two VISAR systems with sensitivities of 4.5241 and 1.9890 km s$^{-1}$ fringes$^{-1}$ and acquisition time windows of 10 and 20 ns (see Fig. \ref{visar}), respectively.





\section{Results}




The key aim of the experiment was to field XRD and RIXS diagnostics simultaneously, for the first time, to probe the changes of electronic and ionic structures of iron oxides under dynamic compression. In this section, the results of the experiment are presented, beginning with a brief discussion of the XRD analysis and the evolution of the ionic structure under pressure. The focus is then shifted onto the RIXS data obtained from compressed samples, providing insights into their electronic structure. 









\subsection{XRD Analysis}




\begin{figure}[t]
     \centering
     \includegraphics[width=\textwidth]{Figures/XRD.pdf}
     \caption[XRD measurements and Hugoniot curve of compressed FeO]{(a) XRD profiles of FeO under different pressure conditions (calculated using the VISAR system) (b) Volume measurements as a function of pressure, derived from the positions of diffraction peaks. These data are compared with various theoretical predictions of the Hugoniot curve in the $v-p$ plane~\cite{fischer2011phase,amouretti2024phase}. The plots are provided courtesy of Dr. Céline Crépisson.}
     \label{XRD}
 \end{figure}

X-ray diffraction was employed to investigate possible phase transitions occurring as a function of pressure and to provide information on the ionic structure of the system. Figure \ref{XRD}a presents the XRD profiles of FeO at various pressure conditions, obtained through azimuthal integrations of 2D diffraction images recorded by the CSPAD detectors. These integrations included corrections for solid angle coverages, detector filter effects, polarisation, and self-attenuation in the sample. When such corrections were not feasible, the integration was performed using the Dioptas software, which relies on the PyFai library~\cite{kieffer2020new}, incorporating solid angle and polarisation corrections.

The XRD pattern recorded at ambient conditions prior to the shock is shown in grey for reference, corresponding to a rock-salt lattice structure (face-centred cubic with one iron and one oxigen atom per unit cell). Considering the orientations of the XFEL and the sample texture, this lattice exhibits two main peaks associated with the (200) and (220) planes. Upon compression, these peaks shift toward higher angles (red stars), due to the reduction of lattice parameters. Compressed samples also exhibit significant peak broadening, indicative of reduced crystallite size and increased micro-strain (lattice parameter fluctuations) under high-pressure conditions~\cite{ichiyanagi2019microstructural}. Due to timing fluctuations and challenges in synchronizing the XFEL and the optical laser, as well as potential x-ray halo irradiating unshocked material, the probed sample includes varying portions of ambient or released material. This accounts for the persistent presence of more or less intense ambient peaks (blue stars), which remain unchanged in position during compression. Minor peaks, such as the one around 43$^{\circ}$ (not labelled), are attributed to ambient and compressed Fe$_3$O$_4$, an impurity resulting from the sample fabrication process.

Le Bail refinements~\cite{le1988ab} were performed on the XRD profiles to determine the lattice parameters and corresponding sample volume under various pressures. Figure \ref{XRD} illustrates the results of this analysis, displaying the $v$-$p$ data points with associated uncertainties. Ambient data points are shown as black crosses, while compressed data are represented by red and yellow crosses, with the latter corrected for residual pressure in the initial sample. This residual pressure arises from stresses induced during the fabrication process, which reduce the FeO unit cell volume. The corrections were quantified by comparing the ambient XRD data with those obtained from a recrystallised (stress-free) FeO crystal. The experimental $v$-$p$ data points were compared with theoretical predictions~\cite{fischer2011phase,amouretti2024phase}, showing good agreement.

As the rock-salt lattice was used as the underlying structure for the Le Bail refinements across all pressures, this analysis demonstrates that FeO retains its ionic structure during dynamic compression up to 136 GPa. The unit cell undergoes compression without any structural phase transitions, indicating an apparent independence between the ionic and electronic transitions, which are examined in the following section.











% Where each iron atom is surrounded by six oxigen atoms and viceversa. phase 

% Rietveld refinement = basically fit theoretical diffraction patterns to real ones!





\subsection{Electronic Structure of Compressed FeO}



\begin{figure}[t]
     \centering
     \includegraphics[width=\textwidth]{Figures/FeO compressed.pdf}
     \caption[Experimental RIXS spectra and extracted electronic structure of compressed FeO]{(a) Averaged RIXS spectra for compressed and ambient conditions. (b) Averaged RIXS spectra with uncertainties indicated by shaded regions, plotted alongside their corresponding STEP reconstructions. For clarity, the spectra for compressed and ambient cases are offset by 2 units. The spurious K$\alpha$ lines were removed from these averaged spectra. (c) Reconstructed $\rho_{\rm eff}$ for both conditions, with uncertainties, including a comparison of the compressed $\rho_{\rm eff}$ with DFT calculations (green solid curve).}
     \label{FeO compressed}
 \end{figure}

Figure \ref{FeO compressed} shows the results of the RIXS analysis conducted on 18 shots of FeO compressed to 100$\pm$5 GPa, as determined using VISAR, alongside 18 reference shots of the uncompressed target. Panel (a) displays the averaged RIXS spectra for the compressed and ambient conditions. Despite the presence of noise, systematic differences between the two curves are evident, reflecting variations in the DOS occurring upon compression. On the other hand, the noise in the averaged spectra highlights that the reconstructed electronic structures are inherently affected by this non-physical disturbances. The spurious K$\alpha$ lines, originating from the XFEL third harmonic, were fitted using the averaged spectra and subsequently subtracted from each individual spectrum. In Fig. \ref{FeO compressed}, the spectra are plotted with the K$\alpha$ lines already subtracted.

Panel (b) shows the averaged spectra with their uncertainties, alongside the corresponding reconstructions, representing the RIXS spectra associated with the extracted $\rho_{\rm eff}$ for both conditions. Unlike the cases analysed in Chapter 4, the uncertainties here are not negligible due to the limited number of shots. These uncertainties, computed using the methods detailed in Chapter 2, must be propagated to the $\rho_{\rm eff}$ extracted from these RIXS spectra. A procedure for this propagation is described in the next section. Notably, the high accuracy achieved in reconstructing the spectra, facilitated by the narrow bandwidth of the seeded beam (the mode employed for collecting these data), also results in noise-fitting, leading to spurious features in the extracted $\rho_{\rm eff}$, particularly in the higher-energy bands, for which the lorentzian factor suppresses the RIXS signal. To address this issue, the reconstructed $\rho_{\rm eff}$, shown in panel (c), are zoomed around the chemical potential (set to 0), as this region is both less affected by noise and critical for observing Mott transitions. At this regard, a systematic closure of the bandgap is observed: repeated fittings of the ambient and compressed RIXS spectra (eight times for each condition) consistently produced a bandgap in the ambient case and none in the compressed case. The standard deviations of these reconstructions, combined with the propagated errors from the experimental RIXS spectra, were used to estimate the uncertainty in $\rho_{\rm eff}$.

The reconstructed $\rho_{\rm eff}$ for the compressed case was compared with DFT calculations. As in previous analyses, to achieve agreement with the experimental curve, a smearing width of 5 eV was applied to the DFT-computed $\rho_{\rm eff}$, and, a translational and multiplicative parameters were used to align and rescale the experimental reconstructions. The details of the DFT calculations are the same provided in Section \ref{sec: dft calculations} for FeO, with the key difference being that the lattice parameters of the underlying ionic structure were modified and calculated by minimizing the system enthalpy at the measured pressure of 100 GPa. This approach is justified by XRD analysis, which indicates that FeO does not change crystalline phase up to this pressure, but only modifies its lattice.

% underlying ---> frozen
% were optimised ---> were modified and calculated

This reconstructed bandgap closure potentially represents the first direct experimental observation of the insulator-to-metal transition in FeO under compression, as predicted by simulations and inferred from indirect measurements. However, the reconstructed bandgap of approximately 6 eV deviates from the documented value of 2.4 eV for ambient FeO~\cite{bowen1975electrical}. This discrepancy likely arises from noise and, ultimately, the limited resolution of our measurements, resulting in significant uncertainties in the reconstructed $\rho_{\rm eff}$. These large uncertainties make the experimentally obtained bandgap value compatible with documented references when error margins are considered. The resolution limitation is inherent to our setup and stems directly from the VH spectrometer’s resolution (6.1 eV), which prevents reconstruction of features below this energy scale like the FeO bandgap. Additionally, the poor SNR of our spectra further degrades the reliability of the reconstructions and precludes effective deconvolution of the instrumental function from the RIXS spectra, which would amplify the noise, making the extracted $\rho_{\rm eff}$ even more unreliable. 

Nonetheless, the recovery of the expected trends and the compatibility of the experimental results with DFT calculations and literature references~\cite{greenberg2023phase}, including bandgap values, show great promises for these experimental techniques. Future experiments with higher-quality data are expected to yield more conclusive and precise results.






\subsubsection{\label{Calculations of the uncertainties} Progagation of the Uncertainties}


To compute the uncertainties on $\rho_{\rm eff}$ due to the errors $\eta$ on the RIXS spectrum $I$, one should process $\eta$ with the operator $P^{-1}_{\rm RIXS}$. Within the STEP framework, this task can simply be obtained by applying the extraction method to the two curves $I\pm \eta = I^{\pm}$. Once the two electronic structures associated with these slightly variated RIXS spectra have been computed - including an averaging over multiple fitting process to remove dependency on the random initialisation of NN - the error bounds on $\rho_{\rm eff}$ is given by these two curves, denoted respectively as $\rho^{+}_{\rm eff}$ and $\rho^{-}_{\rm eff}$. This method assumes that, at least in the region of the function space confined between $I^+$ and $I^-$, the operator $P^{-1}_{\rm RIXS}$ is convex, meaning that for every $I$ confined between $I^+$ and $I^-$, $\rho_{\rm eff}= P^{-1}_{\rm RIXS}(I)$ lies entirely between $\rho^{+}_{\rm eff}$ and $\rho^{-}_{\rm eff}$. This assumption was tested by performing extractions for various $I$ confined within this interval. As a final step, these uncertainties were then summed in quadrature with those coming from the random optimisation process.











\section{Summary}





This chapter discussed the first experiment to simultaneously investigate the evolution of the ionic and electronic structures of dynamically compressed FeO, using x-ray diffraction and resonant inelastic x-ray scattering, respectively. After a brief overview of the experimental setup with a focus on the VISAR system, used to measure pressures within the sample, the XRD data was presented. The analysis of this data revealed that up to $\approx140$ GPa, FeO does not undergo any crystalline phase transitions, but instead experiences only lattice compression. This compression, in turn, induces changes in the electronic structure, which were explored through the RIXS spectra, analysed using the STEP method. The analysis indicated a systematic bandgap closure consistent with theoretical predictions. However, the limited resolution of the reconstructed DOS, primarily due to the small number of RIXS spectra available, makes these results not ready for publication at this stage. Despite these limitations, the preliminary findings highlight the significant potential of these experimental techniques, with future experiments expected to provide more definitive and precise results.

This experiment also served as a proof of concept for employing RIXS as a time-resolved diagnostic tool for irreversible processes in HED science. This is particularly relevant for fusion research, where processes such as capsule implosions are prominent examples. Furthermore, the combination of XRD and RIXS techniques provides a powerful approach to examining the interplay between ionic and electronic structural changes in materials under extreme conditions.