



\chapter{Electronic Structures of Isochorically-Heated Systems}






This chapter demonstrates the application of the previously introduced method to experimental RIXS spectra collected from isochorically-heated samples of Fe, FeO, Fe$_2$O$_3$, and Gd. These spectra were acquired during an experiment conducted in November 2022 at the HED scientific instrument of the European XFEL facility~\cite{zastrau2021high}, where samples were both heated and probed using the same XFEL pulse (``self scattering'' experiment).

The chapter begins by reviewing the experimental setup and design, providing context for the measurements. In the second part, the experimental data alongside the extracted electronic structures and thermodynamic conditions for each material are presented. The analysis of the RIXS spectra is performed across different diagnostics and experimental operational modes to validate the robustness of the results. The analysis of gadolinium is discussed separately in the last part of the chapter, as it employs a slightly distinct flavour of RIXS. This case serves as a valuable example, generalizing the applicability of RIXS techniques in HED physics research.






\section{\label{Experimental setup}Experimental Setup}






\begin{figure}[t]
     \centering
     \includegraphics[width=0.8\textwidth]{Figures/euxfel setup.pdf}
     \caption[Setup of the EuXFEL experiment]{(a) A CAD model of the EuXFEL experimental setup, including a zoom around the FC spectrometer and the target holder (b), and an illustration of the VH spectrometer (c). To ensure precise positioning, a metal pin connects the front of the FC spectrometer to the target location, whereas the VH spectrometer position is adjusted using circular rails (see~\cite{zastrau2021high}). (d) A photograph of the experimental chamber.}
     \label{euxfel setup}
 \end{figure}

Figure \ref{euxfel setup} presents the setup of the experiments along with detailed illustrations of the primary diagnostic tools. These include a flat crystal spectrometer, a von H\'amos spectrometer, and a XRD detector\footnote{The XRD detector was calibrated using CeO$_2$ and LaB$_6$ diffraction profiles.}. The latter captures diffraction profiles for each shot, allowing to cross-check the consistency of the sample's chemical compositions and crystalline integrity. Only the shots that satisfied the consistency checks were retained. Both spectrometers are utilised to record the RIXS spectra, with their complementary characteristics deliberately leveraged at the expense of redundancy: the von H\'amos spectrometer offers a higher single-shot SNR but lower resolution, while the flat crystal spectrometer provides superior resolution at the cost of reduced signal intensity. Both spectrometers were filtered and caged to eliminate background signals while preserving the RIXS signal, which is within the energy range of 6–7 keV. The filters, made of black kapton, had a thickness of 300 $\mu$m, which was calculated using the center for x-ray optics (CRXO) software~\cite{CXRO}.

\begin{figure*}
     \centering
     \includegraphics[width=\textwidth]{Figures/IF and resolution.pdf}
     \caption[Experimental instrument function for the von H\'amos spectrometer]{(a) The Fe K$\alpha$ emission lines (red curve) are convolved with a kernel function to produce a fit (dotted blue line) of the von H\'amos experimental spectrum (black solid line). (b) The extracted kernel function, representing the von H\'amos instrument function. The energy axis centres are arbitrarily chosen, as they are not relevant for these calculations. Working on calibration shots, and hence above the absorption edge, increases the SNR in the experimental spectrum facilitating the fitting. The plot in panel (a) are provided courtesy of Dr. Thomas Gawne, who also performed the fitting.}
     \label{IF and resolution}
 \end{figure*}



The flat crystal spectrometer utilised a Ge (422) reflecting plane (operating in first-order reflection) paired with a ePix100 camera~\cite{epix100}, whereas the von H\'amos one employed a cylindrically bent HAPG crystal with an 80 mm RoC and a 40 $\mu$m thickness coupled to a Jungfrau detector~\cite{preston,jungfrau}. Both cameras digitise signals with 14-bit resolution, with the Jungfrau including two additional bits to encode the gain level. Analysis of spectra for single calibration shots shows that the von H\'amos spectrometer detects roughly 6 times more photons per image than the flat crystal one, albeit with a lower resolution—estimated at about 5.5 eV for the VH spectrometer and 2.3 eV for the FC spectrometer. The resolution of each instrument, comprising the crystal rocking curve, pixel size effects and source broadening, can be assessed by comparing the intrinsic widths of the Fe K$\alpha$ lines (see Table \ref{tab:params}) to their observed widths in the measured spectra, which for calibration shots (where the FEL photon energy is above the absorption edge) is not affected by the bandwidth of the source spectrum. A method for determining the spectrometer resolutions involves identifying the kernel function that, when convolved with the intrinsic Fe K$\alpha$ line profile, reproduces the experimental spectrum. These kernel functions represent the instrument functions of the spectrometers, and their widths provide estimates for the resolutions. Figure \ref{IF and resolution} illustrates the calculated instrument function for the VH spectrometer, along with the corresponding fit. Energy calibration for the two instruments was performed using the methods outlined in Chapter 2, with the Fe K$\alpha$ and Tb L$\alpha$ lines (6238 eV and 6273 eV) serving as reference lines. The detected energy ranges are (5773–6492) eV for the von H\'amos spectrometer and (6110–6458) eV for the flat crystal spectrometer.

% 6238, 6273 5947 eV (calibration lines)
\begin{figure*}[t]
     \centering
     \includegraphics[width=\textwidth]{Figures/FEL spectra.pdf}
     \caption[Example of FEL spectra]{Example of measured XFEL spectra for SASE (a) and seeded (b) mode.}
     \label{Fel spectra}
 \end{figure*}

The XFEL was operated both in SASE and seeded mode at a central photon energy of approximately 7060 eV (only changed for some analysis on gadolinium, see Section \ref{Electronic Structure of Warm-Dense Gadolinium}). This energy lies just below the Fe K-edge (refer to Table \ref{tab:params}) and the Gd L$_3$-edge (7243 eV), ensuring that RIXS is the dominant scattering process. Furthermore, for the iron compounds, the incoming photon energies are sufficiently close to the K$\beta$ (1s-3p) transition energy (7058 eV), enabling the probing of ionisation in the 3p states (M-shell). During the experiment, the iron 3p states can be depopulated due to the heating of the electrons via the intense x-ray irradiation. This suggests that measuring the M-shell RIXS signal, which is proportional to the number of vacancies in the shell, can provide an accurate estimate for the sample's temperature. An example of XFEL spectra for both modes is presented in Figure \ref{Fel spectra}, showing a spectral bandwidth of approximately 20 eV for the SASE mode and a few eV for the seeded mode. In order to apply the STEP method, it is essential to measure the source spectrum on a shot-to-shot basis and correlate it with the RIXS spectra. This was achieved using the Si beamline spectrometer with a resolution of 0.3 eV~\cite{kujala2020hard}. Analysis revealed that, in many shots, the seeded spectrum retained a SASE background (or pedestal), making the deconvolution of the source spectrum an essential task for this case as well. 
 
The total pulse energy in the beam ranged between 500-1000 $\mu$J in the SASE mode, but it was constrained to be between 700-800 $\mu$J for the analysed data in order to limit the sample thermodynamic variations. In constrast, no such restriction was needed for the seeded mode, where the pulse energy varied between 100 and 200 $\mu$J. The pulse energy delivered to the target was measured using diodes integrated into the beamline~\cite{zastrau2021high}.
With an average laser pulse duration of 40 fs and x-rays focused to spot sizes of 7–10 μm (measured using the knife-edge technique~\cite{de2009measurement}), intensities of up to 10$^{17}$ Wcm$^{-2}$ were achieved. The focal spot size was optimised \textit{in-situ} to maximise x-ray heating, diagnosed via the observed RIXS signal from the Fe M-shell. 

The samples employed in the experiment are summarised in the following table.

\bigskip

\begin{table}[h!]
    \renewcommand{\arraystretch}{1.5}
    \centering
    \begin{tabular}{|M{10mm}|M{18mm}|M{15mm}|M{20mm}|}
        \hline
        Material & Dimensions [cm x cm] & Thickness [$\mu$m] & Type\\\hline\hline
        Fe & 2.5 x 2.5 & 20 & Free-standing foil \\\hline
        FeO & 2.5 x 2.5 & 12 & Layer grown on 50 $\mu$m of plastic \\\hline
        Fe$_2$O$_3$ & 2.5 x 2.5 & 10 & Layer grown on 50 $\mu$m of plastic \\\hline
        Gd & 2.5 x 2.5 & 5 & Free-standing foil \\\hline 
    \end{tabular}
    \vspace{20pt}
    \caption{Specifics of samples used in the experiment. The iron oxides were fabricated on a plastic substrate using the physical vapour deposition (PVD) technique. This process introduces impurities into the samples, including other iron oxides of the form Fe$_x$O.}
    \label{tab:targets}
\end{table}

These target thicknesses correspond to roughly half the absorption length at the photon energies used, and they were chosen to maintain an x-ray heated bulk with a uniform temperature. Since the experiment operated at intensities above the damage threshold, samples were rastered, and data were collected at a repetition rate of 10 Hz.







\section{Experimental Results}







In the following sections, the experimental results will be presented, beginning with the extraction of the electronic structure for warm-dense Fe and Fe$_2$O$_3$ using von H\'amos spectroscopic data acquired in SASE mode. This initial analysis demonstrates how vacant density of states can be reconstructed from RIXS spectra with a resolution exceeding the bandwidth of the SASE spectrum, allowing us to discriminate between the valence structures of Fe and Fe$_2$O$_3$. Additionally, the extracted electronic structures enable temperature measurements and provide estimates of M-shell binding energies with smaller uncertainties than those reported in previous studies~\cite{humphries2020probing}.

I will then present the extracted density of states for Fe, obtained either from flat crystal data or using the seeded mode, and compare them with that obtained in the previous section. This comparison will check the consistency of the results and evaluate the quality of reconstruction from dataset with different characteristics, highlighting both the strengths and limitations of the STEP method when applied to experimental data.  

Subsequently, I will discuss the reconstructed density of states for FeO, achieved after having deconvolved the measured instrument function from the RIXS spectra. This analysis demonstrates the capability of achieving very high resolutions, enabling a more detailed exploration of the fine structure of the DOS near the chemical potential, such as bandgap features. Finally, after a brief introduction on the scientific and technological applications of gadolinium, its RIXS data is analysed, emphasizing the differences from iron-based compounds. To assess the reliability of the experimentally extracted electronic structures, all results are compared with FT-DFT simulations, details of which are provided in the next section.





\subsection{\label{sec: dft calculations}DFT Calculations}








The electronic structure of Fe, Fe$_2$O$_3$, FeO and Gd were calculated using finite-temperature density functional theory with a locally-modified version of the \textsc{Abinit} v8.10.3 code~\cite{gonze2016recent,torrent2008implementation}. The relevant modification is the inclusion of the hybrid Kohn-Sham plane-wave-approximation scheme~\cite{zhang2016extended} as described by Ref.~\cite{blanchet2022density} to enable accurate high temperature calculations (see also Introduction). The \textsc{Abinit} code is used to solve the Kohn-Sham system, which provides the KS states and eigenvalues. The DOS is given by the latter using Eq. (\ref{rho}), while the former can be used to calculate the dipole transition matrix elements from the initial electron core state (e.g. 1s for iron compounds) to the calculated valence states~\cite{mazevet2010calculations} (see also Sec. \ref{Electronic Structure of Materials}). The orbital-projected DOS was also calculated with standard techniques~\cite{mazevet2010calculations} to aid in identifying features in the total DOS.

The ion cores were represented using the projector-augmented wave (PAW) scheme~\cite{blochl1994projector, kresse1999ultrasoft}, with Fe, O and Gd PAW potentials generated using the \textit{Atompaw} code~\cite{atompaw}. For efficiency, the Fe $1s$, $2s$, $2p$ orbitals, the O $1s$ orbital, and the Gd orbitals up to the $5p$ states were treated with the frozen-core approximation. This approximation is suitable for the relatively low temperatures ($< 10$~eV) reached in this experiment, as these orbitals are not thermally ionised. Additionally, as mentioned in Section \ref{RIXS theory}, at these energy resolutions one can neglect the core hole interactions, and therefore ignore any core hole in the pseudopontential for the intermediate state of the system.

Calculations were performed at electron temperatures of 300~K, 1~eV, 5~eV, and 10~eV, with the ions frozen in their ambient crystal lattice positions. This further approximation is justified by the femtosecond duration of the XFEL pulses, during which all the spectral emission of interest occurs. This timescale is substantially shorter than the electron-phonon coupling times of several picoseconds, leaving the ion unperturbed during irradiation. The electron temperature range was chosen based on approximate predictions of the sample temperatures, derived from calculations of laser deposition energy.

The Fe calculations were therefore performed in a bcc primitive unit cell containing a single atom. Simulations were carried out with 120 bands and a $30 \times 30\times 30$ $k$-point grid, with a cut-off energy of 50~Ha for the PAW pseudo-wavefunctions and 150~Ha for the all-electron wavefunctions.
For Fe$_2$O$_3$, the calculations were performed in its $\alpha$-phase~\cite{sakurai2009first}, with a unit cell containing 30 atoms. Simulations were performed with 1440 bands and a $6 \times 6 \times 1$ $k$-point grid (the latter direction being the long direction of the unit cell), with a cut-off energy of 20~Ha for the PAW pseudo-wavefunctions and 100~Ha for the all-electron wavefunctions. 
The FeO calculations were performed in its B1 phase, with a unit cell containing 12 atoms. The simulations employed 700 bands and a $12 \times 12 \times 12$ $k$-point grid with a cutoff energy of 40 Ha for the PAW pseudo-wavefunctions, and 140 Ha for the all-electron wavefunctions.
Finally, the Gd calculations were carried out in its hexagonal close-packed $\alpha$-structure, with a unit cell containing 6 atoms, 1800 bands and a $12 \times 12 \times 12$ $k$-point grid, with a cut-off energy of 60~Ha for the PAW pseudo-wavefunctions and 180~Ha for the all-electron wavefunctions.

For the exchange-correlation functional, the PBE form of the generalised gradient approximation was used for the Fe, O and Gd atoms. 
To recover the band gaps in Fe$_2$O$_3$ and FeO, the corrective approach of a Hubbard potential (DFT+$U$) was applied, with $U$ values of 4.0 eV and 4.5 eV for the iron $d$ orbitals in Fe$_2$O$_3$ and FeO, respectively. These values yielded band gaps of approximately 2.0 eV for Fe$_2$O$_3$ and 2.4 eV for FeO, in agreement with experimental results reported in~\cite{xia2013tuning, bowen1975electrical}.
As small adjustments to the value of $U$ at the analysed temperatures did not show meaningful changes to the electronic structure of Fe$_2$O$_3$ and FeO, these $U$ values were chosen for all calculations.













\subsection{Electronic Structures of Warm-Dense Fe and Fe$_2$O$_3$}



\begin{figure*}[t]
     \centering
     \includegraphics[width=\textwidth]{Figures/experimental results1.pdf}
     \caption[Averaged experimental RIXS spectra]{(a) Normalised and averaged RIXS spectra for Fe and Fe$_2$O$_3$ at two different energy ranges on target, measured using the von H\'amos spectrometer and the FEL laser in SASE mode. Each spectrum averages approximately 3,000 shots. (b) Normalised and averaged RIXS spectra for Fe recorded using the various combinations of spectrometers (flat crystal and von H\'amos) and XFEL modes (SASE and seeded). In this case, each spectrum is averaged over approximately 6,000 shots per condition.}
     \label{experimental results1}
 \end{figure*}

%In this case, each spectrum is averaged only over approximately 500 shots per condition to highlight the difference in SNR between the two spectrometers

% Conversely, the flat crystal spectrometer exhibits a noticeable reduction in signal-to-noise ratio, collecting approximately 60 photons per shot on Fe$_2$O$_3$ in SASE mode, compared to the $\approx$350 photons collected by the von H\'amos spectrometer.


% Since the interesting and significant properties of iron oxides under extreme thermodynamic conditions have already been discussed in Section \ref{iron oxides}, here I proceed directly to show the experimental results. 

In Figure \ref{experimental results1} averaged RIXS spectra for various combinations of diagnostic instruments, targets, and photon source configurations are presented. The left panel displays RIXS spectra for Fe and Fe$_2$O$_3$ at two distinct energy ranges delivered to the target. These spectra were recorded using the von H\'amos spectrometer with the FEL laser operating in SASE mode. The differences in the electronic structures of the two materials, reflected in the different RIXS spectra, are evident, as are the subtle increases in the M-shell signal (the two peaks around 6400 eV) caused by the slightly higher laser energy, which heats more the material, further depopulating the M-shell. The small magnitude of such increases suggests a weak dependence of the resulting electronic temperature on the input energy, likely due to the behaviour of the electronic heat capacity in this region of the thermodynamic plane.

The right panel shows Fe RIXS spectra obtained using different combinations of spectrometers and XFEL modes. The broadening of features due to the lower resolution of the von H\'amos spectrometer and the SASE mode is clearly visible. Conversely, the flat crystal spectrometer exhibits a noticeable reduction in signal-to-noise ratio. As discussed earlier with synthetic data and elaborated further in the next section, this low SNR significantly hampers the STEP method, leading to a greater loss of resolution in the reconstructed DOS than the one caused by a poor resolution of the RIXS spectra. Additionally, noise features in the averaged spectra propagate into the extracted DOS, as the algorithm cannot differentiate between noise and actual physical features. For these reasons, except where explicitly stated otherwise, the following analyses for electronic structure extraction focuses exclusively on data obtained with the von H\'amos spectrometer and the FEL operating in SASE mode.



\begin{figure*}[t]
     \centering
     \includegraphics[width=\textwidth]{Figures/Reconstructed.pdf}
     \caption[Measured accessible density of states of Fe and Fe$_2$O$_3$, along with their experimental RIXS spectra]{Measured accessible density of states of Fe and Fe$_2$O$_3$ (a-b), reconstructed from the RIXS measurements, along with their respective experimental spectra (c-d). The grey bands indicate the standard deviation. The measured data are compared with theoretical DFT calculations, for which a smearing width of 7 eV for Fe$_2$O$_3$ and 9 eV for Fe (see Section \ref{Electronic Structure of Materials}) were used. The zero of the energy axis is defined by the chemical potential obtained from DFT simulations, and this convention is maintained throughout the remainder of the thesis. The insets show the reconstruction of the M-shell vacant states, located around (-50,-60)~eV. This feature is only present if the M-shell is thermally ionised by the x-ray beam, thus allowing us to estimate the temperature of the sample. The red dashed vertical line indicates a reference M-shell binding energy, taken from~\cite{lemmon2010nist}.}
     \label{Fig:reconstructed}
\end{figure*}




Examples of single-shot experimental spectra are shown in Fig.~\ref{Fig:reconstructed}, alongside with the corresponding extracted electronic structure. It is observable that the typical level of noise for a single-shot RIXS spectrum is on the order of 25\%, comparable to the cases examined for the synthetic data. The experimental reconstructions of $\rho_{\rm eff}$ are compared with DFT calculations (see Section \ref{sec: dft calculations} for more details), which allow us to identify various features in the experimental data, and to evaluate the resolution with which $\rho_{\rm eff}(\epsilon)$ can be extracted in practice, given our experimental setup. In all comparisons with DFT data, a translational and a multiplicative parameter were used to, respectively, align and rescale the experimental reconstructions with the DFT outputs. The experimental $\rho_{\rm eff}$ have been reconstructed using the dynamic kernel deconvolution scheme, as described in Section \ref{STEP for the RIXS Process}, over approximately 6,000 RIXS shots for each material. Six independent fitting processes were conducted, for both materials, with different random seeds for the NN. The average and standard deviation of the resulting outputs have been taken, respectively, as the best estimate for $\rho_{\rm eff}$ and its error. The error on the experimental RIXS spectra due to noise, which is propagated through to $\rho_{\rm eff}$, becomes negligible because of the large amount of experimental shots. Hence, the error is due only to the stochasticity of the NN optimisation.




\subsubsection{Estimation of the Resolution}



\begin{figure}[t]
     \centering
     \includegraphics[width=\textwidth]{Figures/resolution_studies_methods2.pdf}
     \caption[Methods for the estimate of the extracted DOS resolution]{Two different approaches for the estimate of the extracted DOS resolution. In panel (a) this is carried out by fitting a superposition of multiple Gaussians to the reconstructed DOS, whereas in panel (b) the L$_2$ distance between the experimental reconstruction and the DFT simulations as a function of their smearing width is considered. The error intervals on the reconstructed DOS in panel (a) are identical to those of Fig. \ref{Fig:reconstructed}.}
     \label{Fig:resolution studies_methods}
 \end{figure}


The experimental reconstructions in Fig.~\ref{Fig:reconstructed} agree well with the theoretical predictions, demonstrating the ability to extract densities of states with a fairly complex structure from RIXS spectra with relatively low SNR. The resolution of the extracted $\rho_{\rm eff}$ was estimated with two different methods, shown in Fig.~\ref{Fig:resolution studies_methods}. In panel (a) the reconstructed Fe$_2$O$_3$ $\rho_{\rm eff}$ was fitted with a superposition of multiple Gaussians. This fitting is then used to achieve an estimate of the narrowest feature width that can be reconstructed, and therefore, of the resolution. This width is found to be 6.7 eV. Another method to evaluate the resolution is to consider the L$_2$ distance between the experimental $\rho_{\rm eff}$ and the DFT predictions, as a function of the smearing width applied to the latter. This curve is plotted in Fig.~\ref{Fig:resolution studies_methods}b. The minimum of this curve is found for a smearing width around 8.2 eV, yielding another estimate for the resolution of the extracted $\rho_{\rm eff}$. The small discrepancy between the two methods can be traced back to the different resolutions with which different parts of the DOS seem to be reconstructed (e.g. the continuum slope and the small valence band peak).
This resolution is somewhat lower than the limit of 5.5 eV imposed by the spectrometer resolution in our experimental setup. This small difference is attributed primarily to the SNR of the experimental data, which was fairly low, as the data collection was done in single-shot mode. Nevertheless, the reconstructed electronic structures show resolutions up to three times higher than the spectral FWHM of the XFEL pulse (see Fig. \ref{Fel spectra}), allowing one to distinguish the environment in which a Fe atom is embedded, via differences in the electronic structure of Fe in the insulating Fe$_2$O$_3$ or in the metallic state. Such a distinction cannot be made via RIXS with the normal spectral resolution of the SASE beam of $\sim$20~eV, as it is illustrated in Fig.~\ref{Fig:resolution studies_comparison}, where applying a smearing width of 20 eV to the DFT calculations for Fe$_2$O$_3$ would yield a $\rho_{\rm eff}$ similar to that of metallic Fe.



\begin{figure*}[t]
     \centering
     \includegraphics[width=0.8\textwidth]{Figures/resolution_studies_comparison.pdf}
     \caption[Comparison between the reconstructed $\rho_{\rm eff}$ for Fe$_2$O$_3$ and DFT simulations]{Comparison between the reconstructed $\rho_{\rm eff}$ for Fe$_2$O$_3$ (solid black lines) and the DFT simulations (dashed blue lines) with smearing widths of 0.2 (a), 5 (b), 10 (c), and 20 eV (d) respectively. It is observable that with a smearing of 20 eV, as it would be given by the SASE bandwidth, the characteristic features of Fe$_2$O$_3$ are no longer visible.}
     \label{Fig:resolution studies_comparison}
 \end{figure*}












\subsubsection{Characteristic of the Measured Electronic Structures}






% Maybe plot the orbital-projected DOS




The comparison of the electronic structure with the orbital-projected DOS reveals that the pre-edge feature in Fe$_2$O$_3$, located around -10 eV, contains a mixture of the $2p$ oxygen states and the $3d$ iron states, which are indistinguishable at these resolutions. The band gap between these states and the conduction ones, predicted by experimental measurements~\cite{mohanty1992studies,westre1997multiplet} to have a width of approximately 2 eV, is visible in the reconstruction as a slightly convex plateau located at just below 0 eV. 

In addition to the modulation in the continuum and the pre-edge feature in the Fe$_2$O$_3$ case, this method can reconstruct the M-shell for both materials, as shown in the insets of Figs.~\ref{Fig:reconstructed}a-b. Despite the small area of these features due to the low thermal de-occupation, they are amplified in the RIXS spectra by their vicinity to the resonant energies. Thus, the reconstructed $\rho_{\rm eff}$ gives us a method to experimentally estimate the binding energy (BE) of the M-shell in extreme thermodynamic conditions—a quantity difficult to obtain via theoretical approaches~\cite{stewart1966lowering, ecker1963lowering}. The values found from the reconstructions are $(-56\pm 3)$~eV for Fe and $(-59\pm 5)$~eV for Fe$_2$O$_3$. These values and their uncertainties are determined using Monte Carlo calculations. Specifically, M arrays of the form $y(\epsilon_1), y(\epsilon_2),...,y(\epsilon_N)$ are generated, where each $y(\epsilon_i)$ is sampled from a Gaussian distribution with mean value $\tilde{\rho}_{\rm eff}(\epsilon_i)$ and standard deviation equal to the uncertainty on $\tilde{\rho}_{\rm eff}(\epsilon_i)$. Here, $\epsilon_1, \epsilon_2, ..., \epsilon_N$ are energy points around the M-shell. Then for each of these M array the argmax is computed and their average and standard deviation are then used as the best estimate of the M-shell binding energy and its associated uncertainty, respectively. As finite-temperature effects, such as ionisation potential depression, have a negligible impact on the BE for the temperatures reached in the experiment, these estimates were compared with the ambient M-shell binding energies documented in the NIST database~\cite{lemmon2010nist}. The results were found to be compatible with these reference values.

Similarly to what has been done in Ref.~\cite{humphries2020probing}, the M-shell feature can be exploited to estimate the sample temperature averaged over the beam pulse duration. However, contrary to Ref.~\cite{humphries2020probing}, where the sample temperature is found by fitting the theoretical RIXS spectra to the experimental ones, here the DFT-computed density of accessible states ($\rho|M|^2$) can be directly fitted to the reconstructed $\tilde{\rho}_{\rm eff}$:

\begin{equation}
[1-f_{FD}(\epsilon;T)]\rho_{\small{{\rm DFT}}}(\epsilon)|M_{{\rm DFT}}(\epsilon)|^2 \stackrel{!}{=} \tilde{\rho}_{\rm eff}(\epsilon) \ ,
\label{temperature fit}
\end{equation}

\noindent where the chemical potential must be computed self-consistently. This new procedure enables us to reduce the error bars on the temperatures estimates compared to the previous work, yielding temperatures of $(6.1\pm 0.2)$~eV for Fe and of $(5.2\pm 0.2)$~eV for Fe$_2$O$_3$. These estimates justify the implicit assumption that depopulation of the M-shell is given mainly by thermal collisions (and not by photoionisation), and therefore can be described by the factor $(1-f_{FD})$. The fitting process, based on Eq. (\ref{temperature fit}), is performed using the curve-fitting function from Python's SciPy library~\cite{curvefit}, which also provides uncertainty estimates for the temperatures by propagating the uncertainties in $\tilde{\rho}_{\rm eff}$. As observed in Ref.~\cite{humphries2020probing}, these extracted temperatures provide a good estimate of the peak temperature reached in the target during the irradiation, as this dominates the photoemission. This observation is essentially due to the exponential dependence of the accessible/vacant states on the sample temperature.








\subsubsection{Comparison with Flat Crystal and Seeded Mode Data}







\begin{figure*}[t]
     \centering
     \includegraphics[width=0.8\textwidth]{Figures/comparison different modes.pdf}
     \caption[Comparison of extractions carried out with different spectrometer and XFEL configurations]{A comparison of the iron $\rho_{\rm eff}$ extracted from data obtained using the von H\'amos spectrometer with the XFEL operating in seeded mode (orange) and SASE mode (blue), as well as from the flat crystal spectrometer with the XFEL in SASE mode (green). The shaded bands indicate the standard deviation, as seen for Fig. \ref{Fig:reconstructed}.}
     \label{comparison different modes}
 \end{figure*}

To validate the consistency of the results obtained in the previous section, the DOS for iron was extracted from data acquired using the flat crystal spectrometer with the XFEL in seeded mode, as well as the von H\'amos spectrometer with the XFEL operating in both seeded and SASE modes. The reconstructed Fe $\rho_{\rm eff}$ for these three experimental configurations are shown in Figure \ref{comparison different modes}. 

The extractions and their uncertainties were calculated following the same procedure described for Figure \ref{Fig:reconstructed}, using six independent fitting processes for each configuration. For the FC data, the $\rho_{\rm eff}$ was smoothed over a 2 eV energy scale, corresponding to the typical noise oscillation width in the averaged RIXS spectra of the flat crystal spectrometer (see Figure \ref{experimental results1}).

As observed from the uncertainty bounds, the three curves are consistent and align well with the behaviour of $\rho_{\rm eff}$ predicted by DFT calculations. Notably, the orange curve exhibits the smallest uncertainty bounds and is the only curve that remains within the error bounds of the other two. This observation supports our decision to use RIXS spectra recorded with the VH spectrometer and the XFEL in SASE mode for the following analysis.




 % Justify discrepancies???










\subsection{Electronic Structure of Warm-Dense FeO with ``Ultra-High'' Resolution}




\begin{figure*}[t]
     \centering
     \includegraphics[width=\textwidth]{Figures/FeO_v2.pdf}
     \caption[Electronic structure of FeO]{(a) Reconstructed $\rho_{\rm eff}$ for FeO, obtained after deconvolving the spectrometer instrument function (see Fig.\ref{IF and resolution}) from the RIXS spectra. The reconstructed data is compared with theoretical DFT calculations, for which a smearing width of 3 eV was used, and with XANES measurements reported in~\cite{wilke2001oxidation}. Uncertainty bounds are calculated and presented following the same methodology as in previous figures. (b) Orbital-projected and total DOS of FeO calculated from DFT simulations. Note that the energy range used in the calculations extends up to approximately 10 eV for the projected DOS and 40 eV for the total DOS.}
     \label{FeO extraction}
 \end{figure*}

% Maybe an example of deconvolved spectrum

% (b) The averaged experimental RIXS spectrum for FeO, shown alongside the theoretical spectrum associated with the extracted ρeff . 

% Similar level of accuracy for the fitting of the averaged RIXS spectrum is obtained for all the other analysed cases.


Fig.\ref{FeO extraction}a shows the extracted $\rho_{\rm eff}$ for FeO, obtained by fitting approximately 6,000 RIXS spectra after deconvolving them from the measured von H\'amos instrument function. Since individual RIXS single-shot spectra are too noisy for reliable deconvolution, the spectra were grouped into averaged sets of 20 shots each to improve the signal-to-noise ratio. The deconvolution was performed using Python’s built-in Richardson-Lucy algorithm on this averaged dataset, containing now about 300 spectra. Then, the reconstruction of $\rho_{\rm eff}$ and its uncertainties was carried out following the same procedure described for Figures \ref{Fig:reconstructed}-\ref{comparison different modes} on this ``deconvolved'' dataset. 


% The mean of the deconvolved RIXS spectra is shown in Fig.\ref{FeO extraction}b, along with the theoretical spectrum associated with the extracted $\rho_{\rm eff}$. 

The extracted $\rho_{\rm eff}$ (red curve) is compared with DFT calculations (detailed in Sec.\ref{sec: dft calculations}) and with XANES measurements from~\cite{wilke2001oxidation}. The comparison with XANES effectively demonstrates how the RIXS-based methodology achieves a higher-resolution absorption spectrum than conventional XANES, as anticipated in the theoretical discussion (Sec.~\ref{Alternative Spectroscopic Techniques}). Additionally, the comparison with the DFT-computed $\rho_{\rm eff}$ reveals good agreement with the experimentally-derived electronic structure, providing also estimates for both the sample temperature and the resolution of the reconstructed $\rho_{\rm eff}$. These values were obtained using the methods outlined for Fe and Fe$_2$O$_3$, yielding a temperature of 5.5$\pm$0.3 eV and a resolution of approximately 3 eV - narrower than the spectrometer's instrument function (5.5 eV). This improved resolution highlights the effectiveness of deconvolving the instrument function from the RIXS spectra, surpassing the spectrometer’s intrinsic resolution limits and enabling finer details in the valence bands of warm-dense FeO to be resolved.

Notably, around the chemical potential (set to 0 in Fig.\ref{FeO extraction}), the enhanced resolution reveals the characteristic bandgap of FeO, approximately 2.4 eV~\cite{bowen1975electrical}. While this bandgap is not fully resolved, it manifests as a distinct dip near 0 eV, underscoring the ability of this approach to identify fine structural features that govern the macroscopic properties of the material, such as its conductive or insulating behaviour. Applying this methodology to compressed FeO could potentially enable a direct experimental observation of a Mott transition in compressed iron oxide, which would be a significant result. The analysis of data for compressed FeO is the focus of the next chapter.

Fig.\ref{FeO extraction}b reports the DFT-computed projected and total DOS for FeO. The orbital-projected DOS reveals that the bandgap primarily arises from the energy splitting of the iron 3d orbitals, whose delocalisation is also responsible for the closure of the bandgap under high-pressure conditions. Additionally, the plot shows that the pre-edge peak, appearing approximately 20 eV below the chemical potential, correspond to the oxygen 2p states. The clear differences between the total DOS and the theoretical $\rho_{\rm eff}$, plotted in panel (a) in blue, are due to the modulations introduced by the transition matrix elements $|M|^2$ and the thermal factor.



\subsection{\label{Electronic Structure of Warm-Dense Gadolinium}Electronic Structure of Warm-Dense Gadolinium}





Gadolinium is a ductile and reactive rare earth element, crystallised in a hexagonal close-packed structure at room temperature. With an atomic number of 64 and an electronic configuration of [Xe] 4f$^7$ 5d$^1$ 6s$^2$ gadolinium presents a partially filled f-orbital that contributes to interesting behaviour in various physical and chemical environments. While gadolinium primarily exhibits an oxidation state of +3 in its compounds—utilizing its 5d and 6s electrons for bonding—the seven unpaired 4f electrons give rise to remarkable magnetic characteristics~\cite{dan1998magnetic}. Above 20~\(^\circ\text{C}\), gadolinium is a strong paramagnet, a property that underpins its widespread use in magnetic applications, including magnetic resonance imaging contrast agents~\cite{caravan1999gadolinium} and magnetic refrigeration systems~\cite{tuvsek2013comprehensive}.

Beyond its magnetic properties, gadolinium is notable for its exceptionally high neutron capture cross-section~\cite{leinweber2006neutron}, which suggests potential applications in nuclear technology as a neutron-capturing material, though it has not yet been utilised for this purpose. For this experiment, gadolinium was selected primarily because its absorption edges align closely with those of iron~\cite{menesguen2019precise}, providing a convenient additional benchmark for validating the RIXS-based methodology. However, the potential application of gadolinium in nuclear-related applications adds interest to studying its behaviour under extreme thermodynamic conditions, beyond the realm of fundamental material science.






\subsubsection{RIXS in Gadolinium}



\begin{figure*}[t]
     \centering
     \includegraphics[width=\textwidth]{Figures/Gd RIXS scheme.pdf}
     \caption[RIXS scheme for Gadolinium]{Scheme of the two RIXS channels utilised for gadolinium in the experiment, accompanied by a list of resonant energies involved in the process, as reported in~\cite{menesguen2019precise}.}
     \label{Gd RIXS scheme}
 \end{figure*}


For RIXS in gadolinium, two distinct channels were used in the experiment, illustrated schematically in Figure \ref{Gd RIXS scheme}. In the the first channel (left diagram), the central XFEL photon energy is tuned just below the Gd L$_3$ absoprtion edge (7243 eV) to induce a RIXS process analogous to that employed in iron compounds. The only difference lies in the electronic transitions involved: the excited electron comes from an L$_3$ state, and the resulting hole resides in the M-shell. For this channel, two incoming photon energies were selected: the same value used for iron (7050 eV) and 7150 eV.

The second channel is identical to the first, except that the excited electron originates from the L$_2$ shell instead. Consequently, the XFEL energy was set slightly below the L$_2$ absorption edge (7930 eV), specifically to 7850 eV and 7900 eV. These four sets of RIXS spectra, corresponding to the different incident photon energies, can be jointly analysed within the STEP framework, effectively enhancing the performances of the deconvolution process.

Equations (\ref{finite_FWHM}) and (\ref{ML 1}), used to model the RIXS process, are still valid but incorporate different physical parameters (e.g., $A_f$) specific to gadolinium, as well as modifications to account for the new electronic transitions involved in the scattering. In particular, in this case, there are four possible final states $f$, each corresponding to an L$\alpha$ line (see Fig.\ref{Gd RIXS scheme}). The gadolinium-specific parameters appearing in the RIXS cross-section were taken from Ref.~\cite{menesguen2019precise}.

Figure \ref{Gd RIXS spectra} displays the averaged RIXS spectra for the two lowest central photon energies. The plot highlights how the RIXS features, reflecting the electronic structure, shift along the Lorentzian profiles centred at the resonance lines (in purple in the plot) by an amount corresponding to the change in the incident photon energy. To align the energy window detected by the von H\'amos spectrometer with the energies of the outgoing RIXS photons in gadolinium, the spectrometer's position was adjusted using a motorised stage and the calibration was repeated.

The RIXS process in gadolinium exemplifies the versatility of this scattering technique, demonstrating its applicability to various initial, intermediate, and final states, provided that specific conditions are satisfied.




\begin{figure*}[t]
     \centering
     \includegraphics[width=0.8\textwidth]{Figures/Gd RIXS spectra.pdf}
     \caption[Averaged RIXS spectra of gadolinium]{Averaged RIXS spectra of gadolinium for two central photon energies on target. The plot also shows the relevant resonant lines, including those coming from chromium and iron impurities. Chromium was used for the calibration of the VH spectrometer during the experiment. The iron K$\alpha$ lines have been cut in this plot as part of the image processing steps.}
     \label{Gd RIXS spectra}
\end{figure*}







 
\subsubsection{Experimental Results}







\begin{figure*}[t]
     \centering
     \includegraphics[width=\textwidth]{Figures/Gadolinium.pdf}
     \caption[Electronic structure of Gadolinium]{(a) $\rho_{\rm eff}$ for gadolinium extracted from the RIXS spectra. The reconstructed electronic structure is compared with theoretical DFT calculations, for which a smearing width of 6 eV and a temperature of 4.2 eV were used. Uncertainty bounds are calculated and presented following the same methodology as in previous figures.
     (b) The averaged experimental RIXS spectrum for gadolinium, shown alongside the reconstructed spectrum associated with the extracted $\rho_{\rm eff}$. Only the interval of the spectrum relevant for the fitting is illustrated.}
     \label{Gadolinium}
\end{figure*}



In Fig. \ref{Gadolinium}a, the $\rho_{\rm eff}$ of gadolinium reconstructed from the RIXS spectra is shown. The calculation of $\rho_{\rm eff}$ and its uncertainities were performed with the same methodology used for the other materials, combining Gd RIXS spectra obtained with the different incident photon energies, totaling approximately 4,000 shots. The averaged RIXS spectrum for an incoming photon energy of 7050 eV is shown in Fig.\ref{Gadolinium}b, alongside the reconstructed spectrum corresponding to the extracted $\rho_{\rm eff}$. This plot is representative of the level of fitting accuracy attained on the RIXS spectrum during the extraction process, as it is consistent with the levels achieved across the various materials and conditions.

The extracted $\rho_{\rm eff}$ was compared to DFT calculations, demonstrating strong agreement and confirming the method's capability to reconstruct electronic structures with complex profiles. The sample temperature and the resolution of the extracted $\rho_{\rm eff}$ were estimated using the same methods as for the other materials, yielding a temperature of 4.2$\pm$0.6 eV  and a resolution of approximately 6 eV. The slightly larger uncertainty in the temperature is attributed to the reduced separation between the bound and valence states in gadolinium compared to previous cases, which diminishes the sensitivity of the RIXS signal from these states to temperature variations. In Gd, the bound states are located around -30 eV and correspond to the 5p$_{1/2}$ and 5p$_{3/2}$ orbitals, as indicated by orbital-projected DOS and tabulated data~\cite{henke1982low}.





% Differences from ambient Gadolinium (for another thesis!!!)







\section{Summary}





This chapter presented the analysis of experimental RIXS spectra for warm-dense Fe, FeO, Fe$_2$O$_3$, and Gd. The extracted electronic structures show good agreement with DFT calculations and reveal distinct patterns across the different materials. These distinctions are also made possible by the enhanced resolution provided by the STEP extraction scheme, which overcomes the limitations imposed by the source bandwidth. Ultimately, by deconvolving the instrument function from the RIXS spectra, resolutions of a few eV—exceeding even the spectrometer's intrinsic resolution—were achieved. These high resolutions make it possible to detect fine features near the chemical potential, such as bandgaps, which play a crucial role in determining a material's behaviour. Alongside the valence structure, the reconstructed DOS also include bound states, enabling relatively accurate estimations of sample temperatures and binding energies. Beyond the range of physical quantities accessible with this RIXS-based technique, its successful application to diverse materials underscores the flexibility of this method, which can exploit various electronic excitations.












% Some experimental single shot, with number of photons










