\chapter{Conclusions and Future Perspectives}






\section{Summary}




The main effort of this thesis was to establish a method for extracting high-resolution electronic structures of materials in extreme environments, where the inherent instability and non-reproducibility of thermodynamic conditions pose significant limits to the amount of data that can be collected. This method incorporates both the experimental technique to acquire the data and the methodology for its analysis. The experimental technique is based on the resonant inelastic x-ray scattering process. As shown in Chapter 3, the spectrum of photons scattered by the material encodes rich information about its electronic structure, though in a complex manner. Extracting electronic structures from RIXS spectra involves inverting a complex and ill-conditioned operator—a challenge further compounded by the relatively low SNR of the spectra. To address this, a novel methodology was devised, combining machine learning with differentiable physical modelling. This approach, detailed in Chapter 3, enables the extraction of high-resolution DOS by deconvolving the source spectrum, which must be measured on a shot-to-shot basis and correlated with the corresponding RIXS spectra. This framework, introduced as a generalised method for extracting physical quantities embedded in experimental measurements described by complex but known physical operators, was named STEP (Surrogate Training Embedded in Physics). For the RIXS case, besides the reconstruction of DOS, this methodology also allows for the estimation of scalar quantities like the sample temperature. 

The extraction of density of states and thermodynamic conditions with this method was first tested on synthetic RIXS data. These tests demonstrated good performance, particularly in terms of robustness to noise, when compared to more traditional numerical approaches. Subsequently, the RIXS-based methodology was applied to the analysis of experimental RIXS spectra, as presented in the final two chapters. Chapter 4 detailed the analysis of RIXS spectra collected at the EuXFEL facility from isochorically-heated samples of Fe, FeO, Fe$_2$O$_3$, and Gd. This highlights the flexibility of the method, which is capable of studying a wide range of materials through different electronic excitations. The extracted electronic structures showed good agreement with DFT calculations, revealing distinct patterns across the various materials. These distinctions were made possible by the enhanced resolution achieved by the methodology, which overcomes the limitations imposed by the source bandwidth and, ultimately, the spectrometer’s instrument function, reaching resolutions down to a few eV. Such resolutions enabled the detection of fine features near the chemical potential, including bandgaps, which are critical for determining a material’s behaviour. In addition to valence structures, the reconstructed DOS also included bound states, facilitating relatively accurate estimations of sample temperatures and binding energies.

Chapter 5 presented the analysis of RIXS spectra of compressed and ambient FeO, collected at the LCLS facility during the first experiment to simultaneously study the ionic and electronic structure evolution of dynamically compressed samples using XRD and RIXS. This combination of techniques offers a powerful approach for examining the interplay between ionic and electronic changes under extreme conditions and demonstrates the potential of RIXS as a time-resolved diagnostic tool for irreversible processes in HED science. XRD analysis revealed that, consistent with theoretical predictions~\cite{greenberg2023phase}, FeO undergoes lattice compression without crystalline phase transitions up to $\approx$ 140 GPa. However, this compression drives changes in the electronic structure, evidenced by the extracted DOS from the RIXS, which revealed a systematic bandgap closure. While this behaviour aligns with existing literature~\cite{greenberg2023phase}, the limited resolution of the reconstructed DOS, due to the small dataset, prevents precise and conclusive results. In spite of this, future experiments, leveraging technological advancements, are expected to provide higher-quality data and more definitive insights (see also Section \ref{outlook 2}).

As a complementary subject, the second chapter of this thesis detailed methods for constructing spectra from camera images while enhancing their SNR and possibly their resolution. These techniques, while broadly applicable, are particularly valuable for HED science and our investigations, where low signal levels often characterise the data.






\section{Outlook: Methods for Improving the SNR and Resolution of Camera Images}



A first straightforward improvement to the routine introduced in Chapter 2 for constructing spectra from camera images is the inclusion of  instrument function deconvolution, as described in Section \ref{sec:Resolution}. The IF can either be measured experimentally (see Section \ref{sec:Resolution}) or approximated using theoretical models. This deconvolution would mitigate the resolution losses due to source broadening and finite width of the crystal rocking curve, leaving only pixel-size-related effects, which can be partially addressed using the RL deconvolution, especially in regions with high photon densities.

Another enhancement, which would address pixel-size limitations, could involve incorporating centroid calculations for single-photon clusters, as described in Ref.~\cite{abboud2013sub}, into the photon-counting algorithm. This approach would enable subpixel resolution in regions with low fill fraction $\lambda$, complementing the RL deconvolution. Detectors with small pixel sizes should be chosen to fully exploit these techniques, which could be useful also to refine the energy map calibration, as discussed in Section \ref{flat crystal spectrometer}. 

Within the photon-counting workflow, another avenue for improvement is the implementation of multiphoton cluster identification algorithms. These algorithms could enhance the accuracy of the analysis in regions with intermediate values of $\lambda$ by expanding the set of valid cluster shapes to include those produced by multiple photons or by incorporating more advanced clustering techniques~\cite{xu2005survey}.

Finally, the photon counting algorithms could benefit from optimisation to improve computational efficiency. Currently, the photon counting takes approximately 8 minutes to analyse a 2048×2048 image, which is quite time-consuming, even for post-analysis. Reducing the runtime of these algorithms would significantly enhance their practical utility. One potential approach is to parallelise the analysis by distributing different regions of the image across multiple processors.












\section{\label{outlook 2}Outlook: Improvements and New Applications of the RIXS-based Methodology}


% EXAFS oscillations as well


A natural refinement of the method for extracting the density of states, already mentioned in Chapter 3, is incorporating the measured (or calculated) instrument function into the forward model, similar to the current treatment of the source spectrum. This improvement would enable the deconvolution of the instrument function, further improving the resolution of the reconstructed electronic structures. Additionally, another possible refinement of the forward model would be the gradual relaxation of the approximations introduced in Section \ref{RIXS theory}, particularly concerning the influence of core-hole interactions on the intermediate state. This improvement would lead to a more accurate RIXS model, better capturing the finer energy scales of the spectra.

From a machine learning perspective, exploring alternative network architectures for representing $\rho_{\rm eff}$ would be valuable, especially those that can incorporate physical constraints on the DOS (e.g., positivity). This approach aligns with what is usually done in PINNs methods~\cite{raissi2017physics}.

On the experimental side, I have already emphasised the need for higher-quality RIXS measurements under compressed conditions, which could be obtained by increasing the SNR of single shots and expanding the size of datasets. Leveraging the advanced capabilities of newer facilities—particularly improvements in XFEL pulse energy optical laser energy stability, and repetition rate—would be instrumental in achieving these goals. Our experiment was conducted at LCLS, but facilities such as the European XFEL, with its DIPOLE100X laser~\cite{mason2018development}, offer significantly higher capabilities—delivering up to 100 J per pulse at a maximum repetition rate of 10 Hz, far exceeding the performance of the LCLS optical laser\footnote{Actually, RIXS investigations of dynamically compressed materials were already carried out recently, during a community experiment in August 2024.}. Looking further ahead, the upcoming LCLS-II facility, expected to be operational for experiments in 2027, is anticipated to provide even greater enhancements.

In terms of scientific applications, it would be interesting to test this RIXS-based methodology on new materials relevant to high-pressure physics, such as other iron compounds, nickel, and silicates\footnote{The selected materials must be compatible with our specific RIXS approach, meaning they should contain deep core states that can be excited by x-rays, thereby excluding light elements.}, as well as under different conditions, such as strong magnetic fields. These studies would help characterise the modifications of electronic structures due to the chemical and physical environment.

Finally, the discussion in Section \ref{Electronic Structure of Materials}, along with equations (\ref{finite_FWHM}) and (\ref{rixs cross-section 2}), suggests the intriguing possibility of using RIXS measurements to probe optical conductivities. This objective would require repeated RIXS measurements for different absorption edges of the investigated material.




\section{Outlook: Generalisation of the STEP Method to Other Physical Processes}







Based on discussions with collaborators, where I had the opportunity to present the framework of the STEP method, one of the most promising prospects, in my view, is its generalisation and application to other  physical processes/measurements~\cite{alaa2024step}. Many researchers encounter challenges that could be effectively tackled using our approach. Here, I illustrate some example applications to x-ray diffraction analysis, though others could be made. 

As described in Section \ref{Laser-Compressed Systems}, uniaxial compression generates a shock wave that propagates through the material and reflects off the rear surface, producing a release wave that gradually broadens over time. Consequently, at any given moment, the sample exhibits a longitudinal density profile which corresponds to a variation in lattice parameters. This density profile is therefore encoded in the observed diffraction image, and this relationship can be modelled through a differentiable forward model embedded within the STEP framework~\cite{heighway2020shock}. 

In this approach, the density profile is represented by the neural network, which is iteratively adjusted to ensure that the forward model's output aligns with the experimental XRD images. By performing these measurements at different time steps, the density evolution within the sample can be reconstructed. The resulting density profiles represent averaged values across the transverse planes. However, lattice parameters may vary significantly within these planes. Future applications aim to map the entire strain field, encompassing both longitudinal and transverse directions, as it is also encoded in the diffraction image. This approach would demonstrate the reconstruction of a two-dimensional function\footnote{Certain symmetries in the sample can be exploited.} using the STEP method. Such a method would surpass the commonly used—but occasionally inadequate—\textit{full-constraints model}~\cite{manik2014review}, which assumes a homogeneous strain field. Additionally, another valuable two-dimensional quantity that can be extracted from diffraction images is the orientation distribution function of crystallites, $P(\theta,\phi)$, which characterises the texture of the sample.

Through these examples within the XRD field alone, I hope to have conveyed the potential of the STEP framework in addressing inverse problems—where complex physical quantities must be extracted from experimental measurements.










