



\chapter{\label{Introduction}Introduction}




\section{Quantum Plasmas}




 

%~\cite{saumon1992fluid}




The physical systems studied in this thesis are high-energy density (HED) plasmas, which are ensembles of particles (e.g. atoms, electrons, ions) characterised by high temperatures, ranging from approximately 1 eV to several hundreds eV (1 eV $\approx$ 11600 K), and high densities, going from typical solid state values ($\approx$ 1 g/cm$^3$) up to densities of thousands of g/cm$^3$. In these temperature conditions, atoms are ionised with respect to their ambient condition state, ultimately forming a plasma — an ensemble of electrically charged particles interacting via electromagnetic forces\footnote{There is no sharp phase transition to a plasma state, as the degree of ionisation in a system increases gradually with temperature, and in certain regimes, also with density.}. On the other hand, the high densities strengthen the interactions among particles and amplify quantum effects, making their behaviour resemble that of solid-state systems.  Hence, these systems are often referred to as ``quantum plasmas''. Considering the thermodynamic relationships between pressure, energy, density, and temperature, HED systems are fundamentally characterised by extreme pressures (exceeding 1 Mbar) or, equivalently, energy densities surpassing 10$^{11}$ J/m$^{3}$. In this context, compressed solids are included within the category of quantum plasmas. This consideration is also justified by the fact that, at such pressures and energy densities—comparable to the binding energies of valence electrons—ions in the solid undergo ionisation. Though these states of matter may seem extreme, they are common throughout the universe\footnote{Actually, the plasma state, whether high-density or not, is the most present in the universe, as stars are made of it.}, appearing in the interiors of the stars and giant gaseous planets~\cite{bailey2015higher, guillot1999interiors, kritcher2020measurement}. Thanks to the development of technology, nowadays, it is possible to recreate these states of matter in laboratory, including inertial confinement fusion (ICF)~\cite{atzeni2004physics, nuckolls1972laser, betti2006progress, moses2009national}, dynamic and static compression~\cite{smith2014ramp, jayaraman1983diamond}, and isochoric heating~\cite{vinko2012creation}. As a result, there is substantial interest in studying and modelling HED systems, driven both by fundamental scientific inquiry and practical applications, especially considering the recent advances towards ignition in ICF~\cite{abu2022lawson, abu2024achievement, lindl2014review}.

The theoretical modelling of quantum plasmas, however, presents significant challenges. Due to their high densities, these systems inherit the complexities of solid-state materials, where electrons exhibit quantum behaviour and ions display strong correlations. In addition to this, the elevated temperatures modify the interactions and the local environment experienced by the particles - affecting properties like the mean ionisation state and introducing a range of phenomena, such as collisional excitation and electron recombination~\cite{salzmann1998atomic, pert1990models}. To address these challenges, the modelling of dense plasmas employs a multiscale approach~\cite{payne1992iterative}, which is explored more in detail in Section~\ref{theoretical models}. With this approach, different models are used to study the system at different length scales: atomistic (or more generally, microscopic) models calculate plasma properties, which are then incorporated as input parameters into continuum (or macroscopic) models to predict the system's behaviour for most practical applications~\cite{fryxell2000flash}. Therefore, accurately estimating plasma properties, such as the equation of state and transport coefficients, is crucial for practical models to produce reliable predictions. Given the complexity of these systems, experimental investigations are indispensable for validating and complementing theoretical estimates, which may vary significantly between different models~\cite{stanek2024review}. Thus, experimentally-measured quantities are used as input parameters in simulations, or more frequently, theoretical models are employed to interpret experimental data, producing measurable outputs with plasma properties acting as fitting parameters. In this context, the goal of this thesis is to introduce experimental methods, which, alongside theoretical models, provide a measure of the electronic density of states (DOS) of materials under extreme thermodynamic conditions. The DOS is a fundamental material property that affects its equation of state~\cite{wang2000calculated}, as well as its transport and spectral properties~\cite{kubo1966fluctuation}, underscoring the need for accurate DOS estimation to predict a system's dynamics and behaviour. These connections are explored in more detail in Section~\ref{Electronic Structure of Materials}.



% We begin our discussion with classical plasmas, followed by an analysis of when quantum effects become significant and how they modify the system.

Plasmas can be classed by a few dimensionless parameters, which will now be reviewed. Typically, a plasma consists of free electrons with number density $n_f$ and temperature $T_e$, as well as ions with number density $n_i$ and temperature $T_i$\footnote{In solid-density plasmas, this assumption can break down, as they may contain electrons that are neither fully bound nor entirely free. For such systems, even defining ionisation becomes challenging~\cite{gawne2023investigating}.}. The ion population consists of $N_s$ ion species with charge state $z_j$ and number density $n_j$, with $j = 1, 2, \dots, N_s$. Since free electrons remain in the system, plasmas are globally neutral, meaning that $n_f = \langle z \rangle n_i$, where $\langle z \rangle = \frac{1}{n_i}\sum_{j=1}^{N_s} \langle z_j \rangle n_j$, with $\langle z_j \rangle$ representing the mean charge state of the j-th ion species. For classical plasmas, correlations between particles are mainly quantified by the \textit{Coulomb coupling parameter} $\Gamma_{\alpha \beta}$, which is the ratio of the average electrostatic potential energy between particles of species $\alpha$ and the average thermal energy of particles of species $\beta$. For example, the ion-ion coupling parameter is given by: 

\begin{equation}
\Gamma_{ii} = \frac{\langle z \rangle ^2 e^2}{4 \pi \epsilon_0 k_B T_i a_i} ,
    \label{gamma parameter}
\end{equation}

%\frac{\langle z \rangle ^2 e^2}{4 \pi \epsilon_0 k_B T_i} \biggl(\frac{4\pi n_i}{3}\biggr)^{1/3}

\noindent where $k_B$ is the Boltzmann constant, $e$ is the elementary charge, $\epsilon_0$ is the vacuum permittivity, and $a_i = (4 \pi n_i /3)^{-1/3}$ is the \textit{ion-sphere radius}, which measures the typical distance between ions. Similar expressions can be defined for the electron-electron and the average electron-ion coupling parameters~\cite{ichimaru1982strongly}:

\begin{equation}
\Gamma_{ee} = \langle z \rangle^{-5/3}\frac{T_i}{T_e} \Gamma_{ii} \ \ , \ \  \Gamma_{ei} \sim \langle z \rangle \Gamma_{ee} \ .
    \label{gamma parameter 2}
\end{equation}


% Given the significant mass difference between electrons and ions, if the ions are hotter than the electrons, they will rapidly share their energy, whereas, in the opposite case, energy transfer from electrons to ions would occur at a much slower rate. As a result, in typical scenarios, $T_e \geq T_i$\footnote{Exceptions include systems undergoing nuclear reactions, which continuously supply energy to the ions, maintaining their temperature higher than that of the electrons.} (and often $T_e \approx T_i$), leading to $\Gamma_{ii}\geq \Gamma_{ee}$.


\noindent Given the significant mass difference between electrons and ions—and consequently their differing heat capacities—thermal equilibration between the two occurs at asymmetric rates. If the ions are initially hotter than the electrons, the electrons rapidly equilibrate to the ion temperature, whereas, in the opposite case, the ions heat up at a slower rate. As a result, in typical scenarios, $T_e \geq T_i$\footnote{Exceptions include systems undergoing nuclear reactions, which continuously supply energy to the ions, maintaining their temperature higher than that of the electrons.} (and often $T_e \approx T_i$), leading to $\Gamma_{ii}\geq \Gamma_{ee}$. A plasma is considered weakly coupled if $\Gamma_{ii} \ll 1$, meaning Coulomb interactions are weak and the plasma can be treated similarly to a gas, where almost no structure is present. In this regime, collective effects dominate, effectively screening microscopic electric fields, which decay exponentially over a characteristic length scale known as the \textit{Debye length}, $\lambda_{D,i} \sim a_i/\sqrt{\Gamma_{ii}}$~\cite{chen1984introduction}. Within a sphere of radius $\lambda_{D,i}$, the plasma contains a sufficiently large number of particles (ideal plasma), allowing it to be described as a continuum with correlations occurring only on length scales larger than $\lambda_{D,i}$. 

In contrast, a strongly coupled plasma ($\Gamma_{ii} > 1$) exhibits stronger Coulomb interactions, reducing the shielding and correlation lengths to the scale of the interparticle spacing $a_i$~\cite{stanton2016ionic}, so that, under sufficiently strong coupling, ordered structures can form~\cite{baus1980statistical}. In these strongly coupled systems, characterised by high density and low temperatures, quantum effects often plays a significant role. These effects enter in two primary ways: through the uncertainty principle, which causes the position of particles to be probability distributions over space, and via symmetry effects arising from the indistinguishability of particles, which alter the statistical population of energy levels.

To quantify the first effect, one can start by recalling the thermal De Broglie wavelength of particles $\alpha$,

\begin{equation}
\Lambda_{\alpha} = \biggl(\frac{2\pi\hbar^2}{2\pi m_{\alpha}k_B T_{\alpha}}\biggr)^{1/2},
    \label{De Broglie parameter}
\end{equation}

\noindent where $m_{\alpha}$ is the rest mass of the particle species and $\hbar$ is the reduced Planck constant. This wavelength, which estimates the spatial extension of $\alpha$ particle's wavefunctions, is then compared with the average interparticle spacing $a_{\alpha}\equiv(4 \pi n_{\alpha} /3)^{-1/3}$ by defining the degeneracy parameter $\Chi_{\alpha}=\Lambda_{\alpha}/a_{\alpha}$~\cite{bonitz2019quantum}. If $\Chi_{\alpha}\gtrsim 1$, particles cannot be treated as pointlike objects and quantum mechanics must be employed to describe their dynamics, especially for short-range interactions, where classical collisions become a quantum scattering problem between two particles~\cite{filinov2004temperature}. Given the significantly larger mass of ions compared to electrons, ions in quantum plasmas are typically treated as pointlike particles obeying the classical laws of motions, an assumption that is also adopted in this thesis.

Symmetry constraints - a fundamental requirement of quantum mechanics - further influence particle dynamics, for instance by preventing two identical particles from occupying the same region of space.\footnote{These constraints can be incorporated into semiclassical frameworks by introducing \textit{exchange} or \textit{Pauli potentials} into the particles' equations of motion~\cite{klakow1994semiclassical}.}. However, their primary impact lies in determining the statistical population of electronic energy levels, which follows the Fermi-Dirac distribution\footnote{While this distribution is strictly valid for an ideal fermion gas, its deviations due to interactions are typically small~\cite{daniel1960momentum}, and thus neglected in this thesis.}:

\begin{equation}
f_{FD}(\epsilon;T,\mu) = \frac{1}{1+e^{(\epsilon-\mu)/k_B T}} \ ,
\label{FD}
\end{equation}

\noindent with $\epsilon$ representing the energy of the electronic states and $\mu$ denoting the chemical potential. To quantify the symmetry effects, one can introduce another parameter $\theta_{e}$, defined as follows:

\begin{equation}
\theta_{e} = \frac{T_{e}}{T_{F,e}},  \ \ {\rm with} \ \ T_{F, e} = \frac{\hbar^2}{2 k_B m_e}(3 \pi^2 n_e)^{2/3},
    \label{theta parameter}
\end{equation}

\noindent where $T_{F,e}$ is the Fermi temperature of the electron population. Considering Eq. (\ref{FD}) and the relation between $n$, $T$ and $\mu$ for an ideal fermion gas~\cite{landau1980statistical} (as a first approximation), one can calculate that, if $\theta_{e} \lesssim 1$, $\mu$ is positive, indicating that the system is degenerate and the electrons follow Fermi-Dirac statistics. Conversely, when $\theta_{e} \gg 1$, $\mu/k_B T$ becomes very negative, so that the expression in Eq. (\ref{FD}) reduces to the classical Maxwell-Boltzmann distribution. Considering that $\Chi \approx 1.84  \ \theta^{-1/2}$, the condition $\Chi_e \gtrsim 1$ is equivalent to $\theta_e \lesssim 1$, showing that the statistical and dynamical behaviour of the system transition to the quantum regime simultaneously. By incorporating the Fermi-Dirac distribution in the calculation of the average kinetic energy, a modified definition of the coupling constants for electrons is obtained~\cite{bonitz2024toward}, such that $\Gamma_{ee}$ scales with $n_e^{-1/3}$ and is independent of temperature in the quantum regime. This modification ensures that free electrons in solid systems are weakly correlated, as expected.

\begin{figure}[t]
\begin{center}
  \includegraphics[width=0.7\columnwidth]{Figures/diagram n-T.pdf}  
  \caption[Temperature-density phase diagram]{A temperature-density phase diagram for matter, with regions associated to typical systems indicated by the coloured areas. The plasma is assumed to be in thermal equilibrium, with the temperature here being $T_e = T_i$. Isolines of the ion-ion coupling parameter ($\Gamma_{i,i}=1$, for $\langle z \rangle = 1$) and
  the degeneracy parameter ($\theta_e=1$) are included. The region below these lines is occupied by quantum plasmas, whereas the part above is associated with classical plasmas.} 
   \label{diagram_n_t}
\end{center}
\end{figure}

Figure \ref{diagram_n_t} presents a temperature-density phase diagram of matter, highlighting the regions belonging to the most common plasmas found in nature and in laboratories. For reference, the diagram includes also isolines of the ion-ion coupling parameter $\Gamma_{ii}$ and the electron degeneracy parameter $\theta_e$. Near these lines, there exists an intermediate regime where classical equations describe the general behaviour, but quantum corrections are required. This thesis deals with the regions of the diagram occupied by isochorically-heated matter (in blue) and compressed solid generated by laser-driven planar shocks (in orange), both of which are strongly-coupled and partially-degenerate plasmas.


























\section{Experimental Realisation of HED Systems}




% The origins of shock compression date back to World War II, with groundbreaking research conducted both at Los Alamos National Laboratory and independently in Russia. These studies enabled scientists to explosively shock-load materials to pressures reaching several tens of GPa, allowing the development of the foundational theory behind these phenomena~\cite{zeldovich1967theory}.






Over the years, various methods have been developed to recreate extreme thermodynamic states in laboratory settings, beginning with explosive-driven compressions conducted both at Los Alamos National Laboratory and independently in Russia in the 1930s and 1940s, which were used to study the physics of shock wave propagation~\cite{bleakney1949interaction, zeldovich1967theory}. In the 1950s, the invention of diamond anvil cells, inspired by Bridgman’s pioneering work, enabled the static compression of materials to unprecedented pressures, up to hundreds of GPa~\cite{jayaraman1983diamond, piermarini1962diamond}. In the following years, significant advancements were made in dynamic compression techniques, including the development of z-pinch machines~\cite{filippov1962dense}, gas guns~\cite{jones1966measurement}, and intense optical lasers. The latter played a pivotal role in the then-nascent field of ICF research~\cite{hirsch1967inertial} and flourished in the following years, driven by rapid advancements in laser technology, allowing pressures of a few TPa to be reached~\cite{ragan1982shock}. Only more recently, a new type of HED systems has emerged through isochoric heating, achieved by irradiating targets with modern x-ray free electron lasers (XFELs)~\cite{vinko2012creation, falk2018experimental}, which can drive electronic pressures into the terapascal regime. This section gives an overview of the mechanisms behind the creation of HED matter with optical and x-ray free-electron lasers, which constitute the systems investigated in this thesis. Plasmas generated by powerful lasers are typically very short-lived, lasting only tens of nanoseconds, as extreme thermodynamic conditions quickly dissipate due to processes like heat conduction and hydrodynamic expansion. Consequently, \textit{in situ} techniques, commonly based on diffraction and spectroscopy, are necessary for experimental investigations of these systems.




% or energetic particles




\subsection{\label{Laser-Compressed Systems} Laser-Compressed Systems}



% Maybe put a short review on their functioning
% Solid state laser: standard functioning!!! four level laser with population inversion
% diode-pumped: We used a diode laser to pump the solid state laser
% Diode laser: photons are produced by the photodecay of electrons moving from n to p and filling the holes the p-junction. Stimulated emission occurs with produced photons interacting with photodecaying electrons and producing two coherent photons (avalanche effects). Usually a semiconductor slab is in between the two junctions. First to confine these photoemissions to this region and secondly to confine the emitted radiation to this region thanks to its different refraction index. Then a fabry perot inteferometer (laser cavity) is used to amplify only certain frequencies



To reach the extreme pressures found in the interiors of large planets and stars, laser-driven compression is required. This technique can be classified into two categories: planar compressions~\cite{zeldovich1967theory}, commonly used for pure physical investigations, and spherical compressions~\cite{atzeni2004physics}, mostly employed in nuclear fusion applications. For the purposes of this thesis, only planar shocks are examined below.

In a typical planar shock experiment, the target consists of a series of material slabs glued together. Starting from the side exposed to the laser, the first layer is called the \textit{ablator}, usually made of plastic for their high energy absorption, ease of manufacture, and low cost. Behind it lies the material to be compressed and studied, potentially followed by additional layers for diagnostic purposes and tamper windows designed to sustain the high-pressure conditions within the sample for longer times (see below and Chapter 5 for details). In many cases, a thin layer of suitable material (e.g. aluminium) is placed between the ablator and the sample to shield against laser and ablator radiation, preventing target preheating. Nanosecond optical lasers, such as standard Nd:glass lasers~\cite{thornton1969properties}, or modern diode-pumped lasers~\cite{hanna2021review}, are typically used to compress matter. When the laser strikes the target, it heats the outer layers of the ablator, causing them to vaporise and ionise. These outer layers enter a plasma state, allowing the laser radiation to pass through the solid layers until they reach the so-called \textit{critical density}. For the typical wavelengths, durations and intensities of these lasers, where \textit{inverse bremsstrahlung} is the primary absorption mechanism (see also below), this density is given by~\cite{atzeni2004physics}:

\begin{equation}
n_C = \frac{\pi m_e c^2}{e^2 \lambda_L^2},
    \label{critical density}
\end{equation}

\noindent where $c$ is the speed of light and $\lambda_L$ is the laser wavelength. The laser radiation, along with the emission from the outer plasma layers, can penetrate deeper into the target, ionizing and transforming additional layers into plasma. Heat conduction also contributes to raising the temperature of the inner layers. Using hydrodynamic equations, one can see that the outer plasma layers expand outward and are ablated. Due to momentum conservation, this outward expansion causes the inner, still-solid layers to accelerate in the direction of the beam, driven by the resulting \textit{ablation pressure}. In this absorption regime dominated by inverse bremsstrahlung, this pressure can be estimated with the following expression~\cite{batani2003ablation}:

\begin{equation}
p_{\rm abl} = p_0\biggl(\frac{A}{2Z}\biggr)^{1/3} \biggl(\frac{1}{\lambda_L[\mu{\rm m}]}\biggr)^{2/3} \biggl(\frac{I_L[{\rm W cm}^{-2}]}{10^{14}}\biggr)^{2/3} \ \ {\rm Mbar} \ ,
    \label{ablation pressure}
\end{equation}

\noindent where $I_L$ is the laser intensity, $A$ and $Z$ are the average atomic mass and atomic number of the target material, respectively, and $p_0$ is a dimensionless prefactor, typically around 10. For typical laser wavelengths and intensities (100s of $\mu$m and $\sim$10$^{12}$ W cm$^{-2}$), ablation pressures on the order of $\sim$ 100 GPa can be achieved.

%Note that the ablation pressure increases with frequency of the laser. This is why this frequency is often doubled with anharmonic conversion crystals.



As the compressibility of a solid generally decreases with density, the local sound velocity rises under compression. This effect causes compression waves to steepen as they propagate, eventually transforming into shock waves if they have sufficient distance to travel\footnote{Shocks are defined and theoretically modelled as discontinuities in thermodynamic fields. However, in practice, they always possess a finite width set by the viscosity or stresses of the medium.}. Consequently, the ablation pressure generated by the laser often induces a series of shock waves into the solid layers, with their structure determined by the shape of the laser pulse. These shocks contribute to the overall compression of the material, distorting its lattice and heating its constituents, which eventually leads to melting in some cases. Since the presence of shocks tends to reduce the levels of compression achieved for a given amount of supplied energy, their formation is typically suppressed by carefully shaping the laser pulse~\cite{atzeni2004physics, chen1984introduction}. However, they may be intentionally generated for specific applications or research purposes. A more detailed description of the equations and the mechanics of shock compressions can be found in the next section.


The nanosecond optical lasers used to compress matter interact primarily with free electrons through collective excitations~\cite{harrison1962collective}. The radiation generates electron plasma waves, which are dampened by electron-ion collisions and, to a lesser extent, by kinetic effects~\cite{brambilla1998kinetic}. Laser energy is therefore absorbed primarily through inverse bremsstrahlung (and under certain conditions, through other mechanisms like resonant absorption), and then dissipated in the plasma via electron-ion collisions. 





%Spherical compressions function in a similar manner, but achieve higher levels of compression at the end of the implosion due to the geometrical constraints.












\subsubsection{\label{Mechanisms and Equations of Dynamic Solid Compressions} Mechanisms and Equations of Shock Compressions in Solids}




%\footnote{Rankine and Hugoniot discovered these equations independently for fluids, but the same conservation laws apply to solids.}






In shock compression, the material conditions before and after the shock front must satisfy conservation equations, as established in the 19th century by Rankine and Hugoniot. These equations express the conservation of mass, momentum, and energy across the shock, and link the initial density, pressure\footnote{For simplicity, I consider isotropic stresses in the solid to illustrate the main concepts, while keeping in mind that stresses in solids are often anisotropic.}, and energy of a material to its post-shock state. For a solid, they can be written as (using the subscript 0 for the pre-shock quantities):

\begin{align}
& \rho_0 U_S = \rho (U_S - U_P) \\
& p - p_0 = \rho_0 U_S U_P \\ 
& \varepsilon - \varepsilon_0 = \frac{1}{2} \left( p + p_0 \right) \frac{\left( \rho - \rho_0 \right)}{\rho \rho_0} 
\label{rankine-hugoniot}
\end{align}

\noindent where $U_P$ and $U_S$ are the particle velocity (in the post-shock state) and the shock velocity, respectively, $p$ is the pressure, $\rho$ is the density ($\rho\approx m_in_i$), and $\varepsilon$ is the energy per unit mass. Given that the initial density, pressure, and energy are known, these three equations contain five unknowns ($\rho$, $\varepsilon$, $p$, $U_S$, and $U_P$). To resolve this system, two parameters need to be measured. Typically, the particle and shock velocity are selected, as these can be determined by monitoring the shock transit time and the surface or interface velocity at shock breakout, i.e. when the shock reaches the rear of the target. Today, both quantities are commonly measured using the VISAR (Velocity Interferometer System for Any Reflector) technique~\cite{dolan2006foundations}, which is described in more detail in Chapter 5. In many instances, the equation of state (EOS) of the compressed material is already known and tabulated from previous studies~\cite{johnson1994sesame}, reducing the number of unknowns that has to be measured to one\footnote{When stresses in the material are anisotropic, an additional equation is needed to relate the longitudinal stress along the compression axis to the hydrostatic pressure. This relationship is typically described by a strength model, which is often complex and highly material-specific, making it an active area of ongoing research~\cite{steinberg1980constitutive}.}. If the primary goal is to determine the shock pressure without focusing on energy densities,  it is possible to solve only the first two Hugoniot equations with the aid of a so-called ``shock EOS'', which is an empirical linear relation between $U_S$ and $U_P$:

\begin{equation}
U_S = C_0 + s U_P ,
    \label{shock EOS}
\end{equation}

\noindent where $C_0$ is the ambient bulk sound speed, and $s$ is a material-dependent parameter of order unity. Substituting this relationship into the first two Hugoniot equations and eliminating $U_P$ yields the following commonly used expression for $p$:

\begin{equation}
p - p_0 = \rho_0 C_0^2 \kappa/(1-s\kappa)^2 ,
    \label{hugoniot simple}
\end{equation}

\noindent where $\kappa = (\rho-\rho_0)/\rho$ is the compressibility. While this expression simplifies the calculation of the shocked thermodynamic parameters, it should be applied cautiously, as this linear relationship is not a universal rule.

The locus of states achieved through shock compression is known as the \textit{Hugoniot}. Considering the constraints imposed by the Hugoniot equations and the material's EOS, only one free parameter remains, making the Hugoniot a curve in the thermodynamic plane (e.g. $p=p(\rho;\rho_0,p_0)$ if $\rho$ is chosen as the independent variable). Due to shock heating, for a given compression, the Hugoniot results in temperatures that lie above the isentrope. As an example, in Fig. \ref{iron phase diagram} the Hugoniot and isentrope curves for iron are illustrated in the $p$-$T$ plane, alongside its melting curve.

\begin{figure}[t]
\begin{center}
  \includegraphics[width=0.5\textwidth]{Figures/iron phase diagram.pdf}  
  \caption[Notable curves in high-pressure physics: an example for iron]{Key curves in high-pressure physics for iron are shown on a pressure-temperature diagram, including the melting curve, the Hugoniot, and the isentrope (data from~\cite{jeanloz1980equations, grant2021equation}). A schematic representation of a multishock process, approximating ramp compression, is also included. Smaller individual shocks result in reduced heating of the target, allowing the compression path to remain closer to the isentrope. The Hugoniot in the $p$-$T$ plane is computed considering the additional EOS relating $T$ to $\rho$ and $p$.} 
   \label{iron phase diagram}
\end{center}
\end{figure}

As shown in the figure, due to the sharp increase in entropy during a shock, the Hugoniot intersects the melting curve at low compression levels. To reach higher compression while keeping the material solid, the shock can be fractioned into a series of smaller shocks by properly modulating the laser pulse profile. This multishock process - displayed schematically in Fig. \ref{iron phase diagram} - can achieve greater compression with smaller temperature rises, allowing the investigation of solid states at higher pressures. The limit case of multishock compression is the \textit{ramp compression}, where the laser pulse is precisely shaped to achieve compression without generating shocks~\cite{smith2014ramp}, resulting essentially in an isentropic process.

%Mix compression (shock + ramp) are also possible.


%  (though still less so than they would without the tamper)

A common practice in compression experiments is to add a tamper layer, made of a suitable material, behind the sample to help maintain compression conditions for a longer period. When the shock wave reaches the interface between the sample and the tamper, the inability of the tamper atoms to fully withstand the stresses exerted by the sample allows the outermost layers of the sample to expand more freely. This interaction causes the shock to propagate into the tamper window, while a release wave is reflected back into the sample, further altering its thermodynamic state. Over time, the release wave broadens due to the reduction in sound speed in the decompressed material, without generating additional shocks. In contrast, when no tamper is used, the smaller resistance, compared to the tamper case, offered by air (or vacuum) to the motion of the target's outer layers results in significantly lower pressures in the released state. Additionally, if the Hugoniot curve of the tamper material is known, along with the Hugoniot and the EOS of the sample\footnote{There exist experimental setups including a tamper layer that can perform the impedance matching method without knowing the sample's EOS. This is achieved by measuring also shock velocities, in addition to particles velocities.}, the thermodynamic parameters of the compressed and released sample's states can be found using the impedance matching method~\cite{celliers2005systematic, huser2013hugoniot}. This approach is illustrated in Fig. \ref{impedance matching}, which schematically depicts the thermodynamic history of a shocked sample with a tamper window in both the $x$-$t$ and $\rho$-$p$ planes. 

\begin{figure}[t]
\begin{center}
  \includegraphics[width=0.8\textwidth]{Figures/impedance matching.pdf}  
  \caption[Diagrams for impedance matching method]{Space-time (a) and pressure-velocity (b) diagrams of a shock propagating through a layered target consisting of a sample and a window. These diagrams (adapted from~\cite{wark2022femtosecond}) are helpful for understanding the impedance matching method. The shock transitions the sample from its initial state (0) to a shocked state (S0). Upon reaching the interface with the window, the shock propagates through the window, transitioning it from state (1) to shocked state (S1), while a release wave reflects back into the sample, relaxing it to state (SR0). These states are represented in the $p$–$U_P$, where $p$ and $U_P$ are the pressure and particle velocity of both the shocked window (S1) and the released sample (SR0) for the conservation of physical quantities at the interface. By measuring $U_P$ and knowing the two Hugoniot curves along with the sample's relaxation curve (an isentrope as a first approximation~\cite{heighway2019nonisentropic}), all states can be determined. In practise, the release curve is often approximated as the sample's reflected Hugoniot that, properly translated, goes through the state (SR0). The pressure equations shown in the diagram correspond to the second Hugoniot equation, with initial pressure assumed negligible.} 
   \label{impedance matching}
\end{center}
\end{figure}

% elastic deformation (divided in microrotation plus pure strain deformation)

Understanding the microscopic mechanisms that occur upon solid compression is complex, and many studies, experimentally based on x-ray diffraction, have been performed in this regard~\cite{mcbride2019phase, wehrenberg2017situ, milathianaki2013femtosecond}. Here, a brief description of the main findings is given. If the shock is not too intense, only elastic deformations occur (elastic wave), temporarily distorting the interatomic bonds and increasing the material's potential energy. This stored energy is then released, generating a relatively ordered macroscopic motion, with minimal heat production in the process. 

If the shock strength exceeds the \textit{Hugoniot elastic limit} (HEL), the shear strains induced by elastic deformations reach a critical value (yield point) along specific planes, determined by the type of compression and the crystal structure\footnote{Models such as the simple \textit{Schmidt's law}~\cite{lubliner2008plasticity} are often used to predict which planes will activate.}. Beyond this point, these critical planes begin to slip relative to one another, resulting in plastic deformation of the solid, with the atoms settling into new stable positions through mechanisms such as rotation and twinning~\cite{heighway2020shock, heighway2021kinematics}. These plastic deformations alleviate stresses\footnote{This stress relief actually means that shear stresses remain constant during plastic flow, provided \textit{work hardening} is not considered~\cite{heighway2020shock}.}, which are proportional (in the linearised strain theory) solely to elastic deformations. When plastic flow begins, in fact, the elastic deformations decrease in magnitude ensuring that the total deformation—comprising both elastic and plastic components— satisfies the geometric constraints imposed by the macroscopic setup. For instance, in shock compression, the total deformation is modelled to be purely longitudinal along the compression axis, as the bulk of shocked material is \textit{laterally confined} by the surrounding uncompressed solid.

Since, for energy reasons, it is not possible to break too many interatomic bonds, these plastically deformed regions are localised and are referred to as \textit{dislocations}. The theory of dislocations, first proposed independently in 1934 by E. Orowan, M. Polanyi, and G.I. Taylor~\cite{orowan1949fracture, taylor1934mechanism}, connects the nature and density of dislocations to the rate and type of plastic deformations.

%\footnote{As dislocations tend to grow more easily around pre-existing defects, in materials with a high defect density, stress relief is more effective. However, under shock conditions, where stresses are often intense, homogeneous nucleation of defects becomes necessary.}.

A simple model~\cite{heighway2020shock} demonstrates that when compression exceeds the HEL,  the resulting shock wave separates into two distinct components: a faster elastic wave and a slower plastic wave, with the latter relieving the stresses introduced by the former. This same model predicts that, above a certain shock strength, this two-wave structure disappears, with the plastic and elastic waves propagating together. If the plastic deformations induced by the shock are extensive, they can lead to shock-induced phase transitions in the crystalline structure. 

Sample heating occurs during plastic relaxation, as ions overcome potential barriers to fall into a more stable configuration upon compression. In fact, the energy required to cross these barriers is released as random oscillations around the new equilibrium positions, which manifests macroscopically as a temperature increase. This also explains, on a microscopic level, why stronger shocks lead to greater heating—since more plastic/relaxation work occurs. Additionally, in release waves, defect annihilation contributes to heating as the energy stored in distorted atomic bonds within defects is released as thermal agitation.

Data from shock compression experiments, mostly based on x-ray diffraction, are typically interpreted using elastoplasticity theory~\cite{landau2012theory, lubliner2008plasticity}, or, in a more \textit{ab initio} approach, molecular dynamics (MD) simulations. While computationally intensive, MD simulations can be employed to benchmark or refine analytical elastoplastic models~\cite{heighway2021kinematics}.

















% Pre-existing dislocations relieves stresses, but if we probe on ultrashort timescales and the shock is not too intense, this does not happen. Below strong shock limit, we have an elastic shock wave




% Rule : particle velocity = 2*surface velocity



\subsection{Isochorically-Heated Systems}






\begin{figure}[t]
\begin{center}
  \includegraphics[width=0.9\textwidth]{Figures/xfel scheme.pdf}  
  \caption[Scheme of an XFEL laser]{Scheme of an XFEL laser, adapted from~\cite{ETH}.} 
   \label{XFEL}
\end{center}
\end{figure}

Another way for creating HED matter is by means of x-ray free electron lasers. Below, a brief overview of their key features and operating principles is provided, with a schematic representation shown in Fig. \ref{XFEL}. For a more detailed discussion, readers are encouraged to consult the extensive literature on the subject~\cite{schmuser2014free, pellegrini2016physics,huang2007review}. Unlike conventional lasers, which rely on specific atomic or molecular transitions in a gain medium, FELs generate radiation through the interaction of a beam of high-energy electrons with a magnetic field. The process begins with an electron gun or a linear accelerator (linac) that produces a beam of relativistic electrons. After acceleration, the electron beam passes through an \textit{undulator}, a magnetic structure with alternating poles that forces the electrons to oscillate transversely, and therefore emit synchrotron radiation. This radiation interacts with the electron beam itself, leading to a process called \textit{microbunching}, where the electrons form bunches that are spaced at the wavelength of the emitted radiation~\cite{schmuser2014free} (see Fig. \ref{XFEL}). This microbunching allows the electrons to radiate coherently, thereby amplifying the intensity of the emitted light. In turn, this intensified radiation further increases the bunching effect, creating a positive feedback loop as in standard lasers. This process, which exponentially amplifies the intensity of the x-ray beam, is known as self-amplified spontaneous emission (SASE). Eventually, enough energy is extracted from the electrons that they begin to lag behind, drawing energy out of the radiation field. Beyond this point, the x-ray beam intensity saturates, and the electrons are subsequently removed by an absorber located at the end of the undulator.

To a first approximation, the central wavelength $\lambda_L$ of the laser radiation is related to the undulator period $L$ and the Lorentz factor of the electron beam, $\gamma_e = (1-v_e^2/c^2)^{-1/2}$, by the expression $\lambda_L = L/2\gamma_e^2$~\cite{schmuser2014free}. This relationship can be understood as arising from two relativistic effects: first, from the electron's frame of reference, the undulator appears Lorentz contracted by a factor $\gamma_e$, resulting in emitted radiation with a wavelength of $L/\gamma_e$. When transforming back into the undulator's frame, the radiation emitted in the forward direction is further Doppler shifted by a factor of $2\gamma_e$. Therefore, by accurately gauging the characteristics of the electron bunch (e.g. its initial velocity) and the geometry of the magnetic field, a very short, intense and coherent x-ray pulse is produced, with a tunable central energy and a narrow bandwidth. To give some numbers, the Linac Coherent Light Source (LCLS), the first hard XFEL to become operational, delivers approximately 1 mJ of x-rays with a 0.3\% bandwidth in pulse durations on the order of 10s of femtoseconds~\cite{emma2010first}. This results in a spectral brightness over a billion times higher than that of traditional synchrotron sources~\cite{bilderback2005review}.  Considering also the impressive x-ray-beam divergence of only 2 $\mu$rad, focal spot size of $\sim$ 100 nm are now achievable by means of specific refractive lenses and/or mirrors~\cite{hudec2010kirkpatrick}, allowing spatially localised investigations with radiation intensity of $10^{20}$ Wcm$^{-2}$ on the target. Additionally, thanks to the relative large penetration depths of x-rays, XFELs can volumetrically heat the sample, creating quite uniform thermodynamic conditions. Recent works have demonstrated the capability to generate ultrashort XFEL pulse with durations on the order of attoseconds by properly shaping and manipulating the electron bunch~\cite{franz2024terawatt}, as well as ultrafocused beams with intensities surpassing $10^{22}$ W/cm $^{-2}$, thanks to spot sizes as small as 10 nm~\cite{yamada2024extreme}. These breakthroughs advance the frontiers of x-ray science, unlocking new opportunities such as the study of electron dynamics and non-linear optical phenomena~\cite{vinko2024x}. 

At the onset of the radiation-generation process, the electron emission is spontaneous, resulting in a noisy pattern in the frequency domain, which is preserved during the amplification, yielding a final spiky SASE spectrum with a relative wide bandwidth of approximately 0.3\%. To overcome this limitation, in 2012, \textit{Amann et al.}~\cite{amann2012demonstration} introduced a technique known as \textit{self-seeding}, designed to enhance the coherence and spectral purity of x-ray pulses generated by XFELs. In self-seeding, the undulator is divided into two sections. At the end of the first section, the produced SASE radiation (not yet saturated) is diffracted through a crystal to be monochromatised. Meanwhile, the electron bunch is sent through a magnetic chicane, specifically designed to delay the electron bunch by the same amount of time required for the monochromatised radiation (seed pulse) to emerge from the crystal. This ensures that the electrons and the seed pulse can be spatially and temporally re-aligned at the start of the second undulator section. Interacting with the electrons, the seed pulse imprints an energy modulation onto the electron bunch, which, as with spontaneous emission in the standard SASE mode, creates a periodic structure that matches the wavelength of the seed. The process then proceeds as in the standard SASE, with coherent emission being exponentially amplified. However, due to the imposed initial modulation of the electron bunch, the resulting x-ray pulse exhibits significantly improved temporal coherence and a much narrower bandwidth - down to 0.01\% - compared to typical SASE output. This comes at the expense of reduced pulse energy. This ``self-seeded'' mode is utilised to collect part of the data analysed in Chapter 4.

Contrary to optical lasers, the interaction of the XFEL with matter is more single-particle based. In fact, at x-ray energies, photons interact primarily with bound electrons, causing photoionisation if the photon energy exceeds the electron’s binding energy, or excitation to another bound state if the energy matches a specific transition energy. Since probability of photoionisation decreases with increasing kinetic energy of the ejected electron, deep core electrons are more likely to excited. When photoionisation occurs, the ejected electron thermalises with the surrounding free electrons through collisions in a few fs, raising the overall electron temperature and leaving the ion in an excited state. Ultimately, an electron - whether thermalised or not, bound or free - recombines with the ion via various processes (e.g. radiative emission) filling the core hole. For the materials relevant to this work, the principal process is Auger decay~\cite{salzmann1998atomic}, which ejects another non-thermal electron, which further contributes to the population of free electrons and to the rise of $T_e$. In general, energy is transferred from the laser to the electron population on timescales typical of Auger decay and electron collisions (all within a few fs). This energy transfer continues throughout the FEL irradiation, with modulations due to the pulse shape, the evolution of free electrons temperature and density, and the ion charge distribution. Well after the x-ray pulse - on the timescale of ps - the electrons thermalise with the ion population through electron-phonon interaction, eventually causing the solid to melt and consequently expand. However, if the system is probed on shorter timescales, one can analyse a solid material, with an ordered crystalline structure, but with a hot electron population - representing an example of quantum plasma. This task is achieved by using the XFEL itself, which heats and probe the target simultaneously~\cite{vinko2012creation} (\textit{self-scattering} experiment). Data from such experiments is presented and analysed in Chapter 4. Alternatively, XFELs can serve as probes when coupled with optical lasers to perform time-resolved diffraction or spectroscopy on dynamically compressed systems~\cite{zastrau2021high}. This time-resolved spectroscopy constitutes the ultimate focus of this thesis and is explored in Chapter 5.

















\section{\label{theoretical models}Theoretical Methods: An Overview}



%\setlength{\tabcolsep}{35pt} % Adjust column separation for better spacing
%\renewcommand{\arraystretch}{1} % Adjust row spacing

%\begin{table}[h!]
%\centering
%\begin{tabular}{|M{30mm}|M{20mm}|M{20mm}|}
%\hline
%\textbf{Lengthscales} & \textbf{Model type} & \textbf{Practical methods} \\ \hline\hline
%\addlinespace[34pt] % Adds space after the header row
%\multirow{2}{*}{Macroscopic} & Compressed solid & Hot dense plasma \\
 %       & 2 & -708.5 \\\hline

%\end{tabular}
 %   \vspace{20pt}
 %   \caption{Physical parameters of iron, used for the synthetic spectra. Iron has two L-shell states, implying two possible final states $f=1,2$.}
  %  \label{tab:params}
%\end{table}








Dense plasmas are complex many-body quantum systems, making them challenging to model. As previously mentioned, a multiscale approach is often used for these systems. Although not rigorously exact, this approach is effective due to the clear separation of timescales and lengthscales between the different processes and components in the plasma. 

% Table ... summarizes the models commonly used for HED systems.

At the largest length scales, continuum mechanics models are employed. For compressed materials that are still solid, elastoplasticity models~\cite{clayton2019nonlinear, lubliner2008plasticity} or, even more commonly, fluid models~\cite{fryxell2000flash} — occasionally incorporating corrections to account for material strength - are applied. In these approaches, macroscopic material properties, such as elastic/plastic constants, EOS and heat capacities, are calculated using smaller-scale models or comparing against experimental data. These microscopic models are typically based on molecular dynamics simulations of atoms in the solid~\cite{dickey1969computer}, where the interatomic potentials, often parametrised with the embedded atom method~\cite{daw1984embedded}, can be determined using the same methods applied to dense non-solid plasmas, which is discussed below.

The most widely used methods for describing hot-dense plasmas at macroscopic scales are radiative-magnetohydrodynamic models~\cite{fryxell2000flash, atzeni2005fluid}, particularly in scenarios where a laser compresses a solid, causing it to melt. These models account for interactions with laser radiation, treated semiclassically and often using ray optics approximation~\cite{born2013principles}. The radiation's behaviour is governed by transport equations, with opacity and emissivity coefficients derived from atomic physics~\cite{salzmann1998atomic}, while the dynamics of the plasma is described by the magnetohydrodynamic equations~\cite{davidson2017introduction}. In cases where ion motion and non-uniformities can be neglected, such as in isochoric heating experiments, zero-dimensional radiative codes are used, which evolve ion populations through collision-radiative atomic kinetics equations and the free-electron distribution via a Fokker-Planck equation~\cite{chung2008flychk}. 

%An alternative fluid model for quantum plasmas, derived from low-temperature physics and not too relevant for this thesis, is quantum hydrodynamics~\cite{moldabekov2018theoretical}, which describes the behaviour of a fluid composed of particles interacting quantum mechanically.

To obtain transport coefficients and EOS for the fluid equations, smaller-scale models are necessary. These models generally fall into two categories: kinetic theory approaches\footnote{Standard kinetic theory, like the Vlasov-Fokker-Planck equation, must be modified when quantum effects are relevant (see for example the Wigner model~\cite{manfredi2005model}). Kinetic theory approaches can actually describe the macroscopic length scales as well, replacing the fluid models.}, such as particle-in-cell codes~\cite{fonseca2002osiris}, and, at even smaller length scales, molecular approaches, which can also treat interparticle correlations and includes methods such as Monte Carlo simulations~\cite{gann1979monte} (for statistical properties) and molecular dynamics calculations~\cite{thompson2022lammps}\footnote{When particle interactions are weak, analytical methods like the \textit{hypernetted chain} approach~\cite{hansen2013theory} can also be used to estimate particle correlations.}. The latter is particularly popular for the plasmas studied in this thesis. For these MD simulations, the interatomic potential between ions - evolved using classical equations of motion - is calculated by going to even smaller length scales, and evaluating the electronic wavefunctions, for a frozen ion configurations, with density functional theory (DFT). This approach, known as DFT-MD~\cite{laasonen1993car}, relies on the adiabatic separation of ion and electron motion and is highly effective for modelling quantum plasmas. An alternative method for handling electronic length scales is the path integral Monte Carlo (PIMC)~\cite{gubernatis2016quantum}. While PIMC is more accurate than DFT at higher temperatures, it is less effective at lower temperatures and cannot directly describe dynamic processes~\cite{lang1993monte}. Despite the different timescales of ion and electron dynamics, non-adiabatic effects can still subtly influence certain quantities~\cite{mabey2017strong, angermeier2023disentangling}, particularly in out-of-equilibrium conditions. Hence, approaches that treat simultaneously ion and electron motion have been proposed~\cite{feldmeier2000molecular, filinov2004temperature, svensson2023development}, relying on effective interaction potentials that account for the quantum nature of particles, such as the finite extent of their wavefunctions. Since DFT, with some of its variants, is the primary theoretical framework employed in this thesis to support the experimental findings, its main concepts are now briefly outlined.









\subsection{Density Functional Theory}










Density functional theory is a computational quantum mechanical method used to explore the electronic structure of many-body systems, including atoms, molecules, and solids~\cite{engel2011density}. DFT simplifies the complex 3N-dimensional problem of solving the N-electron time-independent Schrödinger equation by mapping it to a functional minimisation problem based solely on the electron density, $n(\boldsymbol{r})$, of the system (the subscript ``e'' is omitted for simplicity). This approach is founded on the two Hohenberg-Kohn (HK) theorems~\cite{hohenberg1964inhomogeneous}, which establish that the ground-state properties of a many-electron system are uniquely determined by the ground state electron density, which is the unique 3D function that minimises the total energy functional. While the Hohenberg-Kohn theorems only apply to systems at zero temperature without spin or external magnetic fields, various extensions have been developed to handle more complex conditions~\cite{gilbert1975hohenberg, runge1984density, von1972local, vignale1987density}.

In light of the scopes of this thesis, the next section discusses the finite temperature (FT) DFT~\cite{mermin1965thermal}, for which Mermin extended the (HK) theorems. For a fixed electron temperature $T_e = (k_B \beta)^{-1}>0$ and chemical potential $\mu$, the relevant functional is the Mermin grand potential functional $\Omega[n(\boldsymbol{r})]$, rather than the total energy functional. Mermin's theorems state that minimizing $\Omega[n(\boldsymbol{r})]$ over the 3D functions space yields the equilibrium density $n_0(\boldsymbol{r})$, from which many system properties can be determined. At this density, the Mermin functional becomes the grand potential of the grand canonical ensemble, $\Omega_0 = \Omega[n_0(\boldsymbol{r})]$. $\Omega[n(\boldsymbol{r})]$ is composed of a universal functional $G[n(\boldsymbol{r})]$ and the external potential functional $V_{ext}[n(\boldsymbol{r})]$:

\begin{equation}
    \begin{split}
        & \  \Omega[n(\boldsymbol{r})] = G[n(\boldsymbol{r})] + V_{ext}[n(\boldsymbol{r})] = G[n(\boldsymbol{r})] + \int v_{ext}(\boldsymbol{r})n(\boldsymbol{r})d \boldsymbol{r} \\
        & {\rm with} \ \ G[n(\boldsymbol{r})] = T_e[n(\boldsymbol{r})] + V_{ee}[n(\boldsymbol{r})] + \mu N[n(\boldsymbol{r})] - \beta^{-1}S[n(\boldsymbol{r})],
    \end{split}
    \label{Mermin potential}
\end{equation}

\noindent where $v_{ext}$ is the external potential and $T_e[n]$, $V_{ee}[n]$, $N[n]$ and $S[n]$ are, respectively, the kinetic energy, the electron-electron potential energy, the particle number and the entropy functional. While the Mermin theorems simplify the many-body problem and establish that these functionals exist, they do not provide practical forms for some of them. This obstacle has led to various approximation methods, such as orbital-free DFT~\cite{lambert2007properties} and the widely used Kohn-Sham (KS) scheme~\cite{bickelhaupt2000kohn}, which is the one used in this thesis for comparison with experimental data.

This scheme is based on the following ansatz: \textit{The equilibrium density of the fully-interacting system is equal
to the equilibrium density of a non-interacting system with a proper effective potential $v_{eff}$, accounting for the interactions}. Although unproven in general, this assumption has been proven in certain systems and is considered valid in the HED community. Under this assumption, the all-electron wavefunction is factorisable in the product of single particle wavefunctions obeying the following Schrödinger equations:

\begin{equation}
\biggl(-\frac{\hbar^2}{2m_e}\nabla^2 + \overbrace{v_{ext}(\boldsymbol{r}) + \frac{e^2}{2} \int \frac{n(\boldsymbol{r}')}{|\boldsymbol{r}-\boldsymbol{r}'|} d\boldsymbol{r}' + v_{xc}(\boldsymbol{r})}^{v_{eff}(\boldsymbol{r)}} \biggr)\psi_{\alpha}(\boldsymbol{r}) = \epsilon_{\alpha}\psi_{\alpha}(\boldsymbol{r})
\label{schrodinger equation final}
\end{equation}

\noindent where $\alpha$ is the collection of quantum numbers that uniquely identify the eigenstate, and $v_{XC}$ is the exchange-correlation potential, defined as the functional derivative of the famous exchange-correlation functional $V_{XC}$:

\begin{equation}
v_{xc}(\boldsymbol{r}) = \frac{\delta V_{XC}[n(\boldsymbol{r})]}{\delta n(\boldsymbol{r})} \ .
\label{xc potential}
\end{equation}

\noindent This functional contains all the quantum effects – namely exchange and correlation – that are lost during the passage to the non-interating system and not accounted for by the integral term in $v_{eff}$ (Hartree term). Since the exact form of $V_{XC}$ is unknown, various approximations have been developed over the years, which are discussed below.

To close the system of equations, the density $n(\boldsymbol{r})$ must be expressed in terms of the single-particle orbitals. This can be done through the Fermi-Dirac occupation number distribution $f_{FD}(\epsilon_{\alpha};T_e,\mu)$:

\begin{equation}
n(\boldsymbol{r}) = \sum_{\alpha=1}^{\infty} f_{FD}(\epsilon_{\alpha};T_e,\mu) |\psi_{\alpha}(\boldsymbol{r})|^2 \ .
\label{density}
\end{equation}

\noindent Assuming $V_{XC}$ is known, Eqs. (\ref{schrodinger equation final})–(\ref{density}) form a closed system that constitutes the KH-DFT problem, which is solved in a self-consistent manner. First, an initial guess of the equilibrium density is made (often chosen randomly), providing a starting approximation for the total potential. Then, using Eq. (\ref{schrodinger equation final}), the wavefunctions $\psi_{\alpha}$ are computed based on the initial density guess. From these wavefunctions, a new density is derived, and the process is repeated iteratively until certain criteria—such as convergence of the density, potential, wavefunctions, or total energy—are met. The solution of Eq. (\ref{schrodinger equation final}) is the most computationally expensive step in this cycle, and many techniques have been devised to make it efficient. Some of these methods reduce the number of electrons for which wavefunctions must be explicitly computed. Among these, frozen core approximation~\cite{garrity2014pseudopotentials}, which assumes that deep core electrons retain their atomistic wavefunctions, creating a pseudopotential that influences the remaining electrons and must be included in Eq. (\ref{schrodinger equation final}). Another method for this purpose is the so-called hybrid KS orbital-free DFT~\cite{zhang2016extended}, which assumes electrons in the high-energy states to be plane waves. This scheme is useful in systems with high temperatures, and is therefore used in this thesis. Other techniques, like the Projector Augmented-Wave method~\cite{blochl1994projector}, serve instead to narrow the basis set (e.g. plane waves) necessary for decomposing the $\psi_{\alpha}$ function in order to solve the Schrödinger equation\footnote{The \textit{energy cut-off} technique~\cite{payne1992iterative} is also a basis set reduction method in which basis functions that contribute minimally to the expansion are excluded, transforming an infinite discrete basis set into a finite one. The omitted basis functions are typically those associated with the highest kinetic energy~\cite{payne1992iterative}, giving the technique its name.}.


%  the system under study is treated as periodic throughout all space by embedding it in a cell, known as a \textit{supercell}, which is repeated infinitely.

%  allowing each to be expressed in terms of an infinite, discrete basis set.

All the remaining difficulties of the many-body problem have been confined to $V_{XC}$, whose general form is still unknown. 
A significant area of research in DFT focuses on developing approximations for this term, as the choice of XC functional greatly impacts the accuracy of DFT results. Numerous XC functionals have been developed, ranging from generalised ones to those tailored for specific materials.

The most widely used XC functionals are based on the free electron gas model~\cite{fetter2012quantum}. The local density approximation (LDA) is the simplest form, suitable for systems where the electron density varies slowly, and given by:

\begin{equation}
V_{xc}^{LDA}[n] = \int n(\boldsymbol{r}) \epsilon_{XC}^{LDA}(n(\boldsymbol{r})) d\boldsymbol{r} \propto  \int n(\boldsymbol{r})^{4/3} d\boldsymbol{r} .
\label{lda}
\end{equation}

\noindent For systems with larger spatial gradients in electron density, the generalised gradient approximation (GGA) is used, for which the exchange-correlation
energy per unit density $\epsilon_{XC}$ depends on the density gradient as well, $\epsilon_{XC}^{GGA}(n(\boldsymbol{r}), \nabla n(\boldsymbol{r}))$. Many other types of exchange-correlation functionals exists, like higher-order expansions of the density function (e.g. metaGGA), or the so-called "hybrid" functionals. Traditionally, XC functionals ignored explicit temperature dependence, but recent research incorporating quantum Monte Carlo simulations has shown that temperature can have a significant effect~\cite{brown2013path}. This has led to the development of finite-temperature versions of LDA, GGA, hybrid functionals, and ongoing efforts for meta-GGA~\cite{karasiev2022meta,mihaylov2020thermal}. In this thesis, a finite-temperature adaptation of the popular GGA-PBE functional is utilised~\cite{perdew1996generalized, karasiev2022meta}.

Finally, it is worth pointing out that the KS orbitals $\psi_{\alpha}$ and their energies $\epsilon_{\alpha}$ do not have a rigorous physical meaning~\cite{janak1978proof}, but are just introduced as a construction for solving the KS equations. However, depending on the quality of exchange-correlation functional approximation and the strength of the correlations in the system, KS orbitals and energies can resemble the real eigenstates and eigenvalues. In fact, it is common practice (also adopted in this thesis) to derive physical quantities, such as the electronic DOS, from KS eigenstates and eigenvalues, and they generally yield good comparisons with experimental investigations. That said, there is no guarantee that this is true, and therefore caution is needed.  A well-known example is the ``band-gap problem'', a phenomenon where the KS-DFT underestimates the true bandgap~\cite{gunnarsson1986density}, particularly in systems with strong electron correlations~\cite{anisimov1997first}. More generally, binding energies are also typically underestimated by the KS method. To address these issues, various \textit{ad hoc} modifications have been introduced to the DFT framework. These issues are discussed in more detail in the following section, as they are directly relevant to materials investigated in this thesis.





%DFT+U
%DFT+GW
%DMDFT










\subsection{\label{Electronic Structure of Materials} Electronic Structure of Materials and Related Quantities}



% which, among many things, govern the radiative transition probabilities between them. 

% Here, the dependence of $\kappa_R$, $\alpha$ and $n$ on the material's temperature and density is omitted for brevity.




This work focuses on determining the electronic structure of warm-dense materials, specifically examining the electron energy levels $\epsilon_{\alpha}$, summarised by the density of states, and, to a more qualitative extent, the wavefunctions associated with these levels $\psi_{\alpha}$. The electronic structure is a cornerstone from which various material properties can be derived, including the electronic contribution to the equation of state~\cite{wang2000calculated} and the electronic heat capacity~\cite{Ashcroft76}. Furthermore, the real component of the frequency-dependent optical conductivity, $\sigma_R(\omega)$, can be calculated from ($\psi_{\alpha}$,$\epsilon_{\alpha}$) using the Kubo-Greenwood formula~\cite{kubo1957statistical, greenwood1958boltzmann} 

\begin{equation}
\sigma_R(\omega) = \frac{\pi\hbar^3 e^2}{3 m_e \omega V} \sum_{\alpha,\alpha'} [f_{FD}(\epsilon_{\alpha})-f_{FD}(\epsilon_{\alpha'})]|\langle\psi_{\alpha'}|\nabla|\psi_{\alpha}\rangle|^2 \delta(\epsilon_{\alpha'}-\epsilon_{\alpha} -\hbar\omega) \ ,
\label{kubo-greenwood}
\end{equation}

\noindent where $\omega$ is the frequency of the incident radiation and $V$ is the volume of the system. Knowledge of $\sigma_R(\omega)$ enables the calculation of optical properties, such as the refraction index $n(\omega)$ and absorption coefficient $\alpha(\omega)$~\cite{jackson2021classical}, which, in turn, allow for the determination of the Rossland mean opacity~\cite{mazevet2005simulations, perrot1996new}, $\kappa_R$, widely used in radiative-hydrodynamic codes~\cite{atzeni2005fluid}:

\begin{equation}
\frac{1}{\kappa_R} =  \frac{\int_0^{\infty}d\omega n^2(\omega) [\frac{\partial B(\omega, T)}{\partial T}]/\alpha(\omega)}{\int_0^{\infty}d\omega n^2(\omega) B(\omega, T)}  \ ,
\label{Rossland opacity}
\end{equation}

\noindent where $B$ is the normalised Planck function. Additionally, $\sigma_R(\omega)$ also gives access to the electrical conductivity in the limit $\omega\to 0$, which can be related to the electronic thermal conductivity through Kubo's relations~\cite{kubo1957statistical,chester1961law, wang2012equation}.

The electronic structure of materials changes with density and temperature variations~\cite{polsin2022structural, gawne2023investigating}, as these parameters modify the microscopic environment experienced by ions and electrons. Notably, electron binding energies are altered through mechanisms such as Ionisation Potential Depression (IPD)\cite{hoarty2013observations, saumon1992fluid}. In DFT, the electronic density of states $\rho(\epsilon)$ are computed from the eigenvalues $\epsilon_{\alpha}$, where $\alpha$ represent the quantum numbers of spin-1/2 fermions in a periodic system with large unit cells (supercells)~\cite{payne1992iterative}. These quantum numbers include the spin $s=\pm 1/2$, the crystal momentum $\boldsymbol{k}$ varying in the Brillouin Zone (BZ), and the band index $b$. In this case, $\rho(\epsilon)$ has the following expression:

\begin{equation}
\rho(\epsilon) = \frac{1}{\sqrt{2 \pi W_s^2}} \sum_{\boldsymbol{k} \in {\textrm BZ}} \sum_s \sum_{b=1}^{\infty} e^{\frac{(\epsilon-\epsilon_{b,\boldsymbol{k},s})^2}{2 W_s^2}},
\label{rho}
\end{equation}

\noindent where $W_s$ is the so-called smearing width. Typically, only a subset of k-points within the Brillouin zone is chosen to evaluate this sum, with each selected k-point assigned a weight $w_{\boldsymbol{k}}$ to account for its contribution~\cite{makov1995periodic}. The summation over the bands is naturally truncated when the band energies become significantly distant from $\epsilon$. Sufficient computational resources must be available to accurately compute the states up to these bands. 

Another quantity relevant to this thesis, which can also be computed from $\epsilon_{\alpha}$ and $\psi_{\alpha}$, is the absorption cross-section for photons of energy $\hbar\omega$, exciting electrons from a fully occupied core orbital, $\psi_c$. This quantity is expressed as:

\begin{equation}
\sigma_{abs}(\omega) = \frac{1}{\sqrt{2 \pi W_s^2}} \sum_{\boldsymbol{k} \in {\textrm BZ}} \sum_s \sum_{b=1}^{\infty} e^{\frac{(\hbar\omega-\epsilon_{b,\boldsymbol{k},s})^2}{2 W_s^2}} [1-f_{FD}(\epsilon_{b,\boldsymbol{k},s})]|\langle\psi_{b,\boldsymbol{k},s}|\nabla|\psi_c\rangle|^2 ,
\label{theoretical abs}
\end{equation}

\noindent which is effectively a weighted average of the transition probabilities $|\langle\psi|\nabla|\psi_c\rangle|^2$. As expected, this quantity is strictly connected to, and shares a similar mathematical form with, the optical conductivity, given by Eq.~(\ref{kubo-greenwood}).

The energy levels in DFT, and consequently $\rho(\epsilon)$, are not guaranteed to match the true eigenvalues of the system. Discrepancies from reality arise especially in strongly correlated systems, as DFT is essentially a mean field theory, since $V_{XC}$ is approximated and could be very different from the physical one. In such cases, when Coulomb interactions between valence electrons on the same or neighbouring site\footnote{This thesis examines compressed and isochorically heated solids, where relatively ordered ion structures are maintained. Consequently, the concept of ``site'' remains well-defined.} are strong enough, electron hopping between sites is drastically hindered, causing electrons to localise around atoms. This localisation leads to a drop in conductivity and the emergence of a bandgap, characteristic of what is known as a "Mott insulator"~\cite{morosan2012strongly}. A well-known example of such materials are iron oxides, which constitute the principal systems examined in this thesis.

To account for these strong electron-electron interactions, the simplest theoretical approach is the Hubbard Model (HM), which has been studied in various forms and solved using different methods~\cite{arovas2022hubbard}. Several approaches can be used to address strong correlation effects within DFT, such as the GW approximation~\cite{reining2018gw} and Dynamical Mean-Field Theory (DMFT)~\cite{kotliar2006electronic,aoki2014nonequilibrium}. In this work, the DFT+U method~\cite{himmetoglu2014hubbard} is used for comparison with experimental data, while also referencing results from studies involving DFT coupled with DMFT. 

In the DFT+U approach, an additional term is incorporated into the exchange-correlation potential to account for the electron interactions on the same site, along the line of the HM. The strength of this term is parameterised by the Hubbard parameter $U$, which is typically calibrated through comparisons with values of physical quantities reported in the literature due to the complexity of determining it from theory. In contrast, DMFT maps the many-body lattice problem to a local many-body impurity model, where interactions between electrons on different sites are replaced by a single electron interacting with an electronic "bath."


%This term typically takes the form $\sum_i U(n_{i,\downarrow}-n_{i,\uparrow})^2$, where $n_{i,\downarrow}$ and $n_{i,\uparrow}$ represent the number of electrons on atom $i$ with spin down and up, respectively, and $U$ is the Hubbard interaction parameter that must be carefully calibrated.













\section{\label{iron oxides} Iron Oxides under Compressed Conditions}










Iron oxides are important for understanding Earth's core and mantle dynamics. The core, primarily composed of iron and nickel, contains up to 5.3\% oxygen, with smaller percentages for the inner core~\cite{hirose2021light}. Furthermore, iron oxides like Fe$_2$O$_3$ may be present near the core-mantle boundary (CMB) due to subduction of ancient iron formations~\cite{keller2023links}. These oxides can remain stable under extreme pressures and temperatures. Furthermore, several studies suggest that, between 40 and 160 GPa and at temperatures around 0.1 eV, iron oxides undergo a Mott transition, becoming conductive, and that this could affect Earth's global dynamics~\cite{keller2023links, pasternak1999breakdown, greenberg2018pressure, greenberg2023phase, badro2002nature, leonov2015metal}. Therefore, the analysis of the electronic structures of iron oxides in extreme conditions is of great interest both to geophysics, and to the field of strongly correlated systems.

During compression, iron oxides experience metal-insulator, spin and structural transitions, which are essentially due to the deformation of the lattice produced by pressure-induced strains. The distortion of atomic bonds and the cell-volume reduction may change the most favourable lattice structure and modify the localisation of states and the charge transfer, affecting the conduction properties of the material. The change in the atomic bonds may also induce changes in the average electron spin state (spin crossover), as electrostatics interaction between orbitals of neighbouring ions starts to dominate over exchange energy~\cite{hauser2004ligand}. There have been many studies on the correlations between these metal-insulator, structural and spin transitions, but the argument remains still debated~\cite{gavriliuk2008another, lyubutin2009spin, leonov2016magnetic}.

Although the techniques developed in this thesis are applicable to any material, this work focuses specifically on two iron oxides: Fe$_2$O$_3$ (Hematite) and FeO (Wüstite). Theoretical studies using DFT+DMFT have shown that Fe$_2$O$_3$ undergoes a structural transition around 50 GPa from a corundum-type structure to a double-perovskite (DPv) phase~\cite{greenberg2018pressure}. This transition is accompanied by a site-selective spin and Mott transition, where only one of the two iron ions in the basic unit experiences these changes~\cite{greenberg2018pressure, lyubutin2009spin}. The 3d orbitals of this ion delocalise, causing the band gap to close, and simultaneously, the energy splitting between the valence orbitals\footnote{These valence orbitals, for these crystalline structures, are divided into two symmetry groups - $e_g$ and $t_{2g}$ - which have different energies} increases, leading to a collapse of the local spins. These findings are supported by Mossbauer spectroscopy and x-ray diffraction data~\cite{greenberg2018pressure}.

For FeO, analogous studies have revealed a complex phase diagram with several structural transitions, including shifts from the B1 (NaCl) phase to the B8 (NiAs) and B2 (CsCl) phases at various pressures and temperatures~\cite{greenberg2023phase, leonov2015metal, leonov2020emergence}. At approximately 70 GPa and temperatures below 1500 K, the B1 phase of FeO becomes metallic, linked to the delocalisation of the iron 3d orbitals and the consequent collapse of the Mott insulating gap. Interestingly, the high-spin (HS) to low-spin (LS) transition in FeO appears to occur independently of metallisation, with the HS metallic phase persisting up to pressures of 150 GPa, depending on the temperature~\cite{greenberg2023phase}. Beyond this point, FeO transitions to the B8 phase, with the collapse of the iron ions' local magnetic moments. These structural changes have a significant impact on FeO's properties, including its bulk modulus and volume~\cite{greenberg2023phase}.





% Maybe you can put P-T phase diagrams



\section{Overview of the Diagnostic Techniques and Instruments}







As mentioned earlier, due to the short life of experimentally-created HED systems, photon-based techniques are predominantly used to investigate their properties. Over the past few decades, various forms of spectroscopy have been employed to study both the electronic and ionic structures of these systems, while simultaneously diagnosing their thermodynamic conditions. X-ray emission spectroscopy (XES) and x-ray absorption spectroscopy (XAS), in particular, have played a pivotal role in seminal studies by measuring changes in binding energies under extreme thermodynamic conditions~\cite{hoarty2013observations, recoules2009temperature, ping2016laser}. Within XAS, it is possible to distinguish two regimes—x-ray absorption near edge structure (XANES) and extended x-ray absorption fine structure (EXAFS)—which allow for the investigation of the electronic structure around the chemical potential, as well as the oxidation state, coordination state, and interatomic distances of ions~\cite{teo2012exafs,rehr2005progress, wilke2001oxidation, ponader1989rare}. These techniques, particularly relevant to the topics covered in this thesis, are discussed in more detail in Chapter 3. To examine the local environment around ions and crystallographic structures more generally, XRD remains the most widely used method~\cite{bykova2013novel, zachariasen1967general}. Investigating how material properties evolve with changes in temperature and density also requires precise thermodynamic diagnostics. Traditionally, this has been achieved using x-ray Thomson scattering (XRTS)~\cite{glenzer2009x, visco2012measurement, kozlowski2016theory, kritcher2008ultrafast}, though as discussed in Chapter 3, XAS and XES can also directly measure thermodynamic parameters. In all these methods, thermodynamic properties are found by fitting theoretical models to experimental data.

Only in the last years, resonant inelastic x-ray scattering (RIXS) - another typical spectroscopic technique of solid state physics - begun to be applied in the field of warm dense matter~\cite{humphries2020probing, gel2021dynamics}. RIXS is a spectroscopic technique used to probe the electronic, magnetic, and vibrational properties of materials with high sensitivity. In the RIXS process, an incident x-ray photon excites, via an energy non-conserving absorption, a core electron into an unoccupied state, creating a core hole. This hole is subsequently filled by an electron from another shell, resulting in the emission of a photon. The energy of the emitted photon is such that the overall scattering process conserves energy. It is intuitive that this process reveals information about the system's structure of vacant states. Unlike other forms of x-ray scattering, RIXS is resonant, meaning the incident photon energy is tuned close to specific absorption edges of the material, enhancing the sensitivity to particular electronic states. One of RIXS’s key advantages over traditional absorption or emission techniques is its ability to capture the electronic structure of a material across a broad electron energy range in a single spectrum, thereby providing a global picture of this structure. This makes this technique particularly useful for time-resolved studies of dynamic processes. As RIXS is the technique employed in this thesis for the extraction of electronic structures, a detailed description of the theory behind this process is given in Chapter 3. 

Now a brief overview of the typical experimental instruments used in XFEL facilities for HED physics investigations is provided. For further details, refer to the specific documentation~\cite{zastrau2021high, nakatsutsumi2014technical, nagler2015matter}. Before reaching the experimental chamber, the FEL beam travels through several optical components and diagnostic tools, such as x-ray gas monitor (XGM), spectrometers and various imagers that measure the beam characteristics (e.g. energy, spatial and spectral profile)~\cite{zastrau2021high}. Additionally, the beamline includes attenuators, a system of mirrors, refractive lenses, and slits to guide and focus the laser pulse, along with a monochromator~\cite{kujala2020hard}, and a rotating slit to select specific laser pulses, known as pulse picker unit (PPU)~\cite{zastrau2021high}. 

\begin{figure}
\begin{center}
  \includegraphics[width=\columnwidth]{Figures/experimental chamber.pdf}  
  \caption[CAD representation of a typical HED experimental chamber]{(a) Top view of a CAD model of the EuXFEL HED Chamber 1, illustrating the primary laser beam paths and sample position. (b) Perspective view of the same chamber, highlighting the configuration of the facility's main spectrometers, including a von H\'amos spectrometer and diced analysers. The VISAR system, not shown in the images, is located in the separate HED Chamber 2, downstream of Chamber 1. Both images were taken from Ref.~\cite{zastrau2021high}.} 
   \label{experimental chamber}
\end{center}
\end{figure}



Upon entering the experimental hutch, the laser passes through more focusing lenses and diagnostic tools, including a photon arrival monitor (PAM) to synchronise the FEL beam with the optical lasers, and devices for their spatial alignment. The optical lasers are typically diode-pumped lasers, with an optional chirped pulse amplification scheme, capable of delivering up to dozens Js at a repetition rate of 10 Hz, in a pulse duration as short as few ns, and with wavelengths ranging from 500 nm to 1000 nm. The optical laser system, integrated into the HED instrument, includes deformable mirrors for wavefront corrections and off-axis parabolas for focusing. Fig. \ref{experimental chamber} illustrates an example of the laser paths, alongside a CAD model of the experimental chamber at the EuXFEL facility.  The main components of the experimental chamber, which operates under vacuum conditions, include a motorised and highly mobile target mount~\cite{nakatsutsumi2014technical}, a VISAR system, and various pixel detectors and dispersing crystals used for different types of spectroscopy and diffraction experiments~\cite{zastrau2021high}. A detailed description of the typical pixel detectors and spectrometers used in HED experiments is provided in the next chapter.











% Microscopes





%IC2 is a chamber optimized for diffraction experiments involving high photon energies and MHz time structure X-rays, particularly for studying materials under extreme conditions like high pressure and temperature. It hosts two experimental platforms: one for diamond anvil cell (DAC) experiments and another for dynamic laser compression, both of which are highly adaptable for various experimental setups.

%The DAC platform is designed for X-ray and laser heating experiments with precise motorisation and can accommodate multiple DACs. It allows for laser heating, sample alignment, and fast pyrometry measurements. The dynamic laser compression platform supports flexible configurations for high-energy laser experiments and diagnostics, including X-ray diffraction and VISAR velocity measurements.

%Two detector systems are used in IC2: the AGIPD 1M detector, which captures diffraction patterns at high repetition rates, and the Varex twin-detector system, which ensures broad, gapless coverage for DAC and shock experiments. Both detectors are EMP- and debris-resistant, with the AGIPD excelling in high-speed data acquisition and the Varex system offering large-angle coverage in diffraction experiments.

%The system is equipped with advanced diagnostics like VISAR for measuring shock velocities, Streaked Optical Pyrometry (SOP) for temperature measurements, and Fourier Domain Interferometry (FDI) for studying phase shifts and plasma expansion, enabling a wide range of dynamic material studies under extreme conditions.



% The chamber also houses a variety of X-ray spectrometers designed for high-resolution emission and inelastic scattering spectroscopy, as well as compact X-ray detectors compatible with vacuum environments. These tools offer precise diagnostics for studying extreme states of matter.






% consisting of a series of motorized stages for positioning samples with high precision







%The ReLaX laser at the European XFEL is a high-intensity relativistic laser based on a titanium sapphire system. It uses a double chirped pulse amplification scheme with enhancements for temporal contrast and spectral broadening. Capable of delivering up to 300 TW at 5 Hz and 100 TW at 10 Hz with pulse durations as short as 25 fs, the laser operates at a wavelength of 800 nm. It features a robust design with high redundancy and oversized optics, ensuring low fluence. The system is integrated into the HED instrument and includes deformable mirrors for wavefront correction. Low-energy probe beams are available for optical probing or timing correlation with X-rays, and experiments are conducted in the IC1 chamber using off-axis parabola focusing.

%The DiPOLE 100-X laser is a diode-pumped ytterbium
%laser, delivering up to 100 J with a 10 ns pulse or 37 J with a 2 ns pulse. Designed primarily for laser-driven shock and compression experiments, it offers pulse shaping with high precision and can operate at 10 Hz. Frequency doubling to 515 nm has been demonstrated, and synchronisation with the XFEL system ensures minimal timing jitter. The DiPOLE 100-X will be available in both IC1 and IC2 chambers for various irradiation geometries, and it is expected to play a key role in upcoming user experiments alongside other diagnostics like VISAR and XRD. Both lasers were integrated into the European XFEL in 2019, with the first user experiments planned for 2022.


% \subsection{Spectroscopic analysis}




\section{Structure of Thesis and Author Contributions}





% Modify as in Paddy's thesis and thanking ---> contributions



This thesis is organised into six chapters. 

%\textbf{Chapter 1} introduces the field of dense plasmas, outlining their main characteristics and key physical parameters and providing an overview of theoretical models used to describe them. Among these models, particular emphasis is placed on density functional theory, the primary theoretical framework employed in this work to benchmark experimental data. Additionally, this chapter explores the experimental realisation of HED systems relevant to this thesis, along with the typical experimental setups and diagnostic techniques used to study them. Lastly, it discusses the characteristics of the materials under investigation and the specific properties of interest. This chapter serves as a comprehensive introduction, focusing on complementary topics pertinent to the thesis but not covered in its main body, which is dedicated to the development of the theory and techniques central to this work. Serving as a general introduction, this chapter draws its content from the existing literature. The author would like to thank Mr. Pontus Svensson for the useful discussions on the theoretical models for HED systems.


\textbf{Chapter 1} serves as a comprehensive introduction, focusing on complementary topics pertinent to the thesis but not covered in its main body, which is dedicated to the development of the theory and techniques central to this work. Serving as a general introduction, this chapter draws its content from the existing literature. The author would like to thank Mr. Pontus Svensson for the useful discussions on the theoretical models for HED systems and for reviewing this chapter.

\textbf{Chapter 2} provides a step-by-step guide on how to construct a spectrum from the initial camera images, focusing on the spectrometers and detectors commonly used in experiments, and describing their primary challenges. In doing this, the chapter presents methods for enhancing the signal-to-noise ratio and the resolution of a camera image, designed to handle simultaneously regions with high and low photon densities. These methods are tested on synthetic images and experimental data from HED science. While this discussion is motivated by the potential application to the spectroscopic techniques used in HED experiments, the methods presented are generalisable to other scientific fields and hold intrinsic interest. Notably, the content of this chapter has been compiled into a paper published in the \textit{Journal of Applied Physics}.

This paper draws inspiration from the previous work of Dr. Oliver Humphries~\cite{humphries2020isochoric}, to whom the author is highly indebted. Portions of the code for photon-counting algorithms and spectrometers' calibration were developed collaboratively with Dr. Thomas Gawne, Mr. Thomas Campbell, and Mr. Yuanfeng Shi, while the remainder of the code was independently written by the author, who also conducted all simulations. The author further acknowledges Prof. Sam M. Vinko, Dr. Oliver Humphries, and Dr. Thomas Gawne for their valuable discussions and their thoughtful revisions of the paper.

\textbf{Chapter 3} opens with a comprehensive review of the theory behind the RIXS process, tailored to the specific cases relevant to this work and essential for developing the model used to interpret the experimental RIXS data. The chapter then provides a brief overview of alternative spectroscopic techniques within the context of HED physics, highlighting their advantages and, especially, their limitations compared to the RIXS technique. Following this, the method for extracting electronic structures from RIXS measurements—based on neural networks and differentiable physical models—is introduced. This approach is first presented in the broader context of machine learning techniques for solving inverse problems in physics and is then specialised to the RIXS case. The method, labelled as STEP, is validated using synthetic RIXS data, with its performances analysed across various aspects, including comparisons with standard numerical techniques and the extraction of scalar quantities such as temperature. 

This method originated from discussions between the author and Prof. Sam M. Vinko and was subsequently developed and implemented by the author. The author wishes to acknowledge Mr. Karim K. Alaa El-Din for polishing the code and the generalisation of the method's framework to other physical problems. The generation and analysis of the synthetic RIXS data were conducted by the author in collaboration with Mr. Karim K. Alaa El-Din, leading to the publication of a paper in \textit{Royal Society Open Science}, mainly written by Mr. Karim K. Alaa El-Din. The comparison of the method with numerical techniques was carried out solely by the author, while the study on temperature extraction presented here was predominantly conducted by Mr. Karim K. Alaa El-Din.

\textbf{Chapter 4} focuses on the analysis of experimental RIXS spectra collected at the EuXFEL facility for warm-dense iron, iron oxides, and gadolinium (Gd). After introducing the experimental setup and primary diagnostics, the extracted electronic structures of Fe and Fe$_2$O$_3$ are presented, allowing also the determination of sample temperature and electron binding energies. I further show that this method can extract DOSs with a resolution exceeding the x-ray source bandwidth, enabling clear differentiation between the two materials.  The reliability of the reconstructed structures is validated by repeating the extraction process on datasets recorded with different instruments and XFEL operational modes. To further enhance resolution, the spectrometer's instrument function is deconvolved from the FeO RIXS spectra, achieving DOS for this material with resolutions on the order of a few eV. Finally, the RIXS technique is applied to gadolinium, showcasing its flexibility for analysing various materials through different electronic excitations. All extracted electronic structures were compared with finite-temperature DFT calculations, showing excellent agreement.

The analysis of these experimental data was performed entirely by the author. The results for Fe and Fe$_2$O$_3$, along with some synthetic data analysis (see Chapter 3), were published in \textit{Communications Physics}, serving as a proof of principle for the method's effectiveness. This experiment was made possible through the collaborative efforts of many contributors, and the author extends special thanks to Prof. Sam M. Vinko for proposing the experiment and to Dr. Céline Crépisson, Dr. Oliver Humphries, Dr. Thomas Preston and Dr. Thomas Gawne for their significant help. The results for FeO and gadolinium have been drafted for a separate publication, with the choice of journal still under consideration. The DFT calculations presented were performed by Dr. Thomas Gawne.

\textbf{Chapter 5} presents the analysis of experimental RIXS spectra collected at the LCLS facility for dynamically compressed FeO. This study represents the ultimate goal of the thesis, showcasing the application of this RIXS diagnostic technique to dynamical physical processes within the context of HED physics. Similarly to the previous chapter, the discussion begins with an overview of the experimental setup and primary diagnostics, with a focus on the functioning of the VISAR system, which was used to measure pressure conditions. The chapter then delves into the XRD analysis of the compressed targets, which was employed to observe the evolution of the ionic structure under compression, concurrently with changes in the electronic structure, measured using this RIXS-based approach. Applying this method to RIXS spectra collected on both ambient and compressed FeO revealed a systematic bandgap closure upon compression, consistent with the insulator-to-metal transition predicted by theory and earlier studies. 

%However, the limited number of spectra collected for compressed targets resulted in a poor SNR, which constrained the resolution of the DOS. This limitation prevented the attainment of precise and conclusive results suitable for publication. Despite these challenges, the preliminary findings, which successfully captured the expected trends, provide strong motivation to repeat the experiment to obtain enhanced data quality. Such refinements could yield higher-quality results, which would represent a significant achievement in HED physics.

The author expresses deep gratitude to all participants in this experiment, particularly Prof. Sam M. Vinko and Dr. Danae N. Polsin for proposing the study and involving the author. The analysis of VISAR and XRD data, as well as target fabrication, was conducted by Dr. Marion Harmand, Dr. Céline Crépisson, Dr. David McGonegle, Dr. Alexis Amouretti, Dr. David A. Chin, and Dr. Danae N. Polsin. The spectrometer design was a collaborative effort between the author, Dr. Thomas Gawne, and Mr. Thomas Campbell. The DFT calculations presented were performed by Dr. Sam Azadi. All other work presented in this chapter was carried out by the author.

\textbf{Chapter 6} concludes the thesis with a summary of key findings and a discussion of potential next steps to enhance and refine this diagnostic technique and the associated extraction method. It also offers an outlook on the possible generalisations of the STEP method to other physical investigations. On this last point, the author extends his gratitude to Dr. Patrick Heighway for the valuable discussions.

In general, like all theses, much of the work presented here reflects years of development and is collaborative in nature. My foremost acknowledgment goes to my supervisor, Prof. Sam M. Vinko, whose extensive guidance and support have been fundamental to shaping my research. I have also had the opportunity of regularly presenting my work to the members of the Vinko-Wark research group, who provided me with valuable feedback throughout this journey. I am very grateful to every member of this group and especially to Prof. Justin S. Wark, Dr. Thomas Gawne, Mr. Pontus Svensson, Mr. Thomas Campbell, Dr. Sam Azadi, Dr. Céline Crépisson, and Mr. Karim K. Alaa El-Din for their insights and support. A particular acknowledgment goes to Dr. Thomas Gawne and Dr. Sam Azadi for performing the DFT calculations included in this thesis.

% a material with interesting properties under high pressure and significant relevance to geophysical research


\section{Relevant Publications}



\begin{itemize}
    \item \textbf{A. Forte}, T. Gawne, K. K. Alaa El-Din, O. S. Humphries, T. R. Preston, C. Crèpisson, T. Campbell, P. Svensson, S. Azadi, P. Heighway, Y. Shi, D. A. Chin, E. Smith, C. Baehtz, V. Bouffetier, H. Hoppner, A. Amouretti, D. McGonegle, M. Harmand, G. W. Collins, J. S. Wark, D. N. Polsin, and S. M. Vinko, "Resonant inelastic x-ray scattering in warm-dense Fe compounds beyond the SASE FEL resolution limit", \textit{Communications Physics}, \textbf{7}, 266, (2024).

    \item \textbf{A. Forte}, T. Gawne, O. S Humphries, T. Campbell, Y. Shi, S. M Vinko, "Methods for energy dispersive x-ray spectroscopy with photon-counting and deconvolution techniques", \textit{Journal of Applied Physics}, \textbf{7}, 134501, (2025).

    \item K. K. Alaa El-Din, \textbf{A. Forte}, M. F. Kasim, F. Miniati, S. M Vinko, "STEP: extraction of underlying physics with robust machine learning", \textit{Royal Society Open Science}, \textbf{11}, 231374, (2024). \\
    \textbf{Contribution and Relevance:} The paper is based on the methodology developed by the author of this thesis and detailed in Chapter 3. Nearly all of the results were produced collaboratively by the first two authors of the paper, with a portion of them presented in Chapter 3.

    \item A. Amouretti, C. Crèpisson, S. Azadi, D. Cabaret, T. Campbell, D. A. Chin, B. Colin, G.R. Collins, L. Crandall, G. Fiquet, \textbf{A. Forte}, T. Gawne, F. Guyot, P. Heighway, H. J. Lee, D. McGonegle, B. Nagler, J. Pintor, D. N. Polsin, G. Rousse, Y. Shi, E. Smith, J. S. Wark, S. M. Vinko, M. Harmand, "Phase transitions of Fe$_2$O$_3$ under laser shock compression", arXiv:2402.18432, (2024) \\
    \textbf{Contribution and Relevance:} The data presented in this paper were obtained during the LCLS experiment detailed in Chapter 5, in which the author contributed. These results provided deeper insights into the evolution of ionic structure under compression, along with further details on the techniques employed in the investigations. This paper was recently accepted for publication in \textit{Physical Review Letters}.

    \item C. Crèpisson, A. Amouretti, M. Harmand, C. Sanloup, P. Heighway, S. Azadi, D. Mcgonegle, T. Campbell, D. A. Chin, E. Smith, L. Hansen, \textbf{A. Forte}, T. Gawne, H. J. Lee, B. Nagler, Y. Shi, G. Fiquet, F. Guyot, M. Makita, A. Benuzzi-Mounaix, T. Vinci, K. Miyanishi, N. Ozaki, T. Pikuz, H. Nakamura, K. Sueda, T. Yabuuchi, M. Yabashi, J. S. Wark, D. N. Polsin, S. M. Vinko, "Shock-driven amorphization and melt in Fe$_2$O$_3$", \textit{Physical Review B}, \textbf{111}, 024209, (2025). \\
    \textbf{Contribution and Relevance:} Part of the data presented in this paper were obtained during the LCLS experiment detailed in Chapter 5, in which the author contributed. These results provided deeper insights into the evolution of ionic structure under compression, along with further details on the techniques employed in the investigations.

    \item T. Gawne, T. Campbell, \textbf{A. Forte}, P. Hollebon, G. Perez-Callejo, O. Humphries, O. Karnbach, M. F. Kasim, T. R. Preston, H. Ja Lee, A. Miscampbell, Q. Y. Berg, B. Nagler, S. Ren, R. B. Royle, J. S. Wark, S. M. Vinko, "Investigating Mechanisms of State Localization in Highly-Ionized Dense Plasmas", \textit{Physical Review E}, \textbf{108}, 035210, (2023). \\
    \textbf{Contribution:} The author collaborated to the analysis of the experimental data used to benchmark the theoretical model, where some of the techniques described in Chapter 2 were employed.



\end{itemize}













