



\chapter{\label{Introduction}Introduction}




\section{Quantum Plasmas}









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 ionized with respect to their ambient condition state, yielding an ensemble of electrically charged particles interacting via electromagnetic forces, which basically defines the state of plasma. 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''. 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, we are able to recreate these states of matter in laboratory settings, including inertial confinement fusion (ICF)~\cite{atzeni2004physics, nuckolls1972laser, betti2006progress, moses2009national}, dynamic and static compression~\cite{smith2014ramp, jayaraman1983diamond}, and laser 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.

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 particles exhibit quantum behaviour and 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 ionization state and introducing a range of phenomena, such as collisional ionization and electron recombination~\cite{salzmann1998atomic, pert1990models}. To address these challenges, the modelling of dense plasmas employs a multiscale approach, which will be explored more in detail in section~\ref{theoretical models}. With this approach, different models are used to study the system at different length scales: microscopic models calculate plasma properties, which are then incorporated as input parameters into macroscopic models to predict the system's behaviour for most practical applications. This process highlights the importance of accurately estimating key plasma properties, such as the equation of state and transport coefficients. Given the complexity of these systems, experimental investigations are indispensable for validating and complementing theoretical predictions. Experimentally-measured quantities are used as input parameters in simulations, or more frequently, theoretical models are employed to interpret experimental data, 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, governing its equation of state~\cite{wang2000calculated}, as well as its transport and spectral properties~\cite{kubo1966fluctuation}. Estimating the DOS is therefore essential for making accurate predictions about the system's dynamics and behaviour.

\begin{figure}
\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 paramter ($\theta_e=1$) are included.} 
   \label{diagram_n_t}
\end{center}
\end{figure}

Plasmas can be classed by a few dimensionless parameters, which we will now review. 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 $T_i$. The ion population is decomposed in $N_s$ ion species with charge state $z_j$ and number density $n_j$, with $j = 1, 2, \dots, N_s$. Since free electrons are lost by the atoms, plasmas are globally neutral, meaning that $n_e = \langle z \rangle n_i$, where $\langle z \rangle = \frac{1}{n_i}\sum_{j=1}^{N_s} z_j n_j$ is the mean charge state of ions. The correlations between particles are quantified by the \textit{Coulomb coupling parameter} $\Gamma_{\alpha \beta}$, which is the ratio of the average 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} \biggl(\frac{4\pi n_i}{3}\biggr)^{1/3},
    \label{gamma parameter}
\end{equation}
where $k_B$ is the Boltzmann constant, $e$ is the elementary charge, and $\epsilon_0$ is the vacuum permittivity. Similar expressions can be defined for the electron-electron and the electron-ion coupling parameters~\cite{ichimaru1982strongly}. If the plasma species are in thermodynamic equilibrium, these $\Gamma$ values are all similar. A plasma is considered weakly coupled if $\Gamma \ll 1$, meaning Coulomb interactions are weak and the plasma can be treated similarly to a gas, where almost no structure is present. In contrast, a strongly coupled plasma ($\Gamma > 1$) exhibits stronger Coulomb interactions, and under sufficiently strong coupling, ordered structures can form~\cite{baus1980statistical}. 

To quantify the quantum effects instead, we introduce another parameter $\theta_{\alpha}$, defined as follows:

\begin{equation}
\theta_{\alpha} = \frac{T_{\alpha}}{T_{F,\alpha}},  \ \ {\rm with} \ \ T_{F,\alpha} = \frac{h^2}{8 k_B m_\alpha}\biggl(\frac{3 n_\alpha}{\pi}\biggr)^{1/3},
    \label{theta parameter}
\end{equation}
where $m_{\alpha}$ is the rest mass of the particle species, $T_{F,\alpha}$ is the Fermi temperature of the particle population, and $h$ is the Planck constant. One can show~\cite{landau1980statistical} that if $\theta_{\alpha} \lesssim 1$ quantum mechanics must be used to describe the system, as it is degenerate, with the electrons (ions) following Fermi-Dirac (Bose-Einstein) statistics. If $\theta_{\alpha} \gg 1$, the particles behave classically and obey the Maxwell-Boltzmann distribution. This condition can also be interpreted in dynamical terms, recalling the thermal De Broglie wavelength of particles $\alpha$,

\begin{equation}
\Lambda_{\alpha} = \biggl(\frac{h}{2\pi m_{\alpha}k_B}\biggr)^{1/2},
    \label{De Broglie parameter}
\end{equation}
which estimates the spatial extension of $\alpha$ particle's wavefunctions. In fact, it can be shown that $\theta_{\alpha}$ is also the ratio between average interparticle spacing $R_{\alpha}\propto \/n_{\alpha}^{-1/3}$ and $\Lambda_{\alpha}$, suggesting that if $\theta_{\alpha}$ is small, particles cannot be treated as pointlike objects and quantum mechanics must be employed to describe their evolution.

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 $\theta_e$. The region below these lines is occupied by the HED plasmas, whereas the part above is associated with classical plasmas. In this thesis, we will deal with the regions of the diagram occupied by isochorically-heated matter and plasmas generated by laser-driven planar shocks, both of which are strongly-coupled and degenerate plasmas.




















\section{Experimental Realization of HED Systems}








Technological advancements started in the second half of the 20th century have made it possible to create extreme thermodynamic states in laboratory settings. These methods can be broadly categorized into three types: static compression by means of diamond anvil cells~\cite{jayaraman1983diamond}, dynamic compression driven by powerful optical lasers~\cite{wark2022femtosecond} and, only more recently, isochoric heating achieved by irradiating targets with the modern X-ray Free Electron Lasers (XFELs)~\cite{vinko2012creation}. In this section, we will give an overview of the mechanisms behind the creation of HED matter for the latter two methods, as static compression is outside the scope of this thesis. Plasmas generated by powerful lasers are typically very short-lived, lasting only tens of picoseconds, as extreme thermodynamic conditions quickly dissipate due to processes like heat conduction and hydrodynamic expansion. Consequently, \textit{in situ} techniques, almost entirely based on spectroscopy, are necessary for experimental investigations of these systems.









\subsection{Laser-Compressed Systems}








Laser-driven compression of matter can be classified into two categories: planar shocks~\cite{zeldovich1967theory}, commonly used for pure physical investigations, and spherical shocks~\cite{atzeni2004physics}, mostly employed in nuclear fusion applications. Despite the two being quite similar, we will examine only planar shocks, with only brief remarks on spherical ones.

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 the so-called \textit{ablator}, usually made of plastic. Behind it lies the material to be compressed and studied, potentially followed by additional layers for diagnostic purposes (see Chapter 5 for details). Nanosecond optical lasers, such as standard Nd-YAG 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 vaporize and ionize. These outer layers enter a plasma state, allowing the laser radiation to pass through up to the solid layers with critical density:

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

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. 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}. This pressure is build up by a series of shockwaves, determined by the laser pulse shape, which are sent into the solid. The material under study is thus compressed by these shockwaves, which distort its lattice and heat its constituents, eventually leading to melting in some cases. A more detailed description of the equations and the mechanic of these shock compressions can be found in the next section.

Because of the laser's wavelength and pulse duration ($\sim$ 10s of ns and $\sim$ 10s-100s of $\mu$m, respectively), the laser primarily interacts 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 \textit{inverse bremsstrahlung}, 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 Dynamic Solid Compressions}









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

In dynamic shock compression, the material conditions before and after the shock front must satisfy conservation equations, 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{Here, we consider the case of isotropic stresses in the solid.}, 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} 
\end{align}
where $U_P$ and $U_S$ are the particle velocity (in the post-shock state) and the shock velocity, respectively, $p$ is the pressure, 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 VISAR (Velocity Interferometer System for Any Reflector) technique~\cite{dolan2006foundations}, which will be described in more detail in Chapter 5. In many instances, a linear relationship between $U_S$ and $U_P$ is assumed, expressed as $U_S = \alpha + \beta U_P$, where $\alpha$ and $\beta$ are known constants. This simplifies the process, requiring only one parameter to be measured. However, this linearity is not a universal rule and should be applied cautiously.

The locus of states achieved through shock compression is known as the Hugoniot, which, for a given compression, results in pressures that lie above both the isotherm and isentrope due to shock heating\footnote{Typically, even though two parameters must be measured, only one is independent and determines the shock state, as the two are linked by a functional relation specific to the material. In many cases, this relation is linear.}. As an example, the Hugoniot and isentrope curves for iron are illustrated in Fig. \ref{iron phase diagram}, alongside its melting curve.

\begin{figure}
\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.} 
   \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, driven solely by ablation pressure~\cite{smith2014ramp}. Mix compression (shock + ramp) are also possible.

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 compression wave reaches the interface with the tamper window, a shock wave propagates into the window while a release wave is reflected back into the target, further altering its thermodynamic state. The presence of a tamper layer allows higher pressures to be retained in the released state compared to when no tamper is used. Additionally, if the Hugoniot curve of the tamper material is known, the thermodynamic parameters of the compressed and released sample's states can be found using the impedance matching method~\cite{celliers2005systematic}. The thermodynamic history of a shocked sample with a tamper window is schematically illustrated in Fig. \ref{impedance matching}, in both for $x$-$t$ and the $\rho$-P plane.

\begin{figure}
\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 $p–u$, where $p$ and $u$ 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 $u$ and knowing the window's Hugoniot curve along with the sample's relaxation curve, all states can be determined. 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}



Understanding the microscopic mechanisms that occur upon solid compression is complex, and many studies have been performed in this regard~\cite{mcbride2019phase, wehrenberg2017situ, milathianaki2013femtosecond}. Here, we give a brief description of the main findings. If the shock is not too intense (below the strong shock limit, as it is typically the case), an elastic precursor wave travels through the target, deforming elastically the lattice and increasing the shear stresses. These stresses are then released by a plastic wave, which produces permanent lattice distortions, in some cases, leading to shock-induced phase transitions of the crystalline structure. These deformations can occur via various mechanisms such as rotation and twinning~\cite{heighway2021kinematics}. The data from these compression experiments are typically interpreted with flow plasticity and dislocation theory~\cite{lubliner2008plasticity} coupled with molecular dynamics (see also Section \ref{theoretical models}).

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 occur.




















% 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

% Yield under uniaxial compression occurs when plastic deformation occurs.


% Rule : particle velocity = 2*surface velocity



\subsection{Isochorically-Heated Systems}










Another way for creating HED matter is by means of X-ray free electron lasers. Despite a full description of these lasers, which can be easily found in the literature~\cite{schmuser2014free}, is outside the scope of this thesis, we will briefly outline some key features here. 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, making them highly tunable. 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}. This microbunching allows the electrons to radiate coherently, amplifying the intensity of the emitted light through a process of Self-Amplified Spontaneous Emission (SASE). 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 XFEL to become operational in 2009, delivers approximately 1 mJ of X-rays with a 0.3\% bandwidth in pulse durations on the order of 10s of femtoseconds~\cite{emma2010first}. Considering the impressive beam divergence of only 2 $\mu rad$, which is significantly narrower than that of traditional synchrotron sources, focal spot size of $\sim$ 100 nm are achievable by means of specific lenses~\cite{hudec2010kirkpatrick}, allowing spatially localized investigations with radiation intensity of $10^{16}$-$10^{17}$ W$cm^{-2}$ on the target. Additionally, thanks to the large penetration depths of X-rays, XFELs can volumetrically heat the sample, creating quite uniform thermodynamic conditions. 

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 photoionization 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 photoionization decreases with increasing kinetic energy of the ejected electron, deep core electrons are more likely to excited. When photoionization occurs, the ejected electron will thermalize 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 thermalized or not, bound or free - recombine 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 that eventually thermalizes, contributing further to the rise in electron temperature. In general, energy is transferred from the laser to the electron population on timescales typical of radiative and 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 will thermalize with the ion population through electron-phonon interaction, eventually causing the solid to melt and consequently expand. However, if we probe the system on shorter timescales, we can analyse a solid material, with an ordered crystalline structure, but with a hot electron population. This task is achieved by using the XFEL itself, which heats and probe the target simultaneously~\cite{vinko2012creation}. Alternatively, these instruments can serve as probes when coupled with optical lasers to perform time-resolved spectroscopy on dynamically compressed systems~\cite{zastrau2021high}. This experimental technique constitutes the ultimate focus of this thesis.


















\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 are applied~\cite{lubliner2008plasticity, orowan1949fracture}, where macroscopic material properties, such as elastic/plastic constants and heat capacities, are calculated using microscopic models or comparing against experimental data. These microscopic models are based on dislocation theory combined with molecular dynamics (MD) calculations~\cite{heighway2021kinematics}. The interatomic potential for the MD simulations is determined using the same methods applied to compute interatomic forces in dense plasmas and will be 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, while the dynamics of the plasma is described by the magnetohydrodynamic equations. In cases where ion motion and thermodynamic 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 the transport coefficients for the fluid equations, smaller-scale models are necessary. These models can be coarsely divided 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}).}, such as particle-in-cell (PIC) codes~\cite{fonseca2002osiris}, and, at even smaller length scales, molecular approaches, which can also treat interparticle correlations and includes methods such as Monte Carlo methods~\cite{gann1979monte} (for static conditions) or molecular dynamics simulations~\cite{thompson2022lammps}. The latter is particularly popular for the plasmas of our interest. For these MD simulations, the interatomic potential between ions 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 hot quantum plasmas. An alternative method for handling electronic length scales and computing electronically-averaged quantities is the Path Integral Monte Carlo (PIMC)~\cite{gubernatis2016quantum}. While PIMC outperforms DFT at higher temperatures, it is less effective at lower temperatures and unsuitable for dynamic processes. Despite the different timescales of ion and electron dynamics, non-adiabatic effects have been shown to influence certain quantities in out-of-equilibrium conditions~\cite{mabey2017strong}. Hence, approaches that treat simultaneously ion and electron motion have been recently proposed~\cite{feldmeier2000molecular, svensson2023development}. Since DFT, with all its variants, will be the primary theoretical framework employed in this thesis, we will now briefly outline its main concepts.





%\footnote{These methods can actually describe the macroscopic length scales as well, replacing the fluid models.}





\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 Schrödinger equation by mapping it to a functional minimization problem based solely on the electron density, $n(\boldsymbol{r})$, of the system. 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 electron density, which is the unique 3D function that minimizes the total energy functional. While the Hohenberg-Kohn theorems originally 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, we will now discuss the finite temperature (FT) DFT~\cite{mermin1965thermal}, for which Mermin extended the (HK) theorems. Instead of minimizing the total energy functional, the relevant functional at finite temperature is the Mermin grand potential functional, $\Omega[n(\boldsymbol{r})]$. 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 all the system properties can be determined. At this density, the Mermin functional becomes the grand potential of the grand canonical ensemble, $\Omega_0 = \Omega[n_0(\boldsymbol{r})]$. For a fixed electron temperature $T_e = (k_B \beta)^{-1}$ and chemical potential $\mu$, $\Omega[n(\boldsymbol{r})]$ is composed of a universal functional $G[n(\boldsymbol{r})]$ and the external potential $V_{ext}[n(\boldsymbol{r})]$:

\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}
where $v_{ext}$ is the optional external potential and $T_e[n]$, $V_{ee}[n]$, $N[n]$ and $S[n]$ are, respectively, the kinetic energy, the electron-electron interaction, the particle number and the entropy functional. While the Mermin theorems hugely 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 and the widely used Kohn-Sham (KS) scheme, which is the one used in this thesis.

This scheme is based on the following ansatz: \textit{The equilibrium density of the fully-interacting is equal
to the equilibrium density of a non-interacting system with a proper effective potential, accounting for the interactions}. Although unproven in general, this assumption has been proven in certain systems and is adopted in this thesis due to the success of the KS method in practice. Under this assumption, the all-electron wavefunction is factorizable in the product of single particle wavefunctions obeying the following Schrödinger equations:

\begin{equation}
\biggl(-\frac{\hbar^2}{2m_e}\nabla^2 + v_{eff}(\boldsymbol{r})\biggr)\psi_{\alpha}(\boldsymbol{r}) = \epsilon_{\alpha}\psi_{\alpha}(\boldsymbol{r})
\label{schrodinger equation}
\end{equation}
where $v_{eff}$ is the effective potential, which will be examined below, and $\alpha$ is the collection of quantum numbers that uniquely identify the eigenstate. The density of a single-particle system at finite-temperature is given by:

\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}
where $f_{FD}(\epsilon_{\alpha};T_e,\mu)$ is the Fermi-Dirac occupation number distribution:

\begin{equation}
f_{FD}(\epsilon_{\alpha};T_e,\mu) = \frac{1}{1+e^{-\beta(\epsilon_{\alpha}-\mu)}}
\label{fermi-dirac}
\end{equation}
Using this ansatz, the Mermin grand potential can be reformulated as (abbreviating $f_{FD}(\epsilon_{\alpha};T_e,\mu)$ with $f_{FD}(\epsilon_{\alpha})$):

\begin{equation}
\begin{split}
& \Omega[n(\boldsymbol{r})] = \int v_{ext}(\boldsymbol{r})n(\boldsymbol{r}) d\boldsymbol{r} + V_H[n(\boldsymbol{r})] + \mu N[n(\boldsymbol{r})] + T_1 - \beta^{-1}S_1 + V_{XC}[n(\boldsymbol{r})], \\
& V_H[n(\boldsymbol{r})] = \frac{1}{2}\int \frac{n(\boldsymbol{r})n(\boldsymbol{r}')}{|\boldsymbol{r}-\boldsymbol{r}'|}d\boldsymbol{r}d\boldsymbol{r}', \\
& N[n(\boldsymbol{r})] = \int n(\boldsymbol{r}) d\boldsymbol{r}, \\
& T_1 = - \frac{1}{2} \sum_{\alpha=1}^{\infty} f_{FD}(\epsilon_{\alpha}) \bra{\psi_{\alpha}}\nabla^2 \ket{\psi_{\alpha}}, \\
& S_1 =  \sum_{\alpha=1}^{\infty} \{ f_{FD}(\epsilon_{\alpha}) \ln f_{FD}(\epsilon_{\alpha}) + [1-f_{FD}(\epsilon_{\alpha})]\ln[1-f_{FD}(\epsilon_{\alpha})] \}, 
\end{split}
\label{KH grand potential}
\end{equation}
where $V_H[n(\boldsymbol{r})]$ is Hartree interaction functional, which accounts for Coulomb interactions between electrons, and $V_{XC}[n(\boldsymbol{r})]= V_{ee}[n(\boldsymbol{r})] - V_{H}[n(\boldsymbol{r})] + T_e[n(\boldsymbol{r})] - T_1 - \beta^{-1}S[n(\boldsymbol{r})] + \beta^{-1}S_1$ is the famous exchange-correlation functional, which that retains 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 Hartree term. Before discussing this functional more in details, let us have a look at the single-particle Schrödinger equation, which can be derived from Eq. (\ref{KH grand potential}), and in adimensional units is as follows:

\begin{equation}
\biggl(-\frac{1}{2}\nabla^2 + \overbrace{v_{ext}(\boldsymbol{r}) + \frac{1}{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}
with exchange-correlation potential $v_{XC}$ given by the functional derivative of $V_{XC}$:

\begin{equation}
v_{xc}(\boldsymbol{r}) = \frac{\delta V_{XC}[n(\boldsymbol{r})]}{\delta n(\boldsymbol{r})}.
\label{xc potential}
\end{equation}
These last two equations, together with Eq. (\ref{density},\ref{fermi-dirac}), constitute 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. Despite a detailed discussion of these techniques is outside the scope of this thesis, we will briefly highlight a few that were employed in the calculations for our studies. Some of these methods reduce the number of electrons for which wavefunctions must be explicitly computed. Among these, we mention the frozen-in-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 will be therefore used extensively 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. In this context, 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. This periodicity ensures that the set of eigenfunctions is discrete, allowing each to be expressed in terms of an infinite, discrete basis set. As a final step, basis functions that contribute minimally to the expansion are excluded, resulting in a finite basis set~\cite{payne1992iterative}.

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 generalized 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}
For systems with larger spatial gradients in electron density, the generalized gradient approximation (GGA) is used, for which the exchange-correlation
energy per unit density $\epsilon_{XC}$ depends on the density gradient as well, $\epsilon_{XC}^{GGA}(n(\boldsymbol{r}), \nabla n(\boldsymbol{r}))$. A popular example is the GGA-PBE functional~\cite{perdew1996generalized}. 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}.

Finally, we point out that the KS orbitals $\psi_{\alpha}$ and their energies $\epsilon_{\alpha}$ do not have a rigorous physical meaning, but are just introduced as a construction for solving the KS equations. However, depending on the goodness 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 on a good level. In fact, it is common practice 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, we have 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, particularly in systems with strong electron correlations. More generally, binding energies are also typically underestimated by the KS system. To address these issues, various \textit{ad hoc} modifications have been introduced to the DFT framework. We will discuss these problematics more in details in the following, 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}










In this work, our focus is on determining the electronic density of states for warm-dense materials. This is a fundamental quantity from which other important material properties can be derived, such as the equation of state~\cite{wang2000calculated}, the electron heat capacity~\cite{Ashcroft76}, and material opacities and conductivities, calculated through the Kubo-Greenwood formulas\cite{kubo1957statistical, greenwood1958boltzmann}. Numerous studies have shown that electronic structure of materials will change with density and temperature variations, as these parameters will modify the microscopic environment experienced by ions and electrons. In particular, electron binding energies are affected through Ionization Potential Depression (IPD)~\cite{hoarty2013observations} and Pressure Ionization (PI)~\cite{saumon1992fluid}.

In DFT, the electronic density of states $\rho(\epsilon)$ can be 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) = A \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}
where $W_s$ is the so-called smearing width (SW) and A is a normalizing factor. 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 $\epsilon$.

As discussed earlier, the energy levels, and consequently $\rho$, are not guaranteed to match the true eigenvalues of the system. Discrepancies from reality arise with correlated system, as DFT is essentially a mean field theory if we do not consider $V_{XC}$, which could be very different from the physical one. In such cases, when Coulomb interactions between valence electrons on the same site are strong enough, electron hopping between sites is drastically hindered, causing electrons to localize around atoms. This localization leads to a drop in conductivity and the emergence of a bandgap, characteristic of what is known as a "Mott insulator". 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, we primarily use the DFT+U method~\cite{himmetoglu2014hubbard}, 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. 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."


% \subsubsection{Absorption spectrum}






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










Iron oxides are important for understanding Earth's core and mantle dynamics. The outer core 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. Several studies suggest that at extreme pressures 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 changes the most favourable lattice structure, determining a crystalline transitions, and modifies the localization of states and the charge transfer, affecting the conduction properties of the material. The change in the atomic bonds may also induce 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.

Although the techniques developed in this thesis are applicable to any material, we focus 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. The 3d orbitals of this ion delocalize, 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.

For FeO, theoretical 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}. At approximately 70 GPa and temperatures below 1500 K, the B1 phase of FeO becomes metallic, linked to the delocalization 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 metallization, with the HS metallic phase persisting up to pressures of 150 GPa, depending on the temperature. Beyond this, 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.





% 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, spectroscopic 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, we can 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, will be 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 (XTS)~\cite{glenzer2009x, visco2012measurement, kozlowski2016theory, kritcher2008ultrafast}, though as we will discuss, 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 powerful 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 key information about the system's structure of vancant 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 global electronic structure of a material in a single spectrum—making it particularly useful for time-resolved studies of dynamic processes. As RIXS is the technique employed in this thesis for the extraction of electronic structures, we will give a detailed description of the theory behind this process in Chapter 3. 

We will now provide a brief overview of the typical experimental instruments used in XFEL facilities for HED physics investigations. 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, 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 Hamos spectrometer and diced analyzers.} 
   \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 synchronize 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 a few mJs 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 an automated 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 analysis of the typical pixel detectors and spectrometers used in HED experiments will be 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 motorization 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 synchronization 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}








This thesis is organized into six chapters. The first chapter introduces the field of dense plasmas, outlining their main characteristics and key physical parameters, providing an overview of theoretical models used to describe them, and exploring scenarios where dense plasmas occur, both in natural environments and technological applications. Among these models, particular emphasis is placed on density functional theory, the primary theoretical framework employed in this work to benchmark experimental data. Additionally, it explores the experimental realization of HED systems, along with the typical experimental setups and techniques used to study them. Lastly, it discusses the characteristics of the materials investigated and the specific properties of interest. 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.

The second chapter 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 our 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 generalizable to other scientific fields and hold intrinsic interest. Notably, the content of this chapter has been compiled into a paper currently under review at the \textit{Journal of Applied Physics}.

This paper draws inspiration from the previous work of Dr. Oliver Humphries, 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.

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 specialized 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 generalization 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.

The fourth chapter focuses on the analysis of experimental RIXS spectra collected at the EuXFEL facility for warm-dense iron, iron oxides, and gadolinium. After introducing the experimental setup and primary diagnostics, we present the extracted electronic structures of Fe and Fe$_2$O$_3$, which also allow determination of sample temperature and electron binding energies. We further show that this method can extract densities of states 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, we deconvolved the spectrometer's instrument function from the FeO RIXS spectra, achieving DOS for this material with resolutions on the order of a few eV. Finally, we applied the RIXS technique 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, to whom the author is greatly indebted.

Chapter 5 presents the analysis of experimental RIXS spectra collected at the LCLS facility for dynamically compressed FeO, a material with interesting properties under high pressure and significant relevance to geophysical research. This study represents the ultimate goal of the thesis, showcasing the application of this RIXS diagnostic technique to irreversible 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 our 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. Special thanks are extended to Dr. Marion Harmand, Dr. Céline Crépisson, Dr. David McGonegle, Dr. Alexis Amouretti, Dr. David A. Chin, and and the same Dr. Danae N. Polsin for their contributions to designing and conducting the experiment, and especially for their analysis of the VISAR and XRD data. The author also thanks Dr. Thomas Gawne and Mr. Thomas Campbell for collaborating on the design of the RIXS spectrometers. The DFT calculations presented were performed by Dr. Sam Azadi, to whom the author is greatly indebted. All other work presented in this chapter was carried out by the author.

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 generalizations of the STEP method to other physical investigations.

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.

















