# Revealing Non-equilibrium and Relaxation in Warm Dense Matter

Jan Vorberger<EMAIL_ADDRESS>Helmholtz-Zentrum Dresden-Rossendorf
(HZDR), D-01328 Dresden, Germany  Thomas R. Preston  European XFEL, D-22869
Schenefeld, Germany  Nikita Medvedev  Institute of Physics, Czech Academy of
Sciences, 18221 Prague 8, Czech Republic  Institute of Plasma Physics, Czech
Academy of Sciences, 18200 Prague 8, Czech Republic  Maximilian P. Böhme
Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
Technische Universität Dresden, D-01062 Dresden, Germany  Zhandos A.
Moldabekov  Center for Advanced Systems Understanding (CASUS), D-02826
Görlitz, Germany  Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328
Dresden, Germany  Dominik Kraus  Institut für Physik, Universität Rostock,
D-18057 Rostock, Germany  Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328
Dresden, Germany  Tobias Dornheim<EMAIL_ADDRESS>Center for Advanced
Systems Understanding (CASUS), D-02826 Görlitz, Germany  Helmholtz-Zentrum
Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany

###### Abstract

Experiments creating extreme states of matter almost invariably create non-
equilibrium states. These are very interesting in their own right but need to
be understood even if the ultimate goal is to probe high-pressure or high-
temperature equilibrium properties like the equation of state. Here, we report
on the capabilities of the newly developed imaginary time correlation function
(ITCF) technique Dornheim  _et al._ (2022a) to detect and quantify non-
equilibrium in pump-probe experiments fielding time resolved x-ray scattering
diagnostics. We find a high sensitivity of the ITCF even to a small fraction
of non-equilibrium electrons in the Wigner distribution. The behavior of the
ITCF technique is such that modern lasers and detectors should be able to
trace the non-equilibrium relaxation from tens of femto-seconds to several 10s
of picoseconds without the need for a model.

##  I Introduction

Non-equilibrium comes in many shapes and sizes. Sometimes it is desired, often
it needs to be avoided. In any case, its characterization and diagnostics are
difficult Falk (2018); Macchi  _et al._ (2013). Arguably even more difficult
is the accurate theoretical prediction of a non-equilibrium state and its time
evolution Kremp  _et al._ (2005); Bonitz (2016); Arber  _et al._ (2015);
Gamaly (2011); Bonitz  _et al._ (2020).

We present here a new method to detect signatures of non-equilibrium in x-ray
scattering signals. Inspired by a recent model-free temperature diagnostics in
equilibrium Dornheim  _et al._ (2022a), the method leverages the deviation
from the basic detailed balance condition present in equilibrium as measure
for non-equilibrium effects. This deviation is calculated based on the
imaginary time correlation function (ITCF) $F({\bf q},\tau)$, whose symmetry
around its minimum at $\tau=\beta/2$ ($\beta=1/k_{B}T$) is broken in non-
equilibrium. The ITCF can be obtained by a two-sided Laplace transform of the
dynamic structure factor $S({\bf q},\omega)$. The latter is proportional to
the energy resolved x-ray scattering intensity received by a detector
positioned at a scattering angle $\theta=2\arcsin(q/2q_{0})$ with $q_{0}$
denoting the momentum of the incoming photon, ${\bf q}$ and $\omega$
representing the momentum and energy transfer, respectively, during the
scattering event Dornheim  _et al._ (2022a, b).

Performing an analysis of the symmetry of the ITCF to detect non-equilibrium
relies on nothing but first principles. It does not involve the fitting of the
scattering signal with any (approximate) theory nor does it introduce further
however well intended approximations. It works for a wide range of probe beams
as long as density fluctuations and their correlations are probed and the
dispersion of the probe beam in the target is small Dornheim  _et al._
(2022b). This is the case for hard and soft x-rays, electron beams, and even
neutrons. As we will show below, the method can work using a single detector
at a single scattering angle but cross checks with several detectors at a few
different angles provide additional information and certainty in the
equilibrium/non-equilibrium assessment. The shapes of well-characterized
source and probe functions can be taken into account without difficulty.

Leveraging the capabilities of modern optical and x-ray laser systems allows
to not only perform one measurement but indeed track and characterize the
entire time evolution of the non-equilibrium in the driven state and later
during the relaxation phase Tschentscher  _et al._ (2017); Bostedt  _et al._
(2016); Pile (2011). This allows first principles access to relaxation times
and energy transfer rates between subsystems that have hitherto been
restricted by their strong reliance on two-temperature models Waldecker  _et
al._ (2015), amongst others.

The intended use for this method is for experiments involving the creation and
characterization of warm dense matter and other extreme states of matter
Knudson  _et al._ (2015); Fletcher  _et al._ (2015); Kraus  _et al._ (2016,
2017); Celliers  _et al._ (2018); Lazicki  _et al._ (2021); Gaus  _et al._
(2021). Such states are transient in the laboratory but are commonplace in
astrophysical settings like planetary interiors Liu  _et al._ (2019); Brygoo
_et al._ (2021); Militzer  _et al._ (2022), dwarf star interiors Kritcher  _et
al._ (2020); Becker  _et al._ (2014), neutron star crusts Chamel and Haensel
(2008), and the like. It is of paramount importance to know the relaxation
times of a variety of elements and materials after shock absorption or laser
irradiation to study equilibrium properties of said materials to understand
their behaviour at extreme pressures for the earlier mentioned planetary
settings White  _et al._ (2020). In particular, experiments using powerful
x-ray laser beams to isochorically heat and at the same time probe the target
should analyse their scattering signal using the method presented here
Sperling  _et al._ (2015); Kraus  _et al._ (2019).

The investigation of creation and relaxation of hot electrons in materials at
ambient pressure conditions in solid state physics Zahn  _et al._ (2021, 2022)
drives many technological applications in chip manufacturing Marks  _et al._
(2022), medicine Vogel and Venugopalan (2003), surface modification Frischkorn
and Wolf (2006), or generally laser materials processing Bäuerle (2011);
Balling and Schou (2013). From a basic research point of view, we are
interested in the general coupling of the ionic and electronic degrees of
freedom in (non-)equilibrium Vorberger  _et al._ (2010); Benedict  _et al._
(2017); Simoni and Daligault (2020); Medvedev and Milov (2020); Zhang  _et
al._ (2022); Ciricosta  _et al._ (2016a) and a great many experiments have
been performed to test the theoretical predictions for non-equilibrium warm
dense matter states Celliers  _et al._ (1992); White  _et al._ (2012); Zastrau
_et al._ (2014); Hartley  _et al._ (2015); Mo  _et al._ (2018); Fletcher  _et
al._ (2022); Vinko  _et al._ (2012); Ciricosta  _et al._ (2012, 2016b);
Preston  _et al._ (2017). We aim to understand the microscopic dynamics
governing the energy exchange between, for example, $\alpha$-particles and
electrons and between electrons and protons/deuterons/tritons in the fusion
pellet of inertial confinement fusion Edie, D.J.  _et al._ (2013); Zylstra
_et al._ (2022). Such questions are naturally related to the study of response
functions as they are of further importance for the stopping power Malko  _et
al._ (2022), transport quantities like conductivities Ramakrishna  _et al._
(2022), or nonlinear effects like higher harmonics generation Dornheim  _et
al._ (2020, 2021a, 2021b, 2021c); Moldabekov  _et al._ (2022).

To demonstrate the capabilities of our method, we first compute theoretical
dynamic structure factors and ITCFs for very specific well defined non-
equilibrium Wigner distributions. This is done using the technique of real-
time Green’s functions Kremp  _et al._ (2005); Chapman and Gericke (2011);
Vorberger and Chapman (2018). This gives insight into some general behavior of
the ITCF for typical non-equilibrium situations. We then demonstrate how the
time evolution of the non-equilibrium state may be tracked in an experiment
using ITCFs. For this, we use solutions for the time evolution of the Wigner
function from a Boltzmann-Monte Carlo code Medvedev  _et al._ (2022).

Since the set of possible non-equilibrium situations in pump-probe scenarios
is almost infinite, we restrict the discussions in this paper to systems in
non-equilibrium that are homogeneous and isotropic. This is a substantial
restriction but leaves the electron Wigner distribution function free to take
any form. A generalization to in-homogeneous or anisotropic systems Chapman
_et al._ (2014); Kozlowski  _et al._ (2016) is possible and will be covered in
a future publication.

##  II Theory

###  II.1 Theory for the non-equilibrium structure factor

The structure factor $S$ is defined by the correlation function of density
fluctuations $L^{>}$ in non-equilibrium according to Kremp  _et al._ (2005)

$\displaystyle S_{ab}({\bf q},\omega;{\bf R},t)$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}L_{ab}^{>}({\bf q},\omega;{\bf R},t)$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi\sqrt{n_{a}n_{b}}}\int d{\bf
r}ds\;e^{i(\omega s-{\bf q}\cdot{\bf r})}i\hbar L_{ab}^{>}(12)\,,$

where $1=\\{{\bf r}_{1},t_{1},\sigma_{1}\\}$ denotes a complete set of
observables and the species are denoted by the labels $\\{a,b\\}$. Here, a
Fourier transform is performed with respect to the relative coordinates in
space and time ${\bf r}={\bf r}_{1}-{\bf r}_{2}$ and $s=t_{1}-t_{2}$,
respectively. The center of mass variables ${\bf R}=({\bf r}_{1}+{\bf
r}_{2})/2$ and $t=(t_{1}+t_{2})/2$ remain untransformed. Thus, the microscopic
fluctuations are considered in momentum-frequency space whereas the
macroscopic variations in density etc. are treated in real space. For
instance, the system can still be homogeneous macroscopically but the Wigner
distribution may indicate severe deviations from Fermi- or Boltzmann
distributions. In this case, the dependencies on ${\bf R}$ and $t$ are
dropped.

The correlation function of density fluctuations $L_{ab}^{>}$ is given by
Kremp  _et al._ (2005)

$i\hbar L_{ab}^{>}(12)=\langle\delta\rho_{a}(1)\delta\rho_{b}(2)\rangle\,.$
(2)

The average is performed as trace over the fluctuation operators and the non-
equilibrium density operator $\hat{\rho}$,
$\langle\ldots\rangle=\mbox{Tr}\\{\hat{\rho}\ldots\\}$. The density
fluctuations are given in coordinate space as

$\delta\rho(1)=\psi^{\dagger}(1)\psi(1)-\langle\psi^{\dagger}(1)\psi(1)\rangle\,,$
(3)

where we used the creation and annihilation operators $\psi^{\dagger}$ &
$\psi$ in second quantization.

We still define an intermediate scattering function in imaginary time $\tau$
(ITCF) with a relation similar to Eq. (II.1)

$F_{ab}({\bf q},\tau;{\bf
R},t)=\int\limits_{-\infty}^{\infty}d\omega\;e^{-\omega\tau}S_{ab}({\bf
q},\omega;{\bf R},t)\,.$ (4)

This intermediate scattering function has, in equilibrium, all the known
symmetries, in particular detailed balance, but will show significant
deviations from them in a non-equilibrium case which can be used to decide the
degree of non-equilibrium. We introduce

$F_{ab}({\bf q},\tau)/F_{ab}({\bf q},\beta^{\prime}-\tau)$ (5)

as such a measure for non-equilibrium. Here $\beta^{\prime}$ is defined by
$F(\tau=0)=F(\tau=\beta^{\prime})$. The quantity of Eq. (5) will approach
unity the closer the system gets to equilibrium and $\beta^{\prime}$ will be
the inverse temperature then.

Of particular interest is of course the total electron-electron structure
factor $S_{ee}$ and its ITCF as it can be obtained in a variety of scattering
experiment, e.g. via x-ray Thomson scattering Glenzer and Redmer (2009).
However, noise in the experimental signal, a finite energy bandwidth of the
probe beam, and in the detector function, and a finite range of energies
available in the experiment ($[-x,x]$) need to be accounted for

$F_{ee}^{exp}({\bf q},\tau;{\bf
R},t)=\int\limits_{-x}^{x}d\omega\;e^{-\omega\tau}I_{ee}({\bf q},\omega;{\bf
R},t)\,,$ (6)

where

$I_{ee}({\bf q},\omega;{\bf R},t)=R(\omega)\otimes S_{ee}({\bf q},\omega;{\bf
R},t)+\Delta I({\bf q},\omega;{\bf R},t)$ (7)

is the convolution of the detector and source function [summarized as
$R(\omega)$] with the dynamic structure factor and Gaussian noise $\Delta I$
added Sheffield  _et al._ (2010); Dornheim  _et al._ (2022a, b).

Explicit expressions for the dynamic structure in non-equilibrium can be
obtained from the equation of motion for the correlation function of density
fluctuations Vorberger and Chapman (2018). Sometimes these relations are
called generalized fluctuation-dissipation theorems Kremp  _et al._ (2005).
Thus, for a realistic system comprised of electrons and ions, the non-
equilibrium total electron-electron structure factor is in direct polarization
approximation (DPA, neglecting cross-species polarization functions) given by
Vorberger and Chapman (2018)

$\displaystyle S_{ee}({\bf q},\omega;{\bf R},t)=\frac{1}{2\pi
n_{e}}L_{ee}^{>}=$ $\displaystyle\,\frac{{\cal L}_{ee}^{>}+|{\cal
L}_{ee}^{\text{R}}\\!|^{2}V_{ei}^{2}{\cal L}_{ii}^{>}}{|1-V_{ie}{\cal
L}_{ee}^{\text{R}}V_{ei}{\cal L}_{ii}^{\text{R}}|^{2}}\,,$ (8)

Here and below, we omit all the arguments $({\bf q},\omega;{\bf R},t)$ of
every quantity on the right hand sides. The pure species contribution
appearing above are defined by

$\displaystyle{\cal L}_{aa}^{>}({\bf q},\omega;{\bf R},t)=$
$\displaystyle\frac{\Pi_{aa}^{>}}{|1-\Pi_{aa}^{\text{R}}V_{aa}|^{2}}\,,$ (9)

for the greater functions and the retarded functions are given by

$\displaystyle{\cal L}_{aa}^{\text{R}}({\bf q},\omega;{\bf R},t)=$
$\displaystyle\frac{\Pi_{aa}^{\text{R}}}{1-\Pi_{aa}^{\text{R}}V_{aa}}\,.$ (10)

The Coulomb potential is denoted by $V$. The polarization functions $\Pi$ are
needed as input quantities and they can be computed in random phase
approximation (RPA) or using non-equilibrium local field corrections Vorberger
and Chapman (2018). The RPA is sufficient to demonstrate the capabilities of
the intermediate scattering function in imaginary time. The retarded
polarization function $\Pi^{R}$ may be obtained from the correlation functions
and a Kramers-Kronig relation

$\displaystyle\mbox{Im}\Pi^{R}({\bf q},\omega;{\bf R},t)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\left(\Pi^{<}({\bf q},\omega;{\bf R},t)\right.$ (11)
$\displaystyle\qquad\left.-\Pi^{>}({\bf q},\omega;{\bf R},t)\right)\,,$
$\displaystyle\mbox{Re}\Pi^{R}({\bf q},\omega;{\bf R},t)$ $\displaystyle=$
$\displaystyle\int\frac{d\omega^{\prime}}{2\pi}\frac{\mbox{Im}\Pi^{R}({\bf
q},\omega^{\prime};R,t)}{\omega-\omega^{\prime}}\,.$ (12)

The correlation functions themselves can easily be obtained in RPA from

$\displaystyle\Pi^{\lessgtr}({\bf q},\omega;{\bf R},t)$ $\displaystyle=$
$\displaystyle 2\pi i\int\frac{d{\bf k}}{(2\pi\hbar)^{3}}f^{\lessgtr}({\bf
k}+{\bf q},t)f^{\gtrless}(k,t)$ (13)
$\displaystyle\times\delta\left(\hbar\omega-E({\bf q}+{\bf k})+E(k)\right)\,.$

The energy dispersion can be taken from the ideal gas $E(k)=(\hbar k)^{2}/2m$
and the Wigner distributions are $f^{>}=-[1-f(k,t)]$ and $f^{<}=f(k,t)$.

Within a two-temperature model, Eq. (8) can be used to determine not only the
electron temperature but also the ion temperature as the DPA is valid for a
two temperature system. The ITCF defined in Eq. (4) will indicate the correct
electron temperature due to the large mass ratio between electrons and ions.
Knowledge of the electron temperature allows to solve Eq. (8) for the pure ion
structure ${\cal L}_{ii}$ whose temperature can then be determined via the ion
ITCF.

###  II.2 Theory for the Wigner function

The non-equilibrium dynamic structure factor $S({\bf q},\omega;{\bf R},t)$
depends on the knowledge of the Wigner function, which itself is non-trivial
to obtain via a kinetic equation Kremp  _et al._ (2005); Bonitz (2016). For
demonstration purposes, we can model the non-equilibrium (electron) Wigner
function by a bump-on-hot-tail distribution that has been found to occur in
various laser pump-probe setups Fäustlin  _et al._ (2010); Chapman and Gericke
(2011)

$\displaystyle f(k,t)$ $\displaystyle=$ $\displaystyle
A_{c}\left\\{\exp\left[\beta_{c}\left(\frac{\hbar^{2}k^{2}}{2m}-\mu_{c}\right)\right]+1\right\\}^{-1}$
(14) $\displaystyle+$ $\displaystyle
A_{h}\left\\{\exp\left[\beta_{h}\left(\frac{\hbar^{2}k^{2}}{2m}-\mu_{h}\right)\right]+1\right\\}^{-1}$
$\displaystyle+$ $\displaystyle
A_{b}\frac{n\Lambda^{3}}{2}\exp\left(-\beta_{b}\frac{(\hbar
k-p_{b})^{2}}{2m}\right)\,.$

Here, the indices $\\{c;h;b\\}$ stand for cold, hot, and bump, respectively.
The bump is due to direct laser irradiation and is located at momentum
$p_{b}$. The thermal wavelength is $\Lambda^{2}=2\pi\hbar/mk_{B}T_{b}$. Each
part of the Wigner function has its own inverse ’temperature’ $\beta_{i}$, and
chemical potential $\mu_{i}$. The latter two and also the fractions $A_{i}$
can change over time. The total density of electrons is given by the integral
over all momenta, $\hbar k$.

This shape of the transient distribution function in X-ray-irradiated
materials was confirmed by various simulations of the non-equilibrium
electronic system Fäustlin  _et al._ (2010); Medvedev  _et al._ (2011); Hau-
Riege (2013). We use an event-by-event Monte Carlo (MC) simulation to trace
non-equilibrium evolution of the electron distribution function, augmented
with Pauli blocking factors in the scattering probability calculations (which
is essentially identical to a numerical solution of the semi-classical
Boltzmann equation) Medvedev  _et al._ (2011). The method uses an individual
particle simulation to model the response of the electronic system of
aluminium to vacuum ultra-violet or soft X-ray irradiation. The
photoabsorption cross sections are extracted from Ref. Henke  _et al._ (1993).
In case photoabsorption excites a core-shell, its Auger decay then promotes
another electron to the conduction band. The auger decay times are taken from
Ref. Keski-Rahkonen and Krause (1974). All electrons in the conduction band,
excited and existing in aluminium prior to irradiation, are then scattering
with the probabilities depending on the cross section of scattering multiplied
with factors containing the transient distribution function, in accordance
with the Boltzmann collision integral Medvedev  _et al._ (2011). The cross
sections of free-free electron scattering is approximated within the linear
response theory with the Mermin-like complex dielectric function for aluminium
Medvedev  _et al._ (2022). Tracing each scattering event of each electron
within the selected simulation box allows to model the non-equilibrium
evolution of the electron distribution function in time. Despite the fact that
electrons in the Monte Carlo scheme are classical particles, inclusion of the
Pauli blocking factors in the scattering probabilities allows us to obtain a
quantum distribution function (i.e. Fermi-Dirac instead of Maxwell-Boltzmann
in the case of equilibrium), see Ref. Medvedev  _et al._ (2011) for details.
We use 104 MC iterations with approximately 3000 particles in each to obtain
reliable statistics.

##  III Analysis of scattering spectra

We analyze the effects of different types of idealized non-equilibrium
situations. We restrict ourselves to isotropic and homogeneous conditions in
the system. Macroscopic gradients in the density can be taken into account via
a gradient expansion of the microscopic variables Chapman  _et al._ (2014);
Kozlowski  _et al._ (2016).

###  III.1 Study of idealized non-equilibrium situations

Figure 1:  Non-equilibrium structure factor of an electron gas in a VUV-pump
scenario. The Wigner distribution function is shown in panel a); panel b)
shows the Wigner function multiplied with the square of the momentum to better
visualize the fractions of electrons at certain momenta. The dynamic structure
factor is shown in panel c). The intermediate scattering function is shown in
panel d). The electronic density is $n=1.8\times 10^{23}$ cm-3 ($r_{s}=2.07$).
The equilibrium temperature (cold bulk) is $T_{c}=10^{5}$ K. The hot tail has
a temperature of $T_{h}=10^{7}$ K. The wavenumber is $q=0.5/$ aB. In panel d),
the x-axis for the equilibrium case is on top, the non-equilibrium x-axis is
at the bottom.

We first study non-equilibrium effects in an uniform electron gas, perhaps the
most important model system in plasma physics, warm dense matter physics and
solid state physics Dornheim  _et al._ (2016, 2017, 2018a, 2018b, 2019, 2020,
2021d, 2021e, 2021c, 2022c).

A VUV laser of moderate intensity produces a bump in the electron Wigner
distribution function in the vicinity of the contribution of the cold bulk
electrons. Such an example is presented in Fig. 1. We show different
(substantial) fractions of the electrons pumped from the cold bulk into the
bump. The deviations from the equilibrium structure factor (solid green line)
are easily noticeable in panel c) of Fig. 1. As the chosen wavenumber lies
within the collective regime, all structure factors feature plasmon-like
peaks. However, the position of the non-equilibrium plasmon peaks is shifted
to lower energies as compared to the equilibrium case. This is caused by the
lowering of the bulk density when electrons are laser heated into a hot tail
or bump. Hot electrons contribute less to screening and collective effects. In
addition, the ratio of the plasmon peaks is changed considerably as the
detailed balance symmetry is broken in non-equilibrium. The bump causes in
addition further peaks of the dynamic structure factor at higher energies.
These might be identified as beam acoustic modes.

Fig. 1 panel d) displays the ITCFs for all the cases. The green ITCF for the
equilibrium case is symmetric around $\beta/2$ and shows the correct
temperature. Analyzing the ITCFs of the three non-equilibrium cases gives
spurious high temperatures of $T^{*}=2.5\times 10^{5}$ K and above. As the
signatures of non-equilibrium are very obvious when looking at the dynamic
structure factors, the ITCFs seem very equilibrium like in the intervals
$[0,\beta^{*}]$. Mirroring the ITCFs around the $\beta^{*}/2$-axis would only
reveal deviations from the equilibrium symmetry far away from $\beta^{*}/2$.
This means that the collective regime is not well suited to distinguish
equilibrium from non-equilibrium via the ITCF. As long as most of the weight
of the dynamic structure factor is contained in plasmon-like peaks, the ITCF
will always be symmetric showing some spurious temperature stemming from the
ratios of the non-equilibrium plasmon peaks. This can easily be demonstrated
when a simple plasmon model for the dynamic structure factor is Laplace-
transformed into the respective ITCF.

Figure 2:  Non-equilibrium structure factor of an electron gas comparing a VUV
and a XFEL-pump scenario. The Wigner distribution function is shown in panel
a). The dynamic structure factor is shown in panel b). The intermediate
scattering function is shown in panel c). The electronic density is
$n=1.8\times 10^{23}$ cm-3. The equilibrium temperature (cold bulk) is
$T_{c}=10^{5}$ K. The hot tail has a temperature of $T_{h}=10^{7}$ K. The
wavenumber is $q=0.5/$ aB. In panel c), the x-axis for the equilibrium case is
on top, the non-equilibrium x-axis is at the bottom.

Next we want to analyse the effect of different bump energies, i.e., the
effect of pumping the system with harder and harder laser radiation like
x-rays instead of VUV. An example is presented in Fig. 2. The case with the
solid magenta curve is the same as in Fig. 1 featuring a hot tail and a bump
at VUV energies. Retaining only the hot tail, but removing the bump (as might
be the case after the pump laser has switched off, solid cyan line), makes the
additional mode around $\omega=3$ ryd vanish (1 ryd$=13.6$ eV). Only visible
due to the log scale in Fig. 2, the other main difference between equilibrium
and the considered non-equilibrium cases, apart from the differences in
collective mode peaks, is the behaviour of the high energy tails. In the
(solid green) equilibrium case, the dynamic structure factor shows exponential
decay around $\omega=\pm 2$ ryd. In the non-equilibrium cases, the high energy
electrons in the Wigner distribution cause a much slower decay and high energy
plateaus of the dynamic structure factor (the exponential decay sets in around
$\omega=\pm 25$ ryd in the x-ray case (black line). Further, the higher energy
bump in the Wigner distribution at x-ray energies (see blacked dashed line in
panel a) of Fig. 2, $k=25/$aB is equivalent to $8.5$ keV pump laser energy),
even though it is responsible for the large tails, has a lot less influence on
the plasmon peaks than the VUV bump or the hot tail. These electrons are
energetically too far removed to significantly influence the collective
dynamics. We have already seen that under VUV excitation, the asymmetry in the
ITCF is small, such that the collective regime cannot be used to detect non-
equilibrium from the ITCF. However, for the x-ray laser excitation case (black
dashed line in panel c) of Fig. 2), such asymmetry is visible and a single
detector at a single scattering angle should be sufficient.

Figure 3:  Non-equilibrium structure factor of an electron gas in a XFEL-pump
scenario. The Wigner distribution function is shown in panel a). The dynamic
structure factor is shown in panel b). The intermediate scattering function is
shown in panel c). Only the electrons are considered. The electronic density
is $n=1.8\times 10^{23}$ cm-3. The equilibrium temperature (cold bulk) is
$T_{c}=10^{5}$ K. The wavenumber is $q=0.5/$ aB. In panel c), the x-axis for
the equilibrium case is on top, the non-equilibrium x-axis is at the bottom.

The question then naturally arises, how big the fraction of electrons need to
be in the x-ray pump laser scenario for the ITCF symmetry to notice the non-
equilibrium. Such a case study is presented in Fig. 3. Apart from the usual
equilibrium case (solid green line), we show cases with $1\%$, $10\%$, and
$80\%$ of the electrons being contained in the bump at $k=25/a_{B}$,
respectively. Provided the high energy tails of the dynamic structure factor
beyond $\omega=\pm 3$ ryd can be resolved and detected in experiment, even a
$1\%$ population of electrons at $8.5$ keV can be detected in the ITCF in the
collective regime, see panel c). This is even more remarkable, as is clearly
visible in panel b) of Fig. 3 that the dynamic structure is very similar to
the equilibrium case for energies in the range $\omega=\pm 2$ ryd (dash-dotted
violet curve). Naturally, if substantial amounts of electrons are pumped out
of the bulk distribution, the asymmetry in the ITCF becomes more and more
obvious (black dashed and solid cyan curves).

Figure 4:  Non-equilibrium structure factor of an electron gas in a XFEL-pump
scenario for different wavenumbers. The Wigner distribution function is shown
in panel a). The dynamic structure factor is shown in panel b). The
intermediate scattering function is shown in panel c). The electronic density
is $n=1.8\times 10^{23}$ cm-3. The equilibrium temperature (cold bulk) is
$T_{c}=10^{5}$ K. In panel c), the x-axis for the equilibrium case is on top,
the non-equilibrium x-axis is at the bottom.

In case the experimental resolution is insufficient or the asymmetry in the
ITCF is too small to be conclusive, the logical next step would be to check
the dynamic structure and therefore the ITCF at different scattering angles,
i.e., different wavenumbers. This has been done in Fig. 4. Only a single non-
equilibrium distribution as shown in panel a) is evaluated at three different
wavenumbers. These cover the collective, transitional and single particle
regimes. In the case of equilibrium, see the green curves, all ITCFs converge
at the same temperature. Plotting the ITCFs for the x-ray bump scenario (blue,
black, and red curves) gives (apart from any asymmetry in the ITCF) three
totally different values for their spurious temperatures. This constitutes an
unambiguous experimental signature of non-equilibrium. It is again interesting
to observe how the high energy tails behave with increasing wavenumber. Even
though the tail for the largest wavenumber considered here is the lowest of
the three cases in the energy range shown in Fig. 4, it extends farthest out.
In the collective (blue) regime, the high energy tail ends at around
$\omega=\pm 10$ ryd, in the transitional regime (black) it extends to
$\omega=\pm 25$ ryd, but in the single particle regime (red), it stretches to
$\omega=\pm 50$ ryd at the same level of intensity as shown here. In this
case, it would mean that a proper characterization of the target would require
the detector to be highly sensitive in the energy range from $7800$ eV to
$9200$ eV with an x-ray XFEL beam at $8500$ eV. This energy range has been
demonstrated at the HED instrument of the European XFEL Preston  _et al._
(2020).

Figure 5:  Non-equilibrium structure factor of an electron gas in a two-
temperature scenario. The Wigner distribution function is shown in panel a).
The dynamic structure factor is shown for different wavenumbers in panel b).
The intermediate scattering function is shown in panel c). The electronic
density is $n=1.8\times 10^{23}$ cm-3. The equilibrium temperature (cold bulk)
is $T_{c}=10^{5}$ K. The temperature of the tail is $T_{h}=10^{7}$ K. In panel
c), the x-axis for the equilibrium case and the blue non-eq. case is on top,
the non-equilibrium x-axis is at the bottom.

After the pump laser is switched off, the electrons relax back towards
equilibrium. One such intermediate state could be a scenario in which there
are two populations of electrons with different temperatures as presented in
Fig. 5. Again, we consider three different wavenumbers so as to not only
depend on the analysis of the symmetry properties of the ITCF. Again, we can
observe large high energy tails due to the high temperature tail in the Wigner
distribution. As before, even if asymmetries in the ITCFs cannot be resolved,
the differences in the detected temperatures for different wavenumbers are
such that only a single conclusion is possible.

Figure 6:  Non-equilibrium electron structure factor in fully ionized
hydrogen. The total electronic density is $n=1.8\times 10^{23}$ cm-3. The
temperature of the cold bulk electrons is $T_{c}=10^{5}$ K. The protons have a
temperature of $T=10^{5}$ K. The Wigner distribution function is shown in
panel a). The dynamic structure factor for the phonon and plasmon energy
ranges is shown in panels b) & c), respectively. The intermediate scattering
function is shown in panel d). In panel d), the x-axis for the equilibrium
case is on top, the non-equilibrium x-axis is at the bottom.

After analysing the behaviour of the pure electron gas in several different
non-equilibrium situations, we now focus on a more realistic but relatively
simple system, namely fully ionized hydrogen. As a real material, it will not
only feature the signal of the electrons in the dynamic structure factor, such
as plasmon peaks, but the ion acoustic modes. The latter will be modified in
appearance and occurrence from the equilibrium situation. An example is given
in Fig. 6. Panel a) contains the familiar situation of an x-ray pumped
electron distribution. Panel b) focuses on the ion modes. It is clear that the
reduced number of bulk electrons change the screening of the ion modes. We
observe that the modes are more pronounced in the non-equilibrium collective
regime when comparing them to the equilibrium results. On the other hand, the
electron modes and peaks are hardly altered by the presence of the equilibrium
protons, see panel c) and compare to Fig. 4. Nevertheless, the analysis of the
ITCFs for the three different wavenumbers shows strong deviations in behavior
and symmetry. No minimum in the ITCFs can be seen with the naked eye making
the distinction as non-equilibrium simple, even without the use of three
different wavenumbers.

Figure 7:  Non-equilibrium electron structure factor in a two-temperature
scenario in fully ionized hydrogen. The electronic density is $n=1.8\times
10^{23}$ cm-3. The temperature of the cold bulk electrons is $T_{c}=10^{5}$ K.
The temperature of the electrons in the tail is $T_{h}=10^{7}$ K. The protons
have a temperature of $T=10^{5}$ K. The Wigner distribution function is shown
in panels a) & b). The dynamic structure factor for the phonons and plasmons
is shown in panels c) & e), respectively. The intermediate scattering function
is shown in panel d) & f). In panel d), the Laplace transform was only
performed for energies from $-0.1$ ryd to $0.1$ ryd.

As before, we also study the case where the bulk of the electrons and protons
feature a single temperature, but a fraction of the electrons has a higher
temperature, see Fig. 7. The ion modes, as shown in panel c), are hardly
affected by the non-equilibrium electron distribution. Thus, if the ITCF is
determined by integration over the proton energy scale only, one obtains the
temperature of the protons. This is possible here due to the fact that the
bulk of the electrons has the same temperature. We again observe the impact of
the hot electron population on the dynamic structure in panel e). As seen
before, the plasmon position, width and height is altered compared to the
equilibrium case. Of particular importance are again the high energy tails of
the dynamic structure factor. Thus, the non-equilibrium ITCFs lack symmetry
individually and cross the line of unity at different imaginary times $\tau$,
see panel f).

Some remarks are in order for the case of a true two-temperature scenario in
which the electrons have a temperature and the ions have a different
temperature. As already stated in the theory section, the ITCF obtained from
the Laplace transform integrated over the electron energy range will then
always give the electron temperature. Performing a Laplace transform over just
the ion energy range will give a not very useful effective temperature in
between the ion and electron temperature because the electronic screening
changes the true ion modes. However, if the situation is such that a two-
temperature model can be applied in good conscience, then the DPA as
introduced in Eq. (8) is valid. A two-temperature model requires the electron-
ion coupling to be weak such that the energy transfer is slow enough to always
keep the subsystems in their separate equilibriums. This will be the case on
timescales larger than several hundred femtoseconds and up to several hundred
picoseconds. Then, having obtained the total electron structure factor and the
temperature of the electrons (we assume the density to be obtained either from
the ITCF as well or by other means), Eq. (8) can be inverted to obtain the
pure ion structure ${\cal L}_{ii}$. The latter than allows a Laplace transform
for the pure ion ITCF and the ion temperature. This of course assumes that the
ionic energy range of the scattering signal has been resolved sufficiently
which is challenging at the moment Descamps  _et al._ (2020); Wollenweber  _et
al._ (2021).

###  III.2 Time evolution of the scattering signal

In order to demonstrate the potential of our method to resolve the evolution
of the electronic structure via x-ray scattering in a realistic scenario, we
use the solutions of Monte Carlo simulations as input for the calculation of
the time dependent dynamic structure.

Figure 8:  Time evolution of the non-equilibrium structure factor of the
electrons in aluminium under laser irradiation. The electronic density is
$n=1.7\times 10^{23}$ cm-3. The Wigner distribution functions are shown in
panel a). The maximum laser intensity is at $t=0$. The time dependent dynamic
structure factor is shown in panel b). The ITCF is shown in panel c). In panel
d), the ratio between the ITCF of two different imaginary time arguments shows
the deviation from the equilibrium symmetry. For comparison, we show
equilibrium data for temperatures of $0.1$ eV, $1$ eV, and $2$ eV,
respectively (solid, short dashed, and long dashed brown curves).

Fig. 8 shows an example of a realistic laser fluence of $0.2$ J/cm2 onto a
solid slab of aluminium. The laser FWHM is $10$ fs and the laser wavelength is
$135$ Å ($92$ eV) corresponding to the VUV range of the spectrum. Initially,
all electrons are cold and contained in the Fermi function (black) with just
some initial few electrons already being scattered into higher states. The
smallest time step of $5$ fs in panel a) covers typical changes in the Wigner
functions but is smaller than the possible state-of-the-art time resolution
for driver, probe laser and detectors at the European XFEL. We observe that
the VUV pump laser creates a complicated time dependent electron energy
distribution that features several attributes that we have already discussed
in detail in the previous sections. The dominant photo-absorption for 92-eV-
photons in aluminium is by L-shell electrons, which creates the sharp peak at
$30$ eV (considered with broadening) Medvedev  _et al._ (2011). A smaller
contribution of the photo-absorption by the conduction band electrons promotes
photo-electrons to the energies of $\sim 90-100$ eV. The bump in the
population around $80$ eV is produced predominantly by the Auger-decays of
L-shell holes Medvedev  _et al._ (2011). There is a tail in between this bump
and the cold bulk being fed by electrons relaxing down from the bump.

Somewhat surprising, the pump laser also creates a tail of high energy
electrons beyond the nominal pump energy due to photoabsorption by the excited
electrons (sequential multi-photon absorption), and the scattering of
electrons Medvedev  _et al._ (2011). The bump increases in signal strength for
as long as the drive laser is on. The high energy tail continues to be more
populated even when the laser has ceased to drive further electrons into the
bump (purple vs. orange curves). The energy relaxation means that the formerly
steep flank of the Fermi distribution around the Fermi energy $\sim 11~{}eV$
becomes less and less steep for all considered times as the high energy
electrons thermalize. It is clear that even after $100$ fs, the electron
population has not reached a new equilibrium yet. The bump in the distribution
function at $\sim 60-70$ eV is still being fed by the Auger decays of the
L-shell. The complete relaxation of XUV-excited electrons in aluminium is thus
expected on the scale of a few hundred fs, as supported by experiments Bisio
_et al._ (2017).

Panel b) of Fig. 8 displays the time resolved dynamic structure factors for a
wavenumber in the collective regime. In particular for early times, the
dynamic structure contains signatures around $-15$ eV in addition to the
collective mode peaks. These are caused by the population of electrons round
$20$ eV. Compared to the considered equilibrium examples, the plasmon modes
are widened due to the high energy electrons. Temperatures of $1$ eV or $2$ eV
have been suggested as final temperatures after equilibration. Overall, if the
resolution of the detector is good enough, clear distinction between
equilibrium and non-equilibrium is possible. However, the ITCFs show a very
symmetric shape [see panels c) and d)]. Apart from the curious case at $t=-15$
fs, the deviation from an equilibrium like symmetry in the ITCF is less than
$1\%$ at all times. This is caused, as was discussed with Fig. 1 already, by
the dominance of the collective modes at this scattering angle which always
leads to symmetric ITCFs.

Figure 9:  Time evolution of the non-equilibrium ITCF of the electrons in
solid aluminium under laser irradiation. The electronic density is
$n=1.7\times 10^{23}$ cm-3. The Wigner distribution functions are shown in
panel a). In panels b), c) & d), the ratio between the ITCF of two different
imaginary time arguments shows the deviation from the equilibrium symmetry for
different wavenumbers.

Thus, we re-analyze the dynamic structure factor at two different wavenumbers
as shown in Fig. 9. Panels a) and b) repeat the quantities of Fig. 8 for
comparison. We observe that the small change of the scattering angle to
$0.75/$aB or to $1.0/$aB instead of $0.5/$aB is sufficient to see large
deviations from the $F(q,\tau)=F(q,\tau-\beta)$ symmetry of equilibrium. Even
though the black curve for the smallest time again shows the largest deviation
from unity, at later times the ITCF deviation shows the expected behavior:
these curves should relax towards unity for larger and larger times. For
intermediate times, the connection between different non-equilibrium features
of the Wigner distribution and particular deviations from symmetry in the ITCF
is somewhat convoluted such that plateaus in the Wigner distribution lead to
non-monotonous behavior of the ITCFs for different times. Nevertheless, the
power of the ITCF to detect non-equilibrium, especially in the transitional
and single particle regime, is obvious.

Figure 10:  Time evolution of the non-equilibrium ITCF of the electrons in
solid aluminium under laser irradiation of $1$ J/cm2, $10$ fs FWHM, wavelength
$\lambda=135$ Å($92$ eV) . The electronic density is $n=1.7\times 10^{23}$
cm-3, the wavenumber is $q=1/$ aB. The Wigner distribution functions are shown
in panel a). The time dependent averaged ITCF deviations with and without
noise are shown in panel b). The scattering signal (dynamic structure factor)
and the ITCF for various noise levels are shown in panels c) & d) for the case
of averaging over the first $15$ fs. Note that the y-axis is split into linear
and logarithmic scales at $10^{-3}$ in panel c).

For an example of a realistic signal and analysis of an experiment via the
ITCF, we consider the case of a higher laser energy flow onto the target in
Fig. 10. We observe similar features as before in the electron Wigner
distributions. However, due to the higher laser fluence, the cold bulk
population becomes severely depleted and the high energy tail becomes
stronger. The empty energy band between cold bulk and pump bump is filled with
electrons at $t=0$ ($10$ fs earlier than before) and as early as after $t=40$
fs, the Wigner distribution is monotonically decaying for all energies (this
process needed more than $100$ fs in the $0.2$ J/cm2 case).

We have opted to calculate the dynamic structure and the ITCF for a wavenumber
at which it will be possible to observe non-equilibrium effects in the ITCF at
just this single scattering angle. In order to well represent the actual
signal measured in an experiment, we disturb the calculated dynamic structure
factors with Gaussian noise of zero mean and the given standard deviation
$\sigma$ Sheffield  _et al._ (2010); Dornheim  _et al._ (2022b). We also
account for the fact that measurements at modern facilities have a time
resolution limited by the XFEL pulse duration van den Berg  _et al._ (2018);
we assume a minimum here of around $15$ fs and average the obtained dynamic
structure factors accordingly before adding the noise. The energy resolution
of the data has been coarse grained to $0.7$ eV to be of the same order of
magnitude as the dispersion of the HAPG spectrometer at HED Preston  _et al._
(2020).

The dynamic structure for the first time interval considered is shown in panel
c). Several possible noise levels are given. The corresponding ITCFs are shown
in panel d) of Fig. 10. It can be seen that the different levels of noise are
well handled by the Laplace transform. The considered noise does not influence
the ability of the ITCF to detect the non-equilibrium since the asymmetric
shape of the ITCF remains largely unaltered.

We have fixed the integration interval for the Laplace transform to $x=\pm
27.2$ eV. This value coincides with a resolution of the expected scattering
intensity over three to four orders of magnitude in experiment. If the
detection threshold could be expanded to six or seven orders of magnitude,
then a detection of the remnants of the high energy electrons between $60$ and
$80$ eV at $300$ fs would be possible. In any case, the small deviations from
the line at unity still in place in the ITCF at $300$ fs reflects the not
perfect Fermi-/Boltzmann-shape of the Wigner function in the small and medium
energy range, as can be seen in panel a) of Fig. 10 when comparing to the best
fit at $T=3.6$ eV.

The final analysis of the non-equilibrium state and its time evolution for
this particular case of laser irradiation is presented in panel b) of Fig. 10.
It displays the deviation of the ITCFs from the symmetric equilibrium shape.
Solid lines represent the case including noise, dashed lines show results
without noise added. There are small deviations between the curves in the two
cases, as can be expected. However, the deviation from the symmetric
equilibrium case is still well visible when noise is included. Even better,
the shape of the deviation from equilibrium remains largely unaltered when
noise is considered. The relaxation towards equilibrium is apparent when
comparing these three averaging windows. We can compare to the Wigner
distributions of panel a), from which it is clear that after $100$ fs no
thermal distribution has been established yet and we need on the order of
$300$ fs for a final temperature of around $3.6$ eV to be established.

We can also observe that electrons of higher energies seem to scatter towards
energies in the range $60-80$ eV faster than deficiencies in the population at
low energies can be filled. This is the reason that it takes well over $100$
fs to establish a Wigner function that looks even remotely like an equilibrium
distribution.

##  IV Summary and Outlook

We have shown the power of the imaginary time correlation function (ITCF) to
detect signatures of non-equilibrium in the scattering signal obtained in
experiments utilizing VUV, x-ray, or electron probe beams. Further, analyzing
the properties of the ITCF as they change over femto-, pico-, or even nano-
seconds, a diagnostic of the driving and relaxation of the non-equilibrium
state is possible. The entire analysis is model free and is based on the
degree of violation of the detailed balance principle that is valid only in
equilibrium.

We concentrated on non-equilibrium features of the Wigner distribution
function of the electrons. For these cases, we used model Wigner functions
that feature typical artefacts of non-equilibrium, e.g., high energy bumps and
tails. We also used solutions of a Boltzmann-MC code to calculate femto-second
time resolved scattering spectra and ITCFs for the case of laser heated
aluminium. We time-averaged these ITCF spectra and introduced artificial noise
to show the capabilities of the method under typical experimental diagnostics
conditions.

Clear indications for non-equilibrium can be seen in all cases for wavenumbers
(scattering angles) in the single particle or transitional scattering regime.
In such an experimental setup, a single detector is sufficient to trace the
signatures of non-equilibrium. In the non-equilibrium collective regime,
dominating collective peaks in the dynamic structure factor (plasmons,
acoustic modes) will cause the ITCF to be symmetric, even in non-equilibrium,
although with a fake ’temperature’. A non-equilibrium signal might still be
obtained when analyzing not just the interval $\tau\epsilon[0,\beta^{\prime}]$
but the general symmetry around $\beta^{\prime}/2$ outside this interval. A
safer option is in any case the fielding of two or more spectrally resolving
detectors. In this case, if one shows a non-symmetric ITCF, it cannot be
equilibrium. If all detectors show a symmetric ITCF but the obtained values of
$\beta_{i}$ differ, it must be non-equilibrium. In such a way, we imagine,
different local thermodynamic equilibria of distinct parts of the total
scattering volume (as in ICF fusion pellets) will reveal itself.

Thus, we expect this method to become a standard tool in future experimental
scattering setups. Since the probe beam, even a seeded x-ray laser beam, has a
finite bandwidth and the detector has a limited resolution in energy space, it
is absolutely necessary to characterize the source and detector function to at
least the same accuracy and precision as the actual scattering signal. In
particular for two-temperature relaxation, we expect a qualitative and
quantitative improvement in the experimental abilities and characterization of
the target, as nowadays often only the ion temperature is measured but not the
electron temperature Waldecker  _et al._ (2016); Fletcher  _et al._ (2022).

## Acknowledgments

This work was partially supported by the Center for Advanced Systems
Understanding (CASUS) which is financed by Germany’s Federal Ministry of
Education and Research (BMBF) and by the Saxon state government out of the
State budget approved by the Saxon State Parliament. NM gratefully
acknowledges financial support from the Czech Ministry of Education, Youth and
Sports (grants No. LTT17015, LM2018114, and No. EF16_013/0001552).
Computational resources for Boltzmann-MC calculations were supplied by the
project ”e-Infrastruktura CZ” (e-INFRA LM2018140) provided within the program
Projects of Large Research, Development and Innovations Infrastructures.

## References

  * Dornheim  _et al._ (2022a) Tobias Dornheim, Maximilian Böhme, Dominik Kraus, Tilo Döppner, Thomas R. Preston, Zhandos A. Moldabekov, and Jan Vorberger, “Accurate temperature diagnostics for matter under extreme conditions,” Nature Communications 13, 7911 (2022a).
  * Falk (2018) K. Falk, “Experimental methods for warm dense matter research,” High Power Laser Sci. Eng 6, e59 (2018).
  * Macchi  _et al._ (2013) Andrea Macchi, Marco Borghesi, and Matteo Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys. 85, 751–793 (2013).
  * Kremp  _et al._ (2005) D. Kremp, M. Schlanges, and W.-D. Kraeft,  _Quantum Statistics of Nonideal Plasmas_ (Springer, Heidelberg, 2005).
  * Bonitz (2016) M. Bonitz,  _Quantum kinetic theory_ (Springer, Heidelberg, 2016).
  * Arber  _et al._ (2015) T D Arber, K Bennett, C S Brady, A Lawrence-Douglas, M G Ramsay, N J Sircombe, P Gillies, R G Evans, H Schmitz, A R Bell, and C P Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Physics and Controlled Fusion 57, 113001 (2015).
  * Gamaly (2011) E.G. Gamaly, “The physics of ultra-short laser interaction with solids at non-relativistic intensities,” Physics Reports 508, 91–243 (2011).
  * Bonitz  _et al._ (2020) M. Bonitz, T. Dornheim, Zh. A. Moldabekov, S. Zhang, P. Hamann, H. Kählert, A. Filinov, K. Ramakrishna, and J. Vorberger, “Ab initio simulation of warm dense matter,” Physics of Plasmas 27, 042710 (2020).
  * Dornheim  _et al._ (2022b) Tobias Dornheim, Maximilian Böhme, David Chapman, Dominik Kraus, Thomas Preston, Zhandos Moldabekov, Niclas Schlünzn, Attila Cangi, Tilo Döppner, and Jan Vorberger, “Temperature analysis of x-ray thomson scattering data,” arXiv:2212.10510 (2022b).
  * Tschentscher  _et al._ (2017) Thomas Tschentscher, Christian Bressler, Jan Grünert, Anders Madsen, Adrian P. Mancuso, Michael Meyer, Andreas Scherz, Harald Sinn, and Ulf Zastrau, “Photon beam transport and scientific instruments at the european xfel,” Applied Sciences 7 (2017), 10.3390/app7060592.
  * Bostedt  _et al._ (2016) Christoph Bostedt, Sébastien Boutet, David M. Fritz, Zhirong Huang, Hae Ja Lee, Henrik T. Lemke, Aymeric Robert, William F. Schlotter, Joshua J. Turner, and Garth J. Williams, “Linac coherent light source: The first five years,” Rev. Mod. Phys. 88, 015007 (2016).
  * Pile (2011) David Pile, “First light from sacla,” Nature Photonics 5, 456–457 (2011).
  * Waldecker  _et al._ (2015) Lutz Waldecker, Roman Bertoni, and Ralph Ernstorfer, “Compact femtosecond electron diffractometer with 100?kev electron bunches approaching the single-electron pulse duration limit,” Journal of Applied Physics 117, 044903 (2015).
  * Knudson  _et al._ (2015) M. D. Knudson, M. P. Desjarlais, A. Becker, R. W. Lemke, K. R. Cochrane, M. E. Savage, D. E. Bliss, T. R. Mattsson, and R. Redmer, “Direct observation of an abrupt insulator-to-metal transition in dense liquid deuterium,” Science 348, 1455–1460 (2015).
  * Fletcher  _et al._ (2015) L. B. Fletcher, H. J. Lee, T. Döppner, E. Galtier, B. Nagler, P. Heimann, C. Fortmann, S. LePape, T. Ma, M. Millot, A. Pak, D. Turnbull, D. A. Chapman, D. O. Gericke, J. Vorberger, T. White, G. Gregori, M. Wei, B. Barbrel, R. W. Falcone, C.-C. Kao, H. Nuhn, J. Welch, U. Zastrau, P. Neumayer, J. B. Hastings, and S. H. Glenzer, “Ultrabright x-ray laser scattering for dynamic warm dense matter physics,” Nature Photonics 9, 274–279 (2015).
  * Kraus  _et al._ (2016) D. Kraus, A. Ravasio, M. Gauthier, D. O. Gericke, J. Vorberger, S. Frydrych, J. Helfrich, L. B. Fletcher, G. Schaumann, B. Nagler, B. Barbrel, B. Bachmann, E. J. Gamboa, S. Göde, E. Granados, G. Gregori, H. J. Lee, P. Neumayer, W. Schumaker, T. Döppner, R. W. Falcone, S. H. Glenzer, and M. Roth, “Nanosecond formation of diamond and lonsdaleite by shock compression of graphite,” Nature Communications 7, 10970 (2016).
  * Kraus  _et al._ (2017) D. Kraus, J. Vorberger, A. Pak, N. J. Hartley, L. B. Fletcher, S. Frydrych, E. Galtier, E. J. Gamboa, D. O. Gericke, S. H. Glenzer, E. Granados, M. J. MacDonald, A. J. MacKinnon, E. E. McBride, I. Nam, P. Neumayer, M. Roth, A. M. Saunders, A. K. Schuster, P. Sun, T. van Driel, T. Döppner, and R. W. Falcone, “Formation of diamonds in laser-compressed hydrocarbons at planetary interior conditions,” Nature Astronomy 1, 606–611 (2017).
  * Celliers  _et al._ (2018) Peter M. Celliers, Marius Millot, Stephanie Brygoo, R. Stewart McWilliams, Dayne E. Fratanduono, J. Ryan Rygg, Alexander F. Goncharov, Paul Loubeyre, Jon H. Eggert, J. Luc Peterson, Nathan B. Meezan, Sebastien Le Pape, Gilbert W. Collins, Raymond Jeanloz, and Russell J. Hemley, “Insulator-metal transition in dense fluid deuterium,” Science 361, 677–682 (2018).
  * Lazicki  _et al._ (2021) A. Lazicki, D. McGonegle, J. R. Rygg, D. G. Braun, D. C. Swift, M. G. Gorman, R. F. Smith, P. G. Heighway, A. Higginbotham, M. J. Suggit, D. E. Fratanduono, F. Coppari, C. E. Wehrenberg, R. G. Kraus, D. Erskine, J. V. Bernier, J. M. McNaney, R. E. Rudd, G. W. Collins, J. H. Eggert, and J. S. Wark, “Metastability of diamond ramp-compressed to 2 terapascals,” Nature 589, 532–535 (2021).
  * Gaus  _et al._ (2021) Lennart Gaus, Lothar Bischoff, Michael Bussmann, Eric Cunningham, Chandra B. Curry, Juncheng E, Eric Galtier, Maxence Gauthier, Alejandro Laso García, Marco Garten, Siegfried Glenzer, Jörg Grenzer, Christian Gutt, Nicholas J. Hartley, Lingen Huang, Uwe Hübner, Dominik Kraus, Hae Ja Lee, Emma E. McBride, Josefine Metzkes-Ng, Bob Nagler, Motoaki Nakatsutsumi, Jan Nikl, Masato Ota, Alexander Pelka, Irene Prencipe, Lisa Randolph, Melanie Rödel, Youichi Sakawa, Hans-Peter Schlenvoigt, Michal Šmíd, Franziska Treffert, Katja Voigt, Karl Zeil, Thomas E. Cowan, Ulrich Schramm, and Thomas Kluge, “Probing ultrafast laser plasma processes inside solids with resonant small-angle x-ray scattering,” Phys. Rev. Res. 3, 043194 (2021).
  * Liu  _et al._ (2019) Shang-Fei Liu, Yasunori Hori, Simon Müller, Xiaochen Zheng, Ravit Helled, Doug Lin, and Andrea Isella, “The formation of jupiter’s diluted core by a giant impact,” Nature 572, 355–357 (2019).
  * Brygoo  _et al._ (2021) S. Brygoo, P. Loubeyre, M. Millot, J. R. Rygg, P. M. Celliers, J. H. Eggert, R. Jeanloz, and G. W. Collins, “Evidence of hydrogen-helium immiscibility at jupiter-interior conditions,” Nature 593, 517–521 (2021).
  * Militzer  _et al._ (2022) Burkhard Militzer, William B. Hubbard, Sean Wahl, Jonathan I. Lunine, Eli Galanti, Yohai Kaspi, Yamila Miguel, Tristan Guillot, Kimberly M. Moore, Marzia Parisi, John E. P. Connerney, Ravid Helled, Hao Cao, Christopher Mankovich, David J. Stevenson, Ryan S. Park, Mike Wong, Sushil K. Atreya, John Anderson, and Scott J. Bolton, “Juno spacecraft measurements of jupiter’s gravity imply a dilute core,” The Planetary Science Journal 3, 185 (2022).
  * Kritcher  _et al._ (2020) Andrea L. Kritcher, Damian C. Swift, Tilo Döppner, Benjamin Bachmann, Lorin X. Benedict, Gilbert W. Collins, Jonathan L. DuBois, Fred Elsner, Gilles Fontaine, Jim A. Gaffney, Sebastien Hamel, Amy Lazicki, Walter R. Johnson, Natalie Kostinski, Dominik Kraus, Michael J. MacDonald, Brian Maddox, Madison E. Martin, Paul Neumayer, Abbas Nikroo, Joseph Nilsen, Bruce A. Remington, Didier Saumon, Phillip A. Sterne, Wendi Sweet, Alfredo A. Correa, Heather D. Whitley, Roger W. Falcone, and Siegfried H. Glenzer, “A measurement of the equation of state of carbon envelopes of white dwarfs,” Nature 584, 51–54 (2020).
  * Becker  _et al._ (2014) A. Becker, W. Lorenzen, J. J. Fortney, N. Nettelmann, M. Schöttler, and R. Redmer, “Ab initio equations of state for hydrogen (h-reos.3) and helium (he-reos.3) and their implications for the interior of brown dwarfs,” Astrophys. J. Suppl. Ser 215, 21 (2014).
  * Chamel and Haensel (2008) Nicolas Chamel and Pawel Haensel, “Physics of neutron star crusts,” Living Reviews in Relativity 11, 10 (2008).
  * White  _et al._ (2020) S. White, B. Kettle, J. Vorberger, C. L. S. Lewis, S. H. Glenzer, E. Gamboa, B. Nagler, F. Tavella, H. J. Lee, C. D. Murphy, D. O. Gericke, and D. Riley, “Time-dependent effects in melting and phase change for laser-shocked iron,” Phys. Rev. Research 2, 033366 (2020).
  * Sperling  _et al._ (2015) P. Sperling, E. J. Gamboa, H. J. Lee, H. K. Chung, E. Galtier, Y. Omarbakiyeva, H. Reinholz, G. Röpke, U. Zastrau, J. Hastings, L. B. Fletcher, and S. H. Glenzer, “Free-electron x-ray laser measurements of collisional-damped plasmons in isochorically heated warm dense matter,” Phys. Rev. Lett. 115, 115001 (2015).
  * Kraus  _et al._ (2019) D. Kraus, B. Bachmann, B. Barbrel, R. W. Falcone, L. B. Fletcher, S. Frydrych, E. J. Gamboa, M. Gauthier, D. O. Gericke, S. H. Glenzer, S. Göde, E. Granados, N. J. Hartley, J. Helfrich, H. J. Lee, B. Nagler, A. Ravasio, W. Schumaker, J. Vorberger, and T. Döppner, “Characterizing the ionization potential depression in dense carbon plasmas with high-precision spectrally resolved x-ray scattering,” Plasma Phys. Control Fusion 61, 014015 (2019).
  * Zahn  _et al._ (2021) Daniela Zahn, Florian Jakobs, Yoav William Windsor, Hélène Seiler, Thomas Vasileiadis, Tim A. Butcher, Yingpeng Qi, Dieter Engel, Unai Atxitia, Jan Vorberger, and Ralph Ernstorfer, “Lattice dynamics and ultrafast energy flow between electrons, spins, and phonons in a 3d ferromagnet,” Phys. Rev. Research 3, 023032 (2021).
  * Zahn  _et al._ (2022) Daniela Zahn, Florian Jakobs, Hélène Seiler, Tim A. Butcher, Dieter Engel, Jan Vorberger, Unai Atxitia, Yoav William Windsor, and Ralph Ernstorfer, “Intrinsic energy flow in laser-excited $3d$ ferromagnets,” Phys. Rev. Research 4, 013104 (2022).
  * Marks  _et al._ (2022) Michael Raj Marks, Kuan Yew Cheong, and Zainuriah Hassan, “A review of laser ablation and dicing of si wafers,” Precision Engineering 73, 377–408 (2022).
  * Vogel and Venugopalan (2003) Alfred Vogel and Vasan Venugopalan, “Mechanisms of pulsed laser ablation of biological tissues,” Chemical Reviews 103, 577–644 (2003).
  * Frischkorn and Wolf (2006) Christian Frischkorn and Martin Wolf, “Femtochemistry at metal surfaces: nonadiabatic reaction dynamics,” Chemical Reviews 106, 4207–4233 (2006).
  * Bäuerle (2011) D. Bäuerle,  _Laser Processing and Chemistry_ (Springer Verlag, Berlin, 2011).
  * Balling and Schou (2013) P Balling and J Schou, “Femtosecond-laser ablation dynamics of dielectrics: basics and applications for thin films,” Reports on Progress in Physics 76, 036502 (2013).
  * Vorberger  _et al._ (2010) J. Vorberger, D. O. Gericke, Th. Bornath, and M. Schlanges, “Energy relaxation in dense, strongly coupled two-temperature plasmas,” Phys. Rev. E 81, 046404 (2010).
  * Benedict  _et al._ (2017) L. X. Benedict, M. P. Surh, L. G. Stanton, C. R. Scullard, A. A. Correa, J. I. Castor, F. R. Graziani, L. A. Collins, O. Certík, J. D. Kress, and M. S. Murillo, “Molecular dynamics studies of electron-ion temperature equilibration in hydrogen plasmas within the coupled-mode regime,” Phys. Rev. E 95, 043202 (2017).
  * Simoni and Daligault (2020) Jacopo Simoni and Jérôme Daligault, “Calculation of electron-ion temperature equilibration rates and friction coefficients in plasmas and liquid metals using quantum molecular dynamics,” Phys. Rev. E 101, 013205 (2020).
  * Medvedev and Milov (2020) N. Medvedev and I. Milov, “Electron-phonon coupling in metals at high electronic temperatures,” Phys. Rev. B 102, 064302 (2020).
  * Zhang  _et al._ (2022) Jia Zhang, Rui Qin, Wenjun Zhu, and Jan Vorberger, “Energy relaxation and electron-phonon coupling in laser-excited metals,” Materials 15 (2022), 10.3390/ma15051902.
  * Ciricosta  _et al._ (2016a) O Ciricosta, S M Vinko, H. K. Chung, C Jackson, R W Lee, T R Preston, D S Rackstraw, and J S Wark, “Detailed model for hot-dense aluminum plasmas generated by an X-ray free electron laser,” Physics of Plasmas 23 (2016a), 10.1063/1.4942540.
  * Celliers  _et al._ (1992) P. Celliers, A. Ng, G. Xu, and A. Forsman, “Thermal equilibration in a shock wave,” Phys. Rev. Lett. 68, 2305–2308 (1992).
  * White  _et al._ (2012) T. G. White, J. Vorberger, C. R. D. Brown, B. J. B. Crowley, P. Davis, S. H. Glenzer, J. W. O. Harris, D. C. Hochhaus, S. Le Pape, T. Ma, C. D. Murphy, P. Neumayer, L. K. Pattison, S. Richardson, D. O. Gericke, and G. Gregori, “Observation of inhibited electron-ion coupling in strongly heated graphite,” Scientific Reports 2, 889 (2012).
  * Zastrau  _et al._ (2014) U. Zastrau, P. Sperling, M. Harmand, A. Becker, T. Bornath, R. Bredow, S. Dziarzhytski, T. Fennel, L. B. Fletcher, E. Förster, S. Göde, G. Gregori, V. Hilbert, D. Hochhaus, B. Holst, T. Laarmann, H. J. Lee, T. Ma, J. P. Mithen, R. Mitzner, C. D. Murphy, M. Nakatsutsumi, P. Neumayer, A. Przystawik, S. Roling, M. Schulz, B. Siemer, S. Skruszewicz, J. Tiggesbäumker, S. Toleikis, T. Tschentscher, T. White, M. Wöstmann, H. Zacharias, T. Döppner, S. H. Glenzer, and R. Redmer, “Resolving ultrafast heating of dense cryogenic hydrogen,” Phys. Rev. Lett. 112, 105002 (2014).
  * Hartley  _et al._ (2015) N.J. Hartley, P. Belancourt, D.A. Chapman, T. Döppner, R.P. Drake, D.O. Gericke, S.H. Glenzer, D. Khaghani, S. LePape, T. Ma, P. Neumayer, A. Pak, L. Peters, S. Richardson, J. Vorberger, T.G. White, and G. Gregori, “Electron-ion temperature equilibration in warm dense tantalum,” High Energy Density Physics 14, 1–5 (2015).
  * Mo  _et al._ (2018) M. Z. Mo, Z. Chen, R. K. Li, M. Dunning, B. B. L. Witte, J. K. Baldwin, L. B. Fletcher, J. B. Kim, A. Ng, R. Redmer, A. H. Reid, P. Shekhar, X. Z. Shen, M. Shen, K. Sokolowski-Tinten, Y. Y. Tsui, Y. Q. Wang, Q. Zheng, X. J. Wang, and S. H. Glenzer, “Heterogeneous to homogeneous melting transition visualized with ultrafast electron diffraction,” Science 360, 1451–1455 (2018).
  * Fletcher  _et al._ (2022) L. B. Fletcher, J. Vorberger, W. Schumaker, C. Ruyer, S. Goede, E. Galtier, U. Zastrau, E. P. Alves, S. D. Baalrud, R. A. Baggott, B. Barbrel, Z. Chen, T. Döppner, M. Gauthier, E. Granados, J. B. Kim, D. Kraus, H. J. Lee, M. J. MacDonald, R. Mishra, A. Pelka, A. Ravasio, C. Roedel, A. R. Fry, R. Redmer, F. Fiuza, D. O. Gericke, and S. H. Glenzer, “Electron-ion temperature relaxation in warm dense hydrogen observed with picosecond resolved x-ray scattering,” Frontiers in Physics 10 (2022), 10.3389/fphy.2022.838524.
  * Vinko  _et al._ (2012) S. M. Vinko, O. Ciricosta, B. I. Cho, K. Engelhorn, H.-K. Chung, C. R. D. Brown, T. Burian, J. Chalupský, R. W. Falcone, C. Graves, V. Hájková, A. Higginbotham, L. Juha, J. Krzywinski, H. J. Lee, M. Messerschmidt, C. D. Murphy, Y. Ping, A. Scherz, W. Schlotter, S. Toleikis, J. J. Turner, L. Vysin, T. Wang, B. Wu, U. Zastrau, D. Zhu, R. W. Lee, P. A. Heimann, B. Nagler, and J. S. Wark, “Creation and diagnosis of a solid-density plasma with an X-ray free-electron laser,” Nature 482, 59–62 (2012).
  * Ciricosta  _et al._ (2012) O. Ciricosta, S. M. Vinko, H.-K. Chung, B. I. Cho, C. R D Brown, T. Burian, J. Chalupský, K. Engelhorn, R. W. Falcone, C. Graves, V. Hájková, A. Higginbotham, L. Juha, J. Krzywinski, H. J. Lee, M. Messerschmidt, C. D. Murphy, Y. Ping, D. S. Rackstraw, A. Scherz, W. Schlotter, S. Toleikis, J. J. Turner, L. Vysin, T. Wang, B. Wu, U. Zastrau, D. Zhu, R. W. Lee, P. Heimann, B. Nagler, and J. S. Wark, “Direct measurements of the ionization potential depression in a dense plasma,” Physical Review Letters 109 (2012).
  * Ciricosta  _et al._ (2016b) O Ciricosta, S M Vinko, B Barbrel, D S Rackstraw, T R Preston, T Burian, J. Chalupský, B I Cho, H. K. Chung, G L Dakovski, K Engelhorn, V. Hájková, P. Heimann, M. Holmes, L. Juha, J. Krzywinski, R. W. Lee, S. Toleikis, J. J. Turner, U. Zastrau, and J. S. Wark, “Measurements of continuum lowering in solid-density plasmas created from elements and compounds,” Nature Communications 7, 11713 (2016b).
  * Preston  _et al._ (2017) T. R. Preston, S. M. Vinko, O. Ciricosta, P. Hollebon, H.-K. Chung, G. L. Dakovski, J. Krzywinski, M. Minitti, T. Burian, J. Chalupský, V. Hájková, L. Juha, V. Vozda, U. Zastrau, R. W. Lee, and J. S. Wark, “Measurements of the ¡math display=”inline”¿ ¡mi¿K¡/mi¿ ¡/math¿ -Shell Opacity of a Solid-Density Magnesium Plasma Heated by an X-Ray Free-Electron Laser,” Physical Review Letters 119, 085001 (2017).
  * Edie, D.J.  _et al._ (2013) Edie, D.J., Vorberger, J., Rose, S., and Gericke, D.O., “$\alpha$-particle stopping and electron-ion energy relaxation in highly compressed icf fuel,” EPJ Web of Conferences 59, 05018 (2013).
  * Zylstra  _et al._ (2022) A. B. Zylstra, O. A. Hurricane, D. A. Callahan, A. L. Kritcher, J. E. Ralph, H. F. Robey, J. S. Ross, C. V. Young, K. L. Baker, D. T. Casey, T. Döppner, L. Divol, M. Hohenberger, S. Le Pape, A. Pak, P. K. Patel, R. Tommasini, S. J. Ali, P. A. Amendt, L. J. Atherton, B. Bachmann, D. Bailey, L. R. Benedetti, L. Berzak Hopkins, R. Betti, S. D. Bhandarkar, J. Biener, R. M. Bionta, N. W. Birge, E. J. Bond, D. K. Bradley, T. Braun, T. M. Briggs, M. W. Bruhn, P. M. Celliers, B. Chang, T. Chapman, H. Chen, C. Choate, A. R. Christopherson, D. S. Clark, J. W. Crippen, E. L. Dewald, T. R. Dittrich, M. J. Edwards, W. A. Farmer, J. E. Field, D. Fittinghoff, J. Frenje, J. Gaffney, M. Gatu Johnson, S. H. Glenzer, G. P. Grim, S. Haan, K. D. Hahn, G. N. Hall, B. A. Hammel, J. Harte, E. Hartouni, J. E. Heebner, V. J. Hernandez, H. Herrmann, M. C. Herrmann, D. E. Hinkel, D. D. Ho, J. P. Holder, W. W. Hsing, H. Huang, K. D. Humbird, N. Izumi, L. C. Jarrott, J. Jeet, O. Jones, G. D. Kerbel, S. M. Kerr, S. F. Khan, J. Kilkenny, Y. Kim, H. Geppert Kleinrath, V. Geppert Kleinrath, C. Kong, J. M. Koning, J. J. Kroll, M. K. G. Kruse, B. Kustowski, O. L. Landen, S. Langer, D. Larson, N. C. Lemos, J. D. Lindl, T. Ma, M. J. MacDonald, B. J. MacGowan, A. J. Mackinnon, S. A. MacLaren, A. G. MacPhee, M. M. Marinak, D. A. Mariscal, E. V. Marley, L. Masse, K. Meaney, N. B. Meezan, P. A. Michel, M. Millot, J. L. Milovich, J. D. Moody, A. S. Moore, J. W. Morton, T. Murphy, K. Newman, J.-M. G. Di Nicola, A. Nikroo, R. Nora, M. V. Patel, L. J. Pelz, J. L. Peterson, Y. Ping, B. B. Pollock, M. Ratledge, N. G. Rice, H. Rinderknecht, M. Rosen, M. S. Rubery, J. D. Salmonson, J. Sater, S. Schiaffino, D. J. Schlossberg, M. B. Schneider, C. R. Schroeder, H. A. Scott, S. M. Sepke, K. Sequoia, M. W. Sherlock, S. Shin, V. A. Smalyuk, B. K. Spears, P. T. Springer, M. Stadermann, S. Stoupin, D. J. Strozzi, L. J. Suter, C. A. Thomas, R. P. J. Town, E. R. Tubman, P. L. Volegov, C. R. Weber, K. Widmann, C. Wild, C. H. Wilde, B. M. Van Wonterghem, D. T. Woods, B. N. Woodworth, M. Yamaguchi, S. T. Yang, and G. B. Zimmerman, “Burning plasma achieved in inertial fusion,” Nature 601, 542–548 (2022).
  * Malko  _et al._ (2022) S. Malko, W. Cayzac, V. Ospina-Bohórquez, K. Bhutwala, M. Bailly-Grandvaux, C. McGuffey, R. Fedosejevs, X. Vaisseau, An. Tauschwitz, J. I. Apiñaniz, D. De Luis Blanco, G. Gatti, M. Huault, J. A. Perez Hernandez, S. X. Hu, A. J. White, L. A. Collins, K. Nichols, P. Neumayer, G. Faussurier, J. Vorberger, G. Prestopino, C. Verona, J. J. Santos, D. Batani, F. N. Beg, L. Roso, and L. Volpe, “Proton stopping measurements at low velocity in warm dense carbon,” Nature Communications 13, 2893 (2022).
  * Ramakrishna  _et al._ (2022) Kushal Ramakrishna, Mani Lokamani, Andrew Baczewski, Jan Vorberger, and Attila Cangi, “Electrical and thermal conductivity of high-pressure solid iron,”  (2022), 10.48550/ARXIV.2210.10132.
  * Dornheim  _et al._ (2020) Tobias Dornheim, Jan Vorberger, and Michael Bonitz, “Nonlinear electronic density response in warm dense matter,” Phys. Rev. Lett. 125, 085001 (2020).
  * Dornheim  _et al._ (2021a) Tobias Dornheim, Zhandos A. Moldabekov, and Jan Vorberger, “Nonlinear density response from imaginary-time correlation functions: Ab initio path integral monte carlo simulations of the warm dense electron gas,” The Journal of Chemical Physics 155, 054110 (2021a).
  * Dornheim  _et al._ (2021b) Tobias Dornheim, Jan Vorberger, and Zhandos A. Moldabekov, “Nonlinear density response and higher order correlation functions in warm dense matter,” Journal of the Physical Society of Japan 90, 104002 (2021b).
  * Dornheim  _et al._ (2021c) Tobias Dornheim, Maximilian Böhme, Zhandos A. Moldabekov, Jan Vorberger, and Michael Bonitz, “Density response of the warm dense electron gas beyond linear response theory: Excitation of harmonics,” Phys. Rev. Research 3, 033231 (2021c).
  * Moldabekov  _et al._ (2022) Zhandos Moldabekov, Jan Vorberger, and Tobias Dornheim, “Density functional theory perspective on the nonlinear response of correlated electrons across temperature regimes,” Journal of Chemical Theory and Computation 18, 2900–2912 (2022).
  * Chapman and Gericke (2011) D. A. Chapman and D. O. Gericke, “Analysis of thomson scattering from nonequilibrium plasmas,” Phys. Rev. Lett. 107, 165004 (2011).
  * Vorberger and Chapman (2018) J. Vorberger and D. A. Chapman, “Quantum theory for the dynamic structure factor in correlated two-component systems in nonequilibrium: Application to x-ray scattering,” Phys. Rev. E 97, 013203 (2018).
  * Medvedev  _et al._ (2022) N. Medvedev, F. Akhmetov, R.A. Rymzhanov, R. Voronkov, and A.E. Volkov, “Modeling time-resolved kinetics in solids induced by extreme electronic excitation,” Advanced Theory and Simulations , 2200091 (2022).
  * Chapman  _et al._ (2014) D. A. Chapman, D. Kraus, A. L. Kritcher, B. Bachmann, G. W. Collins, R. W. Falcone, J. A. Gaffney, D. O. Gericke, S. H. Glenzer, T. M. Guymer, J. A. Hawreliak, O. L. Landen, S. Le Pape, T. Ma, P. Neumayer, J. Nilsen, A. Pak, R. Redmer, D. C. Swift, J. Vorberger, and T. Döppner, “Simulating x-ray thomson scattering signals from high-density, millimetre-scale plasmas at the national ignition facility,” Physics of Plasmas 21, 082709 (2014).
  * Kozlowski  _et al._ (2016) P. M. Kozlowski, B. J. B. Crowley, D. O. Gericke, S. P. Regan, and G. Gregori, “Erratum: Theory of thomson scattering in inhomogeneous media,” Scientific Reports 6, 26366 (2016).
  * Glenzer and Redmer (2009) S. H. Glenzer and R. Redmer, “X-ray thomson scattering in high energy density plasmas,” Rev. Mod. Phys 81, 1625 (2009).
  * Sheffield  _et al._ (2010) J. Sheffield, D. Froula, S.H. Glenzer, and N.C. Luhmann,  _Plasma Scattering of Electromagnetic Radiation: Theory and Measurement Techniques_ (Elsevier Science, 2010).
  * Fäustlin  _et al._ (2010) R. R. Fäustlin, Th. Bornath, T. Döppner, S. Düsterer, E. Förster, C. Fortmann, S. H. Glenzer, S. Göde, G. Gregori, R. Irsig, T. Laarmann, H. J. Lee, B. Li, K.-H. Meiwes-Broer, J. Mithen, B. Nagler, A. Przystawik, H. Redlin, R. Redmer, H. Reinholz, G. Röpke, F. Tavella, R. Thiele, J. Tiggesbäumker, S. Toleikis, I. Uschmann, S. M. Vinko, T. Whitcher, U. Zastrau, B. Ziaja, and Th. Tschentscher, “Observation of Ultrafast Nonequilibrium Collective Dynamics in Warm Dense Hydrogen,” Physical Review Letters 104, 125002 (2010).
  * Medvedev  _et al._ (2011) N. Medvedev, U. Zastrau, E. Förster, D.O. Gericke, and B. Rethfeld, “Short-time electron dynamics in aluminum excited by femtosecond extreme ultraviolet radiation,” Physical Review Letters 107, 165003 (2011).
  * Hau-Riege (2013) S.P. Hau-Riege, “Nonequilibrium electron dynamics in materials driven by high-intensity x-ray pulses,” Physical Review E 87, 53102 (2013).
  * Henke  _et al._ (1993) B.L. Henke, E.M. Gullikson, and J.C. Davis, “X-Ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92,” Atomic Data and Nuclear Data Tables 54, 181–342 (1993).
  * Keski-Rahkonen and Krause (1974) O. Keski-Rahkonen and M.O. Krause, “Total and partial atomic-level widths,” Atomic Data and Nuclear Data Tables 14, 139–146 (1974).
  * Dornheim  _et al._ (2016) T. Dornheim, S. Groth, T. Sjostrom, F. D. Malone, W. M. C. Foulkes, and M. Bonitz, “Ab initio quantum Monte Carlo simulation of the warm dense electron gas in the thermodynamic limit,” Phys. Rev. Lett. 117, 156403 (2016).
  * Dornheim  _et al._ (2017) Tobias Dornheim, Simon Groth, Fionn D. Malone, Tim Schoof, Travis Sjostrom, W. M. C. Foulkes, and Michael Bonitz, “Ab initio quantum monte carlo simulation of the warm dense electron gas,” Physics of Plasmas 24, 056303 (2017).
  * Dornheim  _et al._ (2018a) T. Dornheim, S. Groth, and M. Bonitz, “The uniform electron gas at warm dense matter conditions,” Phys. Reports 744, 1–86 (2018a).
  * Dornheim  _et al._ (2018b) T. Dornheim, S. Groth, J. Vorberger, and M. Bonitz, “Ab initio path integral Monte Carlo results for the dynamic structure factor of correlated electrons: From the electron liquid to warm dense matter,” Phys. Rev. Lett. 121, 255001 (2018b).
  * Dornheim  _et al._ (2019) T. Dornheim, J. Vorberger, S. Groth, N. Hoffmann, Zh.A. Moldabekov, and M. Bonitz, “The static local field correction of the warm dense electron gas: An ab initio path integral Monte Carlo study and machine learning representation,” J. Chem. Phys 151, 194104 (2019).
  * Dornheim  _et al._ (2021d) Tobias Dornheim, Jan Vorberger, Burkhard Militzer, and Zhandos A. Moldabekov, “Momentum distribution of the uniform electron gas at finite temperature: Effects of spin polarization,” Phys. Rev. E 104, 055206 (2021d).
  * Dornheim  _et al._ (2021e) Tobias Dornheim, Zhandos A. Moldabekov, and Panagiotis Tolias, “Analytical representation of the local field correction of the uniform electron gas within the effective static approximation,” Phys. Rev. B 103, 165102 (2021e).
  * Dornheim  _et al._ (2022c) Tobias Dornheim, Zhandos Moldabekov, Jan Vorberger, Hanno Kählert, and Michael Bonitz, “Electronic pair alignment and roton feature in the warm dense electron gas,” Communications Physics 5, 304 (2022c).
  * Preston  _et al._ (2020) T.R. Preston, S. Göde, J.-P. Schwinkendorf, K. Appel, E. Brambrink, V. Cerantola, H. Höppner, M. Makita, A. Pelka, C. Prescher, K. Sukharnikov, A. Schmidt, I. Thorpe, T. Toncian, A. Amouretti, D. Chekrygina, R.W. Falcone, K. Falk, L.B. Fletcher, E. Galtier, M. Harmand, N.J. Hartley, S.P. Hau-Riege, P. Heimann, L.G. Huang, O.S. Humphries, O. Karnbach, D. Kraus, H.J. Lee, B. Nagler, S. Ren, A.K. Schuster, M. Smid, K. Voigt, M. Zhang, and U. Zastrau, “Design and performance characterisation of the HAPG von Hámos Spectrometer at the High Energy Density Instrument of the European XFEL,” Journal of Instrumentation 15, P11033–P11033 (2020).
  * Descamps  _et al._ (2020) A. Descamps, B. K. Ofori-Okai, K. Appel, V. Cerantola, A. Comley, J. H. Eggert, L. B. Fletcher, D. O. Gericke, S. Göde, O. Humphries, O. Karnbach, A. Lazicki, R. Loetzsch, D. McGonegle, C. A. J. Palmer, C. Plueckthun, T. R. Preston, R. Redmer, D. G. Senesky, C. Strohm, I. Uschmann, T. G. White, L. Wollenweber, G. Monaco, J. S. Wark, J. B. Hastings, U. Zastrau, G. Gregori, S. H. Glenzer, and E. E. McBride, “An approach for the measurement of the bulk temperature of single crystal diamond using an x-ray free electron laser,” Scientific Reports 10, 14564 (2020).
  * Wollenweber  _et al._ (2021) L. Wollenweber, T. R. Preston, A. Descamps, V. Cerantola, A. Comley, J. H. Eggert, L. B. Fletcher, G. Geloni, D. O. Gericke, S. H. Glenzer, S. Göde, J. Hastings, O. S. Humphries, A. Jenei, O. Karnbach, Z. Konopkova, R. Loetzsch, B. Marx-Glowna, E. E. McBride, D. McGonegle, G. Monaco, B. K. Ofori-Okai, C. A. J. Palmer, C. Plückthun, R. Redmer, C. Strohm, I. Thorpe, T. Tschentscher, I. Uschmann, J. S. Wark, T. G. White, K. Appel, G. Gregori, and U. Zastrau, “High-resolution inelastic x-ray scattering at the high energy density scientific instrument at the european x-ray free-electron laser,” Review of Scientific Instruments 92, 013101 (2021).
  * Bisio  _et al._ (2017) Francesco Bisio, Emiliano Principi, Michele Magnozzi, Alberto Simoncig, Erika Giangrisostomi, Riccardo Mincigrucci, Luca Pasquali, Claudio Masciovecchio, Federico Boscherini, and Maurizio Canepa, “Long-lived nonthermal electron distribution in aluminum excited by femtosecond extreme ultraviolet radiation,” Physical Review B 96, 81119 (2017).
  * van den Berg  _et al._ (2018) Q. Y. van den Berg, E. V. Fernandez-Tello, T. Burian, J. Chalupský, H.-K Chung, O. Ciricosta, G. L. Dakovski, V. Hájková, P. Hollebon, L. Juha, J. Krzywinski, R. W. Lee, M. P. Minitti, T. R. Preston, A. G. de la Varga, V. Vozda, U. Zastrau, J. S. Wark, P. Velarde, and S. M. Vinko, “Clocking femtosecond collisional dynamics via resonant x-ray spectroscopy,” Phys. Rev. Lett. 120, 055002 (2018).
  * Waldecker  _et al._ (2016) L. Waldecker, R. Bertoni, R. Ernstorfer, and J. Vorberger, “Electron-phonon coupling and energy flow in a simple metal beyond the two-temperature approximation,” Phys. Rev. X 6, 021003 (2016).