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A prevalent class of challenges in modern physics are inverse problems, where physical quantities must be extracted from experimental measurements. End-to-end machine learning approaches to inverse problems typically require constructing sophisticated estimators to achieve the desired accuracy, largely because they need to learn the complex underlying physical model. Here, we discuss an alternative paradigm: by making the physical model auto-differentiable we can construct a neural surrogate to represent the unknown physical quantity sought, while avoiding having to relearn the known physics entirely. We dub this process surrogate training embedded in physics (STEP) and illustrate that it generalizes well and is robust against overfitting and significant noise in the data. We demonstrate how STEP can be applied to perform dynamic kernel deconvolution to analyse resonant inelastic X-ray scattering spectra and show that surprisingly simple estimator architectures suffice to extract the relevant physical information., Competing Interests: We declare we have no competing interests., (© 2024 The Authors.)

We report on experiments where solid-density Mg plasmas are created by heating with the focused output of the Linac Coherent Light Source x-ray free-electron laser. We study the K-shell emission from the helium- and lithium-like ions using Bragg crystal spectroscopy. Observation of the dielectronic satellites in lithium-like ions confirms that the M-shell electrons appear bound for these high charge states. An analysis of the intensity of these satellites indicates that when modeled with an atomic-kinetics code, the ionization potential depression model employed needs to produce depressions for these ions which lie between those predicted by the well known Stewart-Pyatt and Ecker-Kroll models. These results are largely consistent with recent density functional theory calculations.

Ionization is a problematic quantity in that it does not have a well-defined thermodynamic definition and yet it is a key parameter within plasma modeling. One still therefore aims to find a consistent and unambiguous definition for the ionization state. Within this context we present finite-temperature density functional theory calculations of the ionization state of carbon in CH plasmas using two potential definitions: one based on counting the number of continuum electrons, and another based on the optical conductivity. Differences of up to 10% are observed between the two methods. However, including "Pauli forbidden" transitions in the conductivity reproduces the counting definition, suggesting such transitions are important to evaluate the ionization state.

Free-electron lasers provide bright, ultrashort, and monochromatic x-ray pulses, enabling novel spectroscopic measurements not only with femtosecond temporal resolution: The high fluence of their x-ray pulses can also easily enter the regime of the non-linear x-ray-matter interaction. Entering this regime necessitates a rigorous analysis and reliable prediction of the relevant non-linear processes for future experiment designs. Here, we show non-linear changes in the L 3 -edge absorption of metallic nickel thin films, measured with fluences up to 60 J/cm 2 . We present a simple but predictive rate model that quantitatively describes spectral changes based on the evolution of electronic populations within the pulse duration. Despite its simplicity, the model reaches good agreement with experimental results over more than three orders of magnitude in fluence, while providing a straightforward understanding of the interplay of physical processes driving the non-linear changes. Our findings provide important insights for the design and evaluation of future high-fluence free-electron laser experiments and contribute to the understanding of non-linear electron dynamics in x-ray absorption processes in solids at the femtosecond timescale., Competing Interests: The authors have no conflicts to disclose., (© 2023 Author(s).)

We present experimental observations of K_{β} emission from highly charged Mg ions at solid density, driven by intense x rays from a free electron laser. The presence of K_{β} emission indicates the n=3 atomic shell is relocalized for high charge states, providing an upper constraint on the depression of the ionization potential. We explore the process of state relocalization in dense plasmas from first principles using finite-temperature density functional theory alongside a wave-function localization metric, and find excellent agreement with experimental results.

An extension to the wave packet description of quantum plasmas is presented, where the wave packet can be elongated in arbitrary directions. A generalized Ewald summation is constructed for the wave packet models accounting for long-range Coulomb interactions and fermionic effects are approximated by purpose-built Pauli potentials, self-consistent with the wave packets used. We demonstrate its numerical implementation with good parallel support and close to linear scaling in particle number, used for comparisons with the more common wave packet employing isotropic states. Ground state and thermal properties are compared between the models with differences occurring primarily in the electronic subsystem. Especially, the electrical conductivity of dense hydrogen is investigated where a 15% increase in DC conductivity can be seen in our wave packet model compared with other models. This article is part of the theme issue 'Dynamic and transient processes in warm dense matter'.

Automatic differentiation represents a paradigm shift in scientific programming, where evaluating both functions and their derivatives is required for most applications. By removing the need to explicitly derive expressions for gradients, development times can be shortened and calculations can be simplified. For these reasons, automatic differentiation has fueled the rapid growth of a variety of sophisticated machine learning techniques over the past decade, but is now also increasingly showing its value to support ab initio simulations of quantum systems and enhance computational quantum chemistry. Here, we present an open-source differentiable quantum chemistry simulation code and explore applications facilitated by automatic differentiation: (1) calculating molecular perturbation properties, (2) reoptimizing a basis set for hydrocarbons, (3) checking the stability of self-consistent field wave functions, and (4) predicting molecular properties via alchemical perturbations.

Ultrafast optical excitation of matter leads to highly excited states that are far from equilibrium. In this study, femtosecond x-ray absorption spectroscopy was used to visualize the ultrafast dynamics in photoexcited warm dense Cu. The rich dynamical features related to d vacancies are observed on femtosecond timescales. Despite the success in explaining x-ray absorption data in the picosecond regime, the new femtosecond data are poorly understood through the traditional two-temperature model based on the fast thermalization concept and the static electronic structure for high-temperature metals. An improved understanding can be achieved by including the recombination dynamics of nonthermal electrons and changes in the screening of the excited d block. The population balance between the 4sp and 3d bands is mainly determined by the recombination rate of nonthermal electrons, and the underpopulated 3d block is initially strongly downshifted and recovered in several hundreds of femtoseconds.

Improving the predictive capability of molecular properties in ab initio simulations is essential for advanced material discovery. Despite recent progress making use of machine learning, utilizing deep neural networks to improve quantum chemistry modeling remains severely limited by the scarcity and heterogeneity of appropriate experimental data. Here we show how training a neural network to replace the exchange-correlation functional within a fully differentiable three-dimensional Kohn-Sham density functional theory framework can greatly improve simulation accuracy. Using only eight experimental data points on diatomic molecules, our trained exchange-correlation networks enable improved prediction accuracy of atomization energies across a collection of 104 molecules containing new bonds, and atoms, that are not present in the training dataset.

The development of bright free-electron lasers (FEL) has revolutionized our ability to create and study matter in the high-energy-density (HED) regime. Current diagnostic techniques have been successful in yielding information on fundamental thermodynamic plasma properties, but provide only limited or indirect information on the detailed quantum structure of these systems, and on how it is affected by ionization dynamics. Here we show how the valence electronic structure of solid-density nickel, heated to temperatures of around 10 of eV on femtosecond timescales, can be probed by single-shot resonant inelastic x-ray scattering (RIXS) at the Linac Coherent Light Source FEL. The RIXS spectrum provides a wealth of information on the HED system that goes well beyond what can be extracted from x-ray absorption or emission spectroscopy alone, and is particularly well suited to time-resolved studies of electronic-structure dynamics.

The free-free opacity in plasmas is fundamental to our understanding of energy transport in stellar interiors and for inertial confinement fusion research. However, theoretical predictions in the challenging dense plasma regime are conflicting and there is a dearth of accurate experimental data to allow for direct model validation. Here we present time-resolved transmission measurements in solid-density Al heated by an XUV free-electron laser. We use a novel functional optimization approach to extract the temperature-dependent absorption coefficient directly from an oversampled pool of single-shot measurements, and find a pronounced enhancement of the opacity as the plasma is heated to temperatures of order of the Fermi energy. Plasma heating and opacity enhancement are observed on ultrafast timescales, within the duration of the femtosecond XUV pulse. We attribute further rises in the opacity on ps timescales to melt and the formation of warm dense matter.

The free-free opacity in dense systems is a property that both tests our fundamental understanding of correlated many-body systems, and is needed to understand the radiative properties of high energy-density plasmas. Despite its importance, predictive calculations of the free-free opacity remain challenging even in the condensed matter phase for simple metals. Here we show how the free-free opacity can be modelled at finite-temperatures via time-dependent density functional theory, and illustrate the importance of including local field corrections, core polarization, and self-energy corrections. Our calculations for ground-state Al are shown to agree well with experimental opacity measurements performed on the Artemis laser facility across a wide range of extreme ultraviolet wavelengths. We extend our calculations across the melt to the warm-dense matter regime, finding good agreement with advanced plasma models based on inverse bremsstrahlung at temperatures above 10 eV.

Proton radiography is a technique in high-energy density science to diagnose magnetic and/or electric fields in a plasma by firing a proton beam and detecting its modulated intensity profile on a screen. Current approaches to retrieve the integrated field from the modulated intensity profile require the unmodulated beam intensity profile before the interaction, which is rarely available experimentally due to shot-to-shot variability. In this paper, we present a statistical method to retrieve the integrated field without needing to know the exact source profile. We apply our method to experimental data, showing the robustness of our approach. Our proposed technique allows for the retrieval not only of the path-integrated fields, but also of the statistical properties of the fields.

X-ray emission spectroscopy is a well-established technique used to study continuum lowering in dense plasmas. It relies on accurate atomic physics models to robustly reproduce high-resolution emission spectra, and depends on our ability to identify spectroscopic signatures such as emission lines or ionization edges of individual charge states within the plasma. Here we describe a method that forgoes these requirements, enabling the validation of different continuum lowering models based solely on the total intensity of plasma emission in systems driven by narrow-bandwidth x-ray pulses across a range of wavelengths. The method is tested on published Al spectroscopy data and applied to the new case of solid-density partially-ionized Fe plasmas, where extracting ionization edges directly is precluded by the significant overlap of emission from a wide range of charge states.

Electron-ion collisional dynamics is of fundamental importance in determining plasma transport properties, nonequilibrium plasma evolution, and electron damage in diffraction imaging applications using bright x-ray free-electron lasers (FELs). Here we describe the first experimental measurements of ultrafast electron impact collisional ionization dynamics using resonant core-hole spectroscopy in a solid-density magnesium plasma, created and diagnosed with the Linac Coherent Light Source x-ray FEL. By resonantly pumping the 1s→2p transition in highly charged ions within an optically thin plasma, we have measured how off-resonance charge states are populated via collisional processes on femtosecond time scales. We present a collisional cross section model that matches our results and demonstrates how the cross sections are enhanced by dense-plasma effects including continuum lowering. Nonlocal thermodynamic equilibrium collisional radiative simulations show excellent agreement with the experimental results and provide new insight on collisional ionization and three-body-recombination processes in the dense-plasma regime.

This review is focused on free-electron lasers (FELs) in the hard to soft x-ray regime. The aim is to provide newcomers to the area with insights into: the basic physics of FELs, the qualities of the radiation they produce, the challenges of transmitting that radiation to end users and the diversity of current scientific applications. Initial consideration is given to FEL theory in order to provide the foundation for discussion of FEL output properties and the technical challenges of short-wavelength FELs. This is followed by an overview of existing x-ray FEL facilities, future facilities and FEL frontiers. To provide a context for information in the above sections, a detailed comparison of the photon pulse characteristics of FEL sources with those of other sources of high brightness x-rays is made. A brief summary of FEL beamline design and photon diagnostics then precedes an overview of FEL scientific applications. Recent highlights are covered in sections on structural biology, atomic and molecular physics, photochemistry, non-linear spectroscopy, shock physics, solid density plasmas. A short industrial perspective is also included to emphasise potential in this area.

We present measurements of the spectrally resolved x rays emitted from solid-density magnesium targets of varying sub-μm thicknesses isochorically heated by an x-ray laser. The data exhibit a largely thickness-independent source function, allowing the extraction of a measure of the opacity to K-shell x rays within well-defined regimes of electron density and temperature, extremely close to local thermodynamic equilibrium conditions. The deduced opacities at the peak of the Kα transitions of the ions are consistent with those predicted by detailed atomic-kinetics calculations.

A nonlinear absorber in which the excited state absorption is larger than the ground state can undergo a process called reverse saturable absorption. It is a well-known phenomenon in laser physics in the optical regime, but is more difficult to generate in the x-ray regime, where fast nonradiative core electron transitions typically dominate the population kinetics during light matter interactions. Here, we report the first observation of decreasing x-ray transmission in a solid target pumped by intense x-ray free electron laser pulses. The measurement has been made below the K-absorption edge of aluminum, and the x-ray intensity ranges are 10^{16} -10^{17}  W/cm^{2}. It has been confirmed by collisional radiative population kinetic calculations, underscoring the fast spectral modulation of the x-ray pulses and charge states relevant to the absorption and transmission of x-ray photons. The processes shown through detailed simulations are consistent with reverse saturable absorption, which would be the first observation of this phenomena in the x-ray regime. These light matter interactions provide a unique opportunity to investigate optical transport properties in the extreme state of matters, as well as affording the potential to regulate ultrafast x-ray free-electron laser pulses.

The effect of a dense plasma environment on the energy levels of an embedded ion is usually described in terms of the lowering of its continuum level. For strongly coupled plasmas, the phenomenon is intimately related to the equation of state; hence, an accurate treatment is crucial for most astrophysical and inertial-fusion applications, where the case of plasma mixtures is of particular interest. Here we present an experiment showing that the standard density-dependent analytical models are inadequate to describe solid-density plasmas at the temperatures studied, where the reduction of the binding energies for a given species is unaffected by the different plasma environment (ion density) in either the element or compounds of that species, and can be accurately estimated by calculations only involving the energy levels of an isolated neutral atom. The results have implications for the standard approaches to the equation of state calculations.

The rate at which atoms and ions within a plasma are further ionized by collisions with the free electrons is a fundamental parameter that dictates the dynamics of plasma systems at intermediate and high densities. While collision rates are well known experimentally in a few dilute systems, similar measurements for nonideal plasmas at densities approaching or exceeding those of solids remain elusive. Here we describe a spectroscopic method to study collision rates in solid-density aluminium plasmas created and diagnosed using the Linac Coherent light Source free-electron X-ray laser, tuned to specific interaction pathways around the absorption edges of ionic charge states. We estimate the rate of collisional ionization in solid-density aluminium plasmas at temperatures ~30 eV to be several times higher than that predicted by standard semiempirical models.

High-intensity x-ray pulses from an x-ray free-electron laser are used to heat and probe a solid-density aluminum sample. The photon-energy-dependent transmission of the heating beam is studied through the use of a photodiode. Saturable absorption is observed, with the resulting transmission differing significantly from the cold case, in good agreement with atomic-kinetics simulations.

Here, we report results of an experiment creating a transient, highly correlated carbon state using a combination of optical and x-ray lasers. Scattered x-rays reveal a highly ordered state with an electrostatic energy significantly exceeding the thermal energy of the ions. Strong Coulomb forces are predicted to induce nucleation into a crystalline ion structure within a few picoseconds. However, we observe no evidence of such phase transition after several tens of picoseconds but strong indications for an over-correlated fluid state. The experiment suggests a much slower nucleation and points to an intermediate glassy state where the ions are frozen close to their original positions in the fluid.

An accurate description of the ionization potential depression of ions in plasmas due to their interaction with the environment is a fundamental problem in plasma physics, playing a key role in determining the ionization balance, charge state distribution, opacity and plasma equation of state. Here we present a method to study the structure and position of the continuum of highly ionized dense plasmas using finite-temperature density functional theory in combination with excited-state projector augmented-wave potentials. The method is applied to aluminium plasmas created by intense X-ray irradiation, and shows excellent agreement with recently obtained experimental results. We find that the continuum lowering for ions in dense plasmas at intermediate temperatures is larger than predicted by standard plasma models and explain this effect through the electronic structure of the valence states in these strong-coupling conditions.

The x-ray intensities made available by x-ray free electron lasers (FEL) open up new x-ray matter interaction channels not accessible with previous sources. We report here on the resonant generation of Kα emission, that is to say the production of copious Kα radiation by tuning the x-ray FEL pulse to photon energies below that of the K edge of a solid aluminum sample. The sequential absorption of multiple photons in the same atom during the 80 fs pulse, with photons creating L-shell holes and then one resonantly exciting a K-shell electron into one of these holes, opens up a channel for the Kα production, as well as the absorption of further photons. We demonstrate rich spectra of such channels, and investigate the emission produced by tuning the FEL energy to the K-L transitions of those highly charged ions that have transition energies below the K edge of the cold material. The spectra are sensitive to x-ray intensity dependent opacity effects, with ions containing L-shell holes readily reabsorbing the Kα radiation.

We have used the Linac Coherent Light Source to generate solid-density aluminum plasmas at temperatures of up to 180 eV. By varying the photon energy of the x rays that both create and probe the plasma, and observing the K-α fluorescence, we can directly measure the position of the K edge of the highly charged ions within the system. The results are found to disagree with the predictions of the extensively used Stewart-Pyatt model, but are consistent with the earlier model of Ecker and Kröll, which predicts significantly greater depression of the ionization potential.

Matter with a high energy density (>10(5) joules per cm(3)) is prevalent throughout the Universe, being present in all types of stars and towards the centre of the giant planets; it is also relevant for inertial confinement fusion. Its thermodynamic and transport properties are challenging to measure, requiring the creation of sufficiently long-lived samples at homogeneous temperatures and densities. With the advent of the Linac Coherent Light Source (LCLS) X-ray laser, high-intensity radiation (>10(17) watts per cm(2), previously the domain of optical lasers) can be produced at X-ray wavelengths. The interaction of single atoms with such intense X-rays has recently been investigated. An understanding of the contrasting case of intense X-ray interaction with dense systems is important from a fundamental viewpoint and for applications. Here we report the experimental creation of a solid-density plasma at temperatures in excess of 10(6) kelvin on inertial-confinement timescales using an X-ray free-electron laser. We discuss the pertinent physics of the intense X-ray-matter interactions, and illustrate the importance of electron-ion collisions. Detailed simulations of the interaction process conducted with a radiative-collisional code show good qualitative agreement with the experimental results. We obtain insights into the evolution of the charge state distribution of the system, the electron density and temperature, and the timescales of collisional processes. Our results should inform future high-intensity X-ray experiments involving dense samples, such as X-ray diffractive imaging of biological systems, material science investigations, and the study of matter in extreme conditions.

We have studied a solid-to-plasma transition by irradiating Al foils with the FLASH free electron laser at intensities up to 10(16)  W/cm(2). Intense XUV self-emission shows spectral features that are consistent with emission from regions of high density, which go beyond single inner-shell photoionization of solids. Characteristic features of intrashell transitions allowed us to identify Auger heating of the electrons in the conduction band occurring immediately after the absorption of the XUV laser energy as the dominant mechanism. A simple model of a multicharge state inverse Auger effect is proposed to explain the target emission when the conduction band at solid density becomes more atomiclike as energy is transferred from the electrons to the ions. This allows one to determine, independent of plasma simulations, the electron temperature and density just after the decay of crystalline order and to characterize the early time evolution.

Studies of materials under extreme conditions have relevance to a broad area of research, including planetary physics, fusion research, materials science, and structural biology with x-ray lasers. We study such extreme conditions and experimentally probe the interaction between ultrashort soft x-ray pulses and solid targets (metals and their deuterides) at the FLASH free-electron laser where power densities exceeding 10(17) W/cm(2) were reached. Time-of-flight ion spectrometry and crater analysis were used to characterize the interaction. The results show the onset of saturation in the ablation process at power densities above 10(16) W/cm(2). This effect can be linked to a transiently induced x-ray transparency in the solid by the femtosecond x-ray pulse at high power densities. The measured kinetic energies of protons and deuterons ejected from the surface reach several keV and concur with predictions from plasma-expansion models. Simulations of the interactions were performed with a nonlocal thermodynamic equilibrium code with radiation transfer. These calculations return critical depths similar to the observed crater depths and capture the transient surface transparency at higher power densities.

By use of high intensity XUV radiation from the FLASH free-electron laser at DESY, we have created highly excited exotic states of matter in solid-density aluminum samples. The XUV intensity is sufficiently high to excite an inner-shell electron from a large fraction of the atoms in the focal region. We show that soft-x-ray emission spectroscopy measurements reveal the electronic temperature and density of this highly excited system immediately after the excitation pulse, with detailed calculations of the electronic structure, based on finite-temperature density functional theory, in good agreement with the experimental results.

We investigate ultrafast (fs) electron dynamics in a liquid hydrogen sample, isochorically and volumetrically heated to a moderately coupled plasma state. Thomson scattering measurements using 91.8 eV photons from the free-electron laser in Hamburg (FLASH at DESY) show that the hydrogen plasma has been driven to a nonthermal state with an electron temperature of 13 eV and an ion temperature below 0.1 eV, while the free-electron density is 2.8x10{20} cm{-3}. For dense plasmas, our experimental data strongly support a nonequilibrium kinetics model that uses impact ionization cross sections based on classical free-electron collisions.

In situ x-ray diffraction has been used to measure the shear strain (and thus strength) of single crystal copper shocked to 100 GPa pressures at strain rates over two orders of magnitude higher than those achieved previously. For shocks in the [001] direction there is a significant associated shear strain, while shocks in the [111] direction give negligible shear strain. We infer, using molecular dynamics simulations and VISAR (standing for 'velocity interferometer system for any reflector') measurements, that the strength of the material increases dramatically (to approximately 1 GPa) for these extreme strain rates.

We have focused a beam (BL3) of FLASH (Free-electron LASer in Hamburg: lambda = 13.5 nm, pulse length 15 fs, pulse energy 10-40 microJ, 5 Hz) using a fine polished off-axis parabola having a focal length of 270 mm and coated with a Mo/Si multilayer with an initial reflectivity of 67% at 13.5 nm. The OAP was mounted and aligned with a picomotor controlled six-axis gimbal. Beam imprints on poly(methyl methacrylate) - PMMA were used to measure focus and the focused beam was used to create isochoric heating of various slab targets. Results show the focal spot has a diameter of < or =1 microm. Observations were correlated with simulations of best focus to provide further relevant information.

We have performed short-pulse x-ray scattering measurements on laser-driven shock-compressed plastic samples in the warm dense matter regime, providing instantaneous snapshots of the system evolution. Time-resolved and angularly resolved scattered spectra sensitive to the correlation effects in the plasma show the appearance of short-range order within a few interionic separations. Comparison with radiation-hydrodynamic simulations indicates that the shocked plastic is compressed with a temperature of a few electron volts. These results are important for the understanding of the thermodynamic behavior of strongly correlated matter for conditions relevant to both laboratory astrophysics and inertial confinement fusion research.