Understanding the properties of warm dense hydrogen is important for modeling and predicting the energy flow in many short-lived fusion-relevant platforms such as self-constricted plasmas found in Z-pinches [1], as well as for inertial confinement fusion (ICF) experiments [2]. Critical to the mission of inertial fusion is the knowledge of the electron-ion energy transfer and their temperature equilibration times in a high-energy-density state. The complex interplay of collisional and collective processes driving this relaxation also determines transport properties, like the thermal conductivity and stopping power. Thus, precise knowledge of the electron-ion relaxation strongly furthers our ability to accurately model complex systems such as the hot-spot at stagnation and the deuterium-tritium fuel-ablator interface in an ICF implosion [3–5]. Ultrafast X-ray scattering is a premier way to investigate such processes while yielding additional information on the structural and collective properties of dense matter [6–13].

Temperature relaxation in a dense, moderately coupled and partially degenerate plasma is challenging since it requires consideration of quantum degeneracy of the electron subsystem, the ionic short range structure, strong collisions, and coupling effects from the ions [14–18]. The seminal Landau-Spitzer approach sufficiently describes contributions from binary collisions in weakly interacting, classical plasmas [19, 20], but it completely fails for denser, or colder, systems where the Coulomb logarithm becomes negative. In order to prevent this unphysical behavior, empirically modified or capped forms for the Coulomb logarithm have been suggested without being rigorously tested against experiments [21–24]. Until now, the electron-ion relaxation in dense plasmas have only been inferred, or extrapolated, indirectly from experimental data [25–27] that is measured under equilibrium conditions. Furthermore, determining the energy exchange from electrons to ions and atoms in dense matter requires the preparation of a homogeneous sample and ultrafast probing. The Matter in Extreme Conditions (MEC) end-station at the LCLS provides a unique platform that can isochorically and uniformly heat hydrogen from a cryogenic state to the warm dense matter regime while providing ultrashort, picosecond delayed X-ray pulses that can be scattered and resolved in both energy and wavenumber.

In this article, we present experimental evidence of picosecond resolved electron-ion energy relaxation in dense non-equilibrium hydrogen plasmas using angularly resolved X-ray Thomson scattering. Our results demonstrate that a uniformly heated hydrogen plasma in the warm dense state, approaching an electron temperature of 10 eV at 1 ps, will continue to equilibrate to a temperature state of T _e =T _i =4.5 eV. The results show an electron-ion thermalization time that is shorter than that calculated using a Spitzer-like approach with a generalized Coulomb logarithm, and agrees well with theories that correct the standard Spitzer result to describe the scattering process quantum mechanically [14, 15, 18, 28, 29]. Such models include electron degeneracy and energy transfer via coupled collective modes.

Figure 1 shows a schematic of the experimental setup at the MEC end-station of the LCLS. A 70 fs, frequency doubled Ti:sapphire laser system was used to isochorically heat a 5 μm diameter hydrogen jet to a non-equilibrium state. Initially, liquid hydrogen was held in a helium cooled cryostat at temperatures T = 17 K. It was injected into the MEC vacuum chamber, whereby it evaporatively cooled to cryogenic solid H₂. Using a pulse energy of 120 mJ focused to spot size of 6 μm (w _laser ), vertically polarized along the jet axis, resulted in a peak laser intensity of 5 × 10¹⁷ W/cm² delivered to the hydrogen jet. Separately, the LCLS X-ray beam at 5.5 keV with a FWHM pulse duration of 62 fs (aligned 45°with respect to the laser axis) was delayed at time steps of 1, 2.5, 5, 10, and 25 ps after the fs-laser pulse, and used to measure the plasma properties by applying spectrally and angularly resolved X-ray scattering. The X-ray pulse energy was 4 mJ (3.8 × 10¹² photons per pulse), and was focused using a beryllium lens stack to a focal diameter (w _{F E L} ) of approximately 18 μm.

A fixed angle X-ray scattering spectrometer in the collective scattering regime was used in the forward direction, corresponding to a probing wave vector of k = 4.8 nm⁻¹, to directly monitor changes to the intensity of the elastically scattered signal, as well as the intensity and spectral shift of the inelastic signal to fully account for all the scattered photons detected. Another X-ray scattering spectrometer was placed in the backscattering non-collective geometry (k = 52 nm⁻¹) as a relative calibration measurement to normalize the total scattering cross section following the f-sum rules of particle conservation. Both spectrometers used Highly Annealed Pyrolytic Graphite (HAPG) crystals as dispersive elements in Von-Hamos-geometry with Cornell SLAC Pixel Array Detectors (CSPADs). The angularly resolved diagnostic consisted of a large CSPAD used to measure the solid structure via the Bragg reflections in order to properly normalize and deconvolve the characteristic changes in the elastic scattering contributions to the static structure factor. A high-resolution silicon crystal spectrometer in transmission of the X-ray beam was also used to monitor the input spectrum for each individual X-ray pulse.

The ultrabright properties of the LCLS together with a 1 Hz fs-laser and novel H₂ microjet allow for X-ray scattering from materials with a low scattering cross section. High repetition rate data collection can be used for single photon detection whereby accumulating and averaging photon counts over many scattering events can produce a highly precise, background limited data set with large signal-to-noise ratio [30]. The spectrally resolved X-ray Thomson scattering intensity profiles, in Figure 1, are shown for the forward and backward scattering geometries along with the measured X-ray diffraction patterns. Figure 3 demonstrates the reduction in error of the X-ray Thomson signal strength by accumulating scattering events using this experimental platform.

The total scattered X-ray spectrum demonstrates a unique intensity profile and spectral shape that depends on a combination of scattering contributions that are distinctly related to the temperature, density, and ionization state of the plasma. The total scattered cross section (Eq. 1) can be described in terms of σ _T , the Thomson scattering cross-section, the dynamic electron structure factor S e e t o t ( k , ω ) , and k₁ and k₀, the scattered and incident wave vectors, respectively: d 2 σ d Ω d ω = σ T k 1 k 0 S e e t o t k , ω (1) Simultaneous high-resolution angularly and spectrally resolved X-ray Thomson scattering (XRTS) has proven to be an accurate method to infer the electron density, temperature, and pressure of dense plasmas [8, 31]. Combining direct measurements of the static structure factor with inelastic scattering by free electrons, and in some cases, bound-free or resonant inelastic scattering transitions [32–35] allow for the ability to accurately calculate the ion temperature, the electron temperature, and the ionization of hydrogen throughout the entire relaxation from a non-equilibrium plasma.

Microcanonical DFT-MD simulations were performed to model the ion temperature of the heated system for a given electron temperature determined independently with 3D particle in cell (PIC) simulations using the OSIRIS code [49–51]. At an initial temperature of T _e ∼ 10 eV, it is possible estimate the initial non-thermal crystal mismatch heating of cryogenic molecular hydrogen, see Figure 6. For cryogenic hydrogen, we prepared the initial state of the system as an hcp lattice with randomly oriented hydrogen molecules at the lattice positions. The lattice parameters were a = 4.1 A ̊ , c = 1.64a in line with the mass density of fluid cryogenic hydrogen of ρ = 0.07 g/cm ³. As can be seen in Figure 6, the ion temperature rises to at least ∼ 9000 K within tens of femto-seconds after start of the run.

The amount of ionization may be inferred from computation of the ion-ion and ion-electron static structure factors [49]. The generalized screening cloud ρ(k), i.e., the electronic density around an ion, may be deconvolved from the electron-ion structure via ρ k = S e i k S i i k = 1 − Z ⋅ f k + Z ⋅ q k , (3) and can then be fitted by the average charge state Z, the atomic form factor of hydrogen f(k), and the screening cloud q(k). From these calculations, the average charge state of Z = 0.66 and initial ion temperature of T _i ∼ 1 eV have been derived and were subsequently used in the analysis of the weight of the Rayleigh peak. Using the charge state of the hydrogen system as determined by DFT-MD, hypernetted chain integral equation (HNC) calculations [49] were performed for a wide variety of electron and ion temperatures to obtain the partial static structure factors as needed for the weight of the Rayleigh peak based on the multi-component Chihara formula [38].

The specific ion temperature condition of T _i = 1 eV, measured at t = 1 ps, is consistent with the determined energy release caused by the fs-laser induced ionization of the molecular H-H bond, shown in Figure 6. The instantaneous change of the proton-proton potential to a Coulomb potential gives kinetic thermal motion to the ions at sub-ps times before the thermalization is dominated by the electron-ion energy exchange. This has been modeled and verified using ab initio DFT-MD simulations, highlighted by the red shaded region in Figure 7A. Additionally the electron temperature has been modeled independently with 3D-PIC OSIRIS simulations, as shown in Figure 1, and agrees with the measured conditions shown by the blue shaded region [49] in Figure 7A. The time behavior can be compared to analytical predictions after uniform heating is achieved for t > 1 ps? The measured electron and ion temperatures are plotted as a function of time delay in Figure 7A. The data is plotted together with two energy transfer models using the charge state Z = 0.66. The different theoretical models use an initial electron temperature as observed from the XRTS measurements at t = 1 ps, where the 9 and 12 eV error bounds stem from the error analysis of the spectrally resolved data. One model uses a Landau-Spitzer like theory (HLS) of relaxation via binary collisions, valid in the classical high-temperature region, with a Coulomb logarithm that cannot become negative for high densities [14].

Our results show the first experimentally determined relaxation rates from a dense two-temperature plasma in a non-equilibrium state. The findings allow for a quantitative comparison of various analytical models for the electron-ion energy transfer in the warm dense matter regime where many theories are untested. Such theories can be implemented in radiation-hydrodynamic codes to improve equation of state properties that are important for both ICF and self-constricted plasmas produced in Z-pinches. The results are important for understanding heat transport in dense systems where classical plasma theory breaks down and modified or capped substitutions to the Coulomb logarithm are commonly used.