Unraveling electronic correlations in warm dense quantum plasmas

Abstract

The study of matter at extreme densities and temperatures has emerged as a highly active frontier at the interface of plasma physics, material science and quantum chemistry with relevance for planetary modeling and inertial confinement fusion. A particular feature of such warm dense matter is the complex interplay of Coulomb interactions, quantum effects, and thermal excitations, making its rigorous theoretical description challenging. Here, we demonstrate how ab initio path integral Monte Carlo simulations allow us to unravel this intricate interplay for the example of strongly compressed beryllium, focusing on two X-ray Thomson scattering data sets obtained at the National Ignition Facility. We find excellent agreement between simulation and experiment with a very high level of consistency between independent observations without the need for any empirical input parameters. Our results call into question previously used chemical models, with important implications for the interpretation of scattering experiments and radiation hydrodynamics simulations.

Introduction

Matter at extreme densities and temperatures displays a complex quantum behavior. A particularly intriguing situation emerges when the interaction, thermal, and Fermi energies are comparable. Understanding such warm dense matter (WDM) requires a holistic description taking into account partial ionization, partial quantum degeneracy, and moderate coupling. Indeed, even familiar concepts such as well-defined electronic bound states and ionization break down in this regime.

Interestingly, such conditions are widespread throughout the universe, naturally occurring in a host of astrophysical objects such as giant planet interiors¹, brown and white dwarfs², and, on Earth, meteor impacts³. Moreover, WDM plays a key role in cutting-edge technological applications such as the discovery and synthesis of novel materials⁴. An extraordinary achievement has recently been accomplished in the field of inertial confinement fusion at the National Ignition Facility (NIF)^5,6. In these experiments, both the ablator and the fuel traverse the WDM regime, making a rigorous understanding of such states paramount to reach the reported burning plasma and net energy gain^7,8.

The pressing need to understand extreme states has driven a large leap in the experimental capabilities; the considerable number of remarkable successes includes the demonstration of diamond formation under planetary conditions^4,9, opacity measurements under solar conditions¹⁰, probing atomic physics at Gigabar pressures¹¹, and the determination of energy loss of charged particles^12,13. However, this progress is severely hampered: to diagnose WDM experiments, a thorough understanding of the electronic response is indispensable. Indeed, even the inference of basic parameters such as temperature and density requires rigorous modeling to interpret the probe signal.

Density functional theory combined with classical molecular dynamics for the ions (DFT-MD) has emerged as the de-facto work horse for computing WDM properties. While being formally exact¹⁴, the predictive capability of DFT-MD is limited by the unknown exchange–correlation functional, which has to be approximated in practice, and the application of the Born-Oppenheimer approximation. A potentially superior alternative is given by ab initio path integral Monte Carlo (PIMC) simulations¹⁵, which are in principle capable of providing an exact solution for a given quantum many-body problem without any empirical input. Yet, PIMC simulations of quantum degenerate Fermi systems, such as the electrons in WDM, are afflicted with an exponential computational bottleneck, which is known as the fermion sign problem^16,17. As a consequence, PIMC application to matter under extreme conditions has either been limited to comparably simple systems such as the uniform electron gas model¹⁸, or based on approximations as in the case of restricted PIMC^19,20,21.

Here, we present a solution to this unsatisfactory situation and demonstrate its capabilities on the example of warm dense beryllium (Be). Since our approach is not based on any nodal restriction, we get access to the full spectral information in the imaginary-time domain²². As the capstone of our work, we employ our simulations to re-analyze X-ray Thomson scattering (XRTS) data obtained at the NIF for strongly compressed Be in a backscattering geometry¹¹. In addition, we consider a new data set that has been measured at a smaller scattering angle that is more sensitive to electronic correlations. Our unique access to electron correlation functions allows for alternative ways to interpret the XRTS data, resulting in a very high level of consistency. Interestingly, we find a substantially lower density compared to previously used chemical models¹¹, which has important implications both for our understanding of XRTS measurements and for state-of-the-art radiation hydrodynamics simulations of implosion dynamics.

Results

Simulation approach and capabilities

In principle, the PIMC method allows one to obtain an exact solution to the full quantum-many body problem without any empirical input or approximations. However, the application of PIMC to quantum degenerate electrons is severely hampered by the fermion sign problem^16,17. To circumvent this obstacle, Militzer, Ceperley and others^19,20,21,23 have successfully employed the fixed-node approximation. This restricted PIMC method allows for simulations of large systems without a sign problem, an advantage that comes at the cost of a de-facto uncontrolled approximation in the form of a trial nodal structure^24,25. While RPIMC has been successfully applied to a host of materials²¹ and experimentally verified, e.g. for CH^2,26, the inherent nodal restriction prevents the usual access of PIMC to the full spectral information in the imaginary-time domain^22,27, preventing a corresponding comparison with XRTS measurements.

In this work, we employ a fundamentally different strategy by carrying out a controlled extrapolation over a continuous variable ξ ∈ [ − 1, 1] that is substituted into the canonical partition function^28,29,30, see the Methods Section. This treatment removes the exponential scaling of the computation time with the system size for substantial parts of the WDM regime without the need for any empirical input such as the nodal surface of the density matrix for restricted PIMC or the exchange–correlation (XC) functional for DFT. At the same time, it retains full access to the spectral information about the system encoded in the imaginary-time density-density correlation function (ITCF), thereby allowing for direct comparison between simulations and XRTS measurements. While this approach had been successfully applied to the uniform electron gas model^29,30, we use it here to study the substantially more complex case of electrons and nuclei in WDM.

In Fig. 1a, b, we show snapshots of all-electron PIMC simulations of N_Be = 25 Be atoms (i.e, N_e = 100 electrons) for the mass density ρ = 7.5 g/cc and temperatures of T = 190 eV and T = 100 eV, respectively. The green orbs depict the nuclei, which behave basically as classical point particles, although this is not built into our simulations. The blue paths represent the quantum degenerate electrons; their extension is proportional to the thermal de Broglie wavelength ${\lambda }_{T}=\hslash \sqrt{2\pi /{m}_{{{{\rm{e}}}}}{k}_{{{{\rm{B}}}}}T}$ and serves as a measure for the average extension of a hypothetical single-electron wave function. The interplay of electron delocalization with effects such as Coulomb coupling shapes the physical behavior of the system. In panels c and d, we illustrate PIMC results for the spatially resolved electron density in the external potential of a fixed ion configuration. We find a substantially increased localization around the nuclei for the lower temperature.

Fig. 1: Ab initio PIMC simulations of compressed Be (ρ = 7.5 g/cc).

a Snapshot of a PIMC simulation of N_Be = 25 Be atoms at T = 190 eV. b Same as (a) but for T = 100 eV. The green orbs show the ions and the blue-red paths the quantum degenerate electrons; c, d electronic density in real space for a fixed ion configuration at the same temperatures, with the colors indicating values in units of the mean density in the simulation cell. e Our PIMC simulations give us access to all many-body correlations in the systems, including the spin-resolved electron--electron pair correlation functions g_↑↑(r) and g_↑↓(r), and the ion–ion pair correlation function g_II(r). f Electron--ion and ion--ion static structure factors S_eI(q) and S_II(q), giving us access to the ratio of elastic and inelastic contributions to the full scattering intensity, see the main text; red and blue colors distinguish T = 100 eV and T = 190 eV.

Figure 1e, f shows ab initio PIMC results for the full Be system, where both electrons and nuclei are treated dynamically on the same level. Specifically, panel e shows various pair correlation functions, where the red and blue lines correspond to T = 100 eV and T = 190 eV, respectively. The ion–ion pair correlation function g_II(r) [squares] is relatively featureless in both cases. The same holds for the spin-diagonal electron–electron pair correlation function g_↑↑(r) = g_↓↓(r) [crosses], although the exchange–correlation hole is substantially reduced when compared to g_II(r) mainly due to the weaker Coulomb repulsion. In stark contrast, the spin-offdiagonal pair correlation function g_↑↓(r) [circles] exhibits a nontrivial behavior and strongly depends on the temperature. While being nearly flat for T = 190 eV, g_↑↓(r) markedly increases towards r = 0 for T = 100 eV. This increased contact probability indicates the onset of clustering of two electrons around a single nucleus at the lower temperature, and nicely illustrates the capability of our PIMC simulations to capture the complex interplay of ionization, thermal excitation, and electron–electron correlations. Finally, panel f) shows corresponding results for the ion–ion [squares] and electron-ion [crosses] static structure factor (SSF). These contain important information about the generalized form factor and Rayleigh weight³¹, which are key properties in the interpretation of XRTS experiments¹¹ and a gamut of other applications.

Application to X-ray Thomson scattering

As a demonstration of the present PIMC capabilities, we re-analyze an XRTS experiment with strongly compressed beryllium at the National Ignition Facility (dataset #3 in ref. ¹¹) and repeated the experiment at a smaller scattering angle to focus more explicitly on electronic correlation effects. It can be assumed that the plasma conditions probed in the two measurements are comparable as (1) the probe times are identical within the timing uncertainty of Δt = 0.05 ns and (2) the shot-to-shot fluctuations are small and carefully controlled by high-accuracy laser delivery and high reproducibility in target production. Figure 2a shows an illustration of the experimental set-up using the GBar XRTS platform. 184 optical laser beams (not shown) are used for the hohlraum compression¹¹ of a Be capsule (yellow sphere) which is filled with a core of air. A further 8 laser beams are used to heat a zinc foil generating 9 keV X-rays³² that are used to probe the system (purple ray). By detecting the scattered intensity (blue ray) at an angle θ, we get insight into the microscopic physics of the sample on a specific length scale; the same microscopic physics can be resolved by our new PIMC simulations, a snapshot of which is depicted inside the Be capsule.

Fig. 2: From PIMC to XRTS.

a Schematic illustration of our setup. The Be capsule is compressed and probed by an x-ray source (purple); the scattered photons (blue) are collected by a detector under an angle θ, which is defined as the angle between the purple and blue beams. We use the PIMC method to simulate the quantum degenerate interior of the capsule, allowing for unprecedented comparisons between theory and experiment; b XRTS spectra for θ = 120° (black, dataset #3 in ref. ¹¹) and θ = 75° (red), and the source-and-instrument function (blue dashed); c, f PIMC results for S_ee(q) (symbols) for ρ = 7.5 g/cc (red), ρ = 20 g/cc (green) and ρ = 30 g/cc (yellow) compared to the NIF data point (bold blue, with corresponding uncertainty) and random phase approximation (RPA) results (dotted lines); d, g ITCF F_ee(q, τ) in the q-τ-plane, with the colored surface and dashed blue line corresponding to PIMC simulations for ρ = 20 g/cc and the Laplace transform of the NIF spectra; e, h τ-dependence of F_ee(q, τ) at the probed wavenumber; the shaded blue area quantifies the experimental uncertainty. The center and bottom rows correspond to the θ = 120° and θ = 75° shots, for which we find T = 155.5 eV and T = 190 eV (see the vertical dotted lines in (e, h), respectively, see the Methods Section.

The measured XRTS spectra are shown as the black and red curves in Fig. 2b and have been obtained at scattering angles of θ = 120° (#3 in ref. ¹¹) and θ = 75° (new). They are given by a convolution of the dynamic structure factor S_ee(q, ω) with the combined source-and-instrument function R(ω) [dashed blue]. Since a deconvolution to extract S_ee(q, ω) is unstable, we instead perform a two-sided Laplace transform^22,33,34

$${F}_{ee}(q,\tau )={{{\mathcal{L}}}}\left[{S}_{ee}(q,\omega )\right]=\int_{-\infty }^{\infty }\,{\mbox{d}}\,\omega \,{S}_{ee}(q,\omega ){e}^{-\hslash \omega \tau };$$

(1)

the well-known convolution theorem then gives us direct access to the imaginary-time correlation function (ITCF) F_ee(q, τ) based on the experimental data, with τ ∈ [0, β] being the imaginary time and β = 1/k_BT the inverse temperature, see the Methods Section. The ITCF contains the same information as S_ee(q, ω), but in a different representation²⁷. A particularly important application of F_ee(q, τ) is the model-free estimation of the temperature^33,34, and we find T = 155.5 ± 15 eV and T = 190 ± 20 eV for θ = 120° and θ = 75°, respectively.

A second advantage of Eq. (1) is that it facilitates the direct comparison of the experimental observation with our new PIMC results. As a first point, we consider the electronic static structure factor S_ee(q) = F_ee(q, 0) in Fig. 2c, f for the two relevant temperatures, and the circles and crosses show PIMC results for N_Be = 25 and N_Be = 10 beryllium atoms. Evidently, no finite-size effects can be resolved within the given error bars with the possible exception of the smallest q values. A particular strength of the PIMC method is that it allows us to unambiguously resolve the impact of electronic XC-effects. To highlight their importance for the description of warm dense quantum plasmas even in the high-density regime, we compare the PIMC data with the mean-field based random phase approximation³⁵ (dotted lines). The latter approach systematically underestimates the true S_ee(q) and only becomes exact in the single-particle limit of large wave numbers. The blue circles correspond to S_ee(q) extracted from the NIF data following the procedure introduced in the recent ref. ³⁶. They are consistent with the PIMC results for ρ ≲ 20 g/cc.

The full ITCF F_ee(q, τ) is shown in panels d and g in the q-τ-plane, where the colored surface shows the PIMC results for ρ = 20 g/cc, and the dashed blue lines have been obtained from the experimental data via a two-sided Laplace transform, see the Methods Section. We stress that, while the symmetry of the ITCF had been used for the interpretation of XRTS experiments in previous works^33,37,38, the present study compares exact PIMC simulations with an XRTS measurement. Clearly, the ITCF exhibits a rich structure that is mainly characterized by an increasing decay with τ for larger wave numbers. In fact, this τ-dependence is a direct consequence of the quantum delocalization of the electrons^27,39 and would be absent in a classical system. The NIF data are in very good agreement with our PIMC simulations over the entire τ-range. This can be seen particularly well in panels e and h, where we show the ITCF for the fixed values of q probed in the experiment. We find a more pronounced decay of F_ee(q, τ) with increasing τ for larger q. A second effect is driven by the different temperatures of these separate NIF shots, as a higher temperature leads to a reduction of quantum delocalization and, therefore, a reduced τ-decay. The observed agreement between the PIMC results and the experimental data for different q and temperature is thus nontrivial and constitutes a remarkable level of agreement and consistency between theory and experiment.

In addition, we consider an additional observable that can be directly extracted from the experimental data: the ratio of the elastic to the inelastic contributions to the full scattering intensity I_el/I_inel. In Fig. 3a, the two components are illustrated for the case of θ = 75°. In practice, the elastic signal has the form of the source function [cf. Fig. 2b], and I_inel is given by the remainder. The ratio I_el/I_inel constitutes a distinct measure for the localization of the electrons around the ions on the probed length scale determined by q. Therefore, it is highly sensitive to system parameters such as the density, and, additionally, to the heuristic but often useful concept of an effective ionization degree¹¹. Yet, the prediction of I_el/I_inel from ab initio simulations requires detailed knowledge about correlations between all particle species, see the Methods Section. This requisition is beyond the capabilities of standard DFT-MD, but straightforward with PIMC simulations, cf. Fig. 1.

Fig. 3: Determination of mass density.

a Data from XRTS measurement on strongly driven Beryllium at θ = 75° (q = 5.55 Å⁻¹, green curve) and its elastic (blue) and inelastic (red) contributions; b, c wavenumber dependence of the ratio I_el/I_inel for θ = 75° [θ = 120°]. Solid lines: PIMC results for T = 190 eV [T = 155.5 eV] for ρ = 7.5 g/cc (red), ρ = 20 g/cc (green), and ρ = 30 g/cc (yellow); blue cross: NIF measurements [error bars show fit uncertainty following the procedure used by Döppner et al.¹¹]; shaded colored areas: PIMC simulations at T = 190 ± 20 eV [T = 155.5 ± 15 eV], addressing the inter-dependence of density and temperature. Corresponding values for the reduced temperature Θ = k_BT/E_F are given in the bottom left corners. The insets show magnified segments around the NIF data points with the blue and green lines corresponding to PIMC results for ρ = 21 ± 1 g/cc (Θ = 1.65 ± 0.05) [left] and ρ = 22 ± 2 g/cc (Θ = 1.3 ± 0.1) [right].

In Fig. 3b, c, we show our simulation results for the q-dependence of I_el/I_inel at T = 190 eV and T = 155.5 eV, respectively, for three relevant mass densities. These conditions correspond to the XRTS measurements at 75° and 120°, which we include as the blue crosses. We find very good agreement between PIMC simulation and experimental observation for ρ = 20 g/cc for both scattering angles, which is fully consistent with the independent analysis of S_ee(q) presented in Fig. 2. To quantify the sensitivity of this analysis on the remaining uncertainty in the inferred temperature, we have carried out additional PIMC simulations at T = 190 ± 20 eV and T = 155.5 ± 15 eV and the corresponding results for I_el/I_inel are shown as the shaded areas in panels b and c. Evidently, the ratio of the elastic and inelastic contributions is very robust with respect to ρ. In particular, the value of ρ = 34 ± 4 g/cc obtained in the original ref. ¹¹ is decisively ruled out.

Let us conclude our analysis of the XRTS data by considering the effective degree of ionization Z*. While its precise value depends on a particular definition⁴⁰, it nevertheless constitutes an often useful concept. Moreover, it is an indispensable ingredient, e.g., for integrated radiation hydrodynamics simulations of fusion experiments and other applications. In Fig. 4, we show the effective degree of ionization as a function of the mass density ρ for T ~ 155 eV. The dotted green and dashed blue lines correspond to the Steward-Pyatt model and the popular OPAL EOS, and the black stars to DFT results based on the electrical conductivity¹¹. The yellow square shows the original interpretation of the 120° dataset based on the Chihara model by Döppner et al.¹¹; it is in good agreement with the DFT curve, but inconsistent with the two other models.

Fig. 4: Ionization degree of warm dense Be at T ~ 155 eV as a function of the mass density ρ.

Red area: present PIMC results [with corresponding uncertainty range], see the Methods section for details; black crosses: DFT-MD taken from ref. ¹¹; dotted green: Steward-Pyatt; dashed blue: OPAL; purple bar: DFT-MD analysis of the Rayleigh weight³¹ (see Appendix D of ref. ³¹ for computational details). Using our new PIMC capabilities to interpret the NIF data at 120° (red circle) substantially changes the inferred parameters compared to the chemical model used in the original ref. ¹¹ (dataset #3).

Due to the degree of arbitrariness in its definition, there is no unique way to compute the ionization degree in PIMC; this is in stark contrast to all other observables considered in this work. In some situations, cluster analyses based on spatial correlation functions have been successfully employed^41,42, but their application is expected to become increasingly difficult at strong compression when even the orbitals of bound orbitals start to overlap. Moreover, PIMC simulations do not distinguish between bound electrons and the screening cloud. Here, we follow Böhme et al.⁴³, who suggested to estimate an effective degree of ionization from the electrons capacity to react to an external static perturbation in the linear response regime; see the Methods Section for additional details. The results of this procedure are shown by the red line in Fig. 4 and confirm the underestimation of ionization of the Steward-Pyatt and OPAL models reported by Döppner et al.¹¹. Our final estimate for the conditions of the backscattering XRTS dataset at 120° (75°) is thus given by T = 155.5 ± 15 eV, ρ = 22 ± 2 g/cc, and Z* = 3.6 ± 0.1 (T = 190 ± 20 eV, ρ = 21 ± 2 g/cc, and Z* = 3.88 ± 0.1), see the red circle in Fig. 4. Both the temperature and effective ionization agree (within the given uncertainty intervals) with the original study by Döppner et al.¹¹, whereas the density substantially differs. We further note that a very recent re-analysis of the same XRTS dataset based on DFT-MD results for the Rayleigh weight W_R(q)³¹, see the Methods Section for additional details, has led to an excellent agreement with the present PIMC results; the correspondingly inferred experimental conditions are depicted by the purple bar in Fig. 4.

This leads us to the following conclusions: ab initio simulations (DFT and PIMC) that do not distinguish between bound and free electrons are in very good agreement, whereas the semi-empirical Chihara model significantly deviates. While it is tempting to attribute the latter to its inherent decomposition, the situation is more complicated as Chihara models entail a host of additional approximations for their individual components. For example, the analysis in the original ref. ¹¹ neglected the stimulated de-excitation of a-priori free electrons into energetically lower bound states (denoted as free-bound transitions³⁷) which have been shown to play a significant role at these conditions. Our results thus stress the importance of a holistic and exact treatment of the complicated WDM physics, whereas additional dedicated research will be needed to further understand the source of error in and to potentially improve the computationally cheap Chihara models.

Discussion

We have presented a framework for the highly accurate ab initio PIMC simulation of warm dense quantum plasmas, treating electrons and ions on the same level. As an application, we have analyzed XRTS measurements of strongly compressed Be using existing data¹¹ as well as a new data set that probes larger length scales where electronic XC-effects are more important. Due to their unique access to electronic correlation functions, our PIMC simulations have allowed us to independently analyze various aspects of the XRTS signal. This includes the ITCF F_ee(q, τ) and the ratio of elastic to inelastic contributions for which we have demonstrated a remarkable consistency between simulation and experiment without the need for any empirical parameters. While the estimation of corresponding electronic pair correlation functions is notoriously difficult for DFT-MD simulations, it is also accessible from restricted PIMC simulations; this might extend the proposed PIMC-based interpretation of XRTS experiments to even lower temperatures and more complicated composite materials that remain inaccessible to exact PIMC simulations due to the sign problem.

Our PIMC simulations accurately capture phenomena that manifest over distinctly different length scales due to the simulation of potentially hundreds of electrons and nuclei³⁰. This is particularly important for upcoming XRTS measurements with smaller scattering angles, and for the description of properties that can be probed in the optical limit such as electric conductivity and reflectivity. A key strength is our capability to resolve any type of many-particle correlation function between either electrons or ions. This is in stark contrast with standard DFT-MD simulations, where the computation of electronic pair correlation functions is not possible even if the exact XC functional were known. Moreover, our PIMC simulations are not confined to pair correlation functions and linear response properties such as the dynamic structure factor probed in XRTS²².

Our highly accurate PIMC results will spark a host of developments in the simulation of WDM. Most importantly, they can unambiguously benchmark the accuracy of existing DFT approaches, and provide crucial input for the construction of advanced nonlocal XC functionals⁴⁴. In addition, our results can quantify potential nodal errors in the restricted PIMC approach, support dynamic methods including time-dependent DFT, and test the basic underlying assumptions in widely used theoretical models.

Finally, our simulations will have a direct impact on nuclear fusion and astrophysics. Due to their dependable predictive capability, they provide both key input for integrated modeling such as transport properties and the equation of state and guide the development of experimental set-ups. A case in point is given by XRTS measurements, for which we have demonstrated the capabilities of our PIMC approach to give a highly consistent interpretation of the data for strongly compressed beryllium. This analysis clearly indicates a substantially lower density compared to previously used chemical models, which has important implications for the interpretation of future experiments, and for integrated radiation hydrodynamics simulations.

Having unraveled electronic correlations in warm dense quantum plasmas, we open the path to study light elements and potentially their mixtures for the extreme conditions encountered during inertial confinement fusion implosions and within astrophysical objects. This will be a true game changer for a field that previously lacked predictive capability.

Methods

Model-free analysis of XRTS measurements

To extract the temperature T and normalization S_ee(q) = F_ee(q, 0) from a measured XRTS signal without the need for models and approximations, we switch to the imaginary-time domain^27,33, where the ITCF F_ee(q, τ) is connected to the dynamic structure factor S_ee(q, ω) via a two-sided Laplace transform,

$${F}_{ee}(q,\tau )={{{\mathcal{L}}}}\left[{S}_{ee}(q,\omega )\right]=\int_{-\infty }^{\infty }\,{{{\rm{d}}}}\,\omega \,{S}_{ee}(q,\omega ){e}^{-\hslash \omega \tau }.$$

(2)

Working in the Laplace domain allows one to separate the physical information of interest from the properties of the combined source-and-instrument function R(ω) via the well-known convolution theorem^22,34,

$${{{\mathcal{L}}}}\left[{S}_{ee}(q,\omega )\right]=\frac{{{{\mathcal{L}}}}\left[{S}_{ee}(q,\omega ) \,\circledast \,R(\omega )\right]}{{{{\mathcal{L}}}}\left[R(\omega )\right]}.$$

(3)

Given accurate knowledge of R(ω) based on either source monitoring or additional characterization studies³², it is straightforward to evaluate both the numerator and the denominator of Eq. (3) even in the presence of experimental noise³⁴.

A particularly useful application of Eqs. (2) and (3) is the symmetry relation of the ITCF, F_ee(q, β − τ) = F_ee(q, τ), where β = 1/k_BT is the inverse temperature. It is easy to see that the ITCF is symmetric around τ = β/2, where it attains a minimum. This symmetry property is equivalent to the detailed balance relation of the dynamic structure factor S_ee(q, − ω) = e^−βℏωS_ee(q, ω) that universally holds in thermodynamic equilibrium⁴⁵. This makes it directly possible to infer the temperature from the XRTS signal evaluated in the Laplace domain without any model calculations or approximations.

An additional obstacle is given by the necessarily finite detector range, whereas Eq. (2) in principle requires the integration over an infinite ω-range. In practice, we define the two-sided symmetric incomplete Laplace transform

$${{{{\mathcal{L}}}}}_{x}\left[{S}_{ee}(q,\omega )\right]=\int_{-x}^{x}\,{{{\mathrm{d}}}}\,\omega \,{S}_{ee}(q,\omega ){e}^{ - \hslash \omega \tau},$$

(4)

whose convergence with x is straightforward to check.

In Fig. 5, we show the corresponding ITCF-based temperature analysis of the two XRTS measurements at the NIF. For θ = 120°, we find that the properly deconvolved data (blue) converge around a temperature of T = 155.5 ± 15 eV, in excellent agreement with the improved Chihara model from ref. ³⁷. For θ = 75°, the ITCF analysis gives us a temperature of T = 190 ± 20 eV, with most of the associated uncertainty (shaded blue area) stemming from the influence of the source-and-instrument function in both cases. Following Döppner et al.¹¹, we only assume a single bulk temperature; this is well justified at the probed time interval as the rebound shock has progressed through the majority of the infalling shell material.

Fig. 5: Determination of temperature.

Convergence of the temperature extracted from the ITCF-based analysis of the two NIF shots at a θ = 120° and b θ = 75° that are analyzed in the main text with respect to the integration range x, see Eq. (4). (Green) blue: (not) including the source-and-instrument function R(ω); shaded blue area: uncertainty of the ITCF analysis. The horizontal red lines in both panels show the temperature estimate from a best fit using improved Chihara models³⁷ and have been included as a reference.

Finally, Fig. 6 shows the extraction of the electronic SSF S_ee(q), i.e., the proper normalization of the XRTS signal, via the imaginary-time f-sum rule approach, recently introduced in ref. ³⁶; the a-priori unknown normalization constant C_norm is inferred from the first derivative of the properly deconvolved ITCF with respect to τ around the origin via

$${C}_{{{{\rm{norm}}}}}=-\frac{2{m}_{e}}{{(\hslash q)}^{2}}\frac{\partial }{\partial \tau }\frac{{{{\mathcal{L}}}}\left[I(q,\omega )\right]}{{{{\mathcal{L}}}}\left[R(\omega )\right]}(\tau=0)\,.$$

(5)

Figure 6 shows the convergence of the extracted S_ee(q) with the symmetrically truncated integration range x for the properly deconvolved (blue) and raw data (green). Evidently, the convergence behavior is well reproduced by the simple exponential ansatz

$$f(x)={A}_{x}+{B}_{x}{e}^{ - {C}_{x} x },$$

(6)

which is shown as the red curve. It is noted that Eq. (6) is well justified from a theoretical perspective by the expected exponential decay of the dynamic structure factor for large frequencies. In practice, the final estimate for S_ee(q) and the corresponding uncertainty (shaded red area in Fig. 6) is obtained by performing several exponential fits over different reasonable intervals $x\in [{x}_{\min },{x}_{\max }]$ (we use ${x}_{\min }\equiv 500\,{{\rm{eV}}}$ and ${x}_{\max }=800\,{{\rm{eV}}}$ and $1200\,{{\rm{eV}}}$) and by estimating the spread in the fitting parameter A_x.

Fig. 6: Determination of the normalization.

Convergence of the static structure factor S_ee(q) from the imaginary-time f-sum rule following the method introduced in ref. ³⁶ for a θ = 120° and b θ = 75°. Blue and green correspond to the properly deconvolved and raw results; the red area shows the extrapolation Eq. (6) and associated uncertainty, and the dotted dark horizontal line corresponds to the uncorrelated value of 1 and serves as a guide to the eye.

Ratio of elastic and inelastic contributions

It is common practice to express the full electronic DSF as a sum of elastic and inelastic contributions⁴⁶,

$${S}_{ee}(q,\omega)=\underbrace{{W}_{R}(q)\delta(\omega)}_{{S}_{{{\rm{el}}}}(q,\omega)}+{S}_{{{\rm{inel}}}}(q,\omega).$$

(7)

We note that Eq. (7) does not presuppose any artificial decomposition into effectively bound and free electrons. Instead, the quasi-elastic contribution is due to the longer time scales of the ions due to their heavier mass; it is shaped as the source function R(ω) in the measured XRTS intensity. From a theoretical perspective, it is straightforward to express the integrated ratio of S_el(q, ω) and S_inel(q, ω) as

$$\frac{{I}_{{{{\rm{el}}}}}(q)}{{I}_{{{{\rm{inel}}}}}(q)}=\frac{\int_{-\infty }^{\infty }{{\mbox{d}}}\,\omega \,{S}_{{{{\rm{el}}}}}(q,\omega )}{\int_{-\infty }^{\infty }{{\mbox{d}}}\,\omega \,{S}_{{{{\rm{inel}}}}}(q,\omega )}=\frac{{W}_{R}(q)}{{S}_{ee}(q)-{W}_{R}(q)}={\left(\frac{{S}_{ee}(q){S}_{II}(q)}{{S}_{eI}^{2}(q)}-1\right)}^{-1}.\quad$$

(8)

The theoretical estimation of I_el(q)/I_inel(q) thus requires explicit simulation results for pair correlation functions (evaluated in Fourier space, to get S_ab(q) instead of g_ab(r)) between all types of particles in the system. This is straightforward for PIMC (and also restricted PIMC), whereas no direct estimation of S_ee(q) is possible in DFT-MD. In contrast, the Rayleigh weight ${W}_{R}(q)={S}_{eI}^{2}(q)/{S}_{II}(q)$ does not include S_ee(q) and is thus more suitable for DFT-MD simulations⁴⁶, see also the recent ref. ³¹ for a corresponding comparison between DFT and PIMC.

PIMC estimation of ionization degree

To estimate the effective degree of ionization Z*, we follow Böhme et al.⁴³ and use our new PIMC set-up to solve the electronic problem within the external potential of a fixed ion configuration. Specifically, we apply an additional harmonic external potential of wavenumber q and perturbation amplitude A to the electrons, and study their response in Fourier space. In the limit of infinitesimally weak perturbations, we can expand the density component ${\rho }_{{{{\bf{q}}}}}={\sum }_{l=1}^{N}{e}^{-i{{{{\bf{r}}}}}_{l}{{{\bf{q}}}}}$ as a function of the perturbation amplitude A^22,43

$${\rho }_{{{{\bf{q}}}}}(A)={\rho }_{{{{\bf{q}}}}}(0)+A\chi (q)+{A}^{3}{\chi }^{(3)}(q)+\ldots \quad,$$

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with χ(q) being the well-known static linear response function. We further note that the unperturbed component ρ_q(0) vanishes for uniform systems, but is generally non-zero in the inhomogeneous case, i.e., in the presence of fixed nuclei. Moreover, χ⁽³⁾(q) corresponds to the cubic response at the first harmonic²², which, however, is not of interest for the present work.

In the left panel of Fig. 7, we show ρ_q as a function of the perturbation amplitude A for N_Be = 4 beryllium atoms for four different wave vectors. We note that every data point has been obtained from an independent PIMC simulation for a particular combination of q and A, making this analysis computationally involved. The dashed black lines show cubic fits according to Eq. (9), which are in excellent agreement with the PIMC results. The dotted red lines show the corresponding linear-response limits (i.e., setting χ⁽³⁾(q) ≡ 0). In the right panel, we show the q-dependence of the static linear response function, where the green crosses and red circles depict results for N_Be = 4 and N_Be = 10 beryllium atoms, respectively; we find no significant finite-size effects, as it is expected at these conditions, see ref. ²² and references therein. The dashed blue line has been obtained for a uniform electron gas (UEG) at the same density and temperature, and would correspond to a fully ionized system, i.e., Z* = 4. However, the true electronic density response is somewhat reduced, as the effectively bound electrons will not respond to a sufficiently weak perturbation.

Fig. 7: PIMC based estimation of the effective ionization degree Z* following the approach introduced by Böhme et al.⁴³.

a Fourier component of the density ρ_q [see main text] as a function of the perturbation amplitude A for four different wave vectors q [error bars show the 1σ Monte Carlo uncertainty]. The dashed black lines show cubic fits [Eq. (9)], and the dotted red lines the corresponding linear-response limit. b Linear density response function χ(q) for N_Be = 4 (green) and N_Be = 10 (red) beryllium atoms [error bars show the fit uncertainty in the linear coefficients]. Dashed blue: density response of a uniform electron gas (UEG) at the same conditions, corresponding to a fully ionized system; solid red: density response of a UEG assuming a fraction of free electrons of 3.6/4. The vertical dotted yellow line indicates the Fermi wavenumber q_F = (9π/4)^1/3/r_s. The red mesh indicates the wavenumber region of interest in which the density response is sensitive to electronic localization around the ions.

At this point, we feel that a dedicated discussion of the involved length scales λ = 2π/q is pertinent. In the limit of q → 0, the density response vanishes due to screening effects, effectively masking any differences between bound electrons and the free electron gas. Conversely, even bound electrons can readily react to an external potential in the single-particle limit of λ ≪ r_s; this can be seen for instance in the PIMC data point at q ≈ 14 Å⁻¹ that nearly coincides with the UEG result. To quantify the effective degree of ionization Z*, we thus consider the regime of intermediate wavenumbers q ≲ q_F (with the Fermi wavenumber q_F being indicated by the vertical yellow line), where λ is comparable to the average interparticle distance. Specifically, we match the density response of a UEG with a reduced, effectively free density to the PIMC results; for the present example, we find a best fit at Z* = 3.6, see the solid red curve in Fig. 7.

While this definition of Z* is, by necessity, somewhat arbitrary, it combines a number of advantages. First, it circumvents the need for a strict distinction between bound and free electrons, which is notoriously difficult in PIMC. Indeed, a cluster analysis that only considers spatial correlations between electrons and nuclei cannot easily distinguish bound electrons from the loosely associated screening cloud, which is particularly problematic at the high densities compared in the present work. Instead, our definition measures the degree of electronic localization around the nuclei. Bound electrons will not respond or respond only very little, respectively, whereas loosely associated but effectively unbound electrons contribute to the full density response despite being located in the vicinity of an ion. Therefore, this procedure automatically takes into account medium effects such as ionization potential depression, without the explicit need to resolve them.

PIMC simulation details

We consider a fully spin-polarized system where N_↑ = N_↓ = N/2 with N being the total number of electrons. Moreover, we consider effectively charge neutral systems where N = Z_totN_Be with Z_tot = 4 being the atomic charge and N_Be the total number of atoms. The corresponding Hamiltonian governing the behavior of the thus defined two-component plasma then reads

$$\hat{H}=\, -\frac{{\hslash }^{2}}{2{m}_{e}}{\sum }_{l=1}^{N}{\nabla }_{l}^{2}-\frac{{\hslash }^{2}}{2{m}_{I}}{\sum }_{l=1}^{{N}_{{{{\rm{Be}}}}}}{\nabla }_{l}^{2}\\ +{e}^{2}\left\{{\sum }_{l < k}^{N}{\phi }_{{{{\rm{E}}}}}({\hat{r}}_{l},{\hat{r}}_{k})+{Z}_{{{{\rm{tot}}}}}^{2}{\sum }_{l < k}^{{N}_{{{{\rm{Be}}}}}}{\phi }_{{{{\rm{E}}}}}({\hat{I}}_{l},{\hat{I}}_{k})-{Z}_{{{{\rm{tot}}}}}{\sum }_{k=1}^{N}{\sum }_{l=1}^{{N}_{{{{\rm{Be}}}}}}{\phi }_{{{{\rm{E}}}}}({\hat{I}}_{l},{\hat{r}}_{k})\right\}.$$

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The pair potential ϕ_E(r_a, r_b) is given by the usual Ewald sum, where we follow the definitions of Fraser et al.⁴⁷.

The basic idea of the PIMC method¹⁵ is to express the canonical partition function (i.e., particle number N, volume V, and inverse temperature β are fixed) in coordinate space,

$${Z}_{N,V,\beta }=\frac{1}{{N}^{\uparrow }!{N}^{\downarrow }!}{\sum }_{{\sigma }_{{N}^{\uparrow }}\in {S}_{{N}^{\uparrow }}}{\sum }_{{\sigma }_{{N}^{\downarrow }}\in {S}_{{N}^{\downarrow }}}{\xi }^{{N}_{{{{\rm{pp}}}}}}\int_{V}{\mbox{d}}\,{{{\bf{R}}}}\,\left\langle {{{\bf{R}}}}\parallel \right.{e}^{-\beta \hat{H}}\parallel \left.{\hat{\pi }}_{{\sigma }_{{N}^{\uparrow }}}{\hat{\pi }}_{{\sigma }_{{N}^{\downarrow }}}{{{\bf{R}}}}\right\rangle .$$

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The two sums over all possible permutations ${\sigma }_{{N}^{\uparrow }},{\sigma }_{{N}^{\downarrow }}$ of the respective permutation groups ${S}_{{N}^{\uparrow }},{S}_{{N}^{\downarrow }}$ (with ${\hat{\pi }}_{{\sigma }_{{N}^{\uparrow }}}$ and ${\hat{\pi }}_{{\sigma }_{{N}^{\downarrow }}}$ realizing a particular permutation) take into account that identical quantum particles cannot be distinguished. We further note that the vector R contains the coordinates of all electrons and ions in the system. In addition, N_pp is the number of pair exchanges required to realize a particular permutation, and ξ = 1, ξ = 0, and ξ = − 1 correspond to the physically meaningful cases of Bose-Einstein, Maxwell-Boltzmann, and Fermi-Dirac statistics, with the latter governing the behavior of the electrons in WDM. For completeness, we note that any exchange effects of the ions can safely be neglected at the conditions that are of interest in the present work.

For ξ = − 1, it is well known that positive and negative contributions to any observable cancel to a large degree, which leads to an exponential increase in the required compute time with increasing the number of electrons N = N^↑ + N^↓ or decreasing the temperature T; this is the notorious fermion sign problem^16,17. Here, we follow the approach introduced in refs. ^28,29 and consider the general case of a continuous variable ξ ∈ [ − 1, 1]. The basic idea is to carry out PIMC simulations in the sign-problem free domain of ξ ≥ 0, and to quadratically extrapolate to the fermionic limit of ξ = − 1. Indeed, Dornheim et al.^29,30 have shown very recently that this allows for quasi-exact uniform electron gas results for a range of observables including the SSF and ITCF over substantial parts of the WDM regime. A particular strength of this methodology is that it allows one to rigorously assess its accuracy for the case of a small system for which simulations can be performed even in the fermionic limit of ξ = − 1. Its continued reliability for substantially larger number of particles is then guaranteed both empirically²⁹, and from the principle of electronic nearsightedness⁴⁸. In essence, this approach allows for unprecedented PIMC simulations of fermions without the exponential computational bottleneck, without any uncontrolled approximations or empirical parameters, and without the restrictions on the sampling of the imaginary-time structure inherent to the fixed-node approximation²³.

In Fig. 8, we show this extrapolation for N_Be = 10 Be atoms (i.e., N = 40 electrons) at r_s = 0.93 (ρ = 7.5 g/cc) and T = 155.5 eV. The top left panel shows the q-dependence of the electronic SSF S_ee(q). More specifically, the blue crosses show direct PIMC results for ξ = − 1 that are subject to the full sign problem. We find an average sign of S ≈ 0.098, which means that the simulations are computationally involved, but still feasible. In addition, the shaded gray area encompasses our sign-problem free PIMC results for the range 1 ≥ ξ ≥ 0. Evidently, the bosonic SSF exhibits the opposite trend compared to the fermions with respect to q. Nevertheless, the quadratic extrapolation that is based on the gray area accurately reproduces the fermionic limit, see the red circles. The same trend is discerned for the thermal structure factor ${S}_{{{{\rm{ee}}}}}^{\beta /2}(q)={F}_{{{{\rm{ee}}}}}(q,\beta /2)$, as shown in the top right panel of Fig. 8, although the effects of quantum statistics are less pronounced for τ = β/2.

Fig. 8: Benchmarking the ξ-extrapolation approach for N_Be = 10 Be atoms at T = 155.5 eV and r_s = 0.93 (ρ = 7.5 g/cc).

Blue crosses: exact fermionic PIMC results (i.e., ξ = − 1) [error bars show the 1σ Monte Carlo uncertainty]; shaded gray area: PIMC results in the sign-problem free domain of ξ ∈ [0, 1]; red circles: quadratic extrapolation to the fermionic limit [error bars show the extrapolation uncertainty]. Panels a–d show the electronic static structure factor S_ee(q), the thermal static structure factor F_ee(q, β/2), the spin-diagonal pair correlation function g_↑↑(r) and the spin-offdiagonal pair correlation function g_↑↓(r), respectively.

In the bottom row, we repeat this analysis for the spin-resolved electronic PCF, with the left and right panels showing results for the spin-diagonal and spin-offdiagonal component. For g_↑↑(r), spin effects predominantly shape the behavior for r ≲ 4 Å; fermions exhibit the familiar exchange–correlation hole with g_↑↑(0) = 0, whereas bosons tend to cluster around each other exhibiting the opposite trend. Nevertheless, the extrapolation works exceptionally well and reproduces the fermionic curve. For g_↑↓(r), we find a more subtle behavior and spin-effects play a less important role. At the same time, the ξ-extrapolation method cannot be distinguished from the fermionic benchmark results within the (small) Monte Carlo error bars.

Convergence with number of imaginary-time propagators

An indispensable ingredient to the PIMC method is given by the factorization of the density operator ${e}^{-\beta \hat{H}}\ne {e}^{-\beta \hat{W}}{e}^{-\beta \hat{K}}$, where $\hat{W}$ and $\hat{K}$ are the potential and kinetic energy operators. Here, we use an implementation of the pair approximation, as described in ref. ⁴⁹, that becomes exact in the limit of P → ∞ as  ~ 1/P⁴, where P is the number of imaginary-time propagators, or, equivalently, the number of high-temperature factors.

In Fig. 9, we show results for the electronic SSF S_ee(q) [left] and the electron-ion SSF S_eI(q) [right] for N_Be = 4 Be atoms at r_s = 0.9 (ρ = 7.58 g/cc) and T = 155.5 eV. In particular, the stars, crosses, and circles correspond to the relevant cases of ξ = 1, ξ = 0, and ξ = − 1, respectively, and the blue, red, and green color distinguishes simulation results for P = 50, P = 200, and P = 500. Evidently, we find no systematic dependence even for P = 50. In practice, we use P = 200 throughout this work, as this has the beneficial side effect of a good τ-resolution for the ITCF F_ee(q, τ).

Fig. 9: Checking the convergence of the factorization of the density matrix.

Convergence of S_ee(q) [a] and S_eI (q) [b] with the number of imaginary-time propagators for N_Be = 4, ρ = 7.58 g/cc, and T = 155.5 eV. Error bars show the 1σ Monte Carlo uncertainty.