Model-free Rayleigh weight from x-ray Thomson scattering measurements Open Access

The x-ray Thomson scattering (XRTS) method^1,2 constitutes a powerful experimental technique, which is capable of giving microscopic insights into a probed sample. A particularly important use case for XRTS is the diagnostics of experiments with matter under extreme densities, temperatures, and pressures.³ Such warm dense matter (WDM)^4,5 naturally occurs in various astrophysical objects such as giant planet interiors,^6–8 brown dwarfs,⁹ and the outer crust of neutron stars.¹⁰ Moreover, WDM plays an important role in a variety of technological applications such as inertial confinement fusion (ICF),¹¹ where the fuel capsule has to traverse the WDM regime in a controlled way to reach ignition (⁠ $ρ=500$ g/cc, $T=5×107$ K).¹² The recent breakthrough at the National Ignition Facility (NIF) to reach ignition¹³ has further substantiated the importance of accurately diagnosing WDM states. In the laboratory, these extreme states can be created in the laboratory using a variety of experimental techniques,³ and XRTS is often used to infer a priori unknown system parameters such as the mass density $ρ$⁠, temperature T, and ionization state Z; see, e.g., Ref. 14. These properties can then be used for physical considerations, to inform equation-of-state tables^15–17 and to benchmark integrated multi-scale simulations such as radiation hydrodynamics.¹⁸ 

In the present work, we extend these efforts toward a model-free diagnostics of XRTS measurements by extracting the Rayleigh weight $WR(q)=SeI2(q)/SII(q)$ [where $SeI(q)$ and $SII(q)$ are the electron–ion and ion–ion static structure factor], which describes the degree of electronic localization around the ions,^45,46 directly from the experimental observation. Our idea is based on a combination of the f-sum rule with the ratio of elastic to inelastic scattering $r(q)$⁠, cf. Eq. (5). Therefore, it is generally available and even extends to non-equilibrium situations. As a practical example, we consider an XRTS measurement on strongly compressed Be that has been carried out at the NIF;³³ see also Refs. 24, 25, 47, and 48 for previous experiments with warm dense Be. The proposed direct inference of $WR(q)$ helps to further constrain forward models for the interpretation of XRTS measurements, which is very important in its own right. In addition, we expect $WR(q)$ to be a valuable diagnostic tool. For example, one might first infer the temperature T from a given XRTS signal using the model-free ITCF method³⁷ and then carry out a set of ab initio calculations for $WR(q)$ over a reasonable interval of densities $ρ$⁠; here, we use highly accurate path integral Monte Carlo (PIMC) simulations using the setup described in Ref. 36 and density functional theory molecular dynamics (DFT-MD) simulations by Bethkenhagen et al.⁴⁹ We find excellent agreement between both methods in $WR(q)$⁠, whereas computationally less intensive average-atom models^30,50 and parameterized Chihara fits^1,28 deviate. This is of direct practical consequence for the interpretation of the experimental signal and suggests a significantly lower density of $ρ=(22±2)$ g/cc (see also Ref. 36) compared to the Chihara based estimate of $ρ=(34±4)$ g/cc by Döppner et al.³³ 

From a methodological perspective, we note that $WR(q)$ constitutes a perfect observable for DFT-MD, as it only involves the static single-electron density distribution $ne(r)$ (and the ion density distribution and correlation function, which are both easily accessible within MD). This is in contrast to time-dependent DFT (TD-DFT) calculations that, in practice, contain additional approximations such as the unknown dynamic exchange–correlation kernel in the case of linear-response TD-DFT.^22,42,51,52 Finally, we mention the possibility to use experimental results for $WR(q)$ as a rigorous benchmark for simulations and theoretical models in situations where the density and temperature are already known by other means.

Here, the quasi-elastic nature of the first part is due to the substantially longer ionic timescales and localization of bound and nearly bound electrons around the ions. In practice, treating $Sel(q,E)$ as a delta distribution is appropriate if the SIF $R(E)$ is significantly broader than the actual ion feature. This is usually the case except for dedicated experiments that aim to resolve ionic energy scales.^53,54 Within the chemical picture that assumes a decomposition into effectively bound and free electrons,^1,28,32,33 the inelastic part $Sinel(q,E)$ consists of both transitions between bound and free states (and their reverse process³⁴) and transitions between two free electronic states. The Rayleigh peak, too, is informed by both free electrons (screening cloud) and bound electrons (form factor), and constitutes an indispensable ingredient to Chihara models^23,28,30 that are widely used for the interpretation of XRTS experiments with WDM.^1,32–34 However, it is important to note that we do not have to make such an approximate distinction between bound and free electrons in the present work as the computation of $WR(q)$ only involves correlation functions between all electrons and ions, i.e., $SeI(q)$ and $SII(q)$⁠.

For further insight into the sensitivity of the interpretation of a given XRTS measurement to the employed model or simulation technique, we compare a number of independent methods in Fig. 2, where the top and bottom panels correspond to $ρ=20$ g/cc and $ρ=30$ g/cc, respectively. For both cases, we find very good agreement between PIMC (solid green) and DFT-MD simulations by Bethkenhagen et al.⁴⁹ (dash-dotted purple). The computationally cheaper average-atom model^30,50 (dashed red) is in qualitative, though not quantitative agreement with the ab initio datasets. Instead, it agrees with the experimental data point for $ρ=30$ g/cc. This highlights the sensitivity of equation-of-state measurements to the proper treatment of electronic correlations and other many-body effects in the analysis of the experimental observation. At the same time, we note the comparably high computation effort of PIMC and DFT-MD (⁠ $∼O(105)$ CPUh per calculation) compared to AA (⁠ $∼O(10−1)$ CPUh per calculation). Finally, we include two chemical models for the hypothetical ionization degrees of $Z=3.52$ (dotted black) and $Z=3.73$ (dotted yellow) for both densities, see the supplementary material for additional details. Evidently, the inferred density strongly depends on the inferred ionization state, making the invocation of additional constraints desirable. In Ref. 33, Döppner et al. have used the Saha equation in combination with a semi-empirical form factor lowering model for this purpose, leading to the nominal parameters of $ρ=(34±4)$ g/cc and $Z=3.4±0.1$⁠. A systematic investigation of the origin of this discrepancy between their chemical model and the ab initio datasets from PIMC and DFT-MD is beyond the scope of the present work and will be pursued in a dedicated future study.

Let us conclude with a systematic analysis of the dependence of $WR(q)$ on the density, as shown in Fig. 3 based on extensive DFT-MD results for $ρ=(5−80)$ g/cc at $T=150$ eV. The main trend is given by the systematic increase in the Rayleigh weight with $ρ$⁠, which is to a large degree due to the different length scales in the system; the effect almost vanishes when one adjusts the x axis by the Fermi wavenumber $qF∼1/ρ1/3$⁠. The inset shows a magnified segment around the experimental data point, with the latter being located between the DFT-MD results for $ρ=20$ g/cc (red) and $ρ=24$ g/cc (black); this leads to our final estimate for the density of $ρ=(22±2)$ g/cc (dash-dotted pink).

We have presented a new approach for the model-free extraction of the Rayleigh weight $WR(q)$ from XRTS measurements. Most importantly, $WR(q)$ constitutes an important measure for the electronic localization around the ions, which is related to multiple possible definitions of the average ion charge⁵⁷ and interesting in its own right. In addition, $WR(q)$ is of direct practical value for XRTS diagnostics, as we have demonstrated in detail in a recent experiment with strongly compressed beryllium at the NIF.³³ In particular, we propose to first infer the temperature T from the model-free ITCF thermometry approach^37,38 and to subsequently match the experimental result for $WR(q)$ with simulation results over a reasonable range of densities. We note that $WR(q)$ is a particularly suitable observable for DFT-MD simulations, as it does not involve any dynamic information such as the a priori unknown dynamic exchange–correlation kernel. We find excellent agreement between DFT-MD and PIMC reference data; this is encouraging since direct PIMC simulations are still limited to low-Z materials at moderate to high temperatures,^58,59 whereas DFT-MD and TD-DFT^22,60 are more broadly applicable. An additional advantage of $WR(q)$ over alternative observables such as the ratio of elastic to inelastic scattering $r(q)$ is that the former does not require any explicit information about the electronic static structure factor $See(q)$⁠, which is notoriously difficult for DFT-based methodologies.

In practice, we infer a mass density of $ρ=(22±2)$ g/cc from the beryllium dataset using either PIMC or DFT-MD simulations, which is significantly lower than the nominal value of $ρ=(34±4)$ g/cc that has been reported in the original publication based on a chemical Chihara model; note that the latter must take into account a proper multi-component treatment of differently charged ions.⁶¹ Moreover, we find that the computationally more efficient average-atom model tends to overestimate the density, whereas chemical models generally require additional constraints for parameters such as the ionization degree. Our study thus clearly highlights the importance of an accurate treatment of quantum many-body effects for XRTS based equation-of-state measurements even at relatively high temperatures.

This work opens up new possibilities for the study of warm dense matter and beyond. An important point for future research is given by the sensitivity of the inferred Rayleigh weight to the SIF $R(E)$⁠, which is usually modeled for backlighter setups⁵⁵ but can be known with high precision at modern XFEL facilities.¹⁹ Indeed, recent advances in high-resolution XRTS measurements⁶⁰ at the European XFEL will likely facilitate the application of the model-free ITCF thermometry technique even at moderate temperatures of $T∼1$ eV, which are of relevance for both planetary and materials science; the current model-free framework for the extraction of $WR(q)$ is applicable at any temperature. In combination, these two methods will allow for accurate equation-of-state measurements in previously inaccessible regimes. Finally, we mention the intriguing possibility of performing an XRTS experiment on an isochorically heated sample with an appropriate delay between pump and probe to ensure proper equilibration. Since both T and $ρ$ would be well known in such a scenario, a corresponding measurement of $WR(q)$ using the present framework would constitute a truly unambiguous reference dataset for the rigorous benchmarking of ab initio simulations and chemical models alike.

See the supplementary material for additional information about the extraction of the experimental data point for the Rayleigh weight, as well as a discussion of the range of applicability and limitations of the method. In addition, we provide details on the presented PIMC, DFT-MD, average atom, and chemical model calculations.

This work was partially supported by the Center for Advanced Systems Understanding (CASUS), financed by Germany's Federal Ministry of Education and Research (BMBF) and the Saxon state government out of the State budget approved by the Saxon State Parliament. This work has received funding from the European Union's Just Transition Fund (JTF) within the project Röntgenlaser-Optimierung der Laserfusion (ROLF), Contract No. 5086999001, co-financed by the Saxon state government out of the State budget approved by the Saxon State Parliament. This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2022 research and innovation programme (Grant agreement No. 101076233, “PREXTREME”). The work of Ti.D., M.P.B, S.H., and M.J.M. was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. Ti.D., M.P.B., M.J.M., and D.O.G. were supported by Laboratory Directed Research and Development (LDRD) Grant Nos. 24-ERD-044 and 25-ERD-047. This work was partially supported by the German Research Foundation (DFG) within the Research Unit FOR 2440. Views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. Computations were performed on a Bull Cluster at the Center for Information Services and High-Performance Computing (ZIH) at Technische Universität Dresden and at the Norddeutscher Verbund für Hoch- und Höchstleistungsrechnen (HLRN) under Grant Nos. mvp00018 and mvp00024. S.B.H. was supported by Sandia National Laboratories, a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for DOE's National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. The work of LBF is supported by the DOE Office of Science, Fusion Energy Science under FWP 100866, and supported by the Department of Energy, Laboratory Directed Research and Development program at SLAC National Accelerator Laboratory, under Contract No. DE-AC02-76SF00515.

The authors have no conflicts to disclose.

T. Dornheim: Conceptualization (equal); Investigation (lead); Project administration (equal); Writing – original draft (equal). H. M. Bellenbaum: Investigation (equal); Project administration (equal); Writing – original draft (equal). M. Bethkenhagen: Investigation (equal); Writing – review & editing (equal). S. B. Hansen: Investigation (equal); Writing – review & editing (equal). M. P. Böhme: Investigation (equal); Writing – review & editing (equal). T. Döppner: Investigation (equal); Writing – review & editing (equal). L. B. Fletcher: Investigation (equal); Writing – review & editing (equal). T. Gawne: Investigation (equal); Writing – review & editing (equal). D. O. Gericke: Investigation (equal); Writing – review & editing (equal). S. Hamel: Investigation (equal); Writing – review & editing (equal). D. Kraus: Investigation (equal); Writing – review & editing (equal). M. J. MacDonald: Investigation (equal); Writing – review & editing (equal). Zh. A. Moldabekov: Investigation (equal); Writing – review & editing (equal). T. R. Preston: Investigation (equal); Writing – review & editing (equal). R. Redmer: Investigation (equal); Writing – review & editing (equal). M. Schörner: Investigation (equal); Writing – review & editing (equal). S. Schwalbe: Investigation (equal); Writing – review & editing (equal). P. Tolias: Investigation (equal); Writing – review & editing (equal). J. Vorberger: Investigation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.