Imaginary-time correlation function thermometry: A new, high-accuracy and model-free temperature analysis technique for x-ray Thomson scattering data

Over the last few decades, there has been a surge of interest in the properties of matter at extreme conditions.¹ The phase space representing temperatures of $T ∼ 10 4 – 10 8$ K and pressures of $P ∼ 1 – 10 4$ Mbar is called warm dense matter (WDM), which is ubiquitous throughout our universe and occurs in a variety of astrophysical objects such as giant planet interiors^2–6 and brown dwarfs.^7,8 In addition, WDM plays an important role in a number of cutting-edge technological applications. For example, the fuel capsule in an inertial confinement fusion experiment^9,10 traverses the WDM regime on its pathway toward nuclear fusion.¹¹ Other practical applications include the discovery of novel materials^12–14 and hot-electron chemistry.¹⁵ 

In the laboratory, WDM is generated at large research facilities using a number of techniques, see, e.g., the topical overview by Falk.¹⁶ However, the central obstacle is the rigorous interpretation of the experiment because basic parameters such as the temperature cannot be directly measured. In this situation, the x-ray Thomson scattering (XRTS) approach¹⁷ has emerged as a highly useful method. More specifically, it has become common practice to fit an experimentally observed XRTS signal with a theoretical model to infer system parameters such as the temperature.^18–20 Unfortunately, the rigorous theoretical description of WDM is notoriously difficult.^21–23 In practice, however, uncontrolled approximations, such as the artificial decomposition into bound and free electrons (the cornerstone for Chihara's famous approach^19,24), remain widely used. Consequently, the actual interpretation of an experiment might strongly depend on a particular model, which limits the accuracy of equation of state (EoS) tables²⁰ and other observations.

To overcome this unsatisfactory situation, we have recently introduced a new methodology, referred to as imaginary-time correlation function thermometry (ITCFT),²⁵ which extracts the temperature from a given XRTS signal directly, without the need for any theoretical models or simulations. In particular, we have proposed to compute the two-sided Laplace transform [Eq. (16)] of the measured intensity, which has a number of key advantages: (1) the impact of the instrument function can be completely removed, without the need for a numerically unstable explicit deconvolution, (2) the method is very robust with respect to noise in the experimental data, and (3) the temperature can be measured for arbitrary complex materials without theoretical constraints. The high practical value of this new approach has been demonstrated in Ref. 25 by reevaluating the XRTS measurements of warm dense beryllium by Glenzer et al.,²⁶ aluminum by Sperling et al.,²⁷ and graphite by Kraus et al.¹⁸ 

In the present work, we provide a more detailed introduction to the ITCFT method, including a comprehensive discussion of the underlying theoretical framework. In addition, we present an extensive analysis of synthetic XRTS data over a broad range of temperatures and wave numbers. This allows us to clearly delineate the limitations of this approach and to rigorously predict the required experimental specifications to resolve a given plasma temperature. Finally, we systematically investigate the impact of random noise in the experimentally measured intensity and present an empirical procedure for the quantification of the uncertainty in the temperature extracted using ITCFT.

In addition to its direct value as a diagnostic for WDM, we note that the Laplace domain of the dynamic structure factor has a clear physical interpretation as an imaginary-time correlation function.^23,28,29 The latter naturally emerges in Feynman's path integral formulation of statistical mechanics^30,31 and measures the decay of electron–electron correlations along the imaginary-time axis $τ ∈ [ 0 , ℏ β ]$⁠, where $β = 1 / k B T$ is the inverse thermal energy. More details on imaginary-time correlation functions have been presented in Refs. 28 and 29. We note that the imaginary-time domain contains the same information as the usual frequency representation. In fact, both representations are complementary and tend to emphasize different aspects of the same information about a given system.²⁸ Therefore, our approach has the potential to give novel insights beyond the temperature, such as the excitation energy of quasi-particles or physical effects like the exchange–correlation induced alignment of pairs of electrons at metallic densities.³² 

This paper is organized as follows. In Sec. II, we introduce the theoretical basis for the ITCFT technique, including a brief discussion of XRTS (Sec. II B), the extraction of the temperature in the Laplace domain (Sec. II D), its connection to imaginary-time correlation functions,^33,34 and some practical remarks on the convergence with respect to the experimentally observed frequency range (Sec. II E). Section III is devoted to the analysis of synthetic data and is followed by a new framework for the study of the impact of random noise provided in Sec. IV. In Sec. V, we reanalyze the aforementioned experiments by Kraus et al.¹⁸ and Glenzer et al.²⁶ and, thereby, complement the earlier analysis in Ref. 25 by quantifying the given uncertainties in different properties. The paper is concluded with a summary and an outlook in Sec. VI.

Typical parameters considered in this work are r_s = 2 (electron number density of $n = 2.01 × 10 29 m − 3$⁠) and Θ = 1 (⁠ $T = 12.53 eV = 145.4 kK$⁠).

In Eq. (2), $E s = ℏ ω s$ is the energy of the scattered x rays, $Δ E$ is the energy range associated with the pixel as determined by the properties (crystal orientation and dispersion relation, etc.) of the experimental apparatus, and $∂ P s / ∂ ω s$ is the scattered power per unit frequency as seen by the detector. The meaning of the approximation in Eq. (2) is that the differential scattered power is treated as constant over each pixel and, thus, is evaluated at the mid-point frequency of the energy interval.

If the x-ray source has close-to-uniform spatial and temporal intensity profiles, the volume of plasma probed by the x rays is sufficiently small (relative to its distance from both the source and detector) and is also reasonably homogeneous, and then the volume integration in Eq. (3) can be ignored, and the solid angle integration can be approximated by multiplying by the subtended solid angle element $d Ω$⁠. There are numerous approximate treatments of incorporating spatial inhomogeneity within the target,^40–43 and the incorporation of such effects into the present analysis framework remains an important task for future works. The same is true for the effect of k-blurring, which may be important for large sample volumes in close proximity to a divergent x-ray source. Fortunately, both of these considerations are usually negligible for XFEL experiments. A dedicated discussion related to these restrictions is required and, thus, shall not be addressed further here.

To emphasize this point, Fig. 2 shows the percentage difference between the full (8) and approximate (9) forms of the wave number q as a function of scattered energy $E s$ and scattering angle θ for the example of the collective scattering data taken by Glenzer et al.²⁶ Clearly, the approximate expression (9) is well-fulfilled over the entire dynamic range of the experiment (⁠ $< ± 1 %$ within the central region bounded by the vertical thin black lines). Similar results are found for all other cases considered. Despite the nonlinearity of the physics governing the response of the plasma to the probing radiation (e.g., the Landau damping rate of plasmons³⁸) with respect to q, the small differences between the full and approximate forms of q will have a negligible impact on the shape of the scattered power spectrum. Nevertheless, a correction factor based on a first-order Taylor expansion of the full form of the wave number shift, $q 1 st order = q approx ( 1 − ω / 2 ω i )$⁠, could be incorporated for more accurate results; this will be considered in future work. In contrast to the results for XRTS, however, it should be noted that this approximation often does not hold for low-energy probes (particularly for the visible-wavelength lasers used in optical Thomson scattering to probe samples with the plasmon frequency being only slightly lower than the laser frequency, e.g., see Refs. 44–46), meaning that our diagnostic is currently limited to analyzing XRTS experiments; examination of possible extensions of ITCFT to cases with low-energy probes will be considered in detail in future works.

As a consequence of the foregoing restrictions, the de facto procedure for inferring the temperature (as well as a host of other system parameters, such as the electron number density n_e or ionization degree $⟨ Z ⟩$⁠) from experimental XRTS data has become:^17–19 (1) construct a suitable model $S model [ T ] ( q , ω )$ for the DSF, (2) perform the convolution with the function $R ( ω )$⁠, and (3) compare it to the experimentally measured intensity $I ( q , ω )$⁠, typically, within a nonlinear regression framework, such as Bayesian optimization.⁵⁴ In this way, the originally unknown parameters such as the temperature T are effectively reconstructed from a fit to the XRTS signal. Naturally, this approach strongly relies on the utilized model description for $S ( q , ω )$⁠, which can substantially affect the obtained free parameters. For example, Gregori et al.¹⁹ have suggested using the Chihara decomposition,²⁴ where the total DSF is split into separate contributions from bound electrons, free electrons, and transitions between the two. Yet, the validity of this chemical picture is particularly questionable in the WDM regime, where electrons can be weakly localized around the ions.⁵⁵ 

The present state-of-the-art appeals to the Kubo–Greenwood (KG) formalism, based on eigenvalues and occupations of the Kohn–Sham density functional theory^56–58 for obtaining the dielectric function in the optical limit. It can subsequently be extended to all wave numbers in terms of the Mermin dielectric function, with the required collision frequencies calculated from the KG dielectric function.⁵⁹ 

A more sophisticated alternative is the use of time-dependent density functional theory (TD-DFT) simulations,^55,60,61 an in-principle exact method for determining the quantum dynamics of electrons under external time-dependent perturbations. TD-DFT presupposes neither an artificial decomposition nor a continuation from the optical limit. On the other hand, present implementations of TD-DFT rely on approximations that might limit their utility under WDM conditions. The development of more accurate exchange-correlation approximations beyond the adiabatic approximation is an active area of research.^62–72 Moreover, the considerable computational cost of TD-DFT calculations makes them impracticable as a method for optimizing over a wide range of parameters required for reproducing XRTS signals. Currently, this rules out TD-DFT for on-the-fly interpretation of experiments.

Finally, we note that, despite impressive recent progress, the reliable modeling of $S ( q , ω )$ using potentially more accurate methods such as non-equilibrium Green functions^73,74 or even exact path integral Monte Carlo methods^75–77 is presently not feasible for realistic WDM applications. Moreover, the inevitable systematic errors of less accurate methods such as the Chihara decomposition are expected to become more pronounced for complex materials, such as the ablator coating of an ICF fuel capsule⁹ or complex mixtures of elements that occur in planetary interiors.⁷⁸ 

Note that throughout this section, we plot all (synthetic and experimental) data for the intensity as a function of the energy loss $ℏ ω = ℏ ω 0 − ℏ ω s$ and not directly as a function of the scattered energy $ℏ ω s$⁠. Therefore, the plots appear reflected compared to the original reference material.^18,26 However, we feel that this choice gives the reader a more intuitive connection to the two-sided Laplace transform Eq. (16), in general, and the role of the exponential factor $e − τ ω$⁠, in particular, which are of central importance for the current work.

To begin with, we compute the imaginary-time correlation function from our simple DSF model (see Fig. 3) for relevant values of the wave number q at the electronic Fermi temperature Θ = 1 (⁠ $T = 12.53 eV$⁠) and a metallic density of r_s = 2. By design, all DSFs exhibit the same sharp elastic feature around ω = 0. The yellow curve corresponding to a quarter of the Fermi wave number $q = 0.25 q F$ exhibits a sharp plasmon peak around $ω = 25 eV$⁠. Upon increasing q, the plasmon is first broadened (green curve, $q = 0.5 q F$⁠) and then disappears in a single broad inelastic curve at $q = q F$ (red). Finally, the blue curve computed for a large wave number $q = 3 q F$ in the non-collective, single-particle regime exhibits a broad Gaussian form, and its peak position increases parabolically with q. Figure 3(b) shows the corresponding imaginary-time intermediate scattering function $F ( q , τ ) = L [ S ( q , ω ) ]$⁠, i.e., the two-sided Laplace transform of the DSF defined in Eq. (16). Evidently, the different curves substantially depend on the wave number, thereby reflecting the transition from the collective regime $q ≪ q F$ to the single-particle regime $q ≫ q F$⁠. This has been analyzed in detail in the recent Ref. 29. At the same time, all curves are perfectly symmetric around the same value of $τ 1 / 2$⁠, as expected. Knowledge of the DSF, therefore, clearly allows for a straightforward extraction of the temperature for any value of the wave vector q without any physical assumptions or models, provided that the system is close to thermodynamic equilibrium.

The yellow curve has been obtained for $τ = ℏ β$ and corresponds to the original DSF but mirrored around x = 0. For completeness, we also include a curve for $τ = 2 ℏ β$⁠, which has no physical equivalent in Feynman's imaginary-time path integral picture, but can be easily computed from the two-sided Laplace transform Eq. (16). In this case, the negative frequency range gets substantially enhanced by the exponential factor, whereas, conversely, the positive frequency range gets damped. In practice, the evaluation of Eq. (16) at such large values of τ would require high-quality data of the DSF at very low frequencies, which is unrealistic at present. At the same time, we note that it is not needed to locate the minimum and hence extract the temperature.

To conclude the analysis of the unconvolved DSF, we directly consider the impact of the temperature on the DSF and its Laplace transform. This is shown in Fig. 5 in the collective regime (⁠ $q = 0.5 q F$⁠, left column) and in the single-particle regime (⁠ $q = 3 q F$⁠, right column). In particular, Fig. 5(a) shows the DSF evaluated from the usual UEG model at different values of the temperature; beware that the elastic peak of the depicted synthetic model data does not depend on Θ. The yellow curve has been obtained for $Θ = 0.25$ (⁠ $T = 3.13 eV$⁠) and exhibits sharp plasmon peaks around $ω = ± 20 eV$⁠. Increasing the temperature by a factor of two (⁠ $Θ = 0.5 , T = 6.26 eV$⁠) yields the black curve where the impact of increasing thermal effects is twofold: first, the DSF is broadened overall and decays more slowly for large $| ω |$⁠; second, the plasmon is damped and shifted to significantly larger frequencies both in the positive and negative frequency domains. We note that approximate models for this plasmon shift⁸⁴ have been used to determine the temperature in previous XRTS experiments.⁸⁵ Increasing the temperature further to Θ = 1 (⁠ $T = 12.53 eV$⁠, green) enhances both the broadening and the plasmon shift, until the plasmon is eventually damped out for Θ = 2 (⁠ $T = 25.06 eV$⁠, red) and Θ = 4 (⁠ $T = 50.12 eV$⁠, blue).

Let us next consider the temperature dependence of the DSF in the non-collective regime, i.e., at $q = 3 q F$ depicted in Fig. 5(b). In this regime, all curves exhibit qualitatively similar broad peaks around $ω = 120 eV$⁠. The main impact of the temperature is given by the substantially more slowly vanishing tails for large ω for larger values of Θ and the less pronounced intensities of the DSF at negative frequencies at low Θ due to the detailed balance relation. In Fig. 5(d), we show the corresponding curves for $F ( q , τ )$⁠, which give the same correct values for the (inverse) temperature as in Fig. 5(c). Notably, the minimum in $F ( q , τ )$ at $Θ = 0.25$ is even more shallow than at $q = 0.5 q F$⁠, which makes the usage of Eq. (27) even more essential.

In Sec. III B, we have conclusively demonstrated that knowledge of the dynamic structure factor $S ( q , ω )$ allows a straightforward extraction of the temperature independent of the wave number regime (collective vs single-particle) and without the need for any physical models or simulations. Yet, in a real scattering experiment, we do not have direct access to the DSF because the measured intensity $I ( q , ω )$ is convolved with the instrument function $R ( ω )$ as stated in Eq. (10). We, therefore, analyze in detail the impact of the convolution on extracting the temperature across the relevant range of wave numbers q in Fig. 6.

The top row corresponds to the collective regime, where the inelastic part of the deconvolved DSF [solid yellow, Fig. 6(a)] exhibits a sharp plasmon peak around $± 20 eV$⁠. The dashed lines have been obtained by convolving the yellow curve with Gaussian model instrument functions of different widths σ. Evidently, the main effect of the convolution is a substantial broadening of the sharp features in the original DSF, which becomes more pronounced with increasing σ. Indeed, the convolved intensity appears to consist of a single broad elastic peak for $σ = ω p = 16.67 eV$⁠, and no trace of the plasmon peaks can be recognized with the naked eye. In Fig. 6(b), we show the corresponding results for the two-sided Laplace transform of the intensity. As usual, the solid yellow line corresponds to the exact $F ( q , τ ) = L [ S ( q , ω ) ]$⁠, with a minimum about $τ 1 / 2$ (vertical line). In addition, the dashed curves show results for the Laplace transform of the convolved curves $L [ I ( q , ω ) ]$ for different σ. Evidently, the minimum in the Laplace transforms shifts to smaller τ with increasing width of the instrument function. In other words, the broadening from the convolution makes the extracted temperatures too large.²⁵ Given accurate knowledge of the instrument function $R ( ω )$⁠, it might seem natural to attempt an explicit deconvolution of Eq. (10) to reconstruct the original DSF $S ( q , ω )$⁠. This, in turn, would allow one to subsequently obtain $F ( q , τ ) = L [ S ( q , ω ) ]$ and, thus, to extract the temperature from the location of the unbiased minimum. In practice, such a deconvolution is notoriously unstable with respect to the noise in the input data, which usually prevents the explicit extraction of $S ( q , ω )$⁠. Yet, this obstacle is completely circumvented within the ITCFT methodology due to the convolution theorem in Eq. (20). Particularly, the instrument function and the DSF can be separated in a straightforward way in the Laplace domain. Consequently, we can completely remove the impact of the artificial broadening by dividing the dashed curves by the Laplace transform of the instrument function $L [ R ( ω ) ]$⁠, which gives the original solid yellow curve in all cases.

In addition, we find that the particular value of x for which convergence is reached strongly increases with the width of the instrument function $R σ ( ω )$⁠. In other words, the integral boundaries for which the exact convolution theorem Eq. (20) is recovered scale with σ.

A further interesting point of this analysis is the required accuracy of the intensity needed to extract the exact value of $τ 1 / 2$ and, thus, the temperature. For example, at $σ = 3.33 eV$⁠, convergence is reached around $x = 25 eV$⁠. In this case, the intensity [see Fig. 6(a)] at $ω = − 25 eV$ is reduced by a single order of magnitude compared to the size of the plasmon peak at $ω = 20 eV$⁠. Resolving the inelastic intensity over such a range in a scattering experiment is feasible in modern laser facilities.^47–49 For the broadest instrument function with $σ = 16.67 eV$⁠, the extracted temperature converges around $x = 75 eV$⁠. Yet, here, the convolved intensity has already decayed by more than three orders of magnitude and, therefore, will be difficult to resolve in an actual experiment. This clearly illustrates the importance of a narrow probe function for the accurate and practical analysis of experimental scattering data.

To bring the discussion of Fig. 6(c) to a close, let us consider the dotted curves, which have been obtained by determining the minimum in $L [ I ( q , ω ) ]$ without the correction by $L [ R ( ω ) ]$⁠. We find that the finite width of the instrument function then substantially influences (in fact, decreases) the extracted values of $τ 1 / 2$ even in the case of the relatively narrow Gaussian with $σ = 1.67 eV$⁠.

The bottom three rows of Fig. 6 contain the same analysis, but for increasing values of the wave number q. We, therefore, restrict ourselves here to a concise discussion of the main differences between the different regimes. First, we reiterate our earlier point about the increasing width of the unconvolved DSF with increasing q. This, in turn, means that the impact of the Gaussian instrument function becomes less pronounced for large q. Indeed, the uncorrected curves for both $σ = 1.67$ and $σ = 3.33 eV$ are within 5% of the correct temperature in the single-particle regime [see Fig. 6(l)]. For the narrowest instrument function, this even holds at the Fermi wave number $q = q F$ [see Fig. 6(i)]. As a second observation, we find that the convergence of the extracted temperature with the integration boundary x is shifted to somewhat larger frequencies. This is completely unproblematic for $σ ∈ [ 1.67 , 3.33 , 6.67 ] eV$⁠, as the width of the actual intensity increases similarly. Therefore, the intensity does not have to be resolved over substantially more than one order of magnitude. For $σ = 16.67 eV$⁠, on the other hand, reaching convergence in practice will be difficult.

We further illustrate the impact of the instrument function on the two-sided Laplace transform of the intensity by showing both $I ( q , ω ) e − τ ω$ (dashed) and $S ( q , ω ) e − τ ω$ (solid) for $q = q F$ in Fig. 8 for three relevant values of the imaginary-time τ. The green curves have been obtained for τ = 0 and, thus, show the original intensity and DSF. The red curves correspond to $τ 1 / 2$⁠, where $F ( q , τ )$ attains its minimum. In this case, the contribution to $L [ S ( q , ω ) ]$ is symmetric around ω = 0, whereas the convolution with $R ( ω )$ noticeably skews the corresponding curve to lower frequencies. This trend is even more pronounced for $τ = ℏ β$ (blue curves), where $S ( q , ω ) e − τ ω$ is equal to the solid green curve mirrored around ω = 0, whereas this clearly does not hold for the corresponding dashed curve.

We conclude this section with a more systematic analysis of the impact of the width of the instrument function σ on the extraction of the temperature, which is shown in Fig. 9. We plot the obtained values of $β = 1 / k B T$ as a function of σ for the four wave numbers considered in Fig. 6. The squares show the values where we have corrected for the impact of $L [ R ( ω ) ]$⁠, and we find a perfect agreement with the exact temperature for all combinations of σ and q. The crosses show the extracted raw temperatures without this correction. Overall, all four curves exhibit the same qualitative trend: the error in the uncorrected temperature monotonically decreases with decreasing σ, as is expected. Moreover, the curves are strictly ordered with q, as large wave numbers correspond to broader DSFs, for which the impact of the convolution is less pronounced. The shaded gray area shows an interval of $± 5 %$ around the exact inverse temperature, which can be reached without the correction either for a very narrow instrument function or in the single-particle regime (⁠ $α ≪ 1$⁠). This directly implies that large scattering angles as they can be realized in backscattering experiments make the method more robust against possible uncertainties in the characterization of the instrument function $R ( ω )$⁠.

In Sec. III C, we analyzed in detail the impact of the wave number and the width of the instrument function on the extracted temperature from a convolved scattering intensity signal. In Fig. 10, we extend these considerations by analyzing different values of the temperature Θ. Figure 10(a) shows results for $I ( q , ω )$ at r_s = 2 and $q = 0.5 q F$ for a narrow instrument function with $σ = 1.67 eV$⁠. A comparison with the corresponding deconvolved results for the DSF (see Fig. 5) reveals the substantial broadening of the plasmon peak, in particular at low temperatures. The convergence of the extracted temperature with the integration boundary x is shown in Fig. 10(b). The curves have been rescaled by the respective value of β to allow for a more straightforward comparison. As usual, the shaded gray area indicates an interval of $± 5 %$ and has been included as a reference.

First, we find that the extracted temperature converges toward the exact value for all values of Θ, as is expected; the small deviations from one at large x are a mainly a consequence of the finite ω-resolution in the synthetic data for the intensity, and the finite τ-resolution in our numerical implementation. While the latter can be increased if necessary, the former is determined by the resolution of the employed detector in an experiment. In practice, the attained accuracy from the ITCFT method is limited not by this discretization error but by the experimental noise, cf. Sec. IV.

The values of x for which convergence is reached appear to be nearly independent of Θ. The accurate extraction of the temperature is thus substantially more challenging at low temperatures, where the scattering intensity at negative frequencies can be orders of magnitude smaller than in the positive ω range. For example, the negative plasmon is reduced by three orders of magnitude at $Θ = 0.25$⁠, whereas it is not even reduced by a full order of magnitude for Θ = 1. From a practical perspective, this means that the accurate measurement of the intensity at $ω < 0$ is of prime importance and decisively determines the quality of the extracted temperature for $Θ ≪ 1$⁠, as $ω > 0$ and $ω < 0$ equally contribute to $F ( q , τ 1 / 2 = ℏ β / 2 )$⁠, cf. Fig. 8.

The bottom row of Fig. 10 shows the same analysis for a broader instrument function with $σ = 6.67 eV$⁠. Overall, the conclusions are similar to the previous case, although we do find a more pronounced dependence of the convergence with x on Θ. Still, the importance of the negative frequency range remains the same.

An interesting insight on which we close the current discussion can be made by analyzing the effect of the temperature parameter on the convolved intensity in the single-particle regime. The corresponding results are shown in Fig. 11 for a narrow probe function with $σ = 1.67 eV$⁠. The main difference regarding the extraction of the temperature compared to the smaller wave number shown in Fig. 10 is that we had to use the relation Eq. (27) at $Θ = 0.25$⁠, as the minimum in $F ( q , τ )$ is extremely shallow. Still, we resolve the correct temperature for all temperatures, and the effect of Θ on the value of x for which the convergence is reached is small. From this, one may conclude that the ITCFT technique does not merely amount to an alternative take on the use of detailed balance, as has already been extensively used elsewhere in the literature, since it remains viable across all regimes of the scattering (collectivity) parameter. In contrast, the method of temperature extraction via detailed balance as undertaken elsewhere is restricted in practice to the collective scattering regime where its manifestation on the relative amplitudes of the red- and blue-shifted plasmons can clearly be discerned in experimental data.

In Fig. 12, we analyze how much Gaussian random noise of unit variance contributes to the two-sided Laplace transform as a function of the frequency ω for the synthetic UEG model at conditions characterized by r_s = 2, Θ = 1, and with a wave number of $q = q F$⁠, convolved with a Gaussian probe function of width $σ = 3.33 eV$⁠; the shaded gray area depicts the corresponding $1 σ Δ$ interval determined by $I ( q , ω )$⁠. Figure 12(a) was obtained for τ = 0, where most contributions are due to the positive frequency range. Conversely, Fig. 12(b) corresponds to $τ = ℏ β / 2 = τ 1 / 2$⁠, i.e., the location of the minimum in $L [ S ( q , ω ) ]$⁠. In this case, the contribution of the noise to the Laplace transform of the intensity looks nearly identical to Fig. 12(a) but mirrored at the y-axis. Finally, Fig. 12(c) shows the same information for $τ = ℏ β$⁠. In this case, the noise in the negative frequency range is substantially increased compared to the previous two cases. In practice, it can, thus, be expected that we can resolve $F ( q , τ )$ with higher accuracy in the range of $0 ≤ τ ≤ τ 1 / 2$ compared to $τ > τ 1 / 2$⁠.

To illustrate the remarkable robustness of ITCFT with respect to noise in the experimental data, we perturb synthetic intensities with a series of realistic noise of different pre-factors $σ Δ$ in Fig. 13. The top row was obtained for $σ Δ = 0.01$⁠, and the intensity itself is shown in Fig. 13(a), with the green and red curves showing the perturbed and exact data, respectively. The extraction of the (inverse) temperature from the location of the minimum in $F ( q , τ )$ is shown in Fig. 13(b), where the shaded gray area indicates an interval of $± 5 %$ around the exact value. The solid black line shows the usual convergence with respect to the integration boundary x of the exact intensity, and the dashed blue and green curves have been obtained using two independent sets of random noise. Clearly, both curves attain the correct inverse temperature in the limit of large x despite the perturbation.

The results for this uncertainty in β (given as the $2 σ$-interval) are included in Fig. 13 as the shaded red and green areas, which, indeed, give a real measure for the fluctuation around the exact curve. The corresponding results for the Laplace transform $F ( q , τ )$ are shown in Fig. 13(c), where the solid yellow curve shows the exact result. The dashed black curve has been obtained by taking as input the perturbed data and is close to the former, although a small yet significant deviation is observed. The associated uncertainty in $F ( q , τ )$ computed from Eq. (36) has been included as the shaded gray area and nicely fits the observed difference.

In the center row of Fig. 13, we repeat this analysis for a larger magnitude of the random noise, $σ Δ = 0.05$⁠. Evidently, the larger noise level is directly propagated into larger fluctuations in the extraction of the temperature shown in Fig. 13(e). At the same time, we stress that (1) the error bars from Eq. (36) capture these fluctuations very well, and (2) that the extracted temperature is accurate to $∼ 2 %$ despite the substantial noise level in the input data. The imaginary-time intermediate scattering function depicted in Fig. 13(f) exhibits similar behavior.

Finally, we consider an even higher noise level in $σ Δ = 0.1$ shown in the bottom row of Fig. 13. Still, the extracted temperatures remain within $∼ 4 %$ of the exact result, and our estimated uncertainty measures are accurate both for β and $F ( q , τ )$⁠.

In Secs. III and IV, we have demonstrated the capability of our new approach for extracting the exact temperature from a scattering intensity signal in different situations. Moreover, we have shown that our method is highly robust with respect to noisy input data and have introduced a framework for the empirical quantification of the associated uncertainty both in the temperature and in the imaginary-time intermediate scattering function $F ( q , τ )$⁠. In the following, we turn our attention to actual experimentally measured data and reanalyze (1) an experiment on warm dense graphite by Kraus et al.¹⁸ and (2) the pioneering investigation of plasmons in warm dense beryllium by Glenzer et al.²⁶ 

In Fig. 15, we show our new analysis of the XRTS signal on isochorically heated graphite by Kraus et al.¹⁸ In the left column, we show the measured intensity as the green curve, where accurate data are available over three orders of magnitude. Interestingly, the main source of uncertainty in this experiment is due to the somewhat unclear shape of the instrument function $R ( ω )$⁠. Two plausible possibilities are shown as the blue curves in Figs. 15(a) and 15(d).

From a practical perspective, it is very useful to start the investigation of the experimental dataset by analyzing the distribution of the noise. In Fig. 16(a), we show the corresponding results for Eq. (34) as the red circles, where we have used a smoothening kernel taking into account the nearest five (±2) frequency bins for each value of ω for the evaluation of Eq. (33). Evidently, the deviations fluctuate around the origin, and the overall amplitude of $ξ σ Δ ( ω )$ appears to be approximately constant over the entire ω range. This is a nice empirical validation of the functional form in Eq. (31). For completeness, we note that the elastic feature itself has been omitted from this analysis, as appropriately taking into account the comparably large curvature of $I ( q , ω )$ in this region would require a separate, adaptive smoothing procedure; this is not needed for accurate quantification of the noise level. In Fig. 15(b), we show the corresponding histogram as the red bars, which can be well reproduced by a Gaussian fit (green curve). Both the fit and the direct evaluation of Eq. (35) give a variance of $σ Δ ∼ 10 − 2$⁠, which is used to quantify the uncertainty of both the temperature and $F ( q , τ )$ in the following.

Returning to Figs. 15(b) and 15(d), we find that convergence with the integration boundary x starts around $x = 125 eV$⁠. We note that going beyond $x = 140 eV$ does not make sense in practice, as the experimentally measured intensity vanishes within the given noise level for $ω ≲ − 140 eV$⁠. From these panels, we can clearly see that using either the narrow or the broad instrument function (which we truncate at $ω = 90 eV$ as the constant asymptotes given in the original Ref. 18 are plainly unphysical and would lead to a diverging Laplace transform $L [ R ( ω ) ]$⁠) has a substantial impact on the extracted temperature. Consequently, we gave our final estimate as $T = 18 ± 2 eV$ in our previous investigation.²⁵ At the same time, we stress that the resulting uncertainty due to the somewhat unknown $R ( ω )$ is considerably smaller than in the original; Ref. 18, where the applied Chihara fit gave $T = 21 eV$ with an uncertainty of $∼ 50 %$⁠.

Finally, we show our estimates for $F x ( q , τ ) ≈ F ( q , τ )$ for a converged integration boundary of $x = 126.8 eV$ in the right column of Fig. 15. In particular, the solid green curves show our direct evaluation of Eq. (23), and the shaded gray area indicates the corresponding uncertainty interval. In addition, we also mirror this function around $τ 1 / 2$ (which is estimated from the minimum in the green curve), i.e., $F ( q , ℏ β − τ )$⁠, and the results are shown as the dashed red curve, with the shaded red area indicating the correspondingly mirrored uncertainty interval. Evidently, our extracted results for $F ( q , τ )$ are fairly symmetric within the given uncertainty range for both instrument functions, although the degree of asymmetry [i.e., the difference between $F ( q , τ )$ and $F ( q , ℏ β − τ )$] is somewhat smaller for the narrower instrument function (top). Finally, the dotted blue curves show the two-sided Laplace transformation of the respective $R ( ω )$⁠. Clearly, the impact of the latter is more pronounced for the broader instrument function, as expected.

In conclusion, our present analysis agrees with the less well-controlled Chihara model-based analysis in the original Ref. 18 and highlights the importance of accurate characterization of the instrument function in future scattering experiments.

The second experimental dataset that we reanalyze in the present work is the scattering experiment focusing on the plasmons in warm dense beryllium shown in Fig. 17. Panel (a) shows the experimentally measured XRTS intensity as the green curve, which is clearly afflicted with a much larger noise level compared to the graphite data that we have considered previously. The corresponding reconstruction of the error distribution is shown in Fig. 18, where we find again good agreement with the functional form of Eq. (31); the few outliers for $ω < 0$ are possibly an artifact from a nearly vanishing intensity signal for some ω, leading to increased values for $Δ I ( q , ω ) / I ( q , ω )$ or due to the approximate nature of Eq. (31). We have used the same smoothening kernel to evaluate Eq. (33) as for Fig. 16. Figure 18 shows the corresponding histogram of $ξ σ Δ ( ω )$ as the red bars, which is accurately reproduced by a Gaussian fit with a variance of $σ Δ = 0.21$ (green curve). Note that we omit the outliers with an intensity below an empirical threshold of $I min = 10 4$⁠.

The reconstructed distribution of the random noise is again used to empirically quantify the uncertainty. In Fig. 17(b), we show the convergence of the extracted (inverse) temperature with the integration boundary x, and the shaded gray uncertainty interval very plausibly fits the observed noise in the green curve. As a final result, we obtain a temperature of $T = 14.8 ± 2 eV$⁠. This is close to the value of $T = 12 eV$ given in the original Ref. 26, which was obtained from an approximate Mermin model.⁸⁷ In the context of the present work, the main point of Fig. 17(b) is the remarkable robustness of our methodology even with respect to the considerable noise level in experimental data. Finally, we show our extracted imaginary-time intermediate scattering function in Fig. 17(c). Evidently, the direct evaluation of Eq. (23) (solid green) is in nearly perfect agreement with the mirrored curve (dashed red) over the entire τ-range despite the given uncertainty range (shaded areas).

In this work, we have given a detailed introduction to our new, model-free methodology for extracting the temperature of arbitrary complex materials from XRTS measurements,²⁵ referred to as imaginary-time correlation function thermometry (ITCFT). In particular, we have presented an extensive analysis of synthetic scattering spectra over a wide range of wave numbers and temperatures. Evidently, the method works exceptionally well covering the range from the collective to the single-particle regimes. In addition, we have studied the impact of the width of the instrument function, which decisively determines the minimum temperature that can be extracted from a corresponding XRTS signal. Naturally, these findings have direct implications for the design of future experimental XRTS setups.

Furthermore, we have introduced an empirical framework for quantifying the impact of experimental noise both on the extracted temperature and on $F ( q , τ )$ itself. In practice, the method is well-behaved and highly robust even against substantial noise levels. As a practical demonstration, we have reanalyzed the XRTS experiments by Kraus et al.¹⁸ and Glenzer et al.²⁶ 

On the strength of the analysis presented throughout this paper, we believe that our new ITCFT technique will have a considerable impact on a gamut of applications related to the study of WDM, including the highly active fields of inertial confinement fusion^9,10 and laboratory astrophysics.⁸⁸ On the one hand, we note that our method is particularly suited for modern x-ray free electron laser facilities with a high repetition rate such as LCLS,⁴⁸ SACLA,⁴⁹ and the European XFEL.^47,89 Specifically, its negligible computational cost will open up unprecedented possibilities for the on-the-fly interpretation of XRTS experiments. On the other hand, the robustness with respect to noise makes our approach also the method of choice for less advanced laser diagnostics at other facilities such as NIF.⁹⁰ 

Future developments might include a more rigorous analysis on the impact potential uncertainties in the characterization of the instrument function have on the extracted temperature. Similarly, possible effects of the background subtraction need to be investigated. Extending our framework to take into account the spatial inhomogeneity of a sample such as the fuel capsule in an ICF experiment^41,54 seems promising. Finally, we note that $F ( q , τ )$ contains the same information as $S ( q , ω )$ and, therefore, can be used to extract physical information beyond the temperature such as quasi-particle excitation energies or even more complicated phenomena such as the roton feature³² in the strongly coupled electron liquid.²⁸ 

This work was partially supported by the Center for Advanced Systems Understanding (CASUS), which is financed by Germany's Federal Ministry of Education and Research (BMBF), and by the Saxon State Government out of the State budget approved by the Saxon State Parliament. The work of Ti. D. was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. The PIMC calculations for the UEG were carried out at the Norddeutscher Verbund für Hoch- und Höchstleistungsrechnen (HLRN) under Grant No. shp00026, and on a Bull Cluster at the Center for Information Services and High Performance Computing (ZIH) at Technische Universität Dresden.

The authors have no conflicts to disclose.

Tobias Dornheim: Conceptualization (lead); Investigation (lead); Methodology (lead); Writing – original draft (lead); Writing – review & editing (lead). Jan Vorberger: Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Maximilian Böhme: Methodology (equal); Writing – review & editing (equal). Dave A. Chapman: Investigation (supporting); Methodology (supporting); Writing – original draft (equal); Writing – review & editing (equal). Dominik Kraus: Writing – original draft (equal); Writing – review & editing (equal). Thomas Robert Preston: Writing – original draft (equal); Writing – review & editing (equal). Zhandos Moldabekov: Writing – review & editing (equal). Niclas Schlünzen: Writing – review & editing (equal). Attila Cangi: Writing – review & editing (equal). Tilo Doeppner: Writing – review & editing (equal).

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

As discussed in Sec. II B 1, the form of the scattered power spectrum depends on the differential Thomson cross section $∂ σ ̃ / ∂ Ω | T$⁠. In this expression, the cross section is given a tilde symbol to mark it as a so-called “generalized” cross section. A fact that appears to seldom be addressed in the broader XRTS literature is that there is a subtle distinction between the foregoing expression and that which arises from the Klein–Nishina formula describing the cross section for Compton scattering of a photon from a single, isolated electron. This subject has been discussed extensively and from the perspective of a rigorous quantum-statistical framework by Crowley and Gregori.³⁷ We mention it here only to further emphasize their conclusion.