Correlation function metrology for warm dense matter: Recent developments and practical guidelines

TL;DR

The paper advances a model-free correlation-function metrology for warm dense matter by shifting XRTS analysis into imaginary time through the ITCF $F(k,\tau)$, which is the Laplace transform of the DSF $S(k,\omega)$. By deconvolving the measured spectrum with the source–instrument function (SIF) in Laplace space, temperature is obtained from the detailed-balance symmetry around $\tau=\beta/2$, and the absolute normalization is fixed via the $f$-sum rule, enabling direct access to the static structure factor $S_{ee}(k)$ and the Rayleigh weight $W_R(k)$. The framework provides two independent pathways to determine density and supports non-equilibrium detection by checking symmetry across multiple scattering angles. Practical guidelines include rigorous SIF characterization, convergence checks for the truncated Laplace deconvolution, and multi-angle measurements, with future prospects for analytic continuation to reconstruct the full dynamic structure factor from ITCF data.

Abstract

X-ray Thomson scattering (XRTS) has emerged as a valuable diagnostic for matter under extreme conditions, as it captures the intricate many-body physics of the probed sample. Recent advances, such as the model-free temperature diagnostic of Dornheim et al. [Nat.Commun. 13, 7911 (2022)], have demonstrated how much information can be extracted directly within the imaginary-time formalism. However, since the imaginary-time formalism is a concept often difficult to grasp, we provide here a systematic overview of its theoretical foundations and explicitly demonstrate its practical applications to temperature inference, including relevant subtleties. Furthermore, we present recent developments that enable the determination of the absolute normalization, Rayleigh weight, and density from XRTS measurements without reliance on uncontrolled model assumptions. Finally, we outline a unified workflow that guides the extraction of these key observables, offering a practical framework for applying the method to interpret experimental measurements.

Figures (10)

• Figure 1: Workflow for correlation-function metrology from XRTS. Starting from the measured intensity $I(\mathbf{k},\omega)$ and a measured or modeled source– and– instrument function (SIF) $R(\omega)$, a truncated two-sided Laplace deconvolution yields the (unnormalized) imaginary-time correlation function (ITCF) $F_u(\mathbf{k},\tau)$. Detailed-balance symmetry fixes the temperature via the location of the minimum at $\tau=\beta/2$. The $\tau\!\to\!0$ slope, through the $f$-sum rule, provides the absolute normalization. The normalized ITCF supplies $S_{ee}(\mathbf{k})=F(\mathbf{k},0)$ and, via \ref{['eq:FDT']}, access to the static density response. Combining $S_{ee}(k)$ with the elastic/inelastic ratio gives the Rayleigh weight $W_R(\mathbf{k})$, while comparison of $\chi(\mathbf{k},0)$ to theoretical benchmarks enables density inference. Convergence with respect to the deconvolution window and the $\tau$-grid, together with accurate SIF characterization, are the only analysis prerequisites.
• Figure 2: Schematic picture of the imaginary-time evolution of 4 particles in one dimension. The spatial dimension is depicted here as the $x$-axis while the imaginary time axis $\tau$ is shown here in units of expansion steps $\varepsilon$. Each particle path is closed with respect to $x(0) = x(\beta) = x(6 \epsilon)$. Note that the second and third particles are involved in a so-called permutation cycle Dornheim_permutation_cycles and form a single intertwined path with the same $2\beta$-periodic boundary conditions. Figure taken from Ref. Dornheim2022Physical with the permission of the authors.
• Figure 3: Schematic curve shape of an ITCF shown in blue. The ITCF is a well defined correlation function on the imaginary-time axis $\tau \in [0, \beta]$. The minimum at $\beta/2$ is shown as the red dashed line. An important point of a properly normalized ITCF is at its origin value at $\tau=0$, which coincides with the total static structure factor of the system.
• Figure 4: XRTS spectrum in arbitrary units of heated diamond as reported in Ref. Martin2025. The green curve shows the corresponding source-and-instrument function (SIF) for the reported shot. The spectrum was recorded with a beam energy of 8160 eV at a scattering angle of 170$^{\circ}$.
• Figure 5: Temperature analysis of the heated diamond spectrum reported in Ref. Martin2025. Panel (a) depicts the reported spectrum (red curve) and SIF (green curve) aligned to $\omega=0$ eV and with the energy axis multiplied by $-1$ to correctly align the down- and upshifted energies on the positive and negative axis, respectively. Panel (b) shows the inferred temperature from the ITCF as a function of the truncation. Due to the low SNR of the signal, the deconvolution is only able to resolve detailed balance for truncations between $229$ and $377$ eV. The average inferred temperature within this range is given by $\bar{T} = 85.25 \pm 13.22$ eV and shown as the blue dash-dotted line and the corresponding systematic uncertainties as the blue shaded band. The green dashed line depicts the inferred temperature in Ref. Martin2025 of $T=61^{+27}_{-19}$ eV, where the green shaded band depicts the uncertainties. The sudden drop in the inferred temperature between $376$ and $389$ eV is attributed to the high noise in this interval, which prevents the resolution of detailed balance. Panel (a) depicts the corresponding areas also by the magenta color. Panel (c) shows the calculated ITCF for a truncation of $300$ eV and the vertical blue dashed line represents the location of the minimum.