Spectral Deconvolution without the Deconvolution: Extracting Temperature from X-ray Thomson Scattering Spectra without the Source-and-Instrument Function

TL;DR

This work introduces a ratio-based ITCF approach to extract temperature directly from XRTS spectra without explicit deconvolution or modeling of the DSF or SIF. By comparing Laplace-transformed spectra from multiple scattering angles, the method leverages detailed-balance symmetry to obtain $k_B T$ in thermal equilibrium while remaining robust to spectral noise. Ray-tracing simulations of a mosaic HAPG von Hámos spectrometer demonstrate the method's resilience to detector and crystal variations, misalignment, and spectral broadening, with temperatures converging within ~10% under realistic conditions. The approach also provides a practical handle on identifying non-equilibrium via inconsistencies in temperatures inferred from different angle ratios, offering a promising, model-free diagnostic for high-energy-density physics experiments. The key contributions are (i) a model-free temperature extraction technique that bypasses SIF deconvolution, (ii) a detailed assessment of robustness to instrumental factors including mosaicity and IRC widths, and (iii) guidelines for experimental implementation with three or more scattering angles to enable cross-checks and non-equilibrium detection.

Abstract

X-ray Thomson scattering (XRTS) probes the dynamic structure factor of the system, but the measured spectrum is broadened by the combined source-and-instrument function (SIF) of the setup. In order to extract properties such as temperature from an XRTS spectrum, the broadening by the SIF needs to be removed. Recent work [Dornheim et al. Nature Commun. 13, 7911 (2022)] has suggested that the SIF may be deconvolved using the two-sided Laplace transform. However, the extracted information can depend strongly on the shape of the input SIF, and the SIF is in practice challenging to measure accurately. Here, we propose an alternative approach: we demonstrate that considering ratios of Laplace-transformed XRTS spectra collected at different scattering angles is equivalent to performing the deconvolution, but without the need for explicit knowledge of the SIF. From these ratios, it is possible to directly extract the temperature from the scattering spectra, when the system is in thermal equilibrium. We find the method to be generally robust to spectral noise and physical differences between the spectrometers, and we explore situations in which the method breaks down. Furthermore, the fact that consistent temperatures can be extracted for systems in thermal equilibrium indicates that non-equilibrium effects could be identified by inconsistent temperatures of a few eV between the ratios of three or more scattering angles.

Figures (19)

• Figure 1: (a) Ray traced IFs on the spectrometer used in this work, measured with different single photon energies $E_0$ on the crystal: 8.1 keV (black), 8.3 keV (red), and 8.5 keV (blue). (b) Ray traced IFs at 8.5 keV with the crystal at different positions $\Delta l$ along the dispersion axis. In both plots, the sudden drops in the intensity correspond to the physical edges of a crystal in space being reached, so only part of the crystal reflects photons to the detector.
• Figure 2: A two-dimensional sketch of the von Hámos spectrometer setup used in this work. The crystal is centred at approximately 351 mm and positioned below the dispersion axis at the crystal's radius of curvature, 80 mm. The detector is fixed at 702 mm away from the source and sits in the dispersion plane. This setup corresponds to a central photon energy of 8.31 keV on the crystal and detector. In Section \ref{['sec:Alignment']}, the crystal is moved along the dispersion axis by $\Delta l$ as well, but the detector remains fixed to maintain the spectral range.
• Figure 3: Ray traced XRTS spectra at different temperatures and scattering angles: 10$^\circ$ (red), 20$^\circ$ (blue), and 150$^\circ$ (orange). Also plotted is the SIF computed at 8.5 keV (black).
• Figure 4: Convergence tests of the temperature extracted from the ray traced spectra in Fig. \ref{['fig:Spectra']} via deconvolution of the SIF measured at 8.5 keV (solid lines) for different temperatures (15, 20, 50, 100 eV) and different scattering angles (10$^\circ$ in red, 20$^\circ$ in blue, and 150$^\circ$ in orange), with respect to the integration range $x$ in Eq. (\ref{['eq:define_ITCF']}). The dashed lines show the minima of the Laplace transformed spectra, without removing the SIF. The dotted horizontal line indicates the exact temperature of the system, and the grey area indicates a $\pm10$% interval around these temperatures.
• Figure 5: Convergence tests of the temperature extracted from the ray traced spectra for $T=15$ eV and 20 eV at the different scattering angles (10$^\circ$ in red, 20$^\circ$ in blue, and 150$^\circ$ in orange). Only the deconvolved curves are shown here. Compared to Fig. \ref{['fig:DirectDeconv']}, the Rayleigh weight has been reduced by a factor of 4 here, so more photons are scattered inelastically.