X-ray diagnostics have proven to be a powerful tool to probe high-energy-density matter, in order to access its structure, temperature, ionization state, or density, which are crucial for understanding a variety of plasma processes.1 Techniques such as in situ spectrally resolved X-ray scattering (also called X-ray Thomson scattering, XRTS, in the community) and X-ray diffraction (XRD) have been widely applied as sensitive probes of bulk characteristics.2–7 These diagnostics are complementary to optical methods, which only provide information on the sample surface and fail completely once the matter has become opaque to visible light.

The latter is the case in the warm dense matter (WDM) regime, where densities are comparable to ambient conditions and temperatures are of the order of typical binding energies. The study of WDM is a rapidly expanding field in modern physics, investigating matter in the transient regime between the solid and plasma states. Not only can WDM be found in many astrophysical objects,8–10 it also promises new methods for material synthesis.11,12 Both computational and experimental improvements over the last decades now allow the simulation and measurement of the behavior of these exotic states of matter that have previously been inaccessible.

The generation of WDM states in the laboratory is most commonly based on shocks—utilizing different drivers, ranging from gas guns13 and Z-pinches14 to lasers.15,16 These setups inherently lead to small samples with short lifetimes on the nanosecond scale. As a novel and complementary approach, the use of heavy ions for generating WDM has recently been proposed, in connection with the Facility for Antiproton and Ion Research (FAIR), currently under construction in Darmstadt, Germany.17

With the expected ion numbers, samples of millimeter size can be heated to temperatures of more than ∼10 eV and studied over a significantly longer timescale of hundreds of nanoseconds or even several microseconds. Owing to these longer timescales, the heated region is also expected to reach local thermal equilibrium,17 which has to be carefully verified when analyzing shock-driven WDM states.

For the diagnostics of heavy-ion-heated material, the implementation of X-ray diagnostics for temperature and structural observations is desirable, but this imposes new challenges. In this paper, we report on the first results using laser-generated X rays to perform simultaneous in situ XRTS and XRD measurements on samples heated by heavy ions.

We chose monocrystalline diamond as the system to be investigated in this scenario. In an evacuated oven, a volumetric transition of diamond to graphite can be observed over a timespan exceeding minutes at temperatures above ∼1900 K, with the rate of graphitization rapidly increasing with temperature.18,19 Alternatively, such a change in structure can be induced locally, by depositing over ∼1.2 eV on a single atom.20 Theoretical calculations predict partial graphitization to occur within several hundreds of femtoseconds for such experiments.21 However, the energy threshold for the transition and its speed are subjects of active research.20–23 In our study, we explore the hitherto poorly investigated intermediate-time regime—and also different mechanisms, since the ions interact not only with electrons but also directly with the diamond lattice.

A better understanding of the durability of a heated diamond lattice at ambient pressures on microsecond timescales also has several direct implications. First, the bombardment of diamond with fast ions is a typical scenario at GSI, FAIR, and other accelerator facilities, where this material is used within the sensors of particle detectors owing to its mechanical and electronic properties.24 Information on the material’s stability under these conditions is therefore of practical importance. Second, such knowledge would be beneficial in the case of nanodiamonds from laser-driven shock experiments, where their formation can be probed in situ using XRD10,11 but their recovery has yet to be achieved.25

The experiment was performed at the GSI Helmholtzzentrum für Schwerionenforschung (Darmstadt, Germany), using the heavy ions of the SIS18 heavy ion synchrotron and the nanosecond pulse of the PHELIX high-energy laser facility for generating the X rays for the diagnostics. Using this existing infrastructure allows for testing and improving methods and diagnostics for future operation of FAIR, in what is also termed the FAIR Phase-0 research program.

We infer the temperature of our sample from the ratio of elastic to inelastic scattering in the XRTS diagnostic by comparison with density functional theory molecular dynamics (DFT-MD) simulations (see Sec. III). A simultaneous XRD measurement finds no evidence for graphitization, but might indicate a macroscopic fracture of the sample (Sec. IV). Our results show that with the currently available ion numbers, the sample is heated to temperatures around 1300 K, which is too cold to study WDM. However, with FAIR coming online in the future, more than tenfold higher temperatures are expected to be reached,26 for which the method presented here can also be applied.

The experimental study was carried out in the framework of experiment S489 at GSI in the HHT (high-energy, high-temperature) experiment area. The full experimental setup is described by Hesselbach et al.,27 and the PHELIX capabilities can be found in Major et al.28 Here, we focus on the XRD and XRTS findings of the experiment.

Our full target assembly consisted of three main components enclosed in a milled aluminum housing: a monocrystalline diamond disk of 3 mm diameter and 60 µm thickness, a gold pinhole of 0.5 mm diameter, located a distance of 1.5 mm from the diamond, and finally a 10 µm-thick titanium foil, an additional 1.5 mm behind the pinhole, for creating the probing X rays (see Fig. 1).

To heat the diamond, we used a beam of ²⁰⁸Pb⁶⁷⁺ ions with 450 MeV/u, provided by GSI’s SIS18 accelerator. To increase the energy deposition within the sample, the ions were decelerated using a 15 mm-thick slab of PMMA upstream of the target. Simulations with SRIM29 predict a beam energy of ∼288 MeV/u after this degrader slab. Within a pulse of 250–300 ns full-width half-maximum (FWHM), ∼ 3.4 × 10⁹ ions were focused to a spot of two-dimensional Gaussian shape with FWHM of 0.6 × 0.9 mm².

For probing our target, the PHELIX laser delivered 100–130 J at 527 nm wavelength onto the titanium backlighter foil to generate 4.75 keV He_α emission. We focused the optical beam with a lens of 1.8 m focal length, achieving a spot size of ∼30–40 µm FWHM. The pulse duration amounted to 2 ns, resulting in an intensity of (3–8) × 10¹⁵ W/cm². The X rays were recorded by three diagnostics:(a)A monitor spectrometer, located at an angle of 0°, recording the backlighter source. This spectrometer consisted of a planar highly oriented pyrolytic graphite (HOPG) crystal with a mosaicity of 0.25° and 100 µm thickness.(b)The X-ray Thomson spectrometer at a scattering angle of 135° in Von Hámos geometry,³⁰ using a cylindrically bent HOPG with a radius of 50 mm, a thickness of 100 µm, and a mosaicity of 0.5°. Both spectrally resolved diagnostics used Greateyes CCD detectors to record the signal.(c)An imaging plate (IP; GE Healthcare BAS IP SR 2040 E), positioned at a distance of (422 ± 4) mm from the diamond at an angle of (70.7 ± 1.5)°, to record the diffraction signal around the (111) Laue spot. To ensure that the peak was covered by the ∼15 × 15 cm² area of the IP, we rotated the diamond accordingly, before mounting it on the target, since the cylinder surfaces were oriented in the (100) direction.

As we found that irradiating the target with ions resulted in a notable background, we shielded the direct line of sight of the digital detectors to the target with metal blocks and installed several layers of aluminum and Mylar foils in front of the spectrometers’ entrance windows, the detector chips, and also the IP. To further reduce parasitic signal from charged particles, we deployed magnets with B-field strengths of 0.55 T (XRTS) and 0.4 T (XRD) in front of the X-ray diagnostics.

During the scattering of a photon on an electron, the momentum transfer k = |k_s − k₀| can be approximated by k ≈ (2ω₀/c) sin(θ/2) if the induced change in frequency is small, i.e., ω = ω₀ − ω_s ≪ ω₀. Here, k₀ and ω₀ are the wave vector and angular frequency of the incoming light, k_s and ω_s are those of the scattered light, c is the speed of light, and θ is the scattering angle.

The intensity of the scattered light is proportional to the total static electron structure factor $S_{ee}^{\text{tot}}(k)$, which is commonly decomposed into three contributions:31,32$$S_{ee}^{\text{tot}}(k)=S_{ee}^{\text{ion}}(k)+S_{ee}^{\text{bf}}(k)+S_{ee}^{\text{free}}(k),$$(1)where the three terms describe elastic scattering $(S_{ee}^{\text{ion}})$, inelastic scattering due to bound-free transitions $(S_{ee}^{\text{bf}})$ and inelastic scattering on free—or nearly free—electrons $(S_{ee}^{\text{free}})$, respectively. For the temperatures reached in our experiment, all electrons remain bound to the atom, and so the last term in this equation can be omitted. Neglecting the screening by free electrons for the same reason, and assuming the L shell ionization energy to be small compared with the Compton energy, Eq. (1) can be rewritten as15,31,33,34$$S_{ee}^{\text{tot}}(k)=\underset{\text{elastic}}{\underbrace{|f(k){|}^2{S}_{ii}(k)}}+\underset{\text{inelastic}}{\underbrace{\sum _{n=1}^{Z_{\text{wb}}}[1-f_n^2(k)]}}.$$(2)Here, the ionic form factor f and the ion–ion structure factor S_ii have been introduced. The sum in Eq. (2) runs over all electrons for which the binding energy is smaller than the energy transfer in the scattering process ℏω, namely, the four electrons in the carbon L shell for our setup. f_n denotes the contribution of a single electron to f, which is sometimes approximated as being identical for all electrons.34 In this work, however, we use analytical functions that assume hydrogen-like atoms, giving different contributions depending on the electrons’ principal and azimuthal quantum numbers.35 The ratio of elastic to inelastic scattering x_el/x_inel, a quantity directly accessible in experiments, is hence connected to the static ion–ion structure factor by15$$S_{ii}(k)=\frac{1}{|f(k){|}^2}\left(\sum _{n=1}^{Z_{\text{wb}}}[1-f_n^2(k)]\right)\frac{x_{\text{el}}}{x_{\text{inel}}}.$$(3)

At ambient temperatures, the nearly perfect lattice structure of a monocrystalline sample results in clear Laue spots that contain nearly all coherently scattered light. Hence, an XRTS spectrometer, covering a scattering vector range for which no such peak is expected, will mainly record an inelastic signal. When the target is heated, the described behavior changes. With increased movement of atoms in the lattice, more photons are scattered coherently to scattering angles where no Laue spot is expected, and hence the elastic signal on the detector rises.

Therefore, we can leverage XRTS measurements as a temperature diagnostic by fitting the ratio of elastic to inelastic signal, calculating S_ii using analytical approximations for the f_n.35 This value can then be compared with results of state-of-the-art DFT-MD simulations for the sample material at a given temperature (see Fig. 2). Our method provides valuable benefits over the well-established technique of optical pyrometry, since the X-ray based diagnostic is still sensitive at lower temperatures, where the former approach is severely limited on the fast timescale of dynamic experiments. In contrast to pyrometry—which can only infer the temperature at the sample surface—XRTS provides insight into the bulk and is therefore well suited for the volumetric heating achieved by ion beams.

Applying the method to our setup, DFT-MD calculations predict the ratio of elastic to inelastic scattering from ambient diamond to amount to ∼1.7% for the probed scattering vector lengths around k = 4.45 Å⁻¹. Heating the sample to a temperature of 3000 K increases this value by a factor of ∼10 (shown in the inset of Fig. 2). This sensitivity enables us to investigate temperatures where volumetric graphitization by heating has been observed.18 Furthermore, the simulations predict a linear scaling of the ratio with temperature, simplifying the analysis. For more details concerning our DFT-MD calculations, we refer to Appendix A.

Despite various efforts to shield and deflect particles produced by the ion beam (see Sec. II and Ref. 27), the recorded spectra are dominated by parasitic signal. We attribute this to particles generated by the ion beam interacting with our target, the PMMA degrader, and the entrance window of the chamber. To clean the data, we first subtracted a background by averaging over regions on the detector far from the focal line of the spectrometer crystal. We then removed hard hits and strong, connected signal, guided by a neural network. More details on this step can be found in Appendix C.

The scattering data were fitted as the sum of an inelastic and an elastic contribution. To simulate the former, we calculated the energy shift of inelastically scattered photons using linear response time-dependent DFT (LR-TDDFT), by computing the dielectric function and connecting it to $S_{ee}^{\text{bf}}$ and $S_{ee}^{\text{free}}$ (where the contribution of the latter is small for the reasons mentioned above) via the fluctuation–dissipation theorem.36 To also obtain the instrument-free elastic scattering spectrum, we simulated the emission of the laser-driven titanium plasma using the commercially available PrismSPECT software,37 benchmarked to the monitor spectrometer (see the supplementary material).

To account for the point-spread functions of our diagnostics, we relied on a ray tracing simulation of the setup. The software was written in Python using the package jax,38 enabling us to consider a high number of rays in parallel while keeping the computation time manageable. The simulation of the mosaic crystals in our spectrometers was inspired by the mmpxrt code of Šmíd et al.39 By simulating the full setup, we include the effects of broadening of the signal due to the finite X-ray spot size on the diamond, the dispersion and focusing of the mosaic crystal, and the backlighter spectrum. Supplementary descriptions of the ray tracing simulations are given in Appendix B.

We implemented a Bayesian fitting routine using PyMC40 to obtain the ratio of elastic to inelastic scattering, assuming Gaussian noise with constant but unknown standard deviation. A comparison of the one-dimensional data and the posterior predictions of the fit is presented in Fig. 3 for a representative event.

By changing the delay between the SIS18 beam and the PHELIX pulse, we can investigate how the interaction with the heavy ions changes the scattering spectrum at—virtually—any time during or after the exposure to the ions. In the study presented, we recorded data up to a delay of 20 µs (see Fig. 4). The blue data points were obtained with the analysis procedure described above. The entry denoted with a light blue star shows signal from the backlighter plasma, focused to a different line on the detector, owing to poor shielding. Despite efforts to mask the parasitic signal, this might have resulted in an additional, systematic offset. The errors shown in Fig. 4 were estimated from the uncertainties of the ratio of elastic to inelastic scattering obtained by the Bayesian fitting routine, the sensitivity to the choice of background, and small shifts in the energy calibration, which might be caused, for example, by a misalignment of the laser with respect to the pinhole. The errors in x_el/x_inel were mapped to temperatures by approximating the relationship between the two quantities as linear (as is suggested by the DFT-MD simulations presented in Fig. 2). The notable uncertainties are rooted in the high background and noise ratio of the signal and are planned to be reduced in follow-up studies with improved shielding.

In Fig. 4, we compare the results of our method with temperatures estimated from the number of ions entering our target chamber. To link the latter quantity to temperatures, we calculated the energy deposition of the ions in the sample with SRIM.29 For these simulations, the temporal shape of the ion pulse was approximated by a sin² t profile, fitting the signal measured by a fast current transformer in the beam’s path. The presented calculations account for the inhomogeneous conditions in the target by calculating a spatial average of the temperatures, weighted by the X-ray intensity on the diamond, which was obtained via the ray tracing code. To verify the applicability of the one-dimensional SRIM calculations, we compared the result with those of full 3D simulations using the FLUKA code41,42 for energy deposition. Both methods agree reasonably well, with deviations smaller than 10%. The latter results were additionally used as inputs for calculations applying the ANSYS 2023 R2 software package to model the effects of heat transfer and cooling of the sample. We found that both effects only lead to small corrections, even at the biggest delays investigated for temperatures below 2000 K. Therefore, we neglected these contributions when simulating individual shots. Similar calculations for an identical setup (but different sample materials) found decent agreement with benchmark pyrometry measurements.27 While both methods indicate similar temperatures around, or even exceeding, 1300–1400 K, the temperatures obtained by the XRTS technique are on average slightly lower than those from the SRIM simulations.

Complementary to the technique of XRTS, we fielded spatially resolved XRD. For the same reason that the ratio of elastic to inelastic scattering increases with temperature, the XRD signal is expected to decrease as the sample heats up. This behavior is captured in the Debye–Waller factor.43 However, owing to the high Debye temperature of diamond, our DFT-MD simulations indicate that the intensity of the (111) peak would recede by less than 4% at 1500 K. Hence, for the temperatures reached, we do not expect a notable change of the diffraction signal due to diamond heating alone.

By contrast, a transition to graphite would result in a notable change of the diffraction pattern. For the ion numbers realized, the center of our sample is predicted to reach peak temperatures around 1900 K. Under these conditions, graphitization of diamond has been observed in vacuum.18

Diffraction patterns recorded for different delays between ion and PHELIX pulses are plotted in Fig. 5. Each recording shows three distinct continuous lines which are caused by the spectrum of the backlighter. Because of the finite size of the source on the diamond (∼1 mm in diameter), the Laue spots are smeared out despite the sample being monocrystalline. As the rotation of the diamond—and hence the position of the (111) reflection—could not be perfectly reproduced between targets, the position of the pattern on the IP varies for different shots. Figure 5 accounts for that by displaying the diffraction data shifted so that the recordings align on a common ring. The presented data were scaled by the X-ray intensity on the monitor spectrometer and corrected for the decay of the signal on the IP during the time between recording and scanning. We then removed a constant background.

The collected data show no clear signs of graphitization. For delays up to 500 ns, the signal does not differ noticeably from that of an unheated sample. At longer delays, however, the elongated Laue spots start to break apart. We can reproduce this feature well with our ray tracing software by splitting the cylinder representing the diamond sample in half, along the ion beam direction, and rotating the upper part by few degrees.

While we are currently lacking a conclusive understanding of what is happening to the sample at longer delays, we consider the XRD signal as a hint that our sample might break after irradiation with the ion beam. The fact that the longest delay only shows a faint diffraction pattern might support this hypothesis, as parts of the sample might have moved too far to be captured in the field of view of the IP. Anyway, the change in the XRD signal demonstrates how this method can provide additional insight into the sample that is not apparent in XRTS and is required as a complementary diagnostic when moving to higher temperatures to identify the onset of graphitization.

We have presented a method to leverage the widely used technique of X-ray Thomson scattering as a temperature diagnostic for diamond samples heated by heavy ion beams, relying on DFT-MD and ray tracing simulations. The approach is applicable even under relatively cold conditions, where optical pyrometry fails. Combining this measurement with an XRD diagnostic has allowed us to investigate the microscopic structure of the material, which does not show in situ evidence for graphitization of the diamond and provides hints that our sample breaks on a macroscopic level.

Using heavy ions as a driver yields some unique challenges for the detection of the XRTS signal, since the a substantial background is induced by the ion beam hitting the sample. We have accommodated this by using both physical shielding and post-processing of the data, relying on a neural network to identify faulty pixels. Furthermore, the experiment has probed a wide range of scattering vectors, owing to the large size of the X-ray source on the sample, and so our analysis required that we perform ray tracing simulations of the full setup, from the backlighter to the detector.

The temperatures inferred from the scattering signal appear to be compatible with estimates from ion stopping power simulations. For delays longer than 1 µs, we have recorded a qualitative change in the XRD signal, suggesting a major alteration in our sample. Hence, the temperature measurement for those delays might show a systematic error. While we have not identified graphitization of our sample, raising the number of ions and consequently the temperatures—as will be possible with the SIS100 accelerator—seems a promising route to study this phenomenon, since it is predicted to be vastly accelerated by a small increase in temperature.19

The work described here represents an important step toward a temperature diagnostic for WDM that can provide insight into the bulk of the probed material rather than just the surface properties accessible by optical methods. Such a tool is highly desirable, for example, in view of the novel experimental capabilities being opened up for WDM research by FAIR, allowing the creation of samples of millimeter size.

Acknowledgment. The results presented here are based on experiment S489 performed at the HHT target station in the context of FAIR Phase-0 at GSI, Darmstadt (Germany). J.L. and D.K. were supported by GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt as part of the R&D project SI-Grant No. URDK2224 with the University of Rostock. M.S. and R.R. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Unit FOR 2440. The ab initio calculations were performed at the North-German Supercomputing Alliance (HLRN) facilities. R.B. acknowledges support by the Federal Ministry of Education and Research (BMBF) under Grant No. 05P21RFFA2. O.S.H. and D.K. were supported by the Helmholtz Association under Grant No. ERC-RA-0041.