Creation and characterization of warm dense matter isochorically heated by an intense laser-driven proton beam to temperatures exceeding 100 eV

Abstract

The warm dense matter (WDM) is an exotic state of matter encountered in inertial confinement implosions for fusion energy, as well as the interiors of giant planets like Jupiter, brown dwarfs, the atmospheres of white dwarfs, neutron star crusts, and newly discovered exo-planets. One efficient way to create WDM is to use protons accelerated by a high-intensity short-pulse laser to isochorically heat dense samples to WDM states. Despite its importance, direct temperature measurements within WDM targets are scarce. This study utilizes an intense proton beam generated by the kilojoule EP laser further focused and guided by a curved foil and cone structure to efficiently heat a thin copper sample. A high-resolution streaked spectrometer tuned to copper K_α fluorescence lines provided bulk temperature measurements every  ~2 ps, revealing temperatures exceeding 100 eV in under 50 ps. Particle-in-cell simulations of proton transport and energy deposition closely matched the observed heating dynamics, including transverse temperature gradients revealed by the broadening of K_α lines.

Introduction

The study of Warm Dense Matter (WDM) has emerged as a critical area due to its importance in astrophysics, planetary science, and Inertial Confinement Fusion (ICF)^1,2,3,4,5. This interest is driven not only by the fundamental need to bridge plasma and condensed matter physics but also by the practical necessity of understanding material behavior under extreme conditions. Laser-driven intense proton beams are particularly well-suited for laboratory studies of WDM properties due to their short bunch duration and high rate of energy deposition before significant material expansion occurs⁶. These laboratory experiments are needed to enhance our understanding of the WDM state. Creating and precisely characterizing WDM states is essential for studying ICF implosions or astrophysical and planetary objects such as the crusts of old stars or giant planets’ interiors, where matter exists in this exotic regime that is particularly challenging to model⁷. Moreover, one of the primary challenges in ICF for harnessing the power of fusion energy is to significantly increase the target energy gain from ~2 (as demonstrated in the recent ignition record at the National Ignition Facility⁸) to over 100⁹. A promising approach to achieve this increase in gain is by separating the compression and ignition phases, a concept known as Fast Ignition¹⁰. Coincidentally, proton beams with MeV energies can penetrate deeply into high-density regions of a target with minimal transport instabilities¹¹ and deposit most of their energy at the end of their propagation. This capability is highly desirable for isochoric heating a localized volume to form a hotspot inside dense fusion targets and trigger ignition^12,13. Regardless of the application, a robust characterization of temperature evolution inside the dense heated targets is critically required to improve the modeling of material states and properties in the WDM regime, including ion stopping power¹⁴, equations of state¹⁵, opacity⁵, thermal conductivity¹⁶ and other transport coefficients¹⁷.

Previous experiments have demonstrated various successful setups for generating WDM states. Direct short-pulse laser heating can create ultrathin (≤1 μm) WDM samples¹. However, since the laser energy is deposited at the target surface, the resulting heating is inherently nonuniform in the depth of the target. Additionally, ultrathin samples tend to expand more rapidly, making it challenging to achieve isochoric heating (i.e., heating at constant volume/density). High-energy X-rays offer an alternative by penetrating and depositing energy deep within a dense target, thereby addressing the uniformity issue. However, heating solid matter to a wide range of WDM conditions using X-rays requires a large number of energetic laser beams to irradiate a hohlraum and produce the necessary high flux of X-rays¹⁸. This approach limits diagnostic access and versatility, and the longer timescales of X-ray heating make maintaining isochoric conditions difficult. More recently, experiments have demonstrated the promising capabilities of laser-generated ion beams for achieving isochoric heating of dense targets with high uniformity^16,19,20,21. These experiments have also shown that significantly higher temperatures can be reached by focusing proton beams using curved foils^6,22,23,24. This ability to focus proton beams using curved foils has been a key motivation for proton fast ignition²⁵. Our campaign builds on the well-established method of focusing a picosecond laser onto a solid plastic hemisphere to generate an intense proton beam through Target Normal Sheath Acceleration (TNSA)²⁶ and direct it onto a secondary target. The focusing and guiding of the protons by the curved foil and cone structure unlocked the ability to achieve proton-heated target temperatures exceeding 100 eV for a mid-Z material at solid density. Therefore, beyond WDM studies, the design of our platform with a kJ-class driver, makes it ideally suited to support optimization efforts aimed at advancing the development of proton fast ignition targets.

For all heating techniques, directly characterizing the temperatures of WDM samples remains a significant challenge, with measurements being scarce in the literature. Most previous studies have relied on optical emission to measure surface temperatures^{16,19,20,21,22}. However, determining the temperature within the bulk of the targets typically depends on matching surface temperature data with radiation-hydrodynamics simulations. This approach introduces considerable uncertainties in the WDM target conditions, which significantly complicates the comparison of its properties with theoretical models¹⁴. On the other hand, probing at the higher photon energies of X-rays can give direct access to the temperature and other properties inside the WDM sample, using spectroscopy techniques in emission²⁷, point-projection absorption²⁸, absorption near-edge structure (XANES)^29,30,31,32, as well as X-ray scattering³. In this work we demonstrate the generation and measurement of time-resolved bulk temperatures in WDM samples with temperatures exceeding 20 eV. The capability to generate and accurately characterize high-temperature WDM is critical for advancing ICF modeling, where imploding shell retain WDM properties at temperatures up to several hundreds of eV during compression³³. These conditions also mirror astrophysical systems, such as substellar objects below the hydrogen-burning limit (e.g., brown dwarfs³⁴), where WDM states govern the transport and equation-of-state behavior³⁵. At these elevated temperatures, XANES and X-ray scattering techniques suffer from increased continuum noise and significant thermal broadening, resulting in spectral flattening and potential loss of distinctive features³⁶. While X-ray absorption spectroscopy (XAS) techniques could still be employed, they require a dedicated high-intensity driver to generate a bright and broadband x-ray source for probing the target. Moreover, the presence of ions at various ionization levels complicates XAS analysis due to complex spectra and interpretation challenges. K_α emission spectroscopy does not require a separate driver, and the signal strength directly increases with the number of accelerated fast particles hitting the sample (and thus with heating driver energy), helping to maintain a signal-to-noise ratio greater than one at higher drive energy levels. Additionally, unlike thermal ion-like emissions and absorption lines, the innermost shell K_α lines are only marginally affected by opacity effects, which eliminates the need for radiation transport calculations and enhances the robustness of the technique. This approach is originally applied here to isochorically proton-heated targets. This characterization method is not only a novel contribution to the field but also strengthens future research by improving the ability to accurately characterize the conditions of materials throughout heating into the high-temperature WDM regime.

Specifically, a spectroscopy measurement with high temporal and spectral resolutions is uniquely applied to characterize the heating of solid copper samples subjected to energy deposition from an intense proton beam. Rapid and pronounced shifts in the energies of fluorescence K_α line emissions were observed using the streaked spectrometer HiResSpec³⁷ at OMEGA-EP. These observations are then compared to simulations performed with a collisional-radiative spectral analysis code to derive temperature data. The observed shifts in K_α energies correspond to the sample exceeding temperatures of  ~100 eV within ~50 ps of heating. These results establish a new benchmark for the creation and characterization of dense targets in the 20–100eV temperature range, a critical step toward achieving accurate measurements of the elusive properties of WDM. Furthermore, the heating dynamics observed in the experiment show good agreement with hybrid particle-in-cell simulations, including transverse temperature gradients in the sample which are revealed by a Bayesian analysis of the experimental K_α spectra.

Results

Experimental setup

The experiment was conducted at the OMEGA-EP laser facility, where the EP short-pulse beam was focused onto solid plastic targets to generate a proton beam via TNSA. Two types of proton source targets were used in this experiment: a hemisphere with a cone structure, shown in Fig. 1(a), and a reference flat target, shown in Fig. 1(b). The hemisphere and cone (HC) serve to focus and guide the proton beams to spot diameters comparable to or smaller than the cone tip opening⁶, while the proton beam from the flat target naturally diverges³⁸. It has also been observed that the cone’s benefit for improved proton beam focusing comes at the expense of reduced proton beam energy and temperature compared to smaller targets¹¹. To evaluate the heating capability of the generated proton beams, a 10 μm thick solid copper slab was positioned at the cone tip, or a 200 μm thick copper slab was placed at the same source-to-target standoff distance of 430 μm for the flat target (resulting in on-target proton beam spot diameters on the order of 400 μm instead). Two laser drivers were used alternately: the backlighter (BL) and the sidelighter (SL). Each laser beam delivered ~900 J on target, with pulse durations of  ~6 ps for the BL and  ~12 ps for the SL. With 80% of the total energy contained within a  ~17 μm spot size, the peak intensities were ~2 × 10¹⁹ W/cm² and 1 × 10¹⁹ W/cm² for the BL and SL, respectively. The measured intensity variations across all shots were ~10% for the SL beam and 3% for the BL beam, while the total delivered energy fluctuated by less than 1%. Consequently, shot-to-shot fluctuations in the proton beam energy distribution were minor and did not significantly impact the heating process.

Fig. 1: Sketch of the two target configurations used in the experiment at OMEGA-EP.

a Hemisphere + cone (HC) targets were alternately driven by either the backlighter (BL) beam (~6 ps) or the sidelighter (SL) beam (~12 ps), with a 10-μm thick solid Cu slab positioned at the cone tip. b A flat target was driven by the SL beam, with a 200 μm thick Cu slab placed 430 μm away [same source-to-target standoff distance as in (a)]. The K_α emission was measured by the HiResSpec instrument³⁷, with the line-of-sight indicated in the figure. The source and transmitted proton beams were characterized by a Thomson parabola (TPIE) and radiochromic films (RCF), where the source was obtained using HC targets without a Cu slab. Note that (a, b) are not on the same scale.

The primary diagnostic in our study of intense proton beam energy deposition is the High-Resolution Spectrometer (HiResSpec), a spectrometer coupled with an X-ray streak camera that offers exceptionally high resolution in both spectral energy (E/ΔE~5000) and time (≲2 ps). HiResSpec is configured to measure the Cu K_α lines, spanning 7.97–8.11 keV³⁷. This instrument enables us to monitor changes in the emission spectra as the targets are heated, focusing on shifts in K_α peak energies, relative intensities, and spectral line shapes. K-shell spectroscopy techniques are highly adaptable and can be applied to other elements, including low-Z elements. The campaign also employed several additional diagnostics on OMEGA-EP, with two being particularly critical to our analysis: the Thomson parabola ion energy (TPIE) analyzer and a radiochromic film (RCF) pack. The TPIE analyzer, which uses electric and magnetic fields to separate ions by energy and charge-to-mass ratio³⁹, allows us to measure the proton beam energy distribution and determine its characteristic temperature (Fig. 2a). The RCF, which darkens upon proton exposure, consists of multiple layers to obtain angularly resolved measurements of the energy-dependent proton dose. This diagnostic enables us to characterize the laser-to-proton beam conversion efficiency and the characteristic proton temperature through an analytic fitting of the proton beam energy deposition (Fig. 2b). Both of these secondary diagnostics are essential for characterizing the proton beam and providing empirical parameters for subsequent analysis and initialization of simulations.

Fig. 2: Proton beam measurements along the beam axis without a target at the cone tip.

a Proton energy distribution from the Thomson parabola (TPIE) diagnostic on a BL shot, with its inferred proton characteristic temperature T_p determined from a Maxwell-Boltzmann (MB) distribution fit. The time-of-flight (TOF) of protons from source to target (430 μm) is shown on a secondary x-axis. b Fitting of the dose measured in each layer (HD-1, HD-2, ...) of the RCF stack diagnostic on a SL shot with the dose deposited by a MB energy distribution, indicating inferred proton characteristic temperature and conversion efficiency (CE) from laser to protons for energies >1 MeV (extrapolated).

Experimental results

The isochoric heating of a target by a proton beam is primarily governed by the stopping range of the protons, which is predominantly determined by proton-electron collisions. The primary factors influencing this process are the proton kinetic energy and the density and temperature of the target. Secondary factors, such as beam-driven electromagnetic fields, can modify the proton stopping range when the proton beam current density exceeds  >10¹⁰A/cm² for a length > 50 μm^40,41. For the proton current densities of  ~1.5 × 10¹⁰A/cm² and target thicknesses of ≤25 μm encountered in this study, the influence of beam-driven fields is minor. The proton current density is calculated using Eq. (13) in ref. ⁴¹, employing parameters extracted from the proton energy distribution shown in Fig. 2: total proton energy ε_tot = 15 J (laser-to-proton conversion efficiency of  ~2%) and characteristic proton temperature T_p = 5.7 MeV, as determined from a Maxwell-Boltzmann (MB) distribution fit. The proton beam radius r₀ = 15 μm is further estimated from K_α spectroscopic analysis, and the source-to-target distance of d = 430 μm is dictated by the target geometry. The high-energy protons from the broad TNSA distribution will initially deposit a portion of their energy into the thin target before exiting through its rear side, followed by lower-energy protons, which will deposit all of their energy and stop within the material. The time-of-flight (TOF) of protons can be simply estimated by ${{{\rm{TOF}}}}=d/\sqrt{2{E}_{p}/{m}_{p}}$, where d is the source-to-target distance (here d = 430 μm), E_p is the proton kinetic energy, and m_p is the proton mass. Before the proton heating phase, the faster hot electrons generated by the high-intensity laser interaction rapidly traverse the cone, initiating K_α fluorescence emission and target heating. Given the transient nature of TNSA processes (on the picosecond scale), unraveling the heating history requires measurements with picosecond resolution.

The HiResSpec instrument enables us to capture changes in the X-ray fluorescence spectra on a picosecond time scale. Figure 3 presents time-resolved spectra measured with this instrument for the three configurations. In Fig. 3(c), the spectrum obtained with the flat target displays two streaks corresponding to the cold K_α1 line at 8027.5 eV and the cold K_α2 line at 8047.5 eV. Notably, the intensity of these two K_α lines peaks at ~30 ps, which corresponds to the TOF of  ~1 MeV protons, and remains stable over time without shifting. In contrast, the spectra from the HC targets shown in Fig. 3(a, b) exhibit a blueshift (increase in photon energy) and broadening of the K_α lines after 20 ps, indicating heating of the Cu slab. Beyond 55 ps, the K_α signal vanishes as the last lower-energy protons have stopped within the material.

Fig. 3: K_α time-resolved spectra emitted from the Cu slab and measured with the HiResSpec instrument in the three experimental configurations.

a HC target using the sidelighter driver, b HC target using the backlighter driver, and c flat target using the sidelighter driver. The reference time t = 0 ps corresponds to the first recorded K_α light triggered by fast electrons emitted from the laser interaction region and reaching the target in  <2 ps. At t > 10 ps, protons with kinetic energies  <10 MeV progressively reach the target, which increases the K_α signal intensity and the strong proton heating causes shift and broadening of the K_α lines. Experimental and synthetic X-ray spectra for the three configurations and at three different times are plotted: d SL HC, e BL HC, and f SL flat. All the spectra are integrated over  ~2 ps. The dashed black line is the experimental spectra, and the blue line is the best-fitting synthetic spectra calculated with PrismSPECT assuming a single temperature.

Single temperature analysis

We estimate the temperature of the Cu slabs using the PrismSPECT atomic physics code⁴². The temperature is inferred by fitting the experimental spectra with synthetic spectra calculated by PrismSPECT, assuming a single temperature. Figure 3(d–f) provide examples of the comparison between the data and best fit for the three experimental configurations. To significantly reduce noise, the spectra were integrated over interleaved 2.3 ps windows, preserving the original shapes of the spectral lines without compromising temporal resolution. Figure 3(f) shows the fitting of the X-ray spectra for the flat target configuration. Consistent with Fig. 3(c), the K_α lines are clearly defined and do not exhibit any variations in shape, yielding an estimated temperature ranging from 25 to 38 eV over 20 ps, as shown in Fig. 3(f). For the other two configurations, the PrismSPECT calculations agree well with the experimental spectra at early times, as seen in Fig. 3(d1, e1). However, at later times, the match between PrismSPECT and the experimental data deteriorates, as a single temperature cannot accurately reproduce the observed broadening associated with temperature gradients. Here, the fit is solely driven by the dominant peak (K_α1) shift. Nonetheless, this simple temperature estimate provides valuable insights into the temporal evolution of heating.

Figure 4 shows the temporal evolution of the temperature resulting from the fits presented in Fig. 3. First, we note that the temperature for the flat target remains relatively constant over time, with any observed increase falling within the estimated uncertainty. However, for the HC target, a systematic increase in temperature occurs. These estimations of the temperatures reveal three main heating phases: (i) an initial phase up to 18 ps where plasma temperature spans from 0 eV to 25 eV, (ii) a linear heating phase, and (iii) a saturation phase. Note that for temperatures below 20 eV, the K_α lines do not shift enough to be measured, which poses a lower temperature limit of this spectroscopy technique. Consequently, during phase (i), a more precise temperature estimation is not attainable. We will note finally that results show enhanced heating, with temperatures up to  ~125 eV when using the BL driver, compared to temperatures up to ~80 eV when using the SL driver. This is consistent with the BL driver being at about twice the intensity of the SL driver.

Fig. 4: Temporal evolution of the estimated temperature for the three configurations.

a HC target using the sidelighter driver, b HC target using the backlighter driver, and c flat target using the sidelighter driver. The dashed line represents the posterior mean, and the shaded region corresponds to one standard deviation.

Bayesian analysis

As mentioned earlier, the fit begins to diverge from the experimental data after 20 ps. One factor not accounted for by the single-temperature method is the presence of a temperature gradient within the Cu sample. In our previous work using similar HC targets, imaging of the proton beam’s transverse profile with a spherical crystal revealed a Gaussian-like K_α emission⁶. Therefore, we expect this proton beam profile to induce a transverse temperature gradient within the sample in the present experiment because the transverse sample’s size is larger than the proton beam’s size. To unravel the temperature gradient from the K_α spectra, we employ a Bayesian approach as proposed by Pokornik et al.⁴³. We assume that the temperature profile follows a radial Gaussian function expressed as $T(r)={T}_{\max }{e}^{-{r}^{2}/2{\sigma }^{2}}+{T}_{\min }$, where ${T}_{\max }$ is the maximum temperature at the center of the Cu slab, σ defines the standard deviation of the Gaussian function, and T_min is the baseline temperature. In the rest of the paper, the following three parameters are used to describe the temperature profile: ${T}_{\max }$, ${T}_{\min }$ (the temperature at the edge of the Cu slab), and Φ_FWHM (the Full Width at Half Maximum (FWHM) of the Gaussian function). The total synthetic K_α spectrum is then calculated by summing all the spectra based on the temperature profile. Initial estimates of the three parameters were obtained using the Levenberg–Marquardt (LM) optimization algorithm to fit the total synthetic K_α spectrum to the measured spectrum. Subsequently, the posterior probability distribution was inferred using Bayesian analysis coupled with a Monte Carlo Markov Chain (MCMC) sampling method, with the LM estimated values serving as initial values for the MCMC sampling.

Two examples of the Bayesian analysis are shown in Fig. 5. For comparison, we used the same spectra as those presented in Fig. 3(d). In Fig. 5(a), there is good agreement between the measured K_α spectrum and the Bayesian-inferred spectrum, similar to what was obtained using the single-temperature method. However, the ensemble of accepted spectra exhibits significant variability between 8060 eV and 8080 eV, likely due to the algorithm attempting to fit the noise. This variability translates into large uncertainty for the parameters ${T}_{\max }$ and Φ_FWHM at early times. It is worth noting though that the single-temperature method was both adequate and sufficient at these early times. On the other hand, for the later spectra shown in Fig. 5(b), the Bayesian method provides a better fit than the single-temperature approach. Some discrepancies remain, possibly due to the presence of a longitudinal temperature gradient (discussed further) and the assumption of a Gaussian transverse temperature profile.

Fig. 5: Example of fitting the experimental data using the Bayesian approach.

The red line corresponds to the experimental data, and the blue lines correspond to a sample of accepted spectra. The experimental K_α spectra are from an SL HC shot at two different times corresponding to Fig. 3(d1, d3) for the single temperature method, respectively in (a, b).

We now discuss the values of ${T}_{\max }$, ${T}_{\min }$, and Φ_FWHM obtained from the Bayesian analysis. Figure 6 illustrates the temporal evolution of these parameters for the two HC configurations, excluding the flat target, which does not exhibit a temperature gradient (no significant broadening) and for which a single-temperature analysis was adequate. The heating dynamics observed in the Bayesian analysis are consistent with those from the single-temperature method, with average temperatures reaching up to 100 eV and 75 eV for the BL and SL configurations, respectively. Additionally, maximum temperatures of approximately 200 eV and 130 eV are inferred for the BL and SL configurations, respectively. Note that the average temperatures are ~20% lower compared to the single-temperature analysis as the latter tends to emphasize the higher peak at higher photon energy when fitting K_α1. The FWHM varies between 40 μm and 60 μm, consistent with our previous measurements of focused proton beam profile⁶.

Fig. 6: Temperature evolution estimated from a Bayesian analysis of K_α spectra for HC targets.

The first column (a, b) corresponds to the BL driver, while the second column (c, d) corresponds to the SL driver. In (a, c) the orange line represents ${T}_{\max }$, the blue line represents ${T}_{\min }$, and the black line indicates the average temperature. The temporal evolution of the temperature spatial dispersion Φ_FWHM is shown in (b, d). The uncertainty, represented by the shaded area in (a, c) and by the error bars in (b, d), corresponds to the standard deviation estimated from the Bayesian analysis. The hatched region within the first  ~20 ps corresponds to times when the Bayesian technique is affected by noise, resulting in large uncertainties for ${T}_{\max }$ and Φ_FWHM. The posterior mean temperature remains in agreement with the single-temperature approach (Fig. 4) in this region, which was both accurate and sufficient during these initial times.

LSP simulation

We performed simulations using the hybrid-PIC code LSP to further investigate proton dynamics and their contribution to the sample’s heating. Our methodology follows that of previous work⁶, where an electron source based on laser parameters and experimental measurements is injected into a hemispherical target with a contaminant layer on its rear surface to simulate TNSA acceleration. As the protons are accelerated from the hemispherical target, they are focused by the converging geometry of the TNSA sheath field²². These protons are then transported through the cone structure. Additional guiding and focusing arise from transverse electric fields, created by electrons moving through the cone due to the electron pressure gradient²³. Figure 7 presents the proton number density (a) and the electron number density (b) at 7 ps after the initiation of electron source injection. The corresponding electric fields E_z (d) and E_r (e) are also shown. Electrons and protons reaching the vicinity of the cone tip are collected until t = 22 ps (corresponding to the TOF of 2 MeV protons) to limit computational cost and numerical noise arising from simulation boundaries. These particles are then injected into a Cu slab in a separate simulation to model energy deposition and heating. Temperature values will be compared to experimental results in the next section. Based on the heating information and temperature-dependent cross sections for electron and proton-induced Cu K_α emission^44,45, we can also evaluate the K_α spatial profiles induced by each species. Figure 7(c, f) illustrate the spatial profiles of Cu K_α emission, binned by lateral position, at different times, and driven by electrons and protons, respectively. The simulations offer valuable insights into how electrons and protons contribute to K_α photon generation spatially and temporally. As demonstrated in panels (c) and (f), hot electrons generated by the laser interaction traverse the cone structure and reach the Cu sample earlier than protons, making them the initial dominant source of Cu K_α photons. However, after ~10 ps, protons become the primary contributors, generating a higher photon count in the central region (f) compared to electrons (c). The distinct spatial profiles of K-alpha photons — where protons exhibit a central peak, while electrons yield a more uniform distribution within a 70 μm radius—can be attributed to the ballistic motion of protons, in contrast to the more easily scattered electrons. The higher count of Cu K_α photons induced by electrons on the edge of the sample (70 μm < r < 100 μm) is caused by electrons flowing along the cone.

Fig. 7: Hybrid-PIC simulations of proton acceleration and focusing using an HC target driven by the OMEGA-EP beam.

a, b Illustrate the density maps of protons and electrons at 7 ps, respectively. The electric fields in the z and r directions are depicted in (d, e). The contribution of K_α photons generated in the Cu slab by electrons and protons as a function of radius and time is shown respectively in (c, f).

Comparison LSP/experimental data

These simulations reveal the dynamics of Cu slab heating induced by both electrons and protons, highlighting their respective contributions to the overall heating process. The simulated target heating is compared with experimental measurements obtained through the Bayesian analysis shown in Fig. 6. Specifically, Fig. 8(a) illustrates the temporal evolution of temperature for both the simulations and the experiment. The LSP simulations reveal two distinct phases: an initial electron-dominated heating phase within the first 15 ps, followed by a proton-dominated phase from 15 to 22 ps (corresponding to proton energies ranging from 4.5 to 2 MeV). Simulations were halted at 22 ps to limit computational costs and numerical noise. The agreement between simulations and experimental data, in terms of both timescale and temperature, confirms the heating dynamics driven by non-thermal electrons and TNSA-accelerated protons. Figure 8(b) presents the temporal evolution of the FWHM of the temperature profile. As previously discussed, for early times, while the single-temperature approach provided a reasonable representation, the Bayesian analysis yielded large uncertainties. Nonetheless, the FWHM from the simulation and Bayesian analysis are comparable throughout the time range covered by the simulation. Additionally, the simulated temperature converges within the significantly smaller error bars of the Bayesian analysis near the end of the simulation. It is worth noting that the measurements shown in Figs. 4 and 6 indicate that heating persists beyond  ~25 ps. This prolonged heating can be attributed to low-energy protons (<1.5 MeV) and higher-Z ions, typically carbon ions, accelerated by TNSA. Due to the shorter stopping range of carbon ions and low-energy protons, this late-stage heating will also manifest as a longitudinal temperature gradient within the Cu slab. Further insights into these contributions are provided in the next section, supported by additional simulations in Supplementary Fig. 1.

Fig. 8: Comparison of Cu slab temperature estimates obtained from LSP simulations and Bayesian inference of HiResSpec K_α data.

The average temperature and its standard deviation estimated from the K_α data for the SL HC configuration (see Fig. 6) are shown in (a) as a solid black line and a shaded gray area, respectively. The temperatures resulting from the energy deposition of electrons and protons in the LSP simulations are similarly depicted in a with red and green lines, respectively. The FWHM of the temperature profile is shown in (b), where values inferred from HiResSpec are represented by a blue line (average) with error bars (standard deviation), and LSP simulation values for protons are shown as a green shaded area corresponding to one standard deviation.

Late-time heating and contribution from heavier ions

From  ~25 to 50 ps we measured (Fig. 6) temperatures continuously rising from  ~50 to  100 eV with a transverse temperature gradient increasing from a FWHM of  ~30 μm to a FWHM of  ~50 μm, as well as maximum temperatures reaching up to 200 eV at the end of the heating. First, we note that this late-time heating cannot be solely attributed to energy deposition from low-energy protons (<1.5 MeV) as they would not carry sufficient energy and, moreover, would be readily scattered and deflected during their transport within the cone. Given that the Cu slab sample was positioned directly at the cone tip without a filter, it is likely that heavier ions from TNSA, such as carbon ions, significantly heated the first few microns of the slab at these later times. Therefore, while the assumption of a sole transverse gradient in the Bayesian analysis is valid up to t ~ 25 ps, due to the nearly uniform proton energy deposition along the longitudinal 10 μm depth of the target, the situation at later times is impacted by an additional longitudinal gradient resulting from carbon ion energy deposition. In other words, the maximum temperature and FWHM inferred from the Bayesian analysis at later times also increase due to the development of this longitudinal gradient localized near the front surface. Including a longitudinal gradient in the MCMC model introduced too many concurrent parameters which did not improve the inference of temperature gradients within the sample. Nonetheless, to further explore the formation of a longitudinal gradient, we conducted 1D radiation-hydrodynamics simulations using the HELIOS code. These simulations incorporated time-dependent energy deposition from both a proton beam and a carbon ion beam, with Maxwell-Boltzmann energy distributions based on TPIE and RCF measurements (see “Methods” section). The HELIOS simulations successfully reproduce the temperature evolution presented in this study and confirm the negligible contributions of low-energy protons as well as the longitudinal uniformity along the 10 μm Cu slab depth during proton heating, a longitudinal uniformity that was also observed in the 2D LSP simulations. On the other hand, by leveraging additional data from 25 μm thick Cu slabs and utilizing HELIOS simulations for both protons and carbon ions, we can provide insightful comments on the influence of the longitudinal gradient on the Bayesian analysis as a result of carbon ion energy deposition, as detailed in the Supplementary Discussion. Based on our understanding of temperature gradients and their origin, one could note that the diagnostic technique presented in this paper could achieve even greater precision in measuring WDM target conditions, along with improved temperature uniformity, by employing a filter to block heavier ions and reducing the transverse size of the target to dimensions comparable to or smaller than the proton beam size.

Discussion

We have performed a comprehensive temporal analysis of the isochoric heating of a copper slab induced by intense energy deposition from a TNSA-driven proton beam, which was focused using a cone-enclosed hemisphere target. By employing time-resolved X-ray spectroscopy, we directly measured temperatures within the copper slab exceeding 100 eV in less than 50 ps. In contrast, a proton beam accelerated from a simple flat foil resulted in slab temperatures remaining below 50 eV.

Moreover, through a Bayesian analysis of the experimental data, we could extract information on the temperature gradient developing within the Cu slab. To complement these measurements, we performed numerical simulations using the LSP code. The simulations reveal that nonthermal electrons contribute to guiding TNSA-driven protons by generating a focusing electrostatic field at the cone wall, in agreement with previous findings⁶. Upon reaching the Cu slab, these electrons heat the target ahead of the protons. Electron-driven heating dominates within the first 15 ps, with non-thermal electrons inducing more uniform (hence lower) heating than the focused TNSA-driven protons, due to their greater susceptibility to field deflection and scattering. The average temperature in the Cu slab rises rapidly with proton heating between 10 ps and 15 ps and reaches  ~50 eV at the end of the simulation and proton heating phase at t ~ 25 ps. The average temperature and the gradient scale inferred from LSP agree well with the measurements. At later times (t > 25 ps), the temperature and gradient scale inferred from measurements are influenced by the energy deposition of carbon ions, causing an increase in temperature near the front surface and introducing a longitudinal gradient. These contributions are discussed in light of additional simulations with the HELIOS code in the Supplementary Discussion.

Our findings highlight the potential of using intense proton beams generated by short-pulse lasers to achieve rapid isochoric heating of solid-density materials. Additionally, the use of a curved foil and cone structure to focus and guide the proton beam has proven to be an efficient method for attaining high heating rates. When combined with a Bayesian analysis, the temperature data, resolved every  ~2 ps, can also provide valuable insights into the heating uniformity. The potential of this diagnostic technique to characterize the temperature evolution within the bulk of dense targets heated to temperatures in the 20–100 eV range is considerable. For instance, by employing this technique instead of the surface temperature characterization currently used in proton stopping power measurements in WDM¹⁴, which relies on extensive absolute diagnostic calibration and surface-to-bulk extrapolation using radiation-hydrodynamic simulations, one could significantly reduce error margins and thereby better benchmark stopping power models. Enhancing the characterization of WDM conditions would represent a paradigm shift, greatly benefiting research efforts in nuclear fusion and laboratory astrophysics.

Methods

Experimental

Targets

The HC targets are composed of three main components: (i) a partial hemisphere, (ii) a cone structure, and (iii) an optional sample at the cone tip, in this case, a Cu slab. The partial hemisphere is created by cutting a polar cap from a Glow Discharge Polymer capsule, typically used as ablators in ICF experiments⁴⁶. The hemisphere has a radius of curvature of 430 ± 15 μm, a thickness of 15 ± 1 μm, with surface roughness ≤1 μm, a base outer diameter of 680 ± 50 μm, and a 52° cone half-angle. The cone is 3D-printed by General Atomics using an additive manufacturing technique called two-photon polymerization⁴⁷. The cone has a small inner diameter of 150 μm, a large diameter of 650 μm, a height of 267 μm, and a wall thickness of 30 ± 5 μm. The Cu slab is a laser-cut solid disk with a 100 μm radius and a thickness of either 10 μm or 25 μm. The cone and the Cu slab are both flash-coated with  <1 μm of Al. For the SL flat target, the Cu slab is a machined solid block with a thickness of 200 μm and lateral dimensions of 400 × 400 μm.

Proton beam diagnostics

The electron source used in the LSP simulations was adjusted to approximately match the proton beam energy distribution measured without a target at the cone tip. Specifically, the proton characteristic temperature in the experiment was inferred using RCF stacks and TPIE diagnostics, positioned at distances of 8 cm and 38 cm from the target along the proton beam axis in the SL and BL configurations, respectively. The RCF stack also enabled the estimation of the laser-to-proton conversion efficiency, which was used as an empirical input for initializing the LSP simulation. The RCF stack layers for which the dose was sufficient for measurement were of the Gafchromic^TM HD-v2 type, calibrated according to ref. ⁴⁸ (green color channel) using an Epson 11000 XL scanner cross-calibrated to the scanner used in that work via a step wedge filter comparison. The TPIE diagnostic employed a pinhole diameter of 100 μm, an electric field of 2 kV/cm, and a magnetic field of 1.6 kG to deflect ions along a parabolic trace based on their charge-to-mass ratio, as illustrated in Fig. 9(a). The BAS-TR image plate detector was placed at 10 cm from the pinhole, and its calibrated response to protons was taken from ref. ⁴⁹. In Fig. 2(a), we present the proton energy distribution extracted from the TPIE for a shot in the BL configuration without a target at the cone tip, with the measured characteristic proton temperature being 5.7 ± 0.2 MeV. The dose deposited on the RCF layers is shown in Fig. 9(b), and fitting this dose to a MB distribution [$f(E)\propto {E}^{0.5}{T}_{p}^{-1.5}\exp (-E/{T}_{p})$], using the Stopping-Range-In-Matter (SRIM) model for proton stopping power⁵⁰, yields a proton characteristic temperature T_p of 4.7 ± 0.5 MeV and a laser-to-proton conversion efficiency of 1.7 ± 0.44% for protons with kinetic energies greater than 1 MeV. The quality of this fit is shown in Fig. 2(b), comparing the dose from the MB distribution to the measured dose in each RCF layer. The errors mentioned above are the standard deviations between the fit and the actual measurements, calculated using a covariance analysis.

Fig. 9: Proton beam measurements along the beam axis without a target at the cone tip.

a Raw ion traces on the BAS-SR image plate detector used in the Thomson parabola (TPIE) diagnostic, in photostimulated luminescence (PSL) units. b Dose deposited on the Gafchromic^TM HD-v2 layers in J/kg units.

Modeling

PrismSPECT

Temperature estimation, whether using the single-temperature or Bayesian approaches, was based on synthetic X-ray spectra calculated with the collisional-radiative code PrismSPECT. For these calculations, we performed steady-state non-LTE plasma simulations at a fixed solid density of 8.96 g/cm³, with plasma temperatures ranging from 1 eV to 260 eV. To model the interaction of TNSA-driven particles with the Cu plasma, we employed a dual Maxwellian model. The first Maxwellian distribution represents the bulk Cu plasma, while the second models the nonthermal electrons. These nonthermal electrons were initialized at a temperature of 1 MeV with a total density of 1% of the bulk Cu sample density. We examined the impact of these two key parameters of the nonthermal electron distribution on the K_α spectra. The results indicate that these parameters primarily influence the intensity of the K_α signal, with minimal changes in the shape of the spectra, including the shift and ratio between the K_α1 and K_α2 lines. X-ray spectrum calculations from nonthermal protons are not available in the current released version of the code. To validate the method used in this work using nonthermal electrons, we compared the effects of nonthermal protons and nonthermal electrons in solid-density manganese plasma using an unreleased version of PrismSPECT. The comparison revealed no significant differences in the shapes or shifts of the K_α spectra between the two cases across the range of bulk temperatures examined in this work.

LSP

We employed the hybrid-PIC code LSP to investigate the kinetics of proton acceleration and transport through the main target geometry, which includes a curved foil, a cone structure, and a rear foil. The simulations were conducted in 2D with cylindrical coordinates, at scale, based on the target dimensions detailed in the Targets “Methods” section. By utilizing a hybrid-PIC scheme with an implicit push algorithm and a fluid description for the solid regions of the target, we significantly relaxed the required simulation resolution and macroparticle-per-cell limits compared to standard PIC simulations⁵¹, enabling us to perform large-scale simulations within a 700 × 400 μm simulation box. The particles treated as fluid in simulations follow the same equations of motion as kinetic particles, but with an added collision term incorporating pressure and frictional forces⁵¹. Simultaneously, the particles responsible for the TNSA mechanism—specifically injected hot electrons and surface protons—were treated kinetically, fully resolving the TNSA process. To address the challenge of large-scale and multi-ps time-scale modeling within limited computational resources, we bypassed the laser-target interaction process by injecting a population of electrons to emulate the effects of laser interaction. The electron source was constructed using methods from previous work^6,52, taking into account time-dependent electron energy distribution and current density (electron number). The electron source with a total energy of 15% of the laser energy (900 J) was injected into the front surface of the curved foil with a Gaussian spot size of 20 μm radius for a duration of 10 ps following the laser Gaussian temporal profile. The 15% conversion efficiency from laser to electron was adjusted to approximately match the total proton energy in LSP with measurements from proton beam diagnostics conducted in the experiment without a target at the cone tip (see the Proton beam diagnostics Methods’ section). Specifically, the total proton beam energy in LSP for protons with kinetic energies greater than 1 MeV is 14.4 J, compared to the 15.6 ± 4.1 J measured in the experiment. The time-dependent energy spectrum featured an electron source with a slope temperature increasing over time, reaching 5 MeV at the laser peak, with a 30% Gaussian energy spread in the transverse direction. The electron temperature was determined based on previous studies of electron acceleration in kJ-scale, multi-picosecond laser interactions, which have shown that interaction with developed underdense plasma leads to significantly higher electron temperatures than predicted by ponderomotive scaling^52,53. The peak temperature of 5 MeV employed in the electron source also closely aligns with the Pukhov scaling⁵⁴. This approach of injecting an electron source to simulate laser-driven TNSA in PIC simulations is widely employed^55,56,57, offering good accuracy while enabling large-scale, multi-picosecond interactions with significantly reduced computational requirements and costs. We periodically recorded longitudinal and lateral electric field maps, along with densities, and temperatures. In the first simulation, all particles (protons, hot electrons, and co-moving electrons) that approached the rear foil (z = 400 μm) were recorded every time step. These time-dependent particles were then input into a separate transport simulation, with injection at the front of a Cu slab, preserving each particle’s momentum, lateral position, and relative timing. This method enables multiple transport simulations without the need to re-simulate the entire transport of particles from the hemispherical target. It also allows for the individual injection of either electrons or protons into the Cu sample, facilitating studies of target heating effects from each particle species. Particle stopping calculations in the slab utilized Atzeni’s approach for relativistic electrons⁵⁸, a bound plus free electronic stopping model for protons⁵⁹, and PIXE cross sections^44,45 for electron and proton-induced Cu K_α emission.