Main

The development of short-pulse lasers has been instrumental in bypassing the intermediate destabilizing events that occur during superheating and has allowed experiments to approach the entropy catastrophe threshold4,5,6. Experiments using heating rates exceeding 1014 K s–1 have achieved superheating within the range of T ≈ 1.4Tm – 2.1Tm, where Tm is the melting temperature, nearing but not fully reaching the anticipated limit7,8,9,10,11. Notwithstanding the advancements in heating rates, a direct, model-independent method for accurately measuring ion temperatures in hot, solid-density systems has remained elusive, which constitutes a major roadblock to drawing precise and reliable conclusions about superheating. Indeed, earlier methodologies often relied on indirect measures of ion temperature through analysis of material structure or energy absorption12. In response to this challenge, we utilize inelastic X-ray scattering in the backscattering regime to directly ascertain ion velocity distributions and thus determine the ion temperatures.

Leveraging this pioneering technique, we show that gold subjected to short-pulse laser irradiation unambiguously reaches heating rates exceeding 1015 K s–1, propelling the ions to temperatures of T ≈ 14Tm without losing the solid structure. This measurement not only surpasses the previously predicted bounds of the entropy catastrophe but also suggests a much higher threshold for the superheating of solids, thereby rewriting the fundamental understanding of the stability of the solid phase under extreme conditions.

Measuring the lattice temperature

We use high-resolution inelastic X-ray scattering to accurately measure ion temperatures in gold samples subjected to intense laser heating. Under these conditions, photons scattering off the ions experience an energy shift similar to a Doppler shift, enabling us to determine the velocity distribution and, consequently, the temperature from the resulting spectral broadening of the incident X-rays.

Scattering from phonon modes in crystals is a well-established technique that, in polycrystals, provides information on the phonon density of states13. In our backscattering geometry, the spectrum transitions to a Gaussian profile, a behaviour that emerges in the multiphonon limit at high momentum transfers and temperatures well above the Debye temperature14,15 (see Methods for details). Moreover, the phonon–phonon equilibration rate is an order of magnitude higher than the electron–phonon equilibration rate in gold, allowing us to ascribe a single temperature (Ti) to the phonon subsystem16. As a result, the Gaussian profile has a full-width at half-maximum (FWHM) of ΔEtherm, which determines the temperature from the average velocities of the ions. This relationship is given by

$${T}_{{\rm{i}}}=\frac{1}{{k}_{{\rm{B}}}}\frac{1}{4}\frac{{m}_{{\rm{i}}}{c}^{2}}{8\,{\rm{ln}}\,2}{\left(\frac{\Delta {E}_{\text{therm}}}{{E}_{0}}\right)}^{2},$$
(1)

where mi is the mass of the ion, c is the speed of light, kB is the Boltzmann constant and E0 is the X-ray energy. Notably, this form matches that of a classical gas with a Maxwell–Boltzmann distribution17,18 and is independent of the interatomic potential or the phonon density of states.

The energy shifts of the scattered photons in equation (1) are very small, with a broadening of just approximately 25 meV predicted for gold at a temperature of 1,000 K. In addition, samples are typically very short-lived when heating rates exceed 1015 K s−1. Consequently, measuring the broadening owing to the ion motion necessitates a high-brightness X-ray source paired with a high-resolution spectrometer. Recent advances at the Matter in Extreme Conditions (MEC) instrument19,20 at the Linac Coherent Light Source (LCLS), along with spectrometer technologies developed at the European Synchrotron Radiation Facility21, have enabled experiments with instrument functions of about 50 meV (refs. 22,23,24) and an X-ray probe of 40 fs duration.

The experimental configuration is illustrated in Fig. 1. We used a 50-nm-thick polycrystalline gold sample mounted on a nickel grid irradiated by a 45-fs, 400-nm wavelength short-pulse laser from the MEC facility. The laser was focused to a spot approximately 100 µm in radius with resulting fluences of 1.9 J cm−2 to 4.9 J cm−2. A calibrated camera outside the vacuum chamber recorded the laser fluence during each shot. The LCLS delivered an X-ray beam with a photon energy of 7,491.9 eV, monochromated by a 4-bounce channel-cut Si-533 monochromator set at a Bragg angle of 87.5°. The X-ray beam was focused to a FWHM of 5 µm and aligned to the centre of the optical beam.

Fig. 1: Diagram of the experimental set-up.
figure 1

X-rays produced by the LCLS in the seeded operation mode (E0 = 7,491.9 eV, ΔE ≈ 0.50 eV) are passed through a 4-pass Si (533) monochromator, yielding a total bandwidth of about 32 meV. The X-rays are then used to probe a 50-nm-thick gold foil that is supported on a nickel grid. The primary diagnostic of the probe involves a spectrometer comprising three Si (533) diced analysers and an X-ray camera positioned to capture scattering at an angle of about 170°. A large-area detector (ePix10K) is used to measure the diffraction signal from the (111) and (200) diffraction lines of solid gold.

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The scattered photons were detected using three spherically bent Si(533) diced-crystal analysers positioned to cover scattering angles between 167° and 173°, with the Bragg reflection recorded by an Epix-100a detector. The system, including analysers and detector, was arranged on a 1-m-diameter Rowland circle. In addition, an Epix10k detector captured wide-angle X-ray scattering from 20° to 60°, enabling Laue diffraction analysis of the (111) and (200) planes of gold. During the experiment, we dynamically adjusted the timing between the optical heater and the X-ray probe from −1 ps to 6 ps, approximately the melt time of the sample, and utilized the MEC time tool to measure the on-shot delay between the X-ray and optical lasers25.

The inherently low photon count per shot necessitates single-photon counting techniques, where each detected photon is converted to energy using the dispersion curve of the analysers. Multiple shots are then aggregated to construct a comprehensive spectral profile. The instrument function was determined by scattering from a room-temperature gold target, yielding the data in Fig. 2c. This spectrum was created from 7,486 photons accumulated over 749 shots. As the (422) diffraction peak of gold coincides with the analysers, the resulting spectrum predominantly reflects the elastic zero-phonon line, thus serving as a faithful representation of the instrument function. In this case, the resulting spectrum closely fits a Voigt profile, where the parameters γ and σ were determined using a maximum likelihood estimator and define the width of the Lorentzian and Gaussian components, respectively (see Methods for details). The FWHM of 44.7 meV is in agreement with previously measured widths22,23 and is close to theoretical limits24.

Fig. 2: Illustration of method and example scattering spectra.
figure 2

a,b, An illustration of photon scattering from a cold sample (a) and a heated sample (b). Each X-ray pulse exposes the target to approximately 1010 photons at a central energy of E0 = 7,491.9 eV within a narrow bandwidth (ΔE/E0 = 4.3 × 10−6). Approximately ten scattered photons are detected from each pulse. c, The energy distribution of scattered photons collected by the high-resolution X-ray spectrometer for a cold sample without a pump laser (purple, 7,486 photons) and a heated sample 3 ps after laser irradiation (red,  569 photons) at an incident optical fluence of 4.9 ± 0.5 J cm−2. The distribution from the cold sample, representing the instrument function (IF) of the spectrometer, is well fitted by a Voigt profile with γIF = 9.5 meV and σIF = 14.2 meV, resulting in a FWHM of 44.7 meV. For the heated case, the distribution is also well fit by a Voigt profile with γheated = 9.5 meV and σheated = 47.2 meV, corresponding to a FWHM of 121.6 meV and an ion temperature of 19,000 K. Details of the fitting method can be found in Methods.

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Laser heating of the target causes an increase in ion velocity, leading to Doppler broadening of the spectral lines, as seen by the red distribution in Fig. 2c. Unlike room-temperature experiments, the intense heating from each optical laser pulse destroys the target, necessitating a fresh gold sample for each measurement. This requirement slows the data collection process and, combined with time and target material constraints, limits the number of achievable shots. From our observations, approximately 300 detected photons from about 60 shots are sufficient to produce reasonable spectra and associated error estimates. The spectrum of heated gold, illustrated in Fig. 2c, shows considerable broadening. This broadening is quantitatively analysed using a maximum likelihood estimator, allowing for temperature calculations based on equation (1). Confidence intervals for the broadening, and thus the temperature estimates, are derived through rigorous bootstrapping (see Methods for details).

The result above represents a direct measurement of the ion temperature under high-temperature conditions at solid density, where only indirect methods have been used previously. The straightforward relationship between spectral broadening and temperature in equation (1) capitalizes on the large momentum transfer and elevated temperatures achieved through our backscattering geometry and extreme heating rates. These advancements have transformative potential for high-pressure and high-energy-density research, addressing a long-standing challenge in accurately determining temperature, which has often been a weak point in such studies26,27,28. In this work, we demonstrate its capability by investigating the ultimate stability limits of superheated materials.

Lattice temperature evolution

We directly measured the ion temperature evolution in gold after laser irradiation by varying the delay between the optical pump and the X-ray probe. The optical pump generates ballistic electrons, which volumetrically heat the thermal electrons on a subpicosecond timescale29. Any inhomogeneities in electron heating are rapidly smoothed out by heat conduction of the thermal electrons, given their high thermal conductivity and subpicosecond diffusion timescale30,31 (additional details in Methods). Rapid lattice heating occurs via electron–phonon energy transfer, with hot electrons serving as a thermal reservoir. The energy exchange rates are roughly proportional to the temperature difference, meaning that the energy transfer dynamics can be approximately controlled by the laser fluence. However, this information is not used in our analysis, as our approach allows for direct measurement of ion temperatures, bypassing the complexities of calculating electron–phonon energy exchange.

To allow us to accumulate multiple shots at the same conditions, we carefully measure all relevant experimental parameters—including on-target laser fluence, and the delay between the X-ray and optical pulses. On the basis of laser fluence, the data were divided into two categories, each representing a different heating rate. The lower-fluence scenario, depicted in Fig. 3a, encompasses fluences in the range of 1.9 ± 0.5 J cm−2, resulting in a heating rate of approximately 3.5 × 1015 K s−1; the higher-fluence scenario, shown in Fig. 3b, represents fluences in the range of 4.9 ± 0.5 J cm−2, achieving a heating rate of approximately 6 × 1015 K s−1.

Fig. 3: Ion temperature evolution.
figure 3

a,b, The measured ion temperature of a gold foil, shown by symbols with error bars, after irradiation with a laser fluence of 1.9 ± 0.5 J cm−2 (a) and 4.9 ± 0.5 J cm−2 (b). Error bars for the broadening are derived through a bootstrap resampling method (see Methods for details). These fluences result in heating rates of 3.5 × 1015 K s−1 and 6 × 1015 K s−1, respectively (dashed line). Overlaid are the intensities of the (111) and (220) diffraction peaks whose disappearance can be used to identify the time of melting. Higher intensity is shown in blue. The white arrows indicate the highest temperature at which diffraction lines from a solid are observed. In both cases, the temperature of the superheated solid greatly exceeds the previously proposed limit of three times the melting temperature, shown by the horizontal dashed line.

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The wide-angle X-ray detector, processed using the Dioptas software package32, clearly displays the (111) and (200) Debye–Scherrer rings for time delays of up to 3 ps. The evolution of their azimuthally integrated diffraction intensities is included in Fig. 3. As the ions in the sample heat up, they vibrate increasingly around their equilibrium positions, reducing the intensity of these rings as predicted by Debye and Waller33, with the (200) peak decaying faster than the (111) peak. Upon melting, the sample loses its long-range order, causing the Debye–Scherrer rings to disappear completely. We determine the time of melting from this visible loss of structure. At the heating rate of 3.5 × 1015 K s−1, we are able to heat the gold to a temperature of 13,800 ± 3,200 K, corresponding to about 10Tm. At the higher heating rate of 6 × 1015 K s−1, samples were heated to a temperature of 19,000 ± 4,000 K, corresponding to about 14Tm. Notably, the temperatures exceed the proposed limit of 3Tm in both cases for over 2 ps. This time is approximately an order of magnitude longer than the characteristic phonon oscillation period34 and, thus, much longer than required for homogeneous melting35,36.

Discussion

The prediction of the entropy catastrophe is based on the thermodynamic principle that the entropy of a liquid cannot be lower than that of a solid with identical enthalpy1,37,38. Historically, the analysis of solid and liquid entropies identified two critical temperatures where extrapolated liquid entropy meets crystal entropy, beyond which thermodynamics is violated. The lower temperature, known as the Kauzmann temperature, necessitates a glass transition, triggering the sudden freezing into a crystalline phase38,39,40. The higher temperature, typically at 3Tm and termed the entropy catastrophe temperature, requires a sudden onset of melting. These temperatures delineate the proposed ultimate stability limits for superheating and supercooling.

Figure 4 illustrates the entropy profiles for the liquid and solid phases of gold, as originally proposed by Fecht and Johnson1, who derived the entropy from the available heat capacities via

$$\Delta S={\int }_{{T}_{{\rm{m}}}}^{T}\frac{{C}_{\text{liq}/\text{sol}}}{{T}^{{\prime} }}{\rm{d}}{T}^{{\prime} }.$$
(2)

The heat capacity of the liquid phase has been measured41 as Cliq = 32.97 J mol−1 K−1. For the solid, one finds Csol = 3R + γT, where the coefficient R refers to the gas constant and γ = γexp + γele = 6.6 × 10−3 J mol−1 K−2 incorporates changes owing to the thermal expansion and heating of the electronic component, with expansion clearly dominating41,42: γexp ≈ 10γele. Again, the crossing point occurs at T ≈ 3Tm, a limit we have surpassed by far.

Fig. 4: Calculated entropy for superheated gold.
figure 4

The calculated entropy is shown versus temperature for the solid (blue) and the liquid (red) phases. The results for the solids are calculated in two ways. The light-blue dashed curve shows the entropy including the terms for lattice expansion and changing electronic heat capacity. The dark-blue solid curve does not include these effects, corresponding to rapidly heated gold. For reference, the markers indicate the calculated entropy corresponding to the temperatures we measured in the solid material for the low-fluence (triangles) and high-fluence (squares) cases. The vertical dashed line denotes the originally proposed superheating limit1. Clearly, intense, ultrafast heating without sufficient time for lattice expansion can avoid the entropy catastrophe.

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The superheated states we have observed at temperatures far above the limit of the entropy catastrophe should not exist according to the calculation above (blue dashed line in Fig. 4). Indeed, if such a superheated system were to melt in isolation, it would undergo a spontaneous adiabatic process with a decrease in entropy, in direct violation of the second law of thermodynamics. To resolve this contradiction, we note that the crystal lattice does not undergo appreciable expansion or ablation within the timescales of our experiment. This fact is evident by the fixed location of the Bragg peak in Fig. 3 and is consistent with frequency-domain interferometry results observed in laser-excited gold43,44 (additional details in Methods). Furthermore, the entropy associated with the electronic heat capacity, which already had only a minor role, remains approximately constant across the liquid and solid phases owing to the high electron temperature, which remains relatively stable. Thus, we find γele ≈ 0 and γexp ≈ 0 for the situation created in our experiment. Applying these approximations in equation (2), we obtain the solid-blue line in Fig. 4. This entropy for the solid remains below the liquid entropy for all temperatures, allowing us to surpass the previously proposed limit as long as the heating occurs fast enough to preclude expansion. In essence, the crossing of the two entropy curves is effectively eliminated by ultrafast intense heating, indicating that superheating may not have an upper bound.

Our experimental findings raise an important question about the ultimate stability limit for superheating. In the regime explored, the heat capacity of the solid is governed by universal constants, and its general form remains insufficiently characterized for the liquid45. Therefore, it is unknown whether the entropies of solids and liquids cross at higher temperatures. Moreover, heat capacity variations in non-equilibrium systems and thin films could further alter the entropy balance. Consequently, it is uncertain whether arguments based on the crossing point of entropy curves, similar to the lower-temperature Kauzmann limit, apply here. Nevertheless, our experiments clearly demonstrate that the previously proposed limit of superheating can be exceeded by far if the material is heated fast enough.

Methods

Maximum likelihood estimator

The energy distribution of scattered photons is well described by a Voigt function Vγ,σ(E), which is the convolution of a Gaussian distribution with standard deviation σ and a Lorentzian (Cauchy) distribution with width parameter γ. In this study, we determined all Voigt profiles using a maximum likelihood estimator. Specifically, we maximized the log-likelihood function for the parameters σ and γ, given the observed photon energies {Ei}, where i = 1, 2, …, N and N is the number of detected photons:

$$\log {\mathcal{L}}(\gamma ,\sigma )=\mathop{\sum }\limits_{i=1}^{N}\log [{{V}}_{\gamma ,\sigma }({E}_{i})].$$

For practical application, we employed the maximum likelihood estimator to estimate the σ value of the Voigt profile from the experimental data while keeping γ fixed to the value determined for the instrument function (that is, the known broadening due to the instrument). The thermal broadening ΔEtherm can then be found from the refined σheated value and the instrument function σIF (the standard deviation of the Gaussian component for the cold sample) using:

$$\Delta {E}_{\text{therm}}=2\sqrt{2\,\text{ln}\,2}\sqrt{{\sigma }_{\text{heate}{\text{d}}^{2}}-{\sigma }_{\text{I}{\text{F}}^{2}}}.$$

An example of this application is shown in Fig. 2c, where the experimental data (appropriately binned) are well fitted by the Voigt function with parameters estimated via the maximum likelihood estimator. To assess the reliability of these parameters, we applied bootstrapping methods to generate confidence intervals. By randomly drawing N samples from the observed distribution of photon energies {Ei}, we recalculated the parameters over approximately 1,000 iterations. This process allowed us to construct a distribution for σ, from which we derived the bias and confidence intervals of the estimator. Our analysis confirms that the bias is minimal relative to the width of the confidence intervals for the typical sample size of N ≈ 300 measured photons, and we corrected the results for any detected bias. The 1σ confidence intervals for these measurements are illustrated in Fig. 3.

Applicability of the broadening–temperature relation

Equation (1) in the main text remains applicable in this regime because we are effectively sampling a sufficiently large number of independent scattering events. At large wavenumber Q, the measurement probes individual interactions, and by the central limit theorem, the resulting distribution of energy transfers leads to a well-defined spectral width that is independent of the underlying microscopic details. To model the scattering process more rigorously, we employ a multiphonon expansion14. In this approach, the scattering is treated as a Poisson distribution over multiphonon events, where the contribution from scattering by n phonons is constructed via n convolutions of the single-phonon scattering spectrum, as described in several sources15. Specifically, in our implementation, we approximate the single-phonon scattering spectrum at large Q as g(E)/E2, where g(E) is the generalized phonon density of states applicable in this regime13. As shown in Extended Data Fig. 1, we compare both approaches and find excellent agreement above 2,000 K, reinforcing the validity of equation (1) in this regime. This consistency further supports the simpler formulation used in the main text while ensuring that the key physical effects are accurately captured.

Heating uniformity in 50-nm gold

Understanding the uniformity of electron heating is crucial for our analysis. Here we provide details supporting the assumption of uniform heating within our 50 nm gold samples. The primary factor is the penetration depth of the ballistic electrons used to heat the target, which has been measured to be approximately 100 nm (ref. 29). This ensures that our 50-nm sample is homogeneously heated. In addition, any residual inhomogeneities left by the ballistic electrons will be rapidly smoothed out by heat conduction from thermal electrons, further ensuring a uniform temperature distribution. The electron thermal conductivity at these conditions is between30 3,000 W K−1 m−1 and 5,000 W K−1 m−1, and the electron heat capacity is approximately31 0.3–0.5 J g−1 K−1. From these values, we estimate the thermal diffusion time across the 50-nm sample to be on the order of 0.5 ps. Finally, as additional evidence, we note that previous studies in which gold was heated to energy densities comparable to our lowest-fluence case observed similar expansion and dissociation dynamics in 35-nm films44. This agreement suggests that our slightly thicker targets experience comparable heating conditions and are unlikely to exhibit appreciable temperature gradients.

Wide-angle X-ray detector calibration

All X-ray diffraction patterns were processed using the Dioptas software package. For calibration, polycrystalline LaB6 was used to establish the relationship between pixel positions on the detector and the corresponding scattering angles, as illustrated in Extended Data Fig. 2. To correct for variations among the detector pixels, we subtracted a dark-field image from the dataset. In addition, we corrected for the varying solid angle coverage of each pixel relative to the scattering target. Transmission corrections were applied to account for attenuation due to the layers of 50-µm aluminium and 125-µm polyimide present in the beam path. The data processing also incorporated a polarization correction factor, P(θ, ϕ) = 1 − sin(2θ)cos(2ϕ), where θ is the scattering angle and ϕ is the angle between the plane of the scattering vector and the polarization direction of the incoming X-rays.

Optical laser fluence

To vary the laser fluence in the experiment, we used an iris before the final focusing optics, enabling control over the two laser fluence levels discussed in this work. We calculated the absolute laser fluence for each shot, allowing us to include only the scattered X-ray photons corresponding to specific fluence bands in our analysis. We first measured the laser energy for each shot and accounted for spatial jitter using a calibrated equivalent plane camera. The laser intensity profile was determined by combining these measurements with images of the laser spot taken in the focal plane, as shown in Extended Data Fig. 3. Knowing the precise location of the X-ray beam relative to the laser spot allowed us to calculate the fluence at the point of X-ray interaction for each shot. The X-ray beam position was periodically updated using a yttrium-aluminium-garnet crystal screen, which provided the pixel location of the X-ray beam on the detector. This calibration ensured accurate alignment between the laser spot and the X-ray probe, allowing precise calculation of the laser fluence at the interaction point.

Inelastic X-ray scattering spectrometer dispersion calculation

The diffraction from the (533) plane of each of the three high-resolution silicon diced-crystal analysers was found to lie along an arc with a radius of 1,745.5 pixels (87.3 mm) on the detector (Extended Data Fig. 4). Considering a pixel size of 50 µm, we determined the incident X-ray energy to be 7,491.9 eV and calculated an energy dispersion of 8.187 meV per pixel, or equivalently, 0.164 meV per µm.

Determination of target density

In this work, the diffraction of the Bragg peaks was collected simultaneously with the temperature measurements, enabling experimental determination of the melting point and target expansion. Extended Data Fig. 5a,b illustrates the decay of the (111) Bragg peak for the two fluence cases examined in this study. The decay of the Bragg peak over time reflects the increased thermal motion of atoms around their lattice sites. For both cases, the last solid time delay, as discussed in the main text, was defined by the final delay at which the (111) Bragg peak remained visible, specifically at 3.4 ps and 2.95 ps for the low- and high-fluence cases, respectively. In addition, Extended Data Fig. 5c,d highlights the variation in the position of the peak centre over time. The absence of a shift in the peak centre with time delay indicates that the target density remains constant up to the melting point.