Maximum likelihood estimator

The energy distribution of scattered photons is well described by a Voigt function V γ,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 {E i }, 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 ΔE therm can then be found from the refined σ heated value and the instrument function σ IF (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 {E i }, 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.