White dwarfs represent the final state of evolution for most stars1,2,3. Certain classes of white dwarfs pulsate4,5, leading to observable brightness variations, and analysis of these variations with theoretical stellar models probes their internal structure. Modelling of these pulsating stars provides stringent tests of white dwarf models and a detailed picture of the outcome of the late stages of stellar evolution6. However, the high-energy-density states that exist in white dwarfs are extremely difficult to reach and to measure in the laboratory, so theoretical predictions are largely untested at these conditions. Here we report measurements of the relationship between pressure and density along the principal shock Hugoniot (equations describing the state of the sample material before and after the passage of the shock derived from conservation laws) of hydrocarbon to within five per cent. The observed maximum compressibility is consistent with theoretical models that include detailed electronic structure. This is relevant for the equation of state of matter at pressures ranging from 100 million to 450 million atmospheres, where the understanding of white dwarf physics is sensitive to the equation of state and where models differ considerably. The measurements test these equation-of-state relations that are used in the modelling of white dwarfs and inertial confinement fusion experiments7,8, and we predict an increase in compressibility due to ionization of the inner-core orbitals of carbon. We also find that a detailed treatment of the electronic structure and the electron degeneracy pressure is required to capture the measured shape of the pressure–density evolution for hydrocarbon before peak compression. Our results illuminate the equation of state of the white dwarf envelope (the region surrounding the stellar core that contains partially ionized and partially degenerate non-ideal plasmas), which is a weak link in the constitutive physics informing the structure and evolution of white dwarf stars9.
At pressures of a million to a billion times that of the Earth’s atmosphere (megabar to gigabar), core electrons can affect the compressibility of stellar material by altering both the amount of stored internal energy and the plasma pressure through ionization. At these pressures, the excitation energies of the core electrons and the amount of ionization are difficult to predict10. Depending on how these details are modelled, the equation of state (EOS) along the shock Hugoniot (for example, pressure versus compression in this work) can vary by nearly 10%11, which is important for white dwarf modelling12. Benchmarking EOS models for a hydrocarbon is directly relevant to the modelling of white dwarf stars of the hot DQ class that is characterized by a degenerate carbon and oxygen core surrounded by an envelope of mostly carbon13,14. Uncertainty in EOS and opacity models can affect the extent and properties of mixing within the convection zone, affecting the inferred surface abundances and pulsation properties of pulsating white dwarf stars with carbon-rich envelopes1,13,14,15 (see Methods). In addition, instability growth in inertial confinement fusion experiments in the laboratory7 has been shown to be sensitive to EOS modelling16,17, with more compressible models leading to higher instability growth rates. Until now, no laboratory measurements of the shock Hugoniot have been reported at pressures exceeding ~60 Mbar for any material, and theoretical models in this regime have not been constrained by experimental data. A single dataset at gigabar pressures recorded in underground nuclear explosions for aluminium18 has uncertainty that is too large to distinguish between theoretical models.
We present laboratory EOS measurements in a regime where none currently exist, enabling tests of theoretical models at pressures exceeding 100 million atmospheres. We access the atomic pressure regime >300 Mbar (see Methods), in which extreme temperatures and densities ionize inner-core electron orbitals and can change the shape of the pressure-versus-density function (P−ρ) along the Hugoniot states. We resolve the shape of the Hugoniot response by measuring a continuous sequence of data points in a single shot, using a spherically convergent geometry19,20,21,22. The combination of small scatter and precision of the data effectively constrains EOS models. Alternative techniques such as planar dynamic compression experiments23,24 are restricted to pressures of <60 Mbar and can only access one Hugoniot point in a single shot. Accessing the shock regime in this work required us to increase the laser driver energy relative to that of ref. 19 and use multiple coalescing shock waves to generate a single stronger shock. Operating in an intense X-ray drive environment required mitigation of sources of preheat and background from electrons and X-rays generated from the radiation drive, as well as symmetric shock generation. To interpret the experimental measurements in this high-pressure regime, in which the opacity at the shock front deviates from the cold opacity, we developed a simultaneous density and opacity analysis method25 (see Methods).
These experiments were performed at the National Ignition Facility (NIF)26 where 1 MJ of 351-nm laser light was delivered to the inside of a gold cylinder (hohlraum) creating an X-ray radiation bath with a maximum radiation temperature (Tr) of 294 eV (nearly 3.5 × 106 K); see Fig. 1. The X-ray drive is absorbed by a spherical sample mounted in the centre of the hohlraum, in an outer region called the ablator. The ablator heats and expands, which launches inwardly converging shock waves via the rocket effect towards the centre of the solid sphere. The shocks coalesce into a single stronger shock reaching near-gigabar pressures at radii of ~100 μm. We measure the Hugoniot at the shock front as it travels inward, where the shock-front pressure continuously increases owing to convergence. The shock travels inward faster than the converging material behind it, enabling continuous tracking.
The density and pressure at the shock front are determined from temporally and spatially resolved streaked X-ray radiography measurements. The shock-front speed is determined from the radiograph by tracking the shock-front radial position as a function of time21. The radial density profile was extracted from the transmission of a 9-keV X-ray source (back-lighter) and analysed by profile matching20. At these pressures, the opacity decreases relative to the cold opacity as a result of carbon K-shell ionization. The opacity is simultaneously unfolded from the measured back-lighter transmission25. The mass density is further constrained by conserving the total mass contained within a higher-opacity fiducial layer which is visible in the radiograph (radiographic Ge marker layer; see Fig. 1b) and by matching the density profile to the known density of the unshocked material ahead of the shock. Using Hugoniot jump relations, we calculate the pressure, P, from the measured shock-front speed, D, and the mass density, ρ (refs. 21,25):
where ρ0 is the initial mass density and P0 is the initial pressure of the unshocked material (see Methods).
The experimental data are shown in Fig. 1d, where the shock front is labelled on the image. The shock travels at speeds of up to 150–220 km s−1, traversing the 1-mm sample in ~9 ns. We verify that the shock converges spherically by measuring the shock-front symmetry to be within the radiographic resolution of 25 μm at a shock radius of 200 μm through tracking of the radial density profiles on both sides of the sample in the equatorial plane. We also determine the symmetry through imaging of X-ray self-emission at ~9 ns when the shock wave collapses at the centre of the sample (shock flash). The symmetry of the shock flash was measured using penumbral imaging27 to be within 0.5 ± 0.3 μm at a shock radius of 12 μm in the equatorial direction.
Figure 2 shows the measured opacity at the shock front normalized to the cold material opacity (red curve) as a function of shock-front pressure. Here, the red shaded region and outer red lines correspond to uncertainty contours of ±1σ in the measurement. Also shown are calculations of the normalized opacity with and without trace amounts of fluorine present in the samples (see Methods), using detailed configuration accounting (DCA)28 (black curves). Calculations of the carbon K-shell (inner shell) occupation fraction by DCA (blue circles) show that the calculated drop in opacity is directly correlated to the K-shell occupation, or ionization of the carbon K-shell. The measured opacities in these experiments are consistent with the modelling at pressures up to ~300 Mbar and slightly higher than theory at higher pressures. The measured drop in opacity indicates distortion of the carbon inner shell with ~63% occupation at 450 Mbar.
The measured sequence of data points along the shock Hugoniot is shown in Fig. 3 (red curve) with errors that correspond to uncertainty contours of 1σ (red shaded region). Also shown are previous shock Hugoniot measurements19,29,30,31, theoretical calculations made with molecular dynamics based on Kohn–Sham density functional theory (KS-DFT)32 (orange curve), and models using an average-atom (AA) single ion-in-jellium description with the electronic structure based on Kohn–Sham density functional theory (AA-DFT)33,34,35,36 (solid black curve) and on Thomas–Fermi–Dirac theory (AA-TFD)37 (dot-dashed black curve); see also Methods. These models are commonly used to generate EOS tables for, inertial confinement fusion experiments, for example.
The new extreme-pressure measurements of the P–ρ Hugoniot presented here access the Hugoniot structure associated with the ionization of the carbon core (K-shell) electrons. The shape of the Hugoniot in the range ~100–1,000 Mbar is a result of the ionization of the core electrons of carbon (hydrogen being fully ionized under these conditions). First, ionization absorbs energy from the shock, and the material becomes more compressible by an amount that depends on the interplay between the ionization energy and interactions in the plasma. Then, at higher pressures, the curve swings back to lower compression, owing to reduced conversion of shock-wave energy to internal degrees of freedom. EOS models that include the electronic shell structure (for example, AA-DFT) show a sharper bend in the Hugoniot at these pressures and higher maximum compression than models that lack electronic shells (for example, AA-TFD) and are in better agreement with the measurements. Path-integral Monte Carlo calculations are in general agreement with AA-DFT for hydrocarbons11 and thus also accurately predict the shape of the high-pressure Hugoniot and maximum compressibility for this hydrocarbon.
We provide data that access the conditions deep in the convection zone of hot DQ white dwarf stars. Figure 4 shows a sequence of interior models for the cooling of white dwarfs with a carbon–oxygen core and pure carbon envelope. As the star cools, its structure moves leftward in the diagram. Convectively unstable regions (in red) are associated with the partial ionization of carbon. The experimental points at the highest pressures and temperatures reach the conditions of partial ionization of the core electrons of carbon that are similar to those at the bottom of the convection zone of hot DQ stars. This is the region most responsible for the driving of unstable pulsation modes5,14 and where EOS models show the greatest range of variability. By constraining EOS models in this regime, these data can contribute to more accurate models of hot DQ stars and of their complex origin, thought to be from stellar mergers or through the late helium flash of asymptotic giant branch stars.
Atomic pressure regime
The atomic pressure regime is the pressure greater than that required to significantly distort core electron orbitals, Pa, and can be estimated by
where EH is the Hartree energy and rBohr is the Bohr radius.
Simultaneous mass density and opacity unfold method
The measured transmission radiograph of an X-ray back-lighter source is sensitive to the mass density, or compression, and opacity of the sample material (see Extended Data Fig. 1). When the opacity of a material deviates from the known cold opacity, this must be considered in order to extract density information from the radiograph. As there are no existing experimental measurements of the opacity of any material at the pressures in this work, we simultaneously extract the opacity from the radiograph. Here we use radial profile matching and optimization of the density and opacity to find best fits to the experimental data.
First, an initial guess of the density and opacity profiles are chosen based on physical parameters and radial transmission profiles are calculated. The calculated profiles are compared to the measured transmission profiles and are iterated until a good match to the experimental data are found. Uncertainty contours are determined from the quality of the profile fits to the radiograph. Since the relationships between transmission intensity, mass density and opacity are linear, additional information and assumptions are needed to constrain the range of profiles that provide a fit to the data. This is depicted in Extended Data Fig. 1, with a range of possible compressions and opacities that correspond to a given measured intensity (black dashed curves). Additional information for constraining the compression profile includes the known initial density ahead of the unshocked material and known mass of the sample material inside a Ge radiographic marker layer physically located inside the sample. This constraint on density provides a further constraint on the opacity. This is done continuously as a function of time as the shock traverses the sample. An example shock trajectory is shown in Extended Data Fig. 1 (red dashed curve). We also include an additional constraint that the opacity and density at the shock front vary smoothly in time and do not include discontinuous changes; see ref. 25 for more details on the radiographic unfold analysis method and derived uncertainties.
The uncertainty contours of ±1σ in the measurement shown in Fig. 3 correspond to statistical uncertainty from simultaneously fitting of the parameterized opacity and density profiles to the measured radiograph. The parametrized fits were perturbed around the best fit and a probability distribution was constructed. This analysis includes uncertainty due to noise in the measurement. Systematic uncertainties from the magnification and the sweep speed of the X-ray streak camera correspond to ~2% in compression and ~7% in pressure, and mainly affect the location but not the shape of the Hugoniot. The measurements in Fig. 3 and the additional measurements provided in the next section are consistent in both position and shape of the Hugoniot with AA-DFT.
Extended Data Fig. 2 shows additional data obtained in this study in a separate experiment on NIF (green curves and green shaded region), included to demonstrate consistency of the analysis approach and to support our conclusions. This experiment (shot N130103-009-999) used the same sample material over a similar pressure range and was fielded at cryogenic temperatures (24 K), with a measured initial density of 1.136 g cm−3 consistent with the fielding temperature. The uncertainty contours of ±1σ are larger than the main experiment shown in Fig. 3 and Extended Data Fig. 2 (red curves and red shaded region) mainly as a result of using fewer laser beams to create the X-ray back-lighter.
Extended Data Fig. 2 shows that a slightly higher-density sample probes approximately the same theoretical shock Hugoniot at cryogenic temperatures as accessed at room temperature for both AA-DFT and AA-TFD over our pressure range. In the pressure range of these experiments the theoretical models deviate by roughly 1% in compression at the low-pressure range and overlay at higher pressures. This additional dataset is consistent with the data presented in Fig. 3 with a difference in the best fit compression of <1% at 107 Mbar, ~2% at 300 Mbar and ~3% at 450 Mbar. Both datasets show good agreement with models that include electronic structure in the AA-DFT approach.
Experimental configuration continued
The sample consisted of a solid sphere of poly(alpha methylstyrene) (PAMS, C9H10) with trace amounts of fluorine (1 at.%) from the fabrication process uniformly distributed throughout the sphere, and a measured density of 1.085 ± 0.009 g cm−3 at ambient temperature. The fluorine concentration and uniformity were measured using energy-dispersive X-ray spectroscopy (EDS) and are calculated to have a negligible effect on the theoretical shock Hugoniot (see Extended Data Fig. 3). The spheres were coated with a plastic ablator (glow discharge polymer) that included doped layers of Ge to mitigate X-ray preheat from the hohlraum and act as a radiographic marker for the analysis. The laser drive was a two-shock pulse shape with a total drive energy of 1.1 MJ. The 9-keV, Zn Heα X-ray back-lighter was generated by using up to 16 NIF laser beams incident on a Zn foil delayed in time relative to the drive beams (see Fig. 1c). Our main focus is on the analysis of NIF shot number N130701-002-999, but we also use opacity information from NIF shot N130103-009-99925. Radiation hydrodynamic simulations using HYDRA38 benchmarked to shock timing data39 calculate the shock coalescence of the two shocks inside the ablator to be before the Hugoniot points are extracted. In these experiments, hot electron generation from laser–plasma interactions was inferred from measurements40 to be more than an order of magnitude lower than in previous experiments using higher-pressure gas-fill hohlraums (He density of 0.96–1.6 mg cm−3), for example compared with shot N1401106 of ref. 41. This was achieved by using a near-vacuum hohlraum helium gas-fill density of 0.03 mg cm−3.
The calculated preheat of the sample inside the probed region as a result of the hot electrons is <1 eV. The estimated bulk sample preheat from the measured hohlraum X-ray background42 is <1 eV inside the centre of the sphere and <2.5 eV at the outer probing radius of the sample, which does not cause the calculated shock Hugoniot to deviate outside of the measurement error from the un-preheated Hugoniot22 for these shock strengths. Simulations also indicate that radiative heating of the sample ahead of the shock from the shock itself does not cause deviation from the un-preheated Hugoniot outside of the measurement error for pressures up to ~450 Mbar.
This can be seen in Extended Data Fig. 4, which includes extracted shock-front densities and pressures from radiation hydrodynamic simulations of the experimental configuration (red points). Also plotted are curves of the input Hugoniot to the simulations33,34,35,36 and ±2% deviation from the input Hugoniot. At lower pressures, ~100 Mbar, the extracted Hugoniot deviates from the input Hugoniot owing to bulk preheat from the hohlraum radiation drive at the level of ≤2%. Then, as the shock-front pressure increases, this level of preheat becomes a smaller perturbation on the Hugoniot and the extracted points follow the input Hugoniot more closely. At higher pressures of >450 Mbar, radiative preheat of material ahead of the shock front from the shock front itself starts to play a role. At 450 Mbar, this preheating causes the Hugoniot to deviate from the input Hugoniot at the level of 2% in the direction of lower compression. We restrict the data record in Fig. 3 to ~450 Mbar for this reason. At 720 Mbar, these calculations suggest lower compressions of 4.0, owing to radiative preheat and deviation from the input Hugoniot of >12%. However, the full data range reaching pressures of ~720 Mbar, shown in Extended Data Fig. 5 (red and purple curves and shaded region), does not show this level of reduction in compression at high pressures, which could be an indication that the modelling is overestimating radiative shock-front preheat. Here, the central curves represent the best fit to experimental data, and the shaded regions correspond to ±1σ uncertainty.
Theoretical EOS models
The theoretical Hugoniot calculations for polystyrene presented in Fig. 3 for AA-DFT and AA-TFD are equivalent to widely used EOS models for ICF experiments (see Figure 9 of ref. 8). Equal pressure and temperature additive-volume prescriptions were used to mix the C and H contributions. For AA-TFD, both H and C were treated with Thomas–Fermi–Dirac theory. For AA-DFT, the carbon EOS used ref. 35 and the hydrogen EOS used ref. 36. Here, the AA-DFT model corresponds to EOS table LEOS 5112, which is the CH version of LEOS 540043 (for glow discharge plastic), widely used to model ICF ablators. The AA-TFD model corresponds to SESAME 7593 (see ref. 8 for a more detailed description).
Pulsating white dwarfs
Hot DQ white dwarfs have an envelope composed of mostly carbon mixed with a modest amount of helium13,14. They are thought to result either from a late helium-shell flash in the post-asymptotic giant branch evolution of stars or from stellar mergers. Most are highly magnetic, and photometric variability has been attributed to the relatively rapid rotation of stellar spots in and out of view. However, some hot DQ stars are probably pulsators44,45. Our data will enable probing of the latter types. These data may also lead to an understanding of the unusual low-mass pulsating white dwarf in the system SDSS J1152+0248, which is likely to have a hybrid carbon–oxygen–helium core46.
Source data are provided with this paper. Additional data are available upon request.
Owing to its complexity, the data analysis algorithm that supports the findings of this study is available from D.C.S. upon reasonable request and will be outlined in more detail in a supporting publication.
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This work was performed under the auspices of the US Department of Energy under contract number DE-AC52-07NA27344 and contract number 89233218CNA000001. This work has been supported by Laboratory Directed Research and Development (LDRD) award number 13-ERD-073, and also by the University of California, Office of the President, Lab Fee Grant LFR-17-449059, by the Department of Energy, National Nuclear Security Administration, award DE-NA0003842, and by the Department of Energy, Office of Science, Office of Fusion Energy Sciences, awards DE-SC0018298, DE-SC0019269 and FWP 100182. We also acknowledge the NIF Discovery Science programme for providing access to the facility. This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favouring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.
The authors declare no competing interests.
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Extended data figures and tables
Normalized transmission intensity of an X-ray radiograph versus changes in compression and opacity. A given transmission corresponds to a range of possible compressions and opacities (black dashed curves). Density at the shock front is further constrained through knowledge of the initial material density ahead of the shock front and known mass contained within a region on the radiograph corresponding to a Ge marker layer in the sample. This further constrains the opacity and is included in the analysis as the shock traverses the sample. An example shock trajectory is denoted by a red dashed curve.
Measured pressure versus mass density (ρ) normalized to the initial density (ρ0) along the shock Hugoniot. Green, data from shot N130103-009-999 fielded at 24 K; red (from Fig. 3), shot N130701-002-999 fielded at ambient temperature. Also plotted are the theoretical Hugoniots for AA-DFT and AA-TFD at 24 K (ρ0 = 1.136 g cm−3), indicating that the initial density conditions are predicted to access approximately the same Hugoniot states at high pressure.
Calculations show insensitivity of the theoretical Hugoniot (AA-DFT33,34,35) to fluorobenzene solvent (C6H5F) for concentrations up to 20%, corresponding to 1% atomic fraction of fluorine (green curve). Concentrations of 0.5% F (red curve) and 0% F (blue curve) are also shown, but not visibly distinguishable.
Measured pressure versus mass density (ρ) normalized to the initial density (ρ0) along the shock Hugoniot from this work (red and purple curves and shaded region). The purple curve corresponds to the extended dataset that may be impacted by radiative shock-front preheat. Also plotted are previous experimental data and theoretical modelling of the Hugoniot (see Fig. 3).
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Kritcher, A.L., Swift, D.C., Döppner, T. et al. A measurement of the equation of state of carbon envelopes of white dwarfs. Nature 584, 51–54 (2020). https://doi.org/10.1038/s41586-020-2535-y
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