A measurement of the equation of state of carbon envelopes of white dwarfs.

White dwarfs represent the final state of evolution for most stars1-3. 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.

Electronic dynamics and plasmons of sodium under compression.

Sodium, which has long been regarded as one of the simplest metals, displays a great deal of structural, optical, and electronic complexities under compression. We compressed pure Na in the body-centered cubic structure to 52 GPa and in the face-centered cubic structure from 64 to 97 GPa, and studied the plasmon excitations of both structures using the momentum-dependent inelastic X-ray scattering technique. The plasmon dispersion curves as a function of pressure were extrapolated to zero momentum with a quadratic approximation. As predicted by the simple free-electron model, the square of the zero-momentum plasmon energy increases linearly with densification of the body-centered cubic Na up to 1.5-fold. At further compressions and in face-centered cubic Na above 64 GPa, the linear relation curves progressively toward the density axis up to 3.7-fold densification at 97 GPa. Ab initio calculations indicate that the deviation is an expected behavior of Na remaining a simple metal.

Structural complexity in ramp-compressed sodium to 480 GPa.

The properties of all materials at one atmosphere of pressure are controlled by the configurations of their valence electrons. At extreme pressures, neighboring atoms approach so close that core-electron orbitals overlap, and theory predicts the emergence of unusual quantum behavior. We ramp-compress monovalent elemental sodium, a prototypical metal at ambient conditions, to nearly 500 GPa (5 million atmospheres). The 7-fold increase of density brings the interatomic distance to 1.74 Å well within the initial 2.03 Å of the Na+ ionic diameter, and squeezes the valence electrons into the interstitial voids suggesting the formation of an electride phase. The laser-driven compression results in pressure-driven melting and recrystallization in a billionth of a second. In situ x-ray diffraction reveals a series of unexpected phase transitions upon recrystallization, and optical reflectivity measurements show a precipitous decrease throughout the liquid and solid phases, where the liquid is predicted to have electronic localization. These data reveal the presence of a rich, temperature-driven polymorphism where core electron overlap is thought to stabilize the formation of peculiar electride states.

Measuring the melting curve of iron at super-Earth core conditions.

The discovery of more than 4500 extrasolar planets has created a need for modeling their interior structure and dynamics. Given the prominence of iron in planetary interiors, we require accurate and precise physical properties at extreme pressure and temperature. A first-order property of iron is its melting point, which is still debated for the conditions of Earth's interior. We used high-energy lasers at the National Ignition Facility and in situ x-ray diffraction to determine the melting point of iron up to 1000 gigapascals, three times the pressure of Earth's inner core. We used this melting curve to determine the length of dynamo action during core solidification to the hexagonal close-packed (hcp) structure. We find that terrestrial exoplanets with four to six times Earth's mass have the longest dynamos, which provide important shielding against cosmic radiation.

Interferometric measurements of refractive index and dispersion at high pressure.

We describe a high precision interferometer system to measure the pressure dependence of the refractive index and its dispersion in the diamond anvil cell (DAC). The reflective Fabry-Perot fringe patterns created by both a white light and a monochromatic beam are recorded to determine both the sample thickness and its index at the laser wavelength and to characterize the dispersion in the visible range. Advances in sample preparation, optical setup, and data analysis enable us to achieve [Formula: see text] random uncertainty, demonstrated with an air sample, a factor of five improvement over the best previous DAC measurement. New data on [Formula: see text] liquid water and ice VI up to 2.21 GPa at room temperature illustrate how higher precision measurements of the index and its optical dispersion open up new opportunities to reveal subtle changes in the electronic structure of water at high pressure.

Simultaneous compression and opacity data from time-series radiography with a Lagrangian marker.

Time-resolved radiography can be used to obtain absolute shock Hugoniot states by simultaneously measuring at least two mechanical parameters of the shock, and this technique is particularly suitable for one-dimensional converging shocks where a single experiment probes a range of pressures as the converging shock strengthens. However, at sufficiently high pressures, the shocked material becomes hot enough that the x-ray opacity falls significantly. If the system includes a Lagrangian marker such that the mass within the marker is known, this additional information can be used to constrain the opacity as well as the Hugoniot state. In the limit that the opacity changes only on shock heating, and not significantly on subsequent isentropic compression, the opacity of the shocked material can be determined uniquely. More generally, it is necessary to assume the form of the variation of opacity with isentropic compression or to introduce multiple marker layers. Alternatively, assuming either the equation of state or the opacity, the presence of a marker layer in such experiments enables the non-assumed property to be deduced more accurately than from the radiographic density reconstruction alone. An example analysis is shown for measurements of a converging shock wave in polystyrene at the National Ignition Facility.

Benchmarking boron carbide equation of state using computation and experiment.

Boron carbide (B_{4}C) is of both fundamental scientific and practical interest due to its structural complexity and how it changes upon compression, as well as its many industrial uses and potential for use in inertial confinement fusion (ICF) and high-energy density physics experiments. We report the results of a comprehensive computational study of the equation of state (EOS) of B_{4}C in the liquid, warm dense matter, and plasma phases. Our calculations are cross-validated by comparisons with Hugoniot measurements up to 61 megabar from planar shock experiments performed at the National Ignition Facility (NIF). Our computational methods include path integral Monte Carlo, activity expansion, as well as all-electron Green's function Korringa-Kohn-Rostoker and molecular dynamics that are both based on density functional theory. We calculate the pressure-internal energy EOS of B_{4}C over a broad range of temperatures (∼6×10^{3}-5×10^{8} K) and densities (0.025-50 g/cm^{3}). We assess that the largest discrepancies between theoretical predictions are ≲5% near the compression maximum at 1-2×10^{6} K. This is the warm-dense state in which the K shell significantly ionizes and has posed grand challenges to theory and experiment. By comparing with different EOS models, we find a Purgatorio model (LEOS 2122) that agrees with our calculations. The maximum discrepancies in pressure between our first-principles predictions and LEOS 2122 are ∼18% and occur at temperatures between 6×10^{3}-2×10^{5} K, which we believe originate from differences in the ion thermal term and the cold curve that are modeled in LEOS 2122 in comparison with our first-principles calculations. To account for potential differences in the ion thermal term, we have developed three new equation-of-state models that are consistent with theoretical calculations and experiment. We apply these new models to 1D hydrodynamic simulations of a polar direct-drive NIF implosion, demonstrating that these new models are now available for future ICF design studies.

Theoretical and experimental investigation of the equation of state of boron plasmas.

We report a theoretical equation of state (EOS) table for boron across a wide range of temperatures (5.1×10^{4}-5.2×10^{8} K) and densities (0.25-49 g/cm^{3}) and experimental shock Hugoniot data at unprecedented high pressures (5608±118 GPa). The calculations are performed with first-principles methods combining path-integral Monte Carlo (PIMC) at high temperatures and density-functional-theory molecular-dynamics (DFT-MD) methods at lower temperatures. PIMC and DFT-MD cross-validate each other by providing coherent EOS (difference <1.5 Hartree/boron in energy and <5% in pressure) at 5.1×10^{5} K. The Hugoniot measurement is conducted at the National Ignition Facility using a planar shock platform. The pressure-density relation found in our shock experiment is on top of the shock Hugoniot profile predicted with our first-principles EOS and a semiempirical EOS table (LEOS 50). We investigate the self-diffusivity and the effect of thermal and pressure-driven ionization on the EOS and shock compression behavior in high-pressure and -temperature conditions. We also study the sensitivity of a polar direct-drive exploding pusher platform to pressure variations based on applying pressure multipliers to LEOS 50 and by utilizing a new EOS model based on our ab initio simulations via one-dimensional radiation-hydrodynamic calculations. The results are valuable for future theoretical and experimental studies and engineering design in high-energy density research.

Absolute Hugoniot measurements from a spherically convergent shock using x-ray radiography.

The canonical high pressure equation of state measurement is to induce a shock wave in the sample material and measure two mechanical properties of the shocked material or shock wave. For accurate measurements, the experiment is normally designed to generate a planar shock which is as steady as possible in space and time, and a single state is measured. A converging shock strengthens as it propagates, so a range of shock pressures is induced in a single experiment. However, equation of state measurements must then account for spatial and temporal gradients. We have used x-ray radiography of spherically converging shocks to determine states along the shock Hugoniot. The radius-time history of the shock, and thus its speed, was measured by radiographing the position of the shock front as a function of time using an x-ray streak camera. The density profile of the shock was then inferred from the x-ray transmission at each instant of time. Simultaneous measurement of the density at the shock front and the shock speed determines an absolute mechanical Hugoniot state. The density profile was reconstructed using the known, unshocked density which strongly constrains the density jump at the shock front. The radiographic configuration and streak camera behavior were treated in detail to reduce systematic errors. Measurements were performed on the Omega and National Ignition Facility lasers, using a hohlraum to induce a spatially uniform drive over the outside of a solid, spherical sample and a laser-heated thermal plasma as an x-ray source for radiography. Absolute shock Hugoniot measurements were demonstrated for carbon-containing samples of different composition and initial density, up to temperatures at which K-shell ionization reduced the opacity behind the shock. Here we present the experimental method using measurements of polystyrene as an example.