The properties of hydrogen and helium under extreme conditions

Hydrogen and helium are the most abundant elements in the Universe. They are also, in principle, the most simple. Nonetheless, they display remarkable properties under extreme conditions of pressure and temperature that have fascinated theoreticians and experimentalists for over a century. Advances in computational methods have made it possible to elucidate ever more of their properties. Some of these methods that have been applied in recent years, in particular, those that perform simulations directly from the physical picture of electrons and ions, such as density functional theory and quantum Monte Carlo are reviewed. The predictions from such methods as applied to the phase diagram of hydrogen, with particular focus on the solid phases and the liquid-liquid transition are discussed. The predictions of ordered quantum states, including the possibilities of a low- or zero-temperature quantum fluid and high-temperature superconductivity are also considered. Finally, pure helium and hydrogen-helium mixtures, the latter which has particular relevance to planetary physics, are discussed.

Figure 1

A partial phase diagram of hydrogen. The principal Hugoniot (the pressures and temperatures that can be reached by shocking solid hydrogen initially at atmospheric pressure) is shown as a solid line and the secondary Hugoniot (i.e., points reached with a double shock) is shown branching from it. The isentropes of three giant planets (Jupiter, Saturn, and HD 209458b) and a representative brown dwarf (G1 229B) are shown using dashed lines. The estimated melting temperature of ${\mathrm{H}}_2$ is shown as a solid line as well. Static DAC experiments, shown using a dashed line, are able to probe pressures less than $\sim 300\mathrm{GPa}$ and temperatures less than 1100 K.

Figure 2

Hydrogen phase diagram. Solid lines show the boundaries between the gas, liquid, and solid phases. The solid circles show the (approximate) location of critical or triple points. The dashed lines on the left estimate where fluid hydrogen changes from ${\mathrm{H}}_2$ to H to a classical two-component plasma (TCP). The dotted lines at extreme temperatures estimate where the electrons become degenerate (i.e., the temperatures $T_{\mathrm{F}}$ corresponding to the noninteracting Fermi energy $E_{\mathrm{F}}$ and $0.1E_{\mathrm{F}}$). The precise pressure mechanism by which hydrogen changes from solid ${\mathrm{H}}_2$ to solid H is not well established, so it is shown as a shaded box. The line going vertically away from this shows the separation between the mostly insulating molecular fluid and the mostly conducting atomic fluid; the first-order LLT ends at a critical point, and what is shown at higher temperatures is a crossover. The almost vertical transition line at the extreme right indicates the quantum melting of the protons due to lattice compression.

Figure 3

High-pressure phase diagram of solid molecular hydrogen. Open and filled circles are Raman measurements for hydrogen and deuterium, respectively, and open squares are IR data for the former. Triangles and corresponding dashed lines indicate inferred boundaries for the recently proposed phase IV (171). Note also that the dashed line and the existence of phase I’ are still matters of debate, as discussed in the text. Adapted from 138.

Figure 4

Possible molecular orientations (indicated via arrows) for dense hydrogen (phases II and III), assuming four molecules per unit cell. The solid (empty) arrows represent molecules at $c$ ($c/2$) lattice positions. Molecules are tilted with respect to the $c$ axis at an angle $\alpha$ of $\sim 55°$, and $\delta$ indicates the distance of each molecular center (of mass) from an ideal hcp lattice. From 376.

Figure 5

The most likely candidate for phase III of hydrogen, the $C2/c$ structure, as predicted by 338. This structure essentially consists of rings of three molecules, which are responsible for its strong optical activity.

Figure 6

The candidate structure of phase IV of hydrogen, suggested by 171 and 334. The structure consists of alternating atomic graphenelike and molecular layers.

Figure 7

A single layer of the $Cmca\mathrm{\text{−}}12$ structure (338) at 300 GPa (left) and the $C2/c(2)$ structure (243) at 500 GPa (right). Note that due to the higher compression, $C2/c(2)$ has been enlarged relative to $Cmca\mathrm{\text{−}}12$. Note also that the other predicted high-pressure molecular phase $Cmca$, occurring at pressures intermediate between $Cmca\mathrm{\text{−}}12$ and $C2/c(2)$, is shown in Fig. 4.

Figure 8

Reentrant melting line of phase I of hydrogen. Experimental data: crosses (85), left triangles (142), crosses (86), up triangles (108), and dashed circles (388). Theoretical predictions: BOMD (triangles) (31) and free-energy calculations (solid line) (292). The dashed line at lower temperature is a fit of the Datchi, Gregoryanz and Bonev data to a Kechin equation, while the dashed line at higher temperature is a fit to the Morales data using the same functional form. A stable fluid point predicted by MD using QMC forces (square) (12) is also shown. Below the melting lines are the experimental solid phases (identical to Fig. 3) as well as the recently reported IM transition (109).

Figure 9

Electronic band gaps for molecular hydrogen in a hcp lattice with molecules oriented along the $c$ axis. No ZPM was included in these calculations. From 376.

Figure 10

Predicted ground-state structures of atomic metallic hydrogen: (left) $I{4}_1/amd$ at 500 GPa (300; 338; 270), (middle) $Cmcm$ at 2.5 TPa (243), and (right) $I\mathrm{\text{−}}43d$ at 3.5 TPa (243). Fictitious bonds are drawn between nearest neighbors.

Figure 11

Comparison between the measured and calculated principal Hugoniot for deuterium. The experimental data are plotted as symbols with error bars: $Z$ machine (triangles) (210), gas gun (crosses) (311), explosives (dark squares) (27), (large circles) (33), (diamonds) (143), and laser (open circles) (163), (filled small circles) (207), and (light squares) (84; 73). The continuous line is the Hugoniot from 42. Predictions of various chemical models are reported as dashed lines (from left to right): Kerley (196), FVT (187), Saumon–Chabrier–van Horn (364), and Ross (352). From 42.

Figure 12

Pressure vs temperature along the Hugoniot. Comparison among experimental data and various theoretical predictions. Experiments: gas-gun [shaded circles (166)], Nova laser [diamonds (72)], $Z$ pinch [squares (19)]. Theory: RPIMC [closed circles (280), light shaded circles (197)], BOMD [left triangle (89), up triangle (32), line (42)], direct PIMC [down triangle (117)] and WPMD [right triangle (206)].

Figure 13

Comparison among various theoretical methods of computations of the principal Hugoniot for deuterium. RPIMC (dark closed circles) (280), (light closed circles) (197) and direct PIMC (down triangles) (117). FPMD ground-state electrons (double dot-dashed line) (231), (closed squares) (32) and thermal electrons (dashed line) (168), (continuous line) (89; 42) and (open squares) (290). WPMD (right triangle) (206).

Figure 14

Liquid-liquid transition line and pressure dissociation region. Experimental data for metallic hydrogen: [square (316; 418), diamonds (123), triangles (251)]. Theoretical predictions for the LLPT: [downward triangle BOMD) (292) and (squares CEIMC (292; 236)]. The lines are a guide to the eye through BOMD and CEIMC simulation data, respectively. [Circle (367), dotted line (248)]. Molecular dissociation region according to BOMD: shaded area (393). Primary Hugoniot: thick double-dot-dashed line. QMC prediction of stable fluid state: star (12). Melting curve of the molecular solid (phase I) is represented by two possible curves as discussed in Sec. 4a4: solid (upper dashed) line (292), lower dashed line (31). The expected boundary between phase I and phase IV is also reported (171).

Figure 15

Electrical conductivity of hydrogen as a function of temperature and density. From 167.

Figure 16

Values of $T_c$ for atomic metallic hydrogen calculated using various formulas, as discussed by 271, 272. Adapted from 272.

Figure 17

Values of $T_c$ for metallic molecular hydrogen calculated using SCDFT. See 81 for a discussion of the differences between anisotropic and isotropic SCDFT. Adapted from 81.

Figure 18

Standard three-layer model of Jupiter. From 381.

Figure 19

Measurements of the principle helium Hugoniots for various initial densities. Symbols indicate the measured data, with open and closed symbols indicating whether the shocked state is reflecting or opaque, respectively. The solid line is the SCVH model (364) and the dashed line, ab initio calculations (278). The initial (precompressed) He density, measured relative to the zero-pressure density of the cryogenic liquid and indicated by $\frac{{\rho }_0}{{\rho }_L}$, increases from right to left. From 106.

Figure 20

Comparison of the pressure of liquid helium as a function of density, along an isotherm at $T=6000\mathrm{K}$, between CEIMC (289), DFT-based BOMD (293), and the SCVH (364) and WC (422) chemical models.

Figure 21

Reflectivity measurements of helium using laser-driven dynamic compression (48). Solid diamonds show the observed reflectivity as a function of temperature and final density indicated by the color scale. Curves show the reflectivity obtained from a fit to the data using the semiconductor Drude model for three final state densities, from bottom to top: 0.8, 1.1, and $1.4\mathrm{g}/{\mathrm{cm}}^3$. Triangles are calculated reflectivities by 217 near $1\mathrm{g}/{\mathrm{cm}}^3$ with $a+3\mathrm{eV}$ gap correction. From 48.

Figure 22

Three views of the interior of Saturn. Indicated are the protosolar He/H ratio where color shows less He, and more He. The center is the ice/rock core. The hashed regions indicate molecular hydrogen, while the unhashed regions indicate atomic metallic hydrogen. (1) Saturn at an age of $1.5\times {10}^9$ years, before the onset of He phase separation. (2) The current Saturn according to a previously proposed H-He phase diagram (378). (3) The current Saturn according to a phase diagram derived from new evolutionary models (2). From 119.

Figure 23

The Gibbs free energy of mixing as a function of composition, at a pressure of 400 GPa, for several temperatures, from top to bottom: 5000 K, 6000 K, 8000 K, and 10 000 K. The solid lines represent full free-energy calculations from 293. The dashed lines represent results using the ideal mixing approximation for the entropy from 247.

Figure 24

The effect of increasing concentrations of helium on the proton-proton radial distribution function (293) for various compositions at a temperature of 8000 K and an electronic density given by $r_s=1.05$; helium fractions from top to bottom: $x_{\mathrm{He}}=0.9$, $x_{\mathrm{He}}=0.8$, $x_{\mathrm{He}}=0.6$, and $x_{\mathrm{He}}=0.0$. Inset: Excess entropic contribution of the Gibbs free energy of mixing, in mHa/atom, for several temperatures at a pressure of 800 GPa: $T=6000\mathrm{K}$, $T=8000\mathrm{K}$, and $T=10000\mathrm{K}$.

Figure 25

Demixing temperatures as a function of the helium number fraction, for several pressures, from right to left: 400 (squares), 800 (triangles), and 1000 (diamonds) GPa. Symbols represent results from 293, solid lines are results from 247, dashed lines are results from 333, and the almost vertical solid line at the extreme left is from 205 at 1050 GPa.