Fluid helium at conditions of giant planetary interiors
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Contributed by Raymond Jeanloz, May 16, 2008 (received for review December 19, 2007)
Abstract
As the second mostabundant chemical element in the universe, helium makes up a large fraction of giant gaseous planets, including Jupiter, Saturn, and most extrasolar planets discovered to date. Using firstprinciples molecular dynamics simulations, we find that fluid helium undergoes temperatureinduced metallization at high pressures. The electronic energy gap (band gap) closes at 20,000 K at a density half that of zerotemperature metallization, resulting in electrical conductivities greater than the minimum metallic value. Gap closure is achieved by a broadening of the valence band via increased s–p hydridization with increasing temperature, and this influences the equation of state: The Grüneisen parameter, which determines the adiabatic temperature–depth gradient inside a planet, changes only modestly, decreasing with compression up to the hightemperature metallization and then increasing upon further compression. The change in electronic structure of He at elevated pressures and temperatures has important implications for the miscibility of helium in hydrogen and for understanding the thermal histories of giant planets.
Helium is known to be an electrical insulator at low pressure, with a wide energy gap (19.8 eV) between occupied and unoccupied electron orbitals; it exhibits almost no chemical bonding (1). Under compression, however, helium is predicted to metallize via closure of the energy gap at ≈100 Mbar (10 TPa) (2), a pressure greater than that at Jupiter's center (3). Thus, one might expect helium to be insulating at giantplanetary conditions, for its solubility in metallic hydrogen to be limited and for addition of helium to limit the electrical conductivity of the gaseous envelope (4).
However, recent highpressure results have revealed the role of temperature in metallization, particularly in the fluid state. Fluid hydrogen becomes metallic at 1.4 Mbar at high temperature (>10^{3} K) along the shockwave Hugoniot (5), whereas at low temperature (≤300 K) crystalline hydrogen is expected to metallize only >4 Mbar (6). In a sense, hydrogen at elevated pressures resembles other materials that undergo insulatortometal transitions upon melting, such as silicon and carbon, in which the liquid has a more denselypacked structure than the solid phase. Yet the metallization of fluid hydrogen may also be related to changes in the fluid, from dominantly molecular (H_{2}) at lower pressures to dominantly atomic (H) at higher pressures (7). That ionization and dissociation of the molecule take place across overlapping regimes of density and pressure is a complication that has confounded a full understanding of the metallization of hydrogen. The case of helium is thus revealing in that it effectively isolates the influences of temperature and density on the development of metallic bonding, because both liquid and solid are monatomic and close packed at high pressure.
We performed firstprinciples molecular dynamics simulations and found that the energy gap of fluid helium depends strongly on temperature (Fig. 1). The electronic energy gap can be thought of as the difference in the energies of the highest occupied electronic bonding levels and the lowest unoccupied (nonbonding) electronic levels (valence and conduction bands, respectively, for crystals). Whereas the gap closes at a density of 13 g cm^{−3} at zero temperature, gap closure occurs at 6.6 g cm^{−3} at 20,000 K, where the pressure is 30 Mbar (3 TPa): conditions achieved well within the fluid envelope of Jupiter (3). Our results differ from those of another recent study that found that temperature has a much weaker influence on the energy gap and that the gap closes at the same density virtually independent of temperature (8). We attribute this difference to the more complete sampling of the Brillouin zone used in our computations of the energy gap (see Theoretical Methods).
We find that gap closure originates primarily from a broadening of the valence band, by a factor of nearly two from 0 to 50,000 K and from increased admixture of slike and plike states with increasing temperature (Fig. 2). Comparison with the next divalent element, Be, is instructive. Like He, both solid and liquid states are closely packed, yet the liquid has a density of states at the Fermi level more than twice that of the solid (9). As in He, participation of plike states in the valence band increases with temperature, and metallicity increases with disorder: The reciprocal lattice vectors responsible for the scatteringinduced pseudogap in the solid are smeared out in the liquid structure factor.
The relationship between the liquid structure factor and the electronic structure can be understood on the basis of a nearly freeelectron picture (10). The first sharp peak in the structure factor at V = 1 Å^{3} per atom (Fig. 3), located at q_{P} = 7.4 Å^{−1}, produces a pseudogap at an energy (given by the de Broglie relation) W = ħ^{2}/(2m_{e}) (q_{P}/2)^{2} = 52 eV above the bottom of the valence band (note that Fig. 3 is for a volume intermediate between those shown in Figs. 2 and 4). The magnitude of the first sharp peak S(q_{P}), decreases markedly with increasing temperature, accounting for the closure of the pseudogap Δ with increasing temperature, as Δ = 2(S(q_{p}) − 1)∣w(q_{p})∣, where w is the effective electron–ion interaction (9).
Valenceband broadening in He can be understood on the basis of modifications to the closely packed arrangement of atoms in the liquid state. Inspection of the radial distribution function shows that at V = 1 Å^{3} per atom (6.6 g cm^{−3}) the position of the maximum and halfwidth at halfmaximum of the first peak are 0.98 Å and 0.3 Å, respectively, in the fluid at 50,000 K as compared with the nearestneighbor distance of 1.12 Å in the perfect, hexagonal closely packed (hcp) crystal at the same density (Fig. 3, arrow). In this sense, local structure in the liquid resembles that of the crystal at 1.5–4 times higher densities. The coordination number in the liquid is ≈14 over the entire temperature range considered.
One can derive additional insight into the liquid structure by comparison with a simple model, the onecomponent plasma (OCP) (11). The properties of the OCP depend on a single variable, the unscreened Coulomb coupling parameter, where Z is the nuclear charge, e is the electron charge, k is Boltzmann's constant, T is temperature, and a = (3V/4π)^{1/3} is the ionsphere radius. The structure of our simulated liquid differs, however, in that the heights of the peaks in g(r) and S(q), as well as the distance to the first peak in g(r), are smaller than for the OCP over most of the volume–temperature range of our study. Because local structure is largely determined by repulsive forces (12), this difference indicates a weakened effective ion–ion interaction in our simulated fluid as compared with the OCP. Such weakening, caused by electron screening, is in fact expected.
We have found (Fig. 3) that the structure of our simulated fluid at V = 1 Å^{3} per atom, T = 20,000 K (Γ = 54) is remarkably similar to that found in Monte Carlo simulations of the screened OCP in which screening was approximated by the Lindhard dielectric function (13). For Γ > 50, the structure of our simulated fluid begins to depart substantially from that of the screened OCP (Fig. 3). These departures emphasize the importance of firstprinciples molecular dynamics simulations that include the physics of the electron–ion interaction completely [within the generalized gradient approximation (GGA)]. In our simulated fluid, the height of the first peak in g(r) is greater than that of the screened OCP for Γ > 50 and greater even than that of the unscreened OCP at the highest densities of our study, revealing the importance of nonpointcharge repulsion due to overlap of charge accumulations about the nuclei.
The fluid becomes increasingly metallic with increasing density and is nearly freeelectronlike at 50,000 K and the density of zerotemperature gap closure (Fig. 4). At these conditions, the fluid has a small pseudogap and a density of states at the chemical potential, μ (determined by the number condition), 70% that of the freeelectron value. Thus, the liquid appears to retain features of the crystal's electronic structure, in particular a local minimum in the density of states at μ that persists to the highest densities of our study. The density of states at μ in the fluid at 50,000 K reaches a maximum value at 11 g cm^{−3}, where it is 30% that of the freeelectron gas.
Metallization may occur at densities and temperatures slightly higher than those we find for gap closure, because of localization of the electrons at the band edges (mobility edges) (14). Moreover, because of systematic limitations of the GGA, we anticipate that our results may underestimate the energy gap. Using GW calculations, one study estimates that GGA may underestimate the band gap by a few eV (8). However, the correction was computed at zero temperature. Recent results show that at finite temperature, density functional theory underestimates the gap by considerably less than previously thought (15) so that our computed band gaps may be accurate to better than a few eV. Reasonable agreement (to within 1 eV) with the excitedstate energy of the He atom (16) further indicates that our results should reveal the correct trends.
We find that the electrical conductivity of fluid helium is well described by the Ziman formula (17) modified to account for bandstructure effects (18) (Fig. 5) where the Ziman resisitivity and a_{0}ħ/e^{2} = 0.217 μΩ m, E_{F} and k_{F} are the (freeelectron) Fermi energy and wave vector, respectively, and the effective ion–electron interaction v(q) is that of the screened Coulomb potential with inverse screening length equal to the Thomas–Fermi wave vector k_{TF} (19) The influence of the pseudogap (Fig. 2) appears in the factor the ratio of the temperaturesmoothed density of states at the chemical potential μ where ∂f/∂ε is the derivative of the Fermi–Dirac distribution, to the freeelectron value of the density of states at the Fermi level. The value of g in fluid helium is everywhere less than unity, so that the conductivity is reduced as compared with the Ziman result. This analysis shows that the electronic structure of fluid helium lies in a regime in which the Edward's cancellation theorem (20) no longer applies, and the electron meanfree path is not much larger than the interatomic spacing. The electrical conductivity of fluid helium at the density of hightemperature gap closure is similar to that of the minimum metallic conductivity of Mott (21): 0.026 − 0.333e^{2}/ħ2a (Fig. 5).
The influence of energygap closure on the equation of state is seen in the behavior of the Grüneisen parameter γ, which controls the adiabatic temperature gradient: γ = (∂lnT/∂lnρ)_{S} (Fig. 6A). The value of γ decreases upon compression up to the density of 13 g cm^{−3} and then begins to increase at higher densities. In comparison, a plasma model (22) and the He equation of state from the SESAME tables (23) predict large oscillations in the value of γ associated with pressureinduced ionization transitions; these oscillations are not found in our study. In the plasma or “chemical” picture (22) the fluid is viewed as a collection of electrons, atoms, and singly and doubly charged ions with internal energy levels that are assumed to be unperturbed by interactions with surrounding particles. This picture predicts a rapid increase in the ionization with pressure, which produces a large increase in the density and anomalies in the Grüneisen parameter and other derivatives of the equation of state. We find no evidence of rapid pressure ionization, and the difference in density between our results and the plasma model reaches 50% at conditions of the Jovian gaseous envelope (Fig. 6B).
Firstprinciples molecular dynamics simulations illustrate the limitations of the plasma model at conditions where orbital overlap is large, and electronic states are best described as spatially extended. In particular, the approximation of unperturbed internal energylevels for the electron orbitals of the atom is unlikely to be valid at conditions where the valence bandwidth greatly exceeds the gap, as is the case over most of the pressure range of our study. In the case of hydrogen, the most recent quantum Monte Carlo and densityfunctional molecular dynamics studies also disagree with the plasma model in finding no evidence of a plasma phase transition (24, 25).
Our predictions of the equation of state and electronic properties can be tested with emerging experimental technology (26, 27) (Fig. 7). Shock waves, including multiple shocks and “ramp” waves, generated by powerful lasers in samples precompressed in a diamondanvil cell provide a means of experimentally accessing the entire range of pressure–temperature conditions of giant planets. For example, we predict that the energy gap closes at a density of 2.3 g cm^{−3} along the Hugoniot for 4fold precompression: onefifth the density required for gap closure under static conditions. The influence of temperature on the electronic structure is also illustrated by the carrier concentration, which increases with increasing pressure and increasing temperature, primarily due to closing of the energy gap. We predict that experimental measurements along the 15fold precompressed Hugoniot will be particularly revealing: requiring an initial (precompressed) pressure of 1 Mbar at ambient temperature, this Hugoniot includes pressure–temperature conditions at which our results differ significantly from those of the plasma model (22), and the nonmetaltometal transition should be experimentally detectable via optical absorption and reflectance measurements.
The large influence of temperature on the electronic structure of helium implies that helium rain is unlikely in presentday Jupiter or Saturn. It has been suggested that exsolution and gravitational segregation of helium from hydrogen, upon cooling of the planet, may be responsible for the excess luminosity of Saturn. This argument comes from estimates of the hydrogen–helium miscibility gap that, so far, have been based on calculations performed at low temperature. The results of ref. 28 are based on static calculations with no relaxation about defects. Whereas the miscibility gap computed in ref. 29 is based on static calculations including relaxation and tested against molecular dynamics simulations, the influence of temperature on the electronic structure is slight up to the maximum temperatures of only 3,000 K that were considered. For models of Saturnian evolution, He rainout, if it occurs, would take place at pressures and temperatures in the range of 1–10 Mbar and 5,000–10,000 K, which encompasses the regime (V ≈2 Å^{3}) in which we find that temperature increases the valence bandwidth and decreases the pseudogap by a factor of two as compared with 0 K (Fig. 2). For Jupiter, and for extrasolar planets larger and older than Jupiter, still higher pressures and temperatures become relevant, and the He energy gap may be completely closed, according to our calculations.
Miscibility of He in hydrogen is thus likely to be enhanced in comparison with the predictions of previous lowtemperature calculations, because temperature transforms dense fluid helium from an insulator to a semiconductor and, ultimately, a metal. Enhanced solubility would reduce the critical temperature for miscibility, below which hydrogen and helium are immiscible, to values below those indicated by thermalevolution models for Saturn. Other mechanisms must therefore be found to explain the excess luminosity of Saturn and the helium deficiency of the Jovian and Saturnian atmospheres (30).
The electronic structure of helium may also have an important influence on magneticfield generation. The magnetic diffusivity λ = 1/σμ_{0}, where σ is the electrical conductivity and μ_{0} the magnetic permeability, controls the freedecay time of the magnetic field; it also controls the power of the field and its form, whether it be dipolar or multipolar, via the dimensionless magnetic Ekman number E_{λ} = λ/ΩD^{2} and magnetic Reynolds number Rm = u/λD, where Ω is the rotation rate, D is depth of the fluid, and u is the flow speed (31). Temperatureinduced bandgap closure in helium tends to enhance the electrical conductivity, hence decrease the magnetic diffusivity and increase the magnetic Reynolds number, over the values typically assumed.
Theoretical Methods
Our molecular dynamics simulations are based on density functional theory in the GGA (32), using the projector augmented plane wave (PAW) method (33) as implemented in the VASP code (34). Born–Oppenheimer simulations were performed in the canonical ensemble with a Nosé (35) thermostat with 64 atoms and run for at least 1,000 steps at a 0.1fs time step. We assume thermal equilibrium between ions and electrons via the Mermin functional (36, 37). Tests using larger systems, up to 144 atoms, and greater run durations, up to 3,000 time steps, showed no significant change in equilibrium thermodynamic properties. The Brillouin zone is sampled at zero wave vector (k = 0, Γ point); a basisset size set by the value of the energy cutoff of 600 eV was found sufficient at all but the highest density, where we used an energy cutoff of 1,200 eV. The PAW potentials have an outermost cutoff radius of 1.1 Bohr with no electronic states treated as core states. We have also performed a limited number of static fullpotential linearized augmented plane wave (LAPW) computations (38) to test the limitations of the PAW method at extremely high number densities (>10 Å^{−3}).
The electronic energy gap is, in principle, illdefined at temperatures >0 K. To determine the gap, we locate the range of energies about the chemical potential for which the density of states is <1% of the maximum valence density of states. Properties of the electronic structure, including the electronic density of states, are computed for a series of uncorrelated snapshots by using an enhanced kpoint mesh (39) of 4 × 4 × 4 for lower densities and up to 12 × 12 × 12 for the highest density explored. We found such enhanced kpoint meshes essential for obtaining converged values of the electronic energy gap and concluded that computing the gap using a single kpoint (Γpoint) results in systematically larger values of the gap, accounting for the differences between our results and those of ref. 8.
Acknowledgments
We thank N. W. Ashcroft, G. W. Collins, D. J. Stevenson, and R. M. Wentzcovitch for helpful discussions. This work was supported by the U.S. National Science Foundation and Department of Energy and by the University of California.
Footnotes
 ↵^{†}To whom correspondence may be addressed. Email: l.stixrude{at}ucl.ac.uk or jeanloz{at}berkeley.edu

Author contributions: L.S. and R.J. designed research; L.S. performed research; L.S. and R.J. analyzed data; and L.S. and R.J. wrote the paper.

The authors declare no conflict of interest.

See Commentary on page 11035.
 Received December 19, 2007.
 © 2008 by The National Academy of Sciences of the USA
Freely available online through the PNAS open access option.
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