Burkhard Militzer at UC Berkeley

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Web Name: Burkhard Militzer at UC Berkeley

WebSite: http://militzer.berkeley.edu

ID:227822

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Militzer,Burkhard,at,Berkeley,UC,

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Contact InformationBurkhard Militzer
Professor

University of California, Berkeley
Departments of Earth and Planetary Science and Astronomy
Phone(510) 643-7414Fax(510) 643-9980Emailmilitzer at berkeley dot eduAddress
(Map) 411 McCone Hall, MC 4767
Berkeley, CA 94720, USA
Research Interests

In my research, I use computer simulationsto understand the interior and evolution of giant planets. Materialsin planetary interiors are exposed to extreme temperature and pressureconditions that cannot yet be reached with laboratoryexperiments. Instead we rely on highly accurate first-principles computer simulations techniques. With these methods,we recently explained why neon is depleted in Jupiter's atmosphere andprovided strong, though indirect evidence for helium rain to occur ingiant planets. Our recent simulations predict core erosion to occur in gas giant planets.

Furthermore I study materials in thedeep mantle of our planet and compare my results with static anddynamic high pressure experiments. In some cases, computer simulationsprovide new insight into properties of materials that cannot beobtained with experiments. Inother cases we use them to make predictions for the state of matter atthese extreme pressures. Recent examples include fluid helium and water ice at megabar pressures.

My background is in the field oftheoretical condensed matter physics and I am interested in theory andsimulation of novel materials under extreme conditions. I use avariety of first-principles simulation methods including path integral Monte Carlo, groundstatequantum Monte Carlo, and density functional moleculardynamics.

Research Group

Felipe Gonzalez-Cataldo, postdoctorial researcher.
Sean Wahl, postdoctorial researcher.
Anton Ermakov, postdoctorial researcher.
Jizhou Wu, PhD candidate.
Tanja Kovajevic, PhD student.
Rustin Domingos, PhD student.
Mark Olson, undergraduate student.
Ryu Akiba, undergraduate student.
Formerly in my group at UCB:
Maximilian Böhme, graduate student visiting from Dresden, Germany.
Shefali Bhatia, UCB undergraduate student.
Henry Peterson, UCB undergraduate student.
Kevin Driver, postdoctorial researcher, now staff scientists at LLNL.
Francois Soubiran, postdoctorial researcher, now staff scientists at CEA in France.
Shuai Zhang, PhD student, now staff scientists at LLE in Rochester, NY.
Tanis Leonhardi, UCB graduate student.
Hugh F. Wilson associate specialist, now at CSIRO in Melbourne.
Stephen Stackhouse, now Lecturer at the University of Leeds.
Saad Khairallah, now staff scientists at LLNL.
Mike Wong, UCB undergraduate student, since graduated with PhD degree from Caltech.
Benjamin Sherman, undergraduate student from CSUN visited in 2010 and 2011.
Members of my research group at the Carnegie Institution of Science (2003-2007):
Jan Vorberger, postdoctorial researcher, now staff scientist in HZDR in Rossendorf, Germany
Ken Esler, postdoctoral researcher
Rebekah Graham, Isaac Tamblyn, Seth Jacobsen (all REU summer students)

Teaching

In the fall of2008, I introduced EPS 109 "Computer Simulations with Jupyter Notebooks" as a new course. Anintroduction to computer simulation and data analysis methods is given and studentslearn to write programs with Jupyter notebooks. Have a look the animations that the students made duringthe2008, 2009, 2011, 2012, 2013, 2014,2015,2016,2018,2019, and2020 classes. In spring of 2011, Dino Bellugi and I introduced the graduate class EPS 209 "Matlab Applications in Earth Science". Here is a complication of the final projects.

I teach the course C12 "The Planets". A tour of the mysteries and inner workings of our solarsystem is presented. The class has over 200 students and is directedat nonscience majors. Here are some picturesfrom our class room demonstrations in 2010 and 2012. This course is also offered asan online summer class W12. Here are three examples from our series ofrecorded lectures: a course introduction, one on the Kepler mission, and oneon meteorites. My experiences teaching online are described in an article for the EPS alumni report in 2010.

Here are some pictures from my presentations at UC Berkeley's CalDay events in 2010, 2013, and 2019.I also participated in a field trip to Yosemite National Park.

Open Positions

We have open postdoc positions and opportunities for new Ph.D. students to work in planetary science and on computer simulations of matter at extreme conditions. Alternatively, you may be able to work with us by taking advantage of opportunities in Astronomy.

Ph.D. applicants interested in this research should applyto the department of Earth andPlanetary science. The deadline is in mid-December every year. Applicants are encouraged to contact me in advance to discuss mutual interests and specific research projects.

Calculated Tidal Response of Hot Jupiters Disagrees with Observations

Tidal Love numbers from observations (solid cirlces) and our model predictions (open triangles) are compared for a series of exoplanets. In our latest article, we study how the shape and gravity field of strongly irradiated, giant exoplanets is destorted by the external gravity field of their massive host stars. A planet's response to such external fields is expressed by the tidal Love number, k₂₂, which has been inferred from telescope observations for the exoplanets WASP 4b (green color), HAT-P 13b (red), WASP 18b (blue). In the digram on the left, we compared these observations (solid circles) with our calculated values that we represent with triangles of corresponding colors. The upper triangles represent models without rocky core while the lower triangles show predictios for the largest possible cores. The beige symbols show our predictions for the exoplanets WASP 12b, 103b, 121b, Kepler 57b, and Corot 3b. We find that the static Love number, k₂₂ cannot exceed the value 0.6 (yellow shaded region), which is in contradiction with some of the observations. We suggest additional observations to be made and existing data to be re-analyzed because this discrepancy may imply that the orbits of the observed exoplanets could be affected by other factors like not-yet-detected exoplanets. In our paper, we also derived the gravity harmonics, shape, of moment of inertia for our planet models. First-Principles Equation of State (FPEOS) Database for Warm Dense Matter Computation

States of maximal shock compression of 200 compounds and mixtures predicted by our FPEOS database.With the goal of making warm dense matter computations more reliable and efficient, we make available our first-principles equation of state (FPEOS) database for materials at extreme conditions. We provide our EOS tables the elements H, He, B, C, N, O, Ne, Na, Mg, Al, and Si as well as the compounds LiF, B₄C, BN, CH₄, CH₂, C₂H₃, CH, C₂H, MgO, and MgSiO₃ that are solely based on results from ~5000 path integral Monte Carlo and density functional molecular dynamics simulations. For all these materials, we provide the pressure and internal energy over a density-temperature range from ~0.5 to 50 g/cc and from ~10⁴ to 10⁹ K. In our recent article, we compute isobars, adiabats, and shock Hugoniot curves in the regime of L and K shell ionization. Invoking the linear mixing approximation, we study the properties of mixtures at high density and temperature. We derive the Hugoniot curves for water and alumina as well as for carbon-oxygen, helium-neon, and CH-silicon mixtures. We predict the maximal shock compression ratios of H₂O, H₂O₂, Al₂O₃, CO, and CO₂ to be 4.61, 4.64, 4.64, 4.89, and 4.83, respectively. Finally we use the FPEOS database to determine the points of maximum shock compression for all available binary mixtures (left graph). We provide all FPEOS tables as well as C++ and Python computer codes for interpolation, Hugoniot calculations, and plots of various thermodynamic functions. High-Pressure Phase Diagram of Magnesium Oxide Derived with Computer Simulations

Phase diagram of MgO with liquid and solid B1 and B2 phases. Theoretical and experimental shock Hugoniots curves are compared.With density functional molecular dynamics simulations, we computed the phase diagram of MgO in the pressure range from 50 to 2000 GPa up to temperature of 20000 K. Via thermodynamic integration (TDI), we derive the Gibbs free energies of the B1, B2, and liquid phases and determine their phase boundaries. The B1 structure is a NaCl-type crystal, in which Mg and O nuclei occupy alterating sites. Each atomic species by themselves forms a face-centered cubic lattice. In the B2 structure, is a CsCl-type crystal. Each atom species by themselves form a simple cubic structure. With our computer simulation, we show that anharmonic effects stabilize the B1 phase. We predict the B1-B2-liquid triple point to occur at approximately T = 10000 K and P = 370 GPa, which is higher in pressure than was inferred with quasi-harmonic methods. We predict the principal shock Hugoniot curve to enter the B2 phase stability domain but only over a very small pressure-temperature interval. This may render it difficult to observe this phase with shock experiments because of kinetic effects. Here are a copy of our article and a few slides in PPTX and PDF formats available to download. Simulations and Experiments Reveal Effect of Nanopores on Helium Diffusion in Quartz

Nanopore (yellow) in quartz crystal that may serve as a reservoir of helium atoms. This joint theoretical-experimental project was sparked by a drastic disagreement between laboratory data and results from computer simulations for the diffusion of helium atoms in quartz crystals. This is important because the diffusion of noble gases in minerals is often utilized to reconstruct the thermal histories of rocks. Computer simulations of helium in perfect quartz crystals predicted that already at room temperature, all helium atoms would diffuse out of the crystal because the helium atoms encounter very lower energy barriers along the crystal's z channel. Taken at face value, this would imply all helium would have diffused out of the crystal before the experiments even began. Conversely, however, the lab measurements showed that temperatures between 70 and 220 ^oC were required for most helium atoms diffuse out of quartz crystals. In our article, this discrepancy is resolved by introducing the novel hypothesis that helium atom reside inside nanopores in the quartz crystal. The calculations showed that activation energy for helium atoms to diffuse from the nanopore into the crystal matched experimental data. A consistent effective diffusion model was constructed and the nanopore concentration was estimated to be approximately 10^-5. Excitation Mechanisms in Warm Dense Matter

Four excitation mechanisms that control the shock Hugoniot curve of magnesium are plotted in pressure-density space.In our latest article about warm dense matter, we employ path integral Monte Carlo and density function molecular dynamics simulations to study the properties of hot, dense magensium at high temperature and density. On the left, we show our prediction for the principal shock Hugoniot curve, which will most likely be the first quantity to be measure when laboratory experiments reach these conditions in the future. We identify four main excitation mechanisms that control the density that is reached in these compression experiments. In the low-pressure regime, excitations of L-shell electrons increase the shock compression ratio above the canonical value of 4. At higher pressure, the excitations of K-shell electrons maintain a high compression ratio of about 4.9. The compression ratio starts to decrease again once all K-shell electrons have been ionized. However, this decrease in compensated by the onset of radiative effects. Photons that are spontaneously emitted start making substantial controbutions to the energy and pressure. In compression, relativistic effects, that increase the energy of the electrons, play only a minor role. Polymeric, metallic structure of fluorine predicted to form at high pressure

Novel high-pressure structure of fluorine. Here are two alternate views A and B.At the ambient conditions, fluorine is a highly reactive molecular gas. However, at high pressure, its properties change in a remarkable way. In our latest article, Mark Olson, Shefali Bhatia, Paul Larson and I use computer simulations to predict that fluorine forms a polyermic and metallic structure at 31 Mbar. Instead of the usual diatomic F₂ bonding, the three quarters of the F atoms (blue) in the structure on the left are arranged in a 3D set of chains. The remaining quarter of atoms (red) occupied voids in between these chains. While flourine typically does not conduct electricity, our new high-pressure structure is an excellent conductor.
We obtained these results with a novel computer algorithm that allows us to efficiently predict crystal structures under symmetry and geometric constraints. We compare fluorine, chlorine and iodine and make reference to X-ray diffration experiments. Equilibrium Tidal Response of Jupiter: Detectability by Juno Spacecraft

Signal to noise ratio for dectecting Jupiter's response to tidal forces from Io with the Juno spacecraft.The four Galilean satellites, Io, Europa, Ganemede, and Callisto aswell as the sun all change the shape of Jupiter very slighly throughtheir gravitational forces. In this paper, wecalculate the strengh of this tidal response, which is represented as a series of Love numbers (k_nm).We determine which Love numbers can be detected with the Juno spacecraft in the course of the on-going mission.The graph on the left suggests that Io's k₂₂, k₃₃, k₄₂, and k₃₁ lead tostrong signals and should thus all be detectable. We predict a remarkably small range for Io's equilibrium Love number k₂₂ = 0.58976 ± 0.0001. Any deviation from this prediction can then be attributed with dynamic tidal effects. Effects of K and L Shell Ionization on Shock Hugoniot Curves of Silicates
K shell (blue shade) and L (green shade) shell ionization effect on shock Hugoniot curve of MgSiO₃.Late during solar system formation, rocky planetsgrow through massive impacts. The shock Hugoniot curve characterizehow hot and dense the rocky mantle becomes during an impact. In tworecent papers, we determined the shock Hugoniot curves of MgO and MgSiO₃ with pathintegral and density functional molecular dynamics simulations. In thegraph on the left, we show how the ionizations of K and L shell electronsincreases the density during an impact event. We also provided equation of state tables over a large range of density-temperature conditions for both materials. Saturn's Deep Rotation Period Determined from Cassini and Voyager Data
The graphs shows that only a rotation period of 10:33:34h 55s is compatible with the observed oblateness. Saturn's rotation period cannot be measured directlyand has thus been very uncertain. Estimates vary between 10:32:45 h and10:47:06 h, which is an uncomfortably large range that introducesuncertainties into the analysis of various spacecraft measurements andremote observations. The rotation period cannot be derived fromSaturn's magnetic field because it is perfectly aligned with the planet'saxis of rotation. This is not case for Jupiter and its rotation periodhas been determined precisely to be 9:55:27 h. In our latest article, we combinedgravity data from the Cassini mission and Voyager's measurements ofthe planet's shape to determine a rotation period of 10:33:34 h 55 s. The faster a planet rotates to more oblate it becomes,which enabled us to infer its rotation period.
For this analysis,we developed an accelerated version of the Concentric MacLaurinSpheroid (CMS) method that enabled to constructed Monte Carloensembles of plausible interior models. We currently apply thisapproach to construct models for Jupiter's interior to match gravitymeasurements by the Juno spacecraft. Planet Saturn was born naked but today it has rings and winds 9000 km deep.
Layers in Saturn's interior.
Gravity coefficients of Saturn and Jupiter.
Varying rotation frequency in Saturn's interior. This work is based on a collaborative article published in Sciencethat is entitled "Measurement and implications of Saturn's gravityfield and ring mass". For everyone to use, here are some slides in PPTX and PDF formats as well four graphics files that I prepared. Robert Sanders prepared this press release. Among the news coverage for this work, this Russian report stood out by explaining we had determined when Saturn became "The Lord of the Rings". Also here is an interview on NPR's radio show Science Friday about this work.

During its 13 years in orbit around Saturn, the Cassinispacecraft has made a number of remarkable measurements of the planetand its satellites. But only during its final 22 orbits it dove insideits rings and measured the planet's gravity fields with unprecedentedprecision. Two important findings emerged:

The winds in Saturn's atmosphere are massive and at least 9000 km deep:We had prepared a suite ofmodels for Saturn's interior that included different core massesand amounts of heliumrain. We calculated the expected gravity field and were pretty sureSaturn's gravity coefficent J₈ would fall between -9 and -8x 10^-6. We were completely surprised when the Cassinispacecraft measured J₈ to be -14 x 10^-6, whichimplied something important was missing from all models that we hadconstructed. After we added deep and massive winds to our interiormodels we were able to match all gravity coefficients. The winds needto be at least 9000 km deep. The winds in Saturn atmosphere had beenobserved before but no one had assumed they would reach that deep. Thefirst evidence of very deep winds in giant planets only came late lastyear when measurements of the Juno spacecraft predicted the winds onJupiter to be between 3000 and 5000 km deep.

Saturn's rings are young and only formed 10-100 million yearsago: When I admired Saturn's spectacular rings, I naively assumedthey were as old as the planet itself (4.5 billion years). The firstgravity measurement of the ring mass now tells us otherwise. Theycontain only about 0.4 Mimas masses (2000 Mimas masses = 1 Earth moon)worth of material, which points to a surprisingly young ring age of onlybetween 10 and 100 million years. Before that Saturn presumably didnot have any rings. (On our slides,we explain how one relates ring mass and age.) This tells us adramatic event must have occurred near Saturn in our recent solar systemhistory. 100 million years ago, the dinosaurs still roamed onEarth. They disappeared when a giant impact occured near theYucatan peninsula 65 million years ago. Now we have evidence that adrastic event occurred near the Saturnian system that produced a gazillionpieces of icy rubble that make up the rings today. This suggests that our solar system is not such astable and happy place as one might think. We assume the rings areeither the leftover debris from a comet that was tidally disrupted bySaturn's extreme gravity just like the Shoemaker-Levycomet was pulled apart by Jupiter. Alternatively Saturn originallyhad multiple satellites, their orbits become unstable, and it came to agigantic collision. We cannot tell which scenario is more likely but wedo know something drastic must have happened in the Saturnian systemfairly recently by astronomical standards.Momentum Distribution of Interacting Quantum System ComputedDifferent paths that enter into PIMC simulations of just two particles. Node-avoiding (NA), node-crossing (NX) as well as permuting, node-crossing (PNX) paths are illustrated. The diagonal black line denotes the node the density matrix, =0. Modeling the behavior of interacting quantum systems on a classical computer is challenging. Here we the Feynman's path integral method to map a system of quantum particles onto a system of classical paths. While most thermodynamic properties can be derived from simulations of closed paths, the computation of the momentum distribution requires open paths. In this article, we compute the momentum distribution of the homogeneous electron gas with path integral Monte Carlo (PIMC) simulations.
Since thsi is a fermionic system, we employed the restricted path approach to deal with the fermion sign problem.In the restricted PIMC method, only node-avoiding (NA) paths contribute. For two particles, the nodal restriction prohibits all permutations. However, if simulations with the direct fermion method are performed no restrictions are applied. Nonpermuting paths that cross the nodes (NX) and those that avoid it (NA) both enter with a positive sign. Permuting paths (PNX) are now permitted and enter with a negative weight given by the (-1)^P factor. New Water-Salt Compound Predicted to Form Under Pressure Crystal structure of our new water-salt compound that we predict to form at high pressure. The yellow, green, red, and light spheres denote the positions of the chlorine, sodium, oxygen, and hydrogen atoms. The small arrows denote the dipole of the water molecules that cancel each other out. We developed a new symmetry-driven structure search (SYDSS) algorithmto predict novel materials with ab initio simulations. In ourrecent article, we predict water andsalt form a novel compound at high pressure. While at ambientconditions, water can only incorporate a modest amount of salt, wepredict that both materials form a novel 1:1 stoichiometricH₂O-NaCl compound at high pressure. It is well-known thathigh pressure changes the crystal structure of materials, novelmaterials may form, and immiscible compounds can become miscible. Inthe same article, we also predict twounusual carbon oxides, C₂O and C₄O, to form whileat ambient pressures, only CO₂ and CO are known to exist. How Super-Earths Generate Their Magnetic Fields
Solid silicates (blue line) are semi-conductors that have excitation gap (green region). Liquid silicates (red line) have no gap and are thus semi-metals. They conduct electricity reasonably well. With the Kepler satellite, thousands of new exoplanets werediscovered. Many of them have been described as Super-Earths sincethey are larger than Earth but also have a rocky composition. Theirinteriors are much hotter than Earth's and part of their mantlesare likely to be liquid. In our recent article, weshowed that the electrical conductvity of liquid mantles aresufficiently high so that Super-Earths can generate magnetic fieldswith their mantles. This is a new regime for the generation ofplanetary magnetic fields. Our magnetic field on Earth is generated inthe liquid outer iron core. On Jupiter, it arises from the convectionof liquid metallic hydrogen. On Uranus and Neptune, it is assumed tobe generated in the ice layers. Now we have added molten rocks tothis diverse list of field generating materials. This also impliesthat the magma ocean that existed on the early Earth generated amagnetic field. Aluminum at Extreme Temperature-Pressure Conditions
T-P path of experiments with multiple intermediate shocks.
Valence band gap predicted with our DFT-MD simulations and two semi-analytical models.In our recent article, we studiedaluminum at extreme pressure and temperature conditions with pathintegral Monte Carlo and density functional molecular dynamicssimulations. We derive the equation of state and various electronicproperties. In laboratory experiments, one typically uses shock wavesto reach such extreme conditions. The material becomes very hot if justa single shock is employed. The graph on the left illustrates thatcomparatively low, nearly isentropic temperature conditions can bereached when a number of smallershocks are employed instead.
Aluminum is metal. However, there is gap in the electronic density of states between the 2p and theconduction band. This gap is expected to close at very high density whenthe bound 2p state merge with the free particle states. The twosemi-analytical theories (Stewart-Pyatt and REODP) predict the gap toclose rather rapidly with increasing compression. Conversely, with myDFT-MD simulations, we find the magnitude of the gap hardly changesup to 12-fold compression. This stark disagreement is subject tofurther investigations. Ab initio Simulations of Superionic H₂O, H₂O₂, and H₉O₄ Compounds2O in P2₁/c structure" border="0" height="200">
Superionic water in novel P2₁/c structure.2O₂ compound" border="0" height="200">
Superionic H₂O₂ compound.Deep in the interior of Uranus and Neptune, water has been predictedto occur in a novel, superionicform. In our latest article, we useab initio Gibbs free energy calculations to demonstrate thatsuperionic water changes from face-centeredcubic form to a novel structure with P2₁/c symmetryat 23 Mbar. At even higher pressure of 69 Mbar, superionic water is nolonger stable. It decomposes into two superionicH₂O₂ and H₉O₄ compounds.Simulations of CH pastics, the ablator material in ICF experimentsPolymeric CH structure at high pressure and temperature. The blue and white spheres denote the C and H atoms, respectively. The yellow isosurface denotes the electron density.Density-temperature conditions of our simulations. The black and red triangles label our PIMC and DFT-MD simulations, respectively.In these two articles, (a)and (b), we investigate CHpastic materials at extreme pressure-temperature conditions that arerelevant to inertial confinement fusion experiments. Such hydrocarbonplasmas are of broad interest to laser shock experimentalists, highenergy density physicists, and astrophysicists. Our project has beensupport by a Bluewaters computertime allocation.Jupiter interor models with dilute cores to explain data from the NASA mission Juno
Jupiter interior model with a dilute core. Most models for Jupiter's formation assume it started with a dense core of rock and ice. Once that reached a critical mass of ~10 Earth masses, the run-away accretion of hydrogen-helium gas set in, which lasted until Jupiter had consumed all the gas in its vicinity, leading to a giant planet of 318 Earth masses.
While the temperature and pressure conditions in the planet's center reached ~16000 K and ~40 Mbar, the fate of the core remains ill-understood. Typical core materials like water ice, MgO, SiO₂, and iron are all soluable in hydrogen, which assumes a metallic state under these extreme conditions. It is not unclear, however, if there was sufficent convective energy in Jupiter's early history to spread out the heavy core materials against the forces of gravity.
Here we construct a series of models for Jupiter's interior in order to match the recent gravity measurments of the Juno spacecraft. We demonstrate models with a dilute core match the observations better lending support to the hypothesis that heavy material in Jupiter's core have been redistributed over a substantial fraction of the planet's radius. Different terms ranging diffuse, dilute, expanded and even fuzzy have been invoked to describe such a core. A Jupiter model with a dilute core is shown on the left.Simulations of Calcite V Propeller PhaseCalcite I crystal structure. The blue, brown, and red spheres denote the Ca, C and O ions respectively. The red isosurface denotes the electron density.Calcite V propeller phase. The yellow isosurface denotes the density of oxygen ions that emerges from the propeller rotation of the carbonate CO₃^2- ions.With ab initio computer simulations, we studied the unusualpropeller motion of the carbonate CO₃^2- ions inphase V of calcite (CaCO₃). We found that the ionsperform a tumbling motion and instead of rotating like a perfectlymounted propeller. We also demonstrated that this phase is denser thanthe liquid implying a negative slope of the melting line.Heavy Elements in Giant Planet InteriorsComputer simulation of a hot, dense mixture of hydrogen (white), helium (green) and iron (yellow spheres) atoms. Giant planets are primarily composed of hydrogen and helium but they also contain a small amount of heavier elements. In the atmosphere they make up less than 3% by mass but they dominate the planets opacity. Without their presence we would be able to see through Jupiter's molecular layer and directly observe the planet's metallic interior where its magnetic field is generated. Most scientists assume Jupiter has a core composed of heavy elements. Its size and composition is uncertain but we estimated its mass to be worth 12 Earth masses. The total heavy element fraction in the planet could be as high as 7%.
In this article, Francois Soubiran investigates the properties of various heavy elements in giant planet interiors. The equation of state is computed for C, N, O, Si, Fe, MgO and SiO₂ mixed with hydrogen and helium. Effective mixing rules are derived to make models of giant planet interiors more accurate.Review article entitled "Understanding Jupiter's Interior"This article provides an overview of how models of giant planetinteriors are constructed. We review measurements from past spacemissions that provide constraints for the interior structure ofJupiter. We discuss typical three-layer interior models that consistof a dense central core and an inner metallic and an outer molecularhydrogen-helium layer. These models rely heavily on experiments,analytical theory, and first-principle computer simulations ofhydrogen and helium to understand their behavior up to the extremepressures ~10 Mbar and temperatures ~10,000 K. Wereview the various equations of state used in Jupiter models andcompare them with shock wave experiments. We discuss the possibilityof helium rain, core erosion and double diffusive convection mayhave important consequences for the structure and evolution of giantplanets.
The diagram on the left shows the radius and fractional mass as function of mass for a typical model. The color label various layers. Model for Jupiter's Interior Constructed Before Arrival of Juno SpacecraftTemperature-pressure profiles for Jupiter's interior during the planet's evolution.When the Junospacecraft arrives at Jupiter in July of this year, it will mapout the planet's gravity field with unprecedented precision. What canwe expect to learn about Jupiter's interior? Based on earliermeasurements and on results from ab initio computer simulationsof mixtures of hydrogen, helium, and some heavier elements, Bill Hubbardand I put together a number of different interior models (ApJ, 2016). We predict a massive core of 12 Earth masses consistent with earlier models. Furthermore, wepredict that helium rainhas occurred on this planet for some time, which is a directconsequence of combining the measurements of the Galileoentry probe with results from ab initio calculations forthe hydrogen-helium immiscibilityand adiabats.What is the Composition of the Deep Earth Mantle?
Shear wave splitting strength.Despite a wealth of seismic observations, many questions about thecompositions of the Earth's mantle have remained unanswered. In arecent study (EPSL, 2016) lead by Shuai Zhang, we show thatthe assumption of a pyrolitic composition for the deep Earth is ingood agreement with the preliminary reference Earth model (PREM), which isa 1D seismological representation of the Earth's interior. In collaboration withTao Liu and Stephen Stackhouse (Leeds U.) and Sanne Cottaar (CambridgeU.), we performed ab initio molecular dynamics to calculate theelastic and seismic properties of pure, Fe³⁺ andFe²⁺, and Al³⁺ bearing MgSiO₃perovskite and post-perovskite over a wide range of pressures,temperatures, and Fe/Al compositions.New Path Integral Monte Carlo Simulation Technique for Second-Row ElementsNucleus-electron correlation functions.In our recent publication in PhysicalReview Letters, Kevin Driver and I extended the applicability range offermionic path integral Monte Carlo simulations to heavier elementsand lower temperatures by introducing various localized nodalsurfaces. Hartree-Fock nodes yield the most accurate predictionfor pressure and internal energy that we combine with the results fromdensity functional molecular dynamics simulations to obtain aconsistent equation of state for hot, dense silicon under plasmaconditions and in the regime of warm dense matter (2.3-18.6 g/cc,5x10⁵ - 1.3x10⁸K). The shock Hugoniot curve isderived and the structure of the fluid is characterized with paircorrelation functions. On the left, we estimate the degree ofionization by comparing the integrated nucleus-electron paircorrelation functions from PIMC (symbols) with results for isolatedatoms (black dashed lines).Oxygen and Nitrogen in the Regime of Warm Dense Matter
Phase diagram of nitrogen.In two articles, Kevin Driver, Francois Soubiran, Shuai Zhang, and Icombine path integral Monte Carlo simulations and density functionalmolecular dynamics to study oxygenand nitrogen in the regime of warm dense matter. We characterize both material atextreme pressure and temperature conditions that exist in stellarinteriors and can be probed with shock wave experiments. We use paircorrelation functions and the electronic density of states to describechanges in the structure of the plasma. We compute the shock Hugoniotcurves to compare with laboratory experiments. For nitrogen, wecharacterize the regime of molecular dissociation that leads to aregion of dP/dT<0 at high pressure, which is shown in green in thephase diagram on the left.Do Uranus and Neptune have oceans?Simulation of a H₂-H₂O mixture.Ice giant planets are typically assumed to have a hydrogen-richatmosphere, an intermediate ice layer, and a rocky core. Suchthree-layer models satisfy the observational constraints for Uranusand Neptune. However, it remains unclear whether these planets haveoceans, which would imply the existence of a sharp boundarybetween the hydrogen and water layers. Alternatively, the density andthe water contents of the atmosphere could increase gradually. Recentlaboratory experiments by Bali at el. (2013) favored the oceanhypothesis. In our ApJarticle, Francois Soubiran and I used ab initio computersimulations to determine whether H₂ and H₂O aremissible at high pressure. Contrary to the experimental predictions,we find that both materials are fully miscible under ice giantinterior conditions. We predict that these planets can only haveoceans if icy building blocks were delivered before the gas wasaccreted during planet formation.Do iron and rocks become miscible in the interiors of terrestrial planets?Simulation of a liquid iron-MgO mixture. The brown,green, and red spheres denote Fe, Mg, and O atoms. The grey surfacesshow the electron density.All known terrestrial planets have a separate iron core and a rockymantle because metallic iron has a low solubility in rocky materialsunder typical pressure-temperature conditions in the planetaryinteriors. However, at sufficiently high temperatures, all materialseventually become miscible, even oil and water.
In our recent article,Sean Wahl and I use ab initio computer simulations todetermine what temperature would be required for iron and MgO tobecome miscible in all proportions. We find that the requiredtemperature rises from 4000 to 10,000 K as the pressure is increasedfrom 0 to 500 GPa. Such extreme conditions can be reached during agiant impact on a terrestral planet, implying that not all iron wouldsettle into core during such an event.Recalibration of giant planet mass-radius relationship with ab initio simulationsRevised mass-radius relation
for giant exoplanets. Our new simulation data are shown in red.Using density functional molecular dynamics simulations, we determinethe equation of state for hydrogen-helium mixtures spanningdensity-temperature conditions typical of giant planet interiors. Inour manuscript, a comprehensive equationof state table with 391 density-temperature points isconstructed and the results are presented in form of two-dimensionalfree energy fit for interpolation.
We present a revision to themass-radius relationship which makes the hottest exoplanets increasein radius by ~0.2 Jupiter radii at fixed entropy and for massesgreater than ~0.5 Jupiter mass. This change is large enough tohave possible implications for some discrepant "inflated giantexoplanets".
Our full EOS table as well as our free energy interpolation code has just been made available here.Superionic phase change in water: consequences for Uranus and Neptune In the interiors of Uranus and Neptune (dashed lines in the leftfigure), water is predicted to occur in a superionic state where theoxgyen atoms remain stationary like in a solid while the hydrogenatoms diffuse throughout the crystal like a fluid. Here, weshow that, at 1.0±0.5 Mbar, the oxygen sub-lattice in superionicwater changes from a body-centered cubic lattice (middle) to anface-cented cubic lattice (right). This transformation lead to a moreefficient packing but also reduces the hydrogen diffusion rate, whichmay have further implications for electronic conductivity and magneticdynamo in Uranus and Neptune. Our results were highlighted by Phys.orgNovel chemistry at high pressure: H₄O forms from hydrogen and water ice
Oxygen (red) and hydrogen (blue) atoms in the new H₄O structure.Water and hydrogen at high pressure make up a substantial fraction ofthe interiors of giant planets. Using ab initio randomstructure search methods we investigate the ground-state crystalstructures of water, hydrogen, and hydrogen-oxygen compounds. Here, we find that, at pressuresbeyond 14 Mbar, excess hydrogen is incorporated into the ice phase toform a novel structure with H₄O stoichiometry.
We alsopredict two new ground state structures of water ice with P21/mand I4/mmm symmetry to form at 135 and 330 Mbar,respectively. Here is a slide that summarizes theseven new high pressure ice phases that were recently predicted withab initio calculations.Methane Ice in Uranus and Neptune Assumes a Polymeric and Metallic StateThe four snapshots from our ab initio simulations show howmethane gas at high pressure and temperature forms long hydrocarbonchains. The blue and white spheres denotes the carbon (C) and hydrogenatoms, respectively. The red lines indicate the C-C bonds thatincrease from left to right. In our recent paper, we show that the resultingpolymeric state is metallic and exists in the interiors of Uranus andNeptune. We also predict how such a transformation on the atomisticlevel can be identified with macroscopic shock wave experiments.Path Integral Simulation Technique to Study Plasmas of First-Row ElementsPath integral Monte Carlo simulations are a powerful tool to studyquantum systems at high temperature but applications to elementsbeyond hydrogen and helium with core electrons have so far not been possible. Inour recent PRLarticle, Kevin Driver and I develop a new all-electron path integral Monte Carlotechnique with free-particle nodes for warm dense matter and apply itto water and carbon plasmas. Our results for pressures, internalenergies, and pair correlation functions compare well with densityfunctional molecular dynamics at temperatures of(2.5-7.5)10⁵K. Both methods together form a coherentequation of state over a density-temperature range of 3-12 g/cc and 10⁴-10⁹ K.Erosion of Rocky Cores in Giant Gas PlanetsGas giants are believed to form by the accretion of hydrogen-heliumgas around an initial protocore of rock and ice. The question ofwhether the rocky parts of the core dissolve into the layer ofmetallic fluid hydrogen following formation has significantimplications for planetary structure and evolution. Here we use ab initiocalculations to study rock solubility in fluid hydrogen, choosingmagnesium oxide as a representative example of planetary rockymaterials, and find MgO to be highly soluble in H for temperatures inexcess of approximately 10000 K, implying significant redistributionof rocky core material in Jupiter and larger exoplanets.Hydrogen Equation of State Computed for Fusion ApplicationsUsing path integral Monte Carlo simulations we have derived anequation of state (EOS) table for deuterium that covers typical intertialconfinement fusion conditions at densities ranging from 0.002 to 1596g/cm³and temperatures of 1.35 eV ~ 5.5 keV. The small grey circles in thediagram on the left indicate the temperature-density conditions of oursimulations. The EOS and related results are summarized in an article that has been published in Physical Review B.Bonding Pattern in Ice at High PressureThe bonding properties of water ice at high pressure are studied in this article. By comparing the Wannier orbitals in the Pnma structure (shown in the image on the left), one can tell that they differ substantially from the sp³hybridization in the ice X phase at lower pressures. Most strikingly,the white orbitals are not aligned with any hydrogen bond.Dissolution of Icy Core Materials Gas Giant Planets
Simulations predict water ice to be unstable above 3000 Kelvin when exposed to metallic hydrogenThe four giant planets in our solar system grow so large because icycomets made their cores grow much faster than those of terrestrialplanets, which enabled them to accrete large amounts of gas. With abinitio simulations, Hugh Wilson and I demonstrate in our recent manuscriptthat water ice is not thermodynamically stable at the temperature andpressure conditions where core is exposed to the layer of metallichydrogen above. This implies that the cores in Jupiter and Saturn havebeen eroded over time, with the icy material being redistributedconvectively throughout the planet.
Our work has implications for constraining the interiorstructure and evolution of giant planets and will be relevant for theinterpretation of data from NASA's Juno mission to Jupiter (to be launched inAugust 2011). Core erosion could also provide a significant flux ofheavy elements to the atmosphere of exoplanets and may explain whysome of them have significantly inflated radii.Simulations predict water ice to become a metal at megabar pressures
Four high pressure phases of iceWater ice is one of the most prevalent substances in the solar system,with the majority of it existing at high pressures in the interiors ofgiant planets. The known phase diagram of water is extremely rich, withat least fifteen crystal phases observed experimentally. In our article in Physical Review Letters (seealso cond-mat), HughWilson and I explore the phase diagram of water ice by means of abinitio computer simulations and predict twonew phases to occur at megabar pressures. In the figure fromtop to bottom, you see

1) ice X the highest pressure phase seen in experiments,
2) the Pbcm phase that was predicted with computer simulations in 1996,
3) our new Pbca phase that transforms out of the Pbcm phase via a phonon instability at 7.6 Mbar, and finally
4) our new Cmcm structure that ismetallic and predicted to occur at 15.5 Mbar.

The known high pressure ice phases VII, VIII, X and Pbcm aswell as our Pbca phase are all insulating and composed of twointerpenetrating hydrogen bonded networks, but the Cmcmstructure is metallic and consists of corrugated sheets of H and Oatoms. The H atoms are squeezed into octahedral positions betweennext-nearest O atoms while they occupy tetrahedral positions betweennearest O atoms in the ice X, Pbcm, and Pbca phases.Why is neon missing from Jupiter's atmosphere? Indirect evidence of helium rain
Jupiters interior. Helium rain occurs in the immiscibility layer and depletes the upper layer of both helium and neon.When the Galileo entryprobe entered Jupiter's atmosphere in 1995, it measured thatthe inert gas neon was depleted by afactor of 10 compared to the composition of sun, which represents theconcentrations in nebula that formed our solar system with all itseight planets. So where is all the neon gone that was present inJupiter initially? Using ab initio computersimulations Hugh Wilson and I link the missing neon to anotherprocess that was proposed to occur inside Jupiter: heliumrain.
There is indirect evidence from luminosity measurements that heliumrain occurs on Saturn but it was unclear whether it occurs insideJupiter also. Our calculations now show that neon preferentiallydissolves into helium droplets and it is therefore gradually sequesteredinto the deeper interior as the helium rain falls. The remaining hydrogen-rich envelope isslowly depleted of both neon and helium. The measured concentrationsof both elements agree quantitatively with our calculations.
Read commentary by J. Fortney "Peering into Jupiter", UC Berkeley's press release, Discovery Channel and LA Times articles.
Quantum Monte Carlo Study of the Insulator-to-Metal Transition in Solid Helium

Insulator-to-Metal Transition in Solid Helium at High PressureMetallic solid helium is present in the outer layers of White Dwarfstars. The cooling rate of White Dwarfs is regulated by the heat flowfrom the hot interior to the colder exterior. Theinsulator-to-metal transition is of interest because it marks thepoint where heat transport switches from electronicconductions to photon diffusion. In our paper, theinsulator-to-metal transition in solid helium at high pressure isstudied with different first-principles simulations. Diffusion quantumMonte Carlo (QMC) calculations predict that the band gap closes at adensity of 21.3 g/cc and a pressure of 25.7 terapascals, which is 20%higher in density and 40 higher in pressure than predicted by standarddensity functional calculations. The metallization density derivedfrom GW calculations is found to be in very close agreement with QMCpredictions. Path integral Monte Carlo calculations showed thatthe zero-point motion of the nuclei has no significant effect on themetallization transition. Simulation of Hydrogen-Helium Mixtures in Planetary Interiors
Helium in molecular hydrogen
Helium in metallic hydrogenWe performed density functional molecular dynamics simulation tocharacterize hydrogen-helium mixtures in the interior of solar andextrasolar giant planets. In thisarticle, we address outstanding questions about their structureand evolution e.g. whether Jupiter has a rocky core and if it wasformed by a core accretion process. We describe how the presence ofhelium defers the molecular-to-metallic transition in hydrogen tohigher pressures by stabilizing hydrogen molecules. First Principles Simulation of Fluid Helium at High Pressure

Shock hugoniot curves for precompressed hydrogen and helium.Shock wave experiments allow one to study a material's properties athigh pressure and temperature. In thisarticle, weused first-principles computer simulation to predict the properties ofshock fluid helium at megabar pressures. The simulations show that thecompressibility of helium is substantially increased by electronicexcitations. A maximum compression ratio of 5.24-fold the initialdensity was predicted for 360 GPa and 150000 K. This resultdistinguishes helium from deuterium, for which simulations predicted amaximum compression ratio of 4.3. If the sample are precompressedstatically the compression ratio is reduced, which is shown in theleft graph. Ab Initio Simulations of Liquid Oxygen under Pressure

Spin fluctuations present molecular oxygen (left) are suppressed at high pressures (right).In recent shock wave experiments [Phys. Rev. Lett. 86, 3108 (2001)],the conductivity of liquid oxygen was measured for pressures up to 1.8Mbar and indications for a insulator-metal transition were found. In this article, we report results from density functional molecular dynamics simulations of dense liquid oxygenclose to the metal-insulator transition. We havefound that band gap closure occurs in the molecular liquid, with aslow transition from a semi-conducting to a poor metallic stateoccurring over a wide pressure range. At approximately 80 GPa,molecular dissociation is observed in the metallic fluid. Spinfluctuations play a key role in determining the electronic structureof the low pressure fluid, while they are suppressed at high pressure.Dense Plasma Effects on Nuclear Reaction RatesMany-body enhancement of nuclear reaction rates h(0) as function of the coupling parameter.Dense plasma effects can cause an exponenial change in charge particlenuclear reaction rates important in stellar evolution. In this article, reaction ratesin dense plasmas are examined using path integral Monte Carlo. Quantumeffects causes a reduction in the many body enhancement of thereaction rate, h(0), compared to the classical value. This is shown infigure on the left for different quantum parameters. This reductioncan be attributed to the "quantum smearing" of the Coulomb interactionat the short range resulting in a reduced repulsion between thereacting pair and surrounding particles.Lowering of the Kinetic Energy in Interacting Quantum SystemsTemperature density region of kinetic energy lowering for dense hydrogen and the electron gas.The equilibrium momentum distribution is of fundamental importance tocharacterize many-body systems. In contrast to classical systems wherethe distribution is always Maxwellian, in quantum systems thedistribution depends on particle statistics, bosons or fermions, aswell as on interactions and can display interparticle correlations,which are the basis of superfluidity and superconductivity. In this article, wereport and explain a surprising effect of interactions in quantumsystems on the one particle momentum distribution and kineticenergy. Interactions never lower the ground state kinetic energy of aquantum system. However, at nonzero temperature, where the systemoccupies a thermal distribution of states, interactions can reduce thekinetic energy below the noninteracting value. This isdemonstrated using PIMC simulations for dense hydrogen and the electron gas.Understanding hot dense hydrogen with PIMC simulations Molecular liquidMolecular metallic liquidMetallic liquidThe high temperature phase diagram of hydrogenAt which pressure and density does hydrogen become metallic? At low densities up to about rs=2.6, the properties of hydrogen including the equation of state are well understood. Processes like the thermal dissociation of moleculescan be modelled accurately with PIMC. The resulting proton-proton pair correlation functions are shown. Single and double shock Hugoniot curves from PIMC simulationsSingle shock hugoniot results Single and double shock hugoniot in the phase diagram. Double shock hugoniot results Publications 129. F. Gonzalez-Cataldo, B.K. Godwal, K. Driver, R. Jeanloz, B. Militzer, "A Model of Ramp Compression of Diamond from Ab Initio Simulations", Phys. Rev. B 104 (2021) 134104. DOI: 10.1103/PhysRevB.104.134104 128. S. M. Wahl, D. Thorngren, T. Lu, B. Militzer, "Tidal Response and Shape of Hot Jupiters", Astrophysical J., in press (2021). 127. J. Wu, F. Gonzalez-Cataldo, B. Militzer, "High Pressure Phase Diagram of Beryllium from Ab Initio Free Energy Calculations", Phys. Rev. B 104 (2021) 014103. DOI: 10.1103/PhysRevB.104.014103. Available on the arXiv. 126. T. Dornheim, M. Boehme, B. Militzer, J. Vorberger, "Ab initio path integral Monte Carlo approach to the momentum distribution of the uniform electron gas at finite temperature without fixed nodes", Phys. Rev. B 103 (2021) 205142. DOI: 10.1103/PhysRevB.103.205142. Available on the arXiv. 125. G. Massacrier, M. Boehme, J. Vorberger, F. Soubiran, B. Militzer, "Reconciling Ionization Energies and Band Gaps of Warm Dense Matter Derived with Ab Initio Simulations and Average Atom Models", Phys. Rev. Res. 3 (2021) 023026. DOI: 10.1103/PhysRevResearch.3.023026. 124. H. D. Whitley, G. E. Kemp, C. Yeamans, Z. Walters, B. E. Blue, W.Garbett, M. Schneider, R. S. Craxton, E. M. Garcia, P. W. McKenty, M. Gatu-Johnson, K. Caspersen, J. I. Castor, M. Dane, C. L. Ellison, J. Gaffney, F. R. Graziani, J. Klepeis, N. Kostinski, A. Kritcher, B. Lahmann, A. E. Lazicki, H. P. Le, R. A. London, B. Maddox, M. Marshall, M. E. Martin, B. Militzer, A. Nikroo, J. Nilsen, T. Ogitsu, J. Pask, J. E. Pino, M. Rubery, R. Shepherd, P. A. Sterne, D. C. Swift, L. Yang, S. Zhang "Comparison of ablators for the polar direct drive exploding pusher platform", J. High Energy Density Physics 38 (2021) 100928. DOI: 10.1016/j.hedp.2021.100928 Available on the arXiv. 123. B. Militzer, F. Gonzalez-Cataldo, S. Zhang, K. P. Driver, F. Soubiran, "First-principles equation of state database for warm dense matter computation", Phys. Rev. E 103 (2021) 013203. DOI: 10.1103/PhysRevE.103.013203 Available on the arXiv. Click here to reach our FPEOS webpage. 122. R. Domingos, M. M. Tremblay, D. L. Shuster, B. Militzer, "Simulations and experiments reveal effect of nanopores on helium diffusion in quartz", ACS Earth and Space Chemistry 4 (2020) 1906. DOI: 10.1021/acsearthspacechem.0c00187 121. B. Militzer, F. Gonzalez-Cataldo, S. Zhang, H. D. Whitley, D. C. Swift, M. Millot, "Nonideal Mixing Effects in Warm Dense Matter Studied with First-Principles Computer Simulations", J. Chem. Phys. 153 (2020) 184101. DOI: 10.1063/5.0023232. Available on the arXiv. 120. F. Soubiran, B. Militzer, "Anharmonicity and Phase Diagram of Magnesium Oxide in the Megabar Regime", Phys. Rev. Lett. 125 (2020) 175701. DOI: 10.1103/PhysRevLett.125.175701. 119. F. Gonzalez-Cataldo, F. Soubiran, B. Militzer, "Equation of State of Hot, Dense Magnesium Derived with First-Principles Computer Simulations", Physics of Plasmas 27 (2020) 092706. DOI: 10.1063/5.0017555. Available on the arXiv. 118. M. A. Olson, S. Bhatia, P. Larson, B. Militzer, "Prediction of Chlorine and Fluorine Crystal Structures at High Pressure Using Symmetry Driven Structure Search with Geometric Constraints", J. Phys. Chem. 153 (2020) 094111. Available on the arXiv. 117. S. Zhang, M. C. Marshall, L. H. Yang, P. A. Sterne, B. Militzer, M. Daene, J. A. Gaffney, A. Shamp, T. Ogitsu, K. Caspersen, A. E. Lazicki, D. Erskine, R. A. London, P. M. Celliers, J. Nilsen, H. D. Whitley, "Benchmarking boron carbide equation of state using computation and experiment", Phys. Rev. E 102 (2020) 053203. Available on the arXiv. 116. F. Gonzalez-Cataldo, B. Militzer, "Thermal and Pressure Ionization in Warm, Dense MgSiO₃ Studied with First-Principles Computer Simulations", AIP Conference Proceedings 2272 (2020) 090001. DOI: 10.1063/12.0000793. Available on the arXiv. 115. S. M. Wahl, M. Parisi, W. M . Folkner, W. B. Hubbard, B. Militzer, "Equilibrium Tidal Response of Jupiter: Detectability by Juno Spacecraft", Astrophys. J. 891:42 (2020) 1 (DOI). 114. M. Millot, S. Zhang, D. E. Fratanduono, F. Coppari, S. Hamel, B. Militzer, D. Simonova, S. Shcheka, N. Dubrovinskaia, L. Dubrovinsky, J. H. Eggert, "Recreating giants impacts in the laboratory: Shock compression of MgSiO₃ bridgmanite to 14 Mbar", Geophys. Res. Lett. 47 (2020) e2019GL085476. 113. F. Gonzalez-Cataldo, F. Soubiran, H. Peterson, B. Militzer, "Path Integral Monte Carlo and Density Functional Molecular Dynamics Simulations of Warm, Dense MgSiO₃", Phys. Rev. B 101 (2020) 024107. Available on the arXiv. 112. F. Soubiran, F. Gonzalez-Cataldo, K. P. Driver, S. Zhang, B. Militzer, "Magnesium Oxide at Extreme Temperatures and Pressures Studied with First-Principles Simulations", J. Chem. Phys. 151 (2019) 214104. 111. B. Militzer, S. Wahl, W. B. Hubbard, "Models of Saturn's Interior Constructed with an Accelerated Concentric Maclaurin Spheroid Method", Astrophysical Journal 879 (2019) 78. Available on the arXiv. 110. S. Zhang, A. Lazicki, B. Militzer, L. H. Yang, K. Caspersen, J. A. Gaffney, M. W. Däne, J. E. Pask, W. R. Johnson, A. Sharma, P. Suryanarayana, D. D. Johnson, A. V. Smirnov, P. A. Sterne, D. Erskine, R. A. London, F. Coppari, D. Swift, J. Nilsen, A. J. Nelson, H. D. Whitley, "Equation of state of warm-dense boron nitride combining computation, modeling, and experiment", Phys. Rev. B 99 (2019) 165103. Available on the arXiv. 109. L. Iess, B. Militzer, Y. Kaspi, P. Nicholson, D. Durante, P. Racioppa, A. Anabtawi, E. Galanti, W. Hubbard, M. J. Mariani, P. Tortora, S. Wahl, M. Zannoni, "Measurement and implications of Saturn's gravity field and ring mass", Science 17 Jan 2019:eaat2965. DOI: 10.1126/science.aat2965. 108. B. Militzer, E. L. Pollock, D. Ceperley, "Path Integral Monte Carlo Calculation of the Momentum Distribution of the Homogeneous Electron Gas at Finite Temperature", J. High Energy Density Physics 30 (2019) 13-20. 107. R. Domingos, K. M. Shaik, B. Militzer,"Prediction of Novel High Pressure H₂O-NaCl and Carbon Oxide Compounds with Symmetry-Driven Structure Search Algorithm", Phys. Rev. B 98 (2018) 174107. Also available on the arXiv. 106. F. Soubiran, B. Miltzer,"Electrical conductivity and magnetic dynamos in magma oceans of Super-Earths",Nature Communications 9 (2018) 3883. 105. S. Zhang, B. Militzer, M. C. Gregor, K. Caspersen, L. H. Yang, T. Ogitsu, D. Swift, A. Lazicki, D. Erskine, R. A. London, P. M. Celliers, J. Nilsen, P. A. Sterne, and H. D. Whitley"Theoretical and experimental investigation of the equation of state of boron plasmas",Phys. Rev. E 98 (2018) 023205, available on the arXiv. 104. K. P. Driver, F. Soubiran, B. Militzer, "Path integral Monte Carlo simulations of hot, dense aluminum", Physical Review E 97 (2018) 063207. 103. L. Iess, W. M. Folkner, D. Durante, M. Parisi, Y. Kaspi, E. Galanti, T. Guillot, W. B. Hubbard, D. J. Stevenson, J. D. Anderson, D. R. Buccino, L. Gomez Casajus, A. Milani, R. Park, P. Racioppa, D. Serra, P. Tortora, M. Zannoni, H. Cao, R. Helled, J. I. Lunine, Y. Miguel, B. Militzer, S. Wahl, J. E. P. Connerney, S. M. Levin, S. J. Bolton, "Measurement of Jupiter's asymmetric gravity field", Nature 555 (2018) 220. 102. Y. Kaspi, E. Galanti, W. B. Hubbard, D. J. Stevenson, S. J. Bolton, L. Iess, T. Guillot, J. Bloxham, J. E. P. Connerney, H. Cao, D. Durante, W. M. Folkner, R. Helled, A. P. Ingersoll, S. M. Levin, J. I. Lunine, Y. Miguel, B. Militzer, M. Parisi, S. M. Wahl "Jupiter's atmospheric jet streams extend thousands of kilometres deep", Nature 555 (2018) 223. 101. T. Guillot, Y. Miguel, B. Militzer, W. B. Hubbard, Y. Kaspi, E. Galanti, H. Cao, R. Helled, S. M. Wahl, L. Iess, W. M. Folkner, D. J. Stevenson, J. I. Lunine, D. R. Reese, A. Biekman, M. Parisi, D. Durante, J. E. P. Connerney, S. M. Levin ">104 (2010) 121101. Read commentary by J. Fortney "Peering into Jupiter" in Physics 3 (2010) 26, UC Berkeley's press release, Discovery Channel and LA Times articles.
47. K. P. Esler, R. E. Cohen, B. Militzer, J. Kim, R.J. Needs, and M.D. Towler,"Fundamental high pressure calibration from all-electron quantum Monte Carlo calculations", Phys. Rev. Lett. 104 (2010) 185702. 46. K. P. Driver, R. E. Cohen, Z. Wu, B. Militzer, P. Lopez Rios, M. D. Towler, R. J. Needs, and J. W. Wilkins"Quantum Monte Carlo for minerals at high pressures: Phase stability, equations of state, and elasticity of silica", Proc. Nat. Acad. Sci. 107 (2010) 9519. 45. P. Beck, A.F. Goncharov, J. A. Montoya, V.V. Struzhkin, B. Militzer, R.J.Hemley, and H.-K. Mao,''Response to Comment on Measurement ofthermal diffusivity at high-pressure using a transient heatingtechnique'', Appl. Phys. Lett. 95 (2009) 096101. 44. J. J. Fortney, I. Baraffe, B. Militzer, chapter "Interior Structureand Thermal Evolution of Giant Planets", in "Exoplanets",ed. S. Seager, Arizona Space Science series (2009). 43. B. Militzer, "Correlations in Hot Dense Helium", J Phys. A 42 (2009) 214001, cond-mat/09024281. 42. J. J. Fortney, S. H. Glenzer, M. Koenig, B. Militzer, D. Saumon, and D. Valencia,"Frontiers of the Physics of Dense Plasmas and Planetary Interiors: Experiment, Theory, Applications", Physics of Plasmas 16 (2008) 041003. 41. B. Militzer and W. B. Hubbard,"Comparison of Jupiter Interior Models Derived from First-Principles Simulations", Astrophysics and Space Science 322 (2009) 129, astro-ph/08074266. 40. S. A. Khairallah and B. Militzer,"First-Principles Studies of the Metallization and the Equation of State of Solid Helium", Phys. Rev. Lett. 101 (2008) 106407, physics/08054433. 39. B. Militzer,"Path Integral Monte Carlo and Density Functional Molecular Dynamics Simulations of Hot, Dense Helium", Phys. Rev. B 79 (2009) 155105, cond-mat/08050317. 38. B. Militzer, W. B. Hubbard, J. Vorberger, I. Tamblyn, and S.A. Bonev,"A Massive Core in Jupiter Predicted From First-Principles Simulations",Astrophysical Journal Letters 688 (2008) L45, astro-ph/08074264. 37. P. Beck, A. F. Goncharov, V. Struzhkin, B. Militzer, H.-K. Mao, and R. J. Hemley"Measurement of thermal diffusivity at high pressure using a transient heating technique", Appl. Phys. Lett. 91 (2007) 181914. 36. B. Militzer, W. B. Hubbard,"Implications of Shock Wave Experiments with Precompressed Materials for Giant Planet Interiors", AIP conference proceedings 955 (2007) 1395. 35. J. Vorberger, I. Tamblyn, S.A. Bonev, B. Militzer, "Properties of Dense Fluid Hydrogen and Helium in Giant Gas Planets", Contrib. Plasma Phys. 47 (2007) 375. 34. S. Seager, M. Kuchner, C. A. Hier-Majumder, B. Militzer,"Mass-radius relationship of solid exoplanets", Astrophys. J. 669 (2007) 1279. 33. V. V. Struzhkin, B. Militzer, W. Mao, R. J. Hemley, H.-k. Mao,"Hydrogen Storage in Clathrates", Chem. Rev. 107 (2007) 4133. 32. G. D. Cody, H. Yabuta, T. Araki, L. D. Kilcoyne, C. M. Alexander, H. Ade, P. Dera, M. Fogel, B. Militzer, B. O. Mysen,"An Organic thermometer for Chondritic Parent Bodies", Earth. Planet. Sci. Lett. 272 (2008) 446. 31. J. Vorberger, I. Tamblyn, B. Militzer, S.A. Bonev, "Hydrogen-Helium Mixtures in the Interiors of Giant Planets", Phys. Rev. B 75 (2007) 024206, cond-mat/0609476. 30. B. Militzer, R. J Hemley, "Solid oxygen takes shape", Nature (News Views), 443 (2006) 150. 29. B. Militzer, "First Principles Calculations of Shock Compressed Fluid Helium", Phys. Rev. Lett. 97 (2006) 175501. 28. B. Militzer, R. L. Graham, "Simulations of Dense Atomic Hydrogen in the Wigner Crystal Phase", J. Phys. Chem. Solids, 67 (2006) 2136. 27. B. Militzer, "Hydrogen-Helium Mixtures at High Pressure", J. Low Temp. Phys. 139 (2005) 739. 26. B. Militzer, E. L. Pollock, "Equilibrium Contact Probabilities in Dense Plasmas", Phys. Rev. B, 71 (2005) 134303. 25. J.-F. Lin, B. Militzer, V. V. Struzhkin, E. Gregoryanz, R. J. Hemley, H.-k. Mao, "High Pressure-Temperature Raman Measurements of H₂O Melting to 22 GPa and 900 K", J. Chem. Phys. 121 (2004) 8423. 24. E. L. Pollock, B. Militzer, "Dense Plasma Effects on Nuclear Reaction Rates", Phys. Rev. Lett. 92 (2004) 021101. 23. S. A. Bonev, B. Militzer, G. Galli, "Dense liquid deuterium: Ab initio simulation of states obtained in gas gun shock wave experiments", Phys. Rev. B 69 (2004) 014101. 22. F. Brglez, X.Y. Li, M.F. Stallmann, and B. Militzer, "Evolutionary and Alternative Algorithms: Reliable Cost Predictions for Finding Optimal Solutions to the LABS Problem", Information Sciences, in press, 2004. 21. B. Militzer, F. Gygi, G. Galli, "Structure and Bonding of Dense Liquid Oxygen from First Principles Simulations", Phys. Rev. Lett. 91 (2003) 265503. 20. F. Brglez, X.Y. Li, M.F. Stallmann, and B. Militzer, "Reliable Cost Predictions for Finding Optimal Solutions to LABS Problem: Evolutionary and Alternative Algorithms", Proceedings of The Fifth International Workshop on Frontiers in Evolutionary Algorithms, Cary, NC (2003). 19. B. Militzer, "Path Integral Calculation of Shock Hugoniot Curves of Precompressed Liquid Deuterium", J. Phys. A: Math. Gen. 63 (2003) 6159. 18. B. Militzer, E. L. Pollock, "Lowering of the Kinetic Energy in Interacting Quantum Systems", Phys. Rev. Lett. 89 (2002) 280401. 17. B. Militzer, D. M. Ceperley, J. D. Kress, J. D. Johnson, L. A. Collins, S. Mazevet, "Calculation of a Deuterium Double Shock Hugoniot from Ab Initio Simulations", Phys. Rev. Lett. 87 (2001) 275502. 16. B. Militzer, D. M. Ceperley, "Path Integral Monte Carlo Simulation of the Low-Density Hydrogen Plasma", Phys. Rev. E 63 (2001) 066404. 15. B. Militzer, D. M. Ceperley, "Path Integral Monte Carlo Calculation of the Deuterium Hugoniot", Phys. Rev. Lett. 85 (2000) 1890. 14. B. Militzer, "Path Integral Monte Carlo Simulations of Hot Dense Hydrogen", Ph.D. thesis, University of Illinois at Urbana-Champaign (2000). 13. B. Militzer, E. L. Pollock, "Variational Density Matrix Method for Warm Condensed Matter and Application to Dense Hydrogen", Phys. Rev. E 61 (2000) 3470. 12. B. Militzer, E. L. Pollock, "Introduction to the Variational Density Matrix Method and its Application to Dense Hydrogen", in Strongly Coupled Coulomb Systems 99, ed. by C. Deutsch, B. Jancovici, and M.-M. Gombert, J. Phys. France IV 10 (2000) 315. 11. B. Militzer, W. Magro, and D. Ceperley, "Characterization of the State of Hydrogen at High Temperature and Density", Contr. Plasma Physics 39 (1999) 1-2, 151. 10. W. Magro, B. Militzer, D. Ceperley, B. Bernu, and C. Pierleoni, "Restricted Path Integral Monte Carlo Calculations of Hot, Dense Hydrogen", in Strongly Coupled Coulomb Systems, ed. by G. J. Kalman, J. M. Rommel and K. Blagoev, Plenum Press, New York NY, 1998. 9. W. Ebeling, B. Militzer, and F. Schautz, "Quasi-Classical Theory and Simulation of Two-Component Plasmas", in Strongly Coupled Coulomb Systems, ed. by G. J. Kalman, J. M. Rommel and K. Blagoev, Plenum Press, New York NY, 1998. 8. B. Militzer, W. Magro, and D. Ceperley, "Fermionic Path-Integral Simulation of Dense Hydrogen", in Strongly Coupled Coulomb Systems, ed. by G. J. Kalman, J. M. Rommel and K. Blagoev, Plenum Press, New York NY, 1998. 7. B. Militzer, M. Zamparelli, and D. Beule, "Evolutionary Search for Low Autocorrelated Binary Sequences", IEEE Trans. Evol. Comput. 2 (1998) 34-39. 6. W. Ebeling, B. Militzer, and F. Schautz, "Quasi-classical Theory and Simulations of Hydrogen-like Quantum Plasmas", Contr. Plasma Physics 37 (1997) 2-3, 137. 5. W. Ebeling and B. Militzer, "Quantum Molecular Dynamics of Partially Ionized Plasmas", Phys. Lett. A 226 (1997) 298 4. B. Militzer, "Quanten-Molekular-Dynamik mit reaktiven Freiheitsgraden", in Dynamik, Evolution, Strukturen, ed. J. Freund, Dr. Köster publishing company, Berlin, 1996. 3. B. Militzer, "Quanten-Molekular-Dynamik von Coulomb-Systemen", Logos publishing company, Berlin, 1996, ISBN 3-931216-08-X 2. B.-D. Dörfel and B. Militzer, "Test of Modular Invariance for Finite XXZ Chains", J. Phys. A: Math. Gen. 26 (1993) 4875. 1. A. Richter, G. Kessler, and B. Militzer, "Growth-kinectics of thin-films deposited by laser ablation", 473-478, inLaser Treatment of Materials, ed. by Barry L. Mordike, Oberursel : DGM Informationsgesellschaft, 1992.
Previous interests: low auto-correlated binary sequences (LABS), traffic jams

2003-2007Associate staff member at Geophysical Laboratory of the Carnegie Institution of Washington2000-2003Postdoc in the Quantum Simulations Group at the Lawrence Livermore National Laboratory1996-2000Ph.D. in Prof. Ceperley's group at the University of Illinois at Urbana-Champaign1994-1996Diploma in physics in Prof. Ebeling's group at the Humboldt University at Berlin.
Last updated: 2/4/2021.

TAGS:Militzer Burkhard at Berkeley UC 

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