Research Interests

In my research, I use computer simulations
to understand the interior and evolution of giant planets. Materials
in planetary interiors are exposed to extreme temperature and pressure
conditions that cannot yet be reached with laboratory
experiments. Instead we rely on highly accurate firstprinciples computer simulations techniques. With these methods,
we recently explained why neon is depleted in Jupiter's atmosphere and
provided strong, though indirect evidence for helium rain to occur in
giant planets. Our recent simulations predict core erosion to occur in gas giant planets.
Furthermore I study materials in the
deep mantle of our planet and compare my results with static and
dynamic high pressure experiments. In some cases, computer simulations
provide new insight into properties of materials that cannot be
obtained with experiments. In
other cases we use them to make predictions for the state of matter at
these extreme pressures. Recent examples include fluid helium and water ice at megabar pressures.
My background is in the field of
theoretical condensed matter physics and I am interested in theory and
simulation of novel materials under extreme conditions. I use a
variety of firstprinciples simulation methods including path integral Monte Carlo, groundstate
quantum Monte Carlo, and density functional molecular
dynamics. 
Research Group

Kevin Driver, postdoctorial researcher
Francois Soubiran, postdoctorial researcher
Shuai Zhang, graduate student
Sean Wahl, graduate student
Tanis Leonhardi, graduate student
Formerly in my group at UCB:
Hugh F. Wilson, associate specialist, now at CSIRO in Melbourne.
Felipe Gonzalez, visiting Ph.D. student from the Universidad de Chile.
Stephen Stackhouse, now Lecturer at the University of Leeds.
Saad Khairallah, now at Lawrence Livermore National Laboratory.
Mike Wong, UCB undergraduate student, now PhD student at Caltech.
Benjamin Sherman, visited from CSUN in 2010 and 2011.
Members of my research group at the Carnegie Instiution of Washingon (20032007):
Jan Vorberger, postdoctorial researcher
Ken Esler, postdoctoral researcher
Rebekah Graham, Isaac Tamblyn, Seth Jacobsen (all REU summer students)

Teaching

In the fall of
2008, I introduced EPS 109 "Computer Simulations in
Earth and Planetary Science" as a new course. An
introduction to computer simulation methods will be given and students
learn to program in Matlab. Have a look the animations that the students made during
the 2008,
2009,
2011,
2012,
2013,
2014 and
2015 classes.
In spring of 2011, Dino Bellugi and I introduced a new graduate class EPS 209 "Matlab Applications in Earth Science". Here is a descriptions of the final projects.
I teach the course C12 "The Planets".
A tour of the mysteries and inner workings of our solar
system is presented. The class has over 200 students and is directed
at nonscience majors. Here are some pictures
from our class room demonstrations in 2010 and 2012. This course is also offered as
an online summer class W12. Here are three examples from our series of
recorded lectures: a course introduction,
one on the Kepler mission, and one
on 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 and 2013.
I also participated in a field trip to Yosemite National Park.

Positions

Ph.D. applicants interested in this research should apply
to the department of Earth and
Planetary science or alternatively to the department of Astronomy. The deadline is late in
December every year. Applicants are encouraged to contact me in advance to talk about specific research projects.
I
have no open postdoc positions at the moment but this will change in
the future. You can also work with me by my taking advantage of opportunities in Astronomy.

Heavy Elements in Giant Planet Interiors

Computer 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 SiO_{2} 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 planet
interiors are constructed. We review measurements from past space
missions that provide constraints for the interior structure of
Jupiter. We discuss typical threelayer interior models that consist
of a dense central core and an inner metallic and an outer molecular
hydrogenhelium layer. These models rely heavily on experiments,
analytical theory, and firstprinciple computer simulations of
hydrogen and helium to understand their behavior up to the extreme
pressures ~10 Mbar and temperatures ~10,000 K. We
review the various equations of state used in Jupiter models and
compare them with shock wave experiments. We discuss the possibility
of helium rain, core erosion and double diffusive convection may
have important consequences for the structure and evolution of giant
planets.
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 Spacecraft

Temperaturepressure profiles for Jupiter's interior during the planet's evolution.

When the Juno
spacecraft arrives at Jupiter in July of this year, it will map
out the planet's gravity field with unprecedented precision. What can
we expect to learn about Jupiter's interior? Based on earlier
measurements and on results from ab initio computer simulations
of mixtures of hydrogen, helium, and some heavier elements, Bill Hubbard
and 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, we
predict that helium rain
has occurred on this planet for some time, which is a direct
consequence of combining the measurements of the Galileo
entry probe with results from ab initio calculations for
the hydrogenhelium immiscibility
and adiabats.


What is the Composition of the Deep Earth Mantle?

Shear wave splitting strength.

Despite a wealth of seismic observations, many questions about the
compositions of the Earth's mantle have remained unanswered. In a
recent study (EPSL, 2016) lead by Shuai Zhang, we show that
the assumption of a pyrolitic composition for the deep Earth is in
good agreement with the preliminary reference Earth model (PREM), which is
a 1D seismological representation of the Earth's interior. In collaboration with
Tao Liu and Stephen Stackhouse (Leeds U.) and Sanne Cottaar (Cambridge
U.), we performed ab initio molecular dynamics to calculate the
elastic and seismic properties of pure, Fe^{3+} and
Fe^{2+}, and Al^{3+} bearing MgSiO_{3}
perovskite and postperovskite over a wide range of pressures,
temperatures, and Fe/Al compositions.


New Path Integral Monte Carlo Simulation Technique for SecondRow Elements

Nucleuselectron correlation functions.

In our recent publication in Physical
Review Letters, Kevin Driver and I extended the applicability range of
fermionic path integral Monte Carlo simulations to heavier elements
and lower temperatures by introducing various localized nodal
surfaces. HartreeFock nodes yield the most accurate prediction
for pressure and internal energy that we combine with the results from
density functional molecular dynamics simulations to obtain a
consistent equation of state for hot, dense silicon under plasma
conditions and in the regime of warm dense matter (2.318.6 g/cc,
5x10^{5}  1.3x10^{8}K). The shock Hugoniot curve is
derived and the structure of the fluid is characterized with pair
correlation functions. On the left, we estimate the degree of
ionization by comparing the integrated nucleuselectron pair
correlation functions from PIMC (symbols) with results for isolated
atoms (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 I
combine path integral Monte Carlo simulations and density functional
molecular dynamics to study oxygen
and nitrogen
in the regime of warm dense matter. We characterize both material at
extreme pressure and temperature conditions that exist in stellar
interiors and can be probed with shock wave experiments. We use pair
correlation functions and the electronic density of states to describe
changes in the structure of the plasma. We compute the shock Hugoniot
curves to compare with laboratory experiments. For nitrogen, we
characterize the regime of molecular dissociation that leads to a
region of dP/dT<0 at high pressure, which is shown in green in the
phase diagram on the left.


Do Uranus and Neptune have oceans?

Simulation of a H_{2}H_{2}O mixture.

Ice giant planets are typically assumed to have a hydrogenrich
atmosphere, an intermediate ice layer, and a rocky core. Such
threelayer models satisfy the observational constraints for Uranus
and Neptune. However, it remains unclear whether these planets have
oceans, which would imply the existence of a sharp boundary
between the hydrogen and water layers. Alternatively, the density and
the water contents of the atmosphere could increase gradually. Recent
laboratory experiments by Bali at el. (2013) favored the ocean
hypothesis. In our ApJ
article, Francois Soubiran and I used ab initio computer
simulations to determine whether H_{2} and H_{2}O are
missible at high pressure. Contrary to the experimental predictions,
we find that both materials are fully miscible under ice giant
interior conditions. We predict that these planets can only have
oceans if icy building blocks were delivered before the gas was
accreted during planet formation.


Do iron and rocks become miscible in the interiors of terrestrial planets?

Simulation of a liquid ironMgO mixture. The brown,
green, and red spheres denote Fe, Mg, and O atoms. The grey surfaces
show the electron density.

All known terrestrial planets have a separate iron core and a rocky
mantle because metallic iron has a low solubility in rocky materials
under typical pressuretemperature conditions in the planetary
interiors. However, at sufficiently high temperatures, all materials
eventually become miscible, even oil and water. In our recent article,
Sean Wahl and I use ab initio computer simulations to
determine what temperature would be required for iron and MgO to
become miscible in all proportions. We find that the required
temperature rises from 4000 to 10,000 K as the pressure is increased
from 0 to 500 GPa. Such extreme conditions can be reached during a
giant impact on a terrestral planet, implying that not all iron would
settle into core during such an event.


Recalibration of giant planet massradius relationship with ab initio simulations

Revised massradius relation for giant exoplanets. Our new simulation data are shown in red.

Using density functional molecular dynamics simulations, we determine
the equation of state for hydrogenhelium mixtures spanning
densitytemperature conditions typical of giant planet interiors. In
our manuscript, a comprehensive equation
of state table with 391 densitytemperature points is
constructed and the results are presented in form of twodimensional
free energy fit for interpolation. We present a revision to the
massradius relationship which makes the hottest exoplanets increase
in radius by ~0.2 Jupiter radii at fixed entropy and for masses
greater than ~0.5 Jupiter mass. This change is large enough to
have possible implications for some discrepant "inflated giant
exoplanets".
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 left
figure), water is predicted to occur in a superionic state where the
oxgyen atoms remain stationary like in a solid while the hydrogen
atoms diffuse throughout the crystal like a fluid. Here, we
show that, at 1.0±0.5 Mbar, the oxygen sublattice in superionic
water changes from a bodycentered cubic lattice (middle) to an
facecented cubic lattice (right). This transformation lead to a more
efficient packing but also reduces the hydrogen diffusion rate, which
may have further implications for electronic conductivity and magnetic
dynamo in Uranus and Neptune. Our results were highlighted by Phys.org

Novel chemistry at high pressure: H_{4}O forms from hydrogen and water ice

Oxygen (red) and hydrogen (blue) atoms in the new H_{4}O structure.

Water and hydrogen at high pressure make up a substantial fraction of
the interiors of giant planets. Using ab initio random
structure search methods we investigate the groundstate crystal
structures of water, hydrogen, and hydrogenoxygen compounds. Here, we find that, at pressures
beyond 14 Mbar, excess hydrogen is incorporated into the ice phase to
form a novel structure with H_{4}O stoichiometry. We also
predict two new ground state structures of water ice with P21/m
and I4/mmm symmetry to form at 135 and 330 Mbar,
respectively. Here is a slide that summarizes the
seven new high pressure ice phases that were recently predicted with
ab initio calculations.


Methane Ice in Uranus and Neptune Assumes a Polymeric and Metallic State


The four snapshots from our ab initio simulations show how
methane gas at high pressure and temperature forms long hydrocarbon
chains. The blue and white spheres denotes the carbon (C) and hydrogen
atoms, respectively. The red lines indicate the CC bonds that
increase from left to right. In our recent paper, we show that the resulting
polymeric state is metallic and exists in the interiors of Uranus and
Neptune. We also predict how such a transformation on the atomistic
level can be identified with macroscopic shock wave experiments.


Path Integral Simulation Technique to Study Plasmas of FirstRow Elements


Path integral Monte Carlo simulations are a powerful tool to study
quantum systems at high temperature but applications to elements
beyond hydrogen and helium with core electrons have so far not been possible. In
our recent PRL
article, Kevin Driver and I develop a new allelectron path integral Monte Carlo
technique with freeparticle nodes for warm dense matter and apply it
to water and carbon plasmas. Our results for pressures, internal
energies, and pair correlation functions compare well with density
functional molecular dynamics at temperatures of
(2.57.5)·10^{5}K. Both methods together form a coherent
equation of state over a densitytemperature range of 312 g/cc and 10^{4}10^{9} K.


Erosion of Rocky Cores in Giant Gas Planets


Gas giants are believed to form by the accretion of hydrogenhelium
gas around an initial protocore of rock and ice. The question of
whether the rocky parts of the core dissolve into the layer of
metallic fluid hydrogen following formation has significant
implications for planetary structure and evolution. Here we use ab initio
calculations to study rock solubility in fluid hydrogen, choosing
magnesium oxide as a representative example of planetary rocky
materials, and find MgO to be highly soluble in H for temperatures in
excess of approximately 10000 K, implying significant redistribution
of rocky core material in Jupiter and larger exoplanets.


Hydrogen Equation of State Computed for Fusion Applications


Using path integral Monte Carlo simulations we have derived an
equation of state (EOS) table for deuterium that covers typical intertial
confinement fusion conditions at densities ranging from 0.002 to 1596
g/cm^{3}
and temperatures of 1.35 eV ~ 5.5 keV. The small grey circles in the
diagram on the left indicate the temperaturedensity conditions of our
simulations. The EOS and related results are summarized in an article that has been published in Physical Review B.


Bonding Pattern in Ice at High Pressure


The 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 sp^{3}
hybridization 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 hydrogen

The four giant planets in our solar system grow so large because icy
comets made their cores grow much faster than those of terrestrial
planets, which enabled them to accrete large amounts of gas. With ab
initio simulations, Hugh Wilson and I demonstrate in our recent manuscript
that water ice is not thermodynamically stable at the temperature and
pressure conditions where core is exposed to the layer of metallic
hydrogen above. This implies that the cores in Jupiter and Saturn have
been eroded over time, with the icy material being redistributed
convectively throughout the planet.
Our work has implications for constraining the interior
structure and evolution of giant planets and will be relevant for the
interpretation of data from NASA's Juno mission to Jupiter (to be launched in
August 2011). Core erosion could also provide a significant flux of
heavy elements to the atmosphere of exoplanets and may explain why
some of them have significantly inflated radii.


Simulations predict water ice to become a metal at megabar pressures

Four high pressure phases of ice

Water ice is one of the most prevalent substances in the solar system,
with the majority of it existing at high pressures in the interiors of
giant planets. The known phase diagram of water is extremely rich, with
at least fifteen crystal phases observed experimentally. In our article in Physical Review Letters (see
also condmat), Hugh
Wilson and I explore the phase diagram of water ice by means of ab
initio computer simulations and predict two
new phases to occur at megabar pressures. In the figure from
top 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 is
metallic and predicted to occur at 15.5 Mbar.
The known high pressure ice phases VII, VIII, X and Pbcm as
well as our Pbca phase are all insulating and composed of two
interpenetrating hydrogen bonded networks, but the Cmcm
structure is metallic and consists of corrugated sheets of H and O
atoms. The H atoms are squeezed into octahedral positions between
nextnearest O atoms while they occupy tetrahedral positions between
nearest O atoms in the ice X, Pbcm, and Pbca phases.


Why is neon missing from Jupiter's atmosphere? Indirect evidence of helium rain

Jupiter’s interior. Helium rain occurs in the immiscibility layer and depletes the upper layer of both helium and neon.

When the Galileo entry
probe entered Jupiter's atmosphere in 1995, it measured that
the inert gas neon was depleted by a
factor of 10 compared to the composition of sun, which represents the
concentrations in nebula that formed our solar system with all its
eight planets. So where is all the neon gone that was present in
Jupiter initially? Using ab initio computer
simulations Hugh Wilson and I link the missing neon to another
process that was proposed to occur inside Jupiter: helium
rain.
There is indirect evidence from luminosity measurements that helium
rain occurs on Saturn but it was unclear whether it occurs inside
Jupiter also. Our calculations now show that neon preferentially
dissolves into helium droplets and it is therefore gradually sequestered
into the deeper interior as the helium rain falls. The remaining hydrogenrich envelope is
slowly depleted of both neon and helium. The measured concentrations
of 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 InsulatortoMetal Transition in Solid Helium

InsulatortoMetal Transition in Solid Helium at High Pressure

Metallic solid helium is present in the outer layers of White Dwarf
stars. The cooling rate of White Dwarfs is regulated by the heat flow
from the hot interior to the colder exterior. The
insulatortometal transition is of interest because it marks the
point where heat transport switches from electronic
conductions to photon diffusion. In our paper, the
insulatortometal transition in solid helium at high pressure is
studied with different firstprinciples simulations. Diffusion quantum
Monte Carlo (QMC) calculations predict that the band gap closes at a
density 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 standard
density functional calculations. The metallization density derived
from GW calculations is found to be in very close agreement with QMC
predictions. Path integral Monte Carlo calculations showed that
the zeropoint motion of the nuclei has no significant effect on the
metallization transition.


Simulation of HydrogenHelium Mixtures in Planetary Interiors

Helium in molecular hydrogen

Helium in metallic hydrogen

We performed density functional molecular dynamics simulation to
characterize hydrogenhelium mixtures in the interior of solar and
extrasolar giant planets. In this
article, we address outstanding questions about their structure
and evolution e.g. whether Jupiter has a rocky core and if it was
formed by a core accretion process. We describe how the presence of
helium defers the moleculartometallic transition in hydrogen to
higher 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 at
high pressure and temperature. In this
article (accepted for publication in Physical Review Letters), we
used firstprinciples computer simulation to predict the properties of
shock fluid helium at megabar pressures. The simulations show that the
compressibility of helium is substantially increased by electronic
excitations. A maximum compression ratio of 5.24fold the initial
density was predicted for 360 GPa and 150000 K. This result
distinguishes helium from deuterium, for which simulations predicted a
maximum compression ratio of 4.3. If the sample are precompressed
statically the compression ratio is reduced, which is shown in the
left 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 1.8
Mbar and indications for a insulatormetal transition were found.
In this article, we report
results from density functional molecular dynamics simulations of dense liquid oxygen
close to the metalinsulator transition. We have
found that band gap closure occurs in the molecular liquid, with a
slow transition from a semiconducting to a poor metallic state
occurring over a wide pressure range. At approximately 80 GPa,
molecular dissociation is observed in the metallic fluid. Spin
fluctuations play a key role in determining the electronic structure
of the low pressure fluid, while they are suppressed at high pressure.


Dense Plasma Effects on Nuclear Reaction Rates

Manybody enhancement of nuclear reaction rates h(0) as function of the coupling parameter.

Dense plasma effects can cause an exponenial change in charge particle
nuclear reaction rates important in stellar evolution.
In this article, reaction rates
in dense plasmas are examined using path integral Monte Carlo. Quantum
effects causes a reduction in the many body enhancement of the
reaction rate, h(0), compared to the classical value. This is shown in
figure on the left for different quantum parameters. This reduction
can be attributed to the "quantum smearing" of the Coulomb interaction
at the short range resulting in a reduced repulsion between the
reacting pair and surrounding particles.


Lowering of the Kinetic Energy in Interacting Quantum Systems

Temperature density region of kinetic energy lowering for dense hydrogen and the electron gas.

The equilibrium momentum distribution is of fundamental importance to
characterize manybody systems. In contrast to classical systems where
the distribution is always Maxwellian, in quantum systems the
distribution depends on particle statistics, bosons or fermions, as
well as on interactions and can display interparticle correlations,
which are the basis of superfluidity and superconductivity.
In this article, we
report and explain a surprising effect of interactions in quantum
systems on the one particle momentum distribution and kinetic
energy. Interactions never lower the ground state kinetic energy of a
quantum system. However, at nonzero temperature, where the system
occupies a thermal distribution of states, interactions can reduce the
kinetic energy below the noninteracting value. This is
demonstrated using PIMC simulations for dense hydrogen and the electron gas.


Understanding hot dense hydrogen with PIMC simulations




Molecular liquid 
Molecular metallic liquid 
Metallic liquid 

The high temperature phase diagram of hydrogen

At 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 molecules
can be modelled accurately with PIMC. The resulting protonproton pair correlation functions are shown.


Single and double shock Hugoniot curves from PIMC simulations



90. 
S. Zhang, K. P. Driver, F. Soubirana, B. Militzer,
"Equation of states and shock compression of warm dense sodium a firstprinciples study",
J. High Energy Density Physics 21 (2016) 16.

89. 
B. Militzer,
"Supercell Design for FirstPrinciples Simulations of Solids and Application to Diamond, Silica, and Superionic Water",
J. High Energy Density Physics 21 (2016) 8.

88. 
S. X. Hu, B. Militzer, L. A. Collins, K. P. Driver, and J. D. Kress,
"FirstPrinciples Prediction of the Softening of the Silicon Shock Hugoniot Curve",
Phys. Rev. B 94 (2016) 094109. Also available on the arXiv. 
87. 
B. Militzer, F. Soubiran, S. M. Wahl, W. Hubbard,
"Understanding Jupiter's Interior",
J. Geophysical Research, 121 (2016) 1552. Also available on the arXiv. 
86. 
S. M. Wahl, W. B. Hubbard, B. Militzer,
"Tidal response of preliminary Jupiter model",
Astrophysical J., in press (2016). Also available on the arXiv. 
85. 
F. Soubiran, B. Militzer,
"The properties of heavy elements in giant planet envelopes",
Astrophysical J. 829 (2016) 14. Also available on the arXiv.
 84. 
B. Militzer,
"Computation of the High Temperature Coulomb Density Matrix in Periodic Boundary Conditions", Computer Physics Communications 204 (2016) 88. Also available on the
arXiv.
 83. 
S. M. Wahl, W. B. Hubbard, B. Militzer,
"The Concentric Maclaurin Spheroid method with tides and a rotational enhancement of Saturn's tidal response",
Icarus, in press (2016). Also available on the arXiv. 
82. 
W. B. Hubbard, B. Militzer,
"A preliminary Jupiter model",
Astrophysical Journal 820 (2016) 80.
Also available on the arXiv. 
81. 
K. P. Driver, B. Militzer,
"FirstPrinciples Equation of State Calculations of Warm Dense Nitrogen",
Phys. Rev. B 93 (2016) 064101. 
80. 
S. Zhang, S. Cottaar, T. Liu, S. Stackhouse, B. Militzer,
"Highpressure, temperature elasticity of Fe and Albearing MgSiO_{3}: implications for the Earth's lower mantle",
Earth and Planetary Science Letters 434 (2016) 264. 
79. 
B. Militzer, K. P Driver,
"Development of Path Integral Monte Carlo Simulations with Localized Nodal Surfaces for SecondRow Elements",
Phys. Rev. Lett. 115 (2015) 176403. 
78. 
K. P. Driver, F. Soubiran, Shuai Zhang, and B. Militzer
"Firstprinciples equation of state and electronic properties of warm dense oxygen",
J. Chem. Phys. 143 (2015) 164507. 
77. 
F. Soubiran, B. Militzer,
"Miscibility calculations for water and hydrogen in giant planets",
Astrophys. J. 806 (2015) 228. 
76. 
S. X. Hu, V. N. Goncharov, T. R. Boehly, R. L. McCrory, S. Skupsky, L. A. Collins, J. D. Kress, B. Militzer,
"Impact of FirstPrinciples Properties of DeuteriumTritium on Inertial Confinement Fusion Target Designs",
Physics of Plasmas 22 (2015) 056304. 
75. 
F. Soubiran, B. Militzer,
"HydrogenWater Mixtures in Giant Planet Interiors Studied with Ab Initio Simulations",
J. High Energy Density Physics 17 (2015) 157. 
74. 
K. Driver, B. Militzer,
"Firstprinciples simulations and shock Hugoniot calculations of warm dense neon",
Phys. Rev. B 91 (2015) 045103. 
73. 
S. Wahl, B. Militzer,
"Hightemperature miscibility of iron and rock during terrestrial planet formation",
Earth and Planetary Science Letters 410 (2015) 25. 
72. 
Y. Lin, R. E. Cohen, S. Stackhouse, K. P. Driver, B. Militzer, L. Shulenburger and J. Kim,
"Equations of state and stability of MgSiO_{3} perovskite and postperovskite phases from quantum Monte Carlo simulations",
Phys. Rev. B 90 (2014) 184103, available on the archive. 
71. 
H. F. Wilson, B. Militzer,
"Interior phase transformations and massradius relationships of siliconcarbon planets",
Astrophys. J. 973 (2014) 34. 
70. 
F. GonzalezCataldo, H. F. Wilson, B. Militzer,
"Solubility of silica in metallic hydrogen: implications for the stability of rocky cores in giant planets",
Astrophys. J. 787 (2014) 79. 
69. 
P. Kaercher, B. Militzer, H.R. Wenk,
"Ab initio calculatios of elastic constants in plagioclase feldspars",
American Mineralogist 99 (2014) 2344. 
68. 
L. X. Benedict, K. P. Driver, S. Hamel, B. Militzer, T. Qi, A. A. Correa, A. Saul, E. Schwegler,
"A multiphase equation of state for carbon addressing high pressures and temperatures",
Phys. Rev. B 89 (2014) 224109, available on condmat. 
67. 
S. M. Wahl, H. F. Wilson, B. Militzer,
"Solubility of iron in metallic hydrogen and stability of dense cores in giant planets",
Astrophysical Journal 773 (2013) 95, available on astroph. 
66. 
B. Militzer, W. B. Hubbard,
"Ab Initio Equation of State for HydrogenHelium Mixtures with Recalibration of the GiantPlanet MassRadius Relation",
Astrophysical Journal 774 (2013) 148, available on astroph. 
65. 
B. K. Godwal, S. Stackhouse, J. Yan, S. Speziale, B. Militzer, R. Jeanloz,
"CoDetermination
of Crystal Structures at High Pressure: Combined Application of Theory
and Experiment to the Intermetallic Compound AuGa_{2}",
Phys. Rev. B Rapid Comm. 87 (2013) 100101. 
64. 
B. Militzer,
"Equation of state calculations of hydrogenhelium mixtures in solar and extrasolar giant planets",
Physical Review B 87 (2013) 014202. 
63. 
H. F. Wilson, M. L. Wong, B. Militzer,
"Superionic to superionic phase change in water: consequences for the interiors of Uranus and Neptune",
Physical Review Letters 110 (2013) 151102, also available on astroph. 
62. 
S. Zhang, H. F. Wilson, K. P. Driver, B. Militzer,
"H_{4}O and other hydrogenoxygen compounds at giantplanet core pressures",
Physical Review B 87 (2013) 024112, also available on condmat. 
61. 
B. L. Sherman, H. F. Wilson, D. Weeraratne, and B. Militzer,
"Ab Initio Simulations of Hot, Dense Methane During Shock Experiments",
Physical Review B 86 (2012) 224113,
also available on condmat. 
60. 
B. Militzer,
"Ab Initio Investigation of a Possible LiquidLiquid Phase Transition in MgSiO_{3} at Megabar Pressures",
Journal of High Energy Density Physics 9 (2013) 152, available also on condmat. 
59. 
K. P. Driver, B. Militzer,
"AllElectron Path Integral Monte Carlo Simulations of Warm Dense Matter: Application to Water and Carbon Plasmas",
Phys. Rev. Lett. 108 (2012) 115502.
 58. 
H. F. Wilson, B. Militzer,
"Rocky core solubility in Jupiter and giant exoplanets",
Phys. Rev. Lett. 108 (2012) 111101. Suggested read by PRL editor. Also available on astroph. 
57. 
S. X. Hu, B. Militzer, V. N. Goncharov, and S. Skupsky,
"FPEOS: A FirstPrinciples Equation of State Table of Deuterium for Inertial Confinement Fusion Applications",
Phys. Rev. B, 84 (2011) 224109, also available on condmat. 
56. 
H. F. Wilson, B. Militzer,
"Solubility of water ice in metallic hydrogen: consequences for core erosion in gas giant planets",
Astrophys. J. 745 (2012) 54, also available on astroph.
 55. 
B. Militzer,
"Bonding and Electronic Properties of Ice at High Pressure",
Intern. J. Quantum Chemistry 112 (2011) 314,
also available on condmat.
 54. 
L. Miyagi, W. Kanitpanyacharoen, S. Stackhouse, B. Militzer, H.R. Wenk,
"The Enigma of PostPerovskite Anisotropy: Deformation versus Transformation Textures",
Physics and Chemistry of Minerals 38 (2011) 665, DOI: 10.1007/10.1007/s002690110439y.
 53. 
B. Militzer, H. F. Wilson,
"New Phases of Water Ice Predicted at Megabar Pressures",
Phys. Rev. Lett. 105 (2010) 195701, available on condmat.
 52. 
S. X. Hu, B. Militzer, V. N. Goncharov, and S. Skupsky,
"StrongCoupling and Degeneracy Effects in Inertial Confinement Fusion Implosions",
Phys. Rev. Lett. 104 (2010) 235003.
 51. 
B. Militzer, H.R. Wenk, S. Stackhouse, and L. Stixrude,
"FirstPrinciples Calculation of the Elastic Moduli of Sheet Silicates and their Application to Shale Anisotropy", American Mineralogist 96 (2011) 125.
 50. 
A. R. Rhoden, B. Militzer, E. M. Huff, T. A. Hurford, M. Manga, and M. A. Richards,
"Constraints on Europa's rotational dynamics from modeling of tidallydriven fractures",
Icarus 210 (2010) 770.
 49. 
H. F. Wilson and B. Militzer,
"Sequestration of noble gases in giant planet interiors", Phys. Rev. Lett. 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.
 48. 
K. P. Esler, R. E. Cohen, B. Militzer, J. Kim, R.J. Needs, and M.D. Towler,
"Fundamental high pressure calibration from allelectron quantum Monte Carlo calculations", Phys. Rev. Lett. 104 (2010) 185702.
 47. 
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. 
46. 
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 of
thermal diffusivity at highpressure using a transient heating
technique’”'', Appl. Phys. Lett. 95 (2009) 096101. 
45. 
J. J. Fortney, I. Baraffe, B. Militzer, chapter "Interior Structure
and Thermal Evolution of Giant Planets", in "Exoplanets",
ed. S. Seager, Arizona Space Science series (2009).
 44. 
B. Militzer,
"Correlations in Hot Dense Helium", J Phys. A 42 (2009) 214001, condmat/09024281.
 43. 
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.
 42. 
B. Militzer and W. B. Hubbard,
"Comparison of Jupiter Interior Models Derived from FirstPrinciples Simulations", Astrophysics and Space Science 322 (2009) 129, astroph/08074266.
 41. 
S. A. Khairallah and B. Militzer,
"FirstPrinciples Studies of the Metallization and the Equation of State of Solid Helium", Phys. Rev. Lett. 101 (2008) 106407, physics/08054433.
 40. 
B. Militzer,
"Path Integral Monte Carlo and Density Functional Molecular Dynamics Simulations of Hot, Dense Helium", Phys. Rev. B 79 (2009) 155105, condmat/08050317.
 39. 
B. Militzer, W. B. Hubbard, J. Vorberger, I. Tamblyn, and S.A. Bonev,
"A Massive Core in Jupiter Predicted From FirstPrinciples Simulations",
Astrophysical Journal Letters 688 (2008) L45, astroph/08074264.
 38. 
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.
 37. 
B. Militzer, W. B. Hubbard,
"Implications of Shock Wave Experiments with Precompressed Materials for Giant Planet Interiors",
AIP conference proceedings 955 (2007) 1395.
 36. 
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.
 35. 
S. Seager, M. Kuchner, C. A. HierMajumder, B. Militzer,
"Massradius relationship of solid exoplanets", Astrophys. J. 669 (2007) 1279.
 34. 
V. V. Struzhkin, B. Militzer, W. Mao, R. J. Hemley, H.k. Mao,
"Hydrogen Storage in Clathrates",
Chem. Rev. 107 (2007) 4133.
 33. 
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.
 32. 
J. Vorberger, I. Tamblyn, B. Militzer, S.A. Bonev,
"HydrogenHelium Mixtures in the Interiors of Giant Planets",
Phys. Rev. B 75 (2007) 024206, condmat/0609476.
 31. 
B. Militzer, R. J Hemley,
"Solid oxygen takes shape", Nature (News & Views), 443 (2006) 150.
 30. 
B. Militzer,
"First Principles Calculations of Shock Compressed Fluid Helium",
Phys. Rev. Lett. 97 (2006) 175501.
 29. 
B. Militzer, R. L. Graham,
"Simulations of Dense Atomic Hydrogen in the Wigner Crystal Phase", J. Phys. Chem. Solids, 67 (2006) 2136.
 28. 
B. Militzer,
"HydrogenHelium Mixtures at High Pressure", J. Low Temp. Phys. 139 (2005) 739.
 27. 
B. Militzer, E. L. Pollock,
"Equilibrium Contact Probabilities in Dense Plasmas", Phys. Rev. B, 71 (2005) 134303.
 26. 
J.F. Lin, B. Militzer, V. V. Struzhkin, E. Gregoryanz, R. J. Hemley, H.k. Mao,
"High PressureTemperature Raman Measurements of H_{2}O Melting to 22 GPa and 900 K", J. Chem. Phys. 121 (2004) 8423.
 25. 
B. Militzer, E. L. Pollock, D. Ceperley,
"Path Integral Monte Carlo Calculation of the Momentum Distribution of the Homogeneous Electron Gas at Finite Temperature", submitted to Phys. Rev. B (2003).
 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 LowDensity 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 UrbanaChampaign (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) 12, 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,
"QuasiClassical Theory and Simulation
of TwoComponent 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 PathIntegral 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) 3439. 
6. 
W. Ebeling, B. Militzer, and F. Schautz,
"Quasiclassical Theory and
Simulations of Hydrogenlike Quantum Plasmas",
Contr. Plasma Physics 37 (1997) 23, 137. 
5. 
W. Ebeling and B. Militzer,
"Quantum Molecular Dynamics
of Partially Ionized Plasmas",
Phys. Lett. A 226 (1997) 298 
4. 
B. Militzer,
"QuantenMolekularDynamik mit reaktiven Freiheitsgraden",
in Dynamik, Evolution, Strukturen,
ed. J. Freund, Dr. Köster
publishing company, Berlin, 1996. 
3. 
B. Militzer,
"QuantenMolekularDynamik
von CoulombSystemen",
Logos publishing company, Berlin, 1996, ISBN 393121608X 
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,
"Growthkinectics of thinfilms deposited by laser ablation", 473478, in
Laser Treatment of Materials, ed. by Barry L. Mordike, Oberursel : DGM Informationsgesellschaft, 1992.

