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Research Interests
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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 first-principles 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 first-principles simulation methods including path integral Monte Carlo, groundstate
quantum Monte Carlo, and density functional molecular
dynamics. |
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Research Group
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Hugh F. Wilson, associate specialist
Kevin Driver, postdoctorial researcher
Shuai Zhang, graduate student
Sean Wahl, graduate student
Formerly in my group:
Stephen Stackhouse, Lecturer in the School of Earth and Environment, University of Leeds.
Saad Khairallah, now at Lawrence Livermore National Laboratory
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Teaching
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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.
This course is now 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.
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 movies that the students made during
the 2008 and
2009 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.
Here are some pictures from my presentation at UC Berkeley's CalDay event 2010.
I also participated in a field trip to Yosemite National Park.
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Positions
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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 my taking advantage of oportunities in Astronomy.
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Erosion of Rocky Cores in Giant Gas Planets
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Gas giants are believed to form by the accretion of hydrogen-helium
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.
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Hydrogen Equation of State Computed for Fusion Applications
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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/cm3 and temperatures of 1.35 eV ~ 5.5 keV. The small grey circles in the diagram on the left indicate the temperature-density conditions of our simulations. The EOS and related results are summarized in an article that has been published in Physical Review B.
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Bonding Pattern in Ice at High Pressure
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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 sp3
hybridization in the ice X phase at lower pressures. Most strikingly,
the white orbitals are not aligned with any hydrogen bond.
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Dissolution of Icy Core Materials Gas Giant Planets
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Simulations predict water ice to be unstable above 3000 Kelvin when exposed to metallic hydrogen
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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.
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Simulations predict water ice to become a metal at megabar pressures
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Four high pressure phases of ice
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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 cond-mat), 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
next-nearest O atoms while they occupy tetrahedral positions between
nearest O atoms in the ice X, Pbcm, and Pbca phases.
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Why is neon missing from Jupiter's atmosphere? Indirect evidence of helium rain
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Jupiter’s interior. Helium rain occurs in the immiscibility layer and depletes the upper layer of both helium and neon.
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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 hydrogen-rich 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.
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Quantum Monte Carlo Study of the Insulator-to-Metal Transition in Solid Helium
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Insulator-to-Metal Transition in Solid Helium at High Pressure
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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
insulator-to-metal transition is of interest because it marks the
point where heat transport switches from electronic
conductions to photon diffusion. In our paper, the
insulator-to-metal transition in solid helium at high pressure is
studied with different first-principles 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 zero-point motion of the nuclei has no significant effect on the
metallization transition.
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Simulation of Hydrogen-Helium Mixtures in Planetary Interiors
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Helium in molecular hydrogen
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Helium in metallic hydrogen
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We performed density functional molecular dynamics simulation to
characterize hydrogen-helium 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 molecular-to-metallic transition in hydrogen to
higher pressures by stabilizing hydrogen molecules.
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First Principles Simulation of Fluid Helium at High Pressure
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Shock hugoniot curves for precompressed hydrogen and helium.
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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 first-principles 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 5.24-fold 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.
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Ab Initio Simulations of Liquid Oxygen under Pressure
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Spin fluctuations present molecular oxygen (left) are suppressed at high pressures (right).
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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 insulator-metal transition were found.
In this article, we report
results from density functional molecular dynamics simulations of dense liquid oxygen
close to the metal-insulator transition. We have
found that band gap closure occurs in the molecular liquid, with a
slow transition from a semi-conducting 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.
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Dense Plasma Effects on Nuclear Reaction Rates
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Many-body enhancement of nuclear reaction rates h(0) as function of the coupling parameter.
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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.
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Lowering of the Kinetic Energy in Interacting Quantum Systems
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Temperature density region of kinetic energy lowering for dense hydrogen and the electron gas.
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The equilibrium momentum distribution is of fundamental importance to
characterize many-body 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.
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Understanding hot dense hydrogen with PIMC simulations
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| Molecular liquid |
Molecular metallic liquid |
Metallic liquid |
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The high temperature phase diagram of hydrogen
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At which pressure and density does hydrogen become metallic?
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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 proton-proton pair correlation functions are shown.
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Single and double shock Hugoniot curves from PIMC simulations
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| 58. |
H. F. Wilson, B. Militzer,
"Rocky core solubility in Jupiter and giant exoplanets",
submitted to Phys. Rev. Lett. (2011), available on astro-ph. |
| 57. |
S. X. Hu, B. Militzer, V. N. Goncharov, and S. Skupsky,
"FPEOS: A First-Principles Equation of State Table of Deuterium for Inertial Confinement Fusion Applications",
Phys. Rev. B, 84 (2011) 224109, also available on cond-mat. |
| 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 astro-ph.
| | 55. |
B. Militzer,
"Bonding and Electronic Properties of Ice at High Pressure",
Intern. J. Quantum Chemistry 112 (2011) 314,
also available on cond-mat.
| | 54. |
L. Miyagi, W. Kanitpanyacharoen, S. Stackhouse, B. Militzer, H.-R. Wenk,
"The Enigma of Post-Perovskite Anisotropy: Deformation versus Transformation Textures",
Physics and Chemistry of Minerals 38 (2011) 665, DOI: 10.1007/10.1007/s00269-011-0439-y.
| | 53. |
B. Militzer, H. F. Wilson,
"New Phases of Water Ice Predicted at Megabar Pressures",
Phys. Rev. Lett. 105 (2010) 195701, available on cond-mat.
| | 52. |
S. X. Hu, B. Militzer, V. N. Goncharov, and S. Skupsky,
"Strong-Coupling 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,
"First-Principles 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 tidally-driven 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 all-electron 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 high-pressure using a transient heating
technique’”'', Appl. Phys. Lett. 95 (2009) 096101. |
| 45. |
B. Militzer,
"Computation of the High Temperature Coulomb Density Matrix in Periodic Boundary Conditions" (2009),
cond-mat/09044282.
| | 44. |
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).
| | 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. |
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).
| | 23. |
E. L. Pollock, B. Militzer,
"Dense Plasma Effects on Nuclear Reaction Rates",
Phys. Rev. Lett. 92 (2004) 021101. |
| 22. |
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. |
| 21. |
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.
| | 20. |
B. Militzer, F. Gygi, G. Galli,
"Structure
and Bonding of Dense Liquid Oxygen from First Principles Simulations",
Phys. Rev. Lett. 91 (2003) 265503. |
| 19. |
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). |
| 18. |
B. Militzer,
"Path
Integral Calculation of Shock Hugoniot Curves of Precompressed Liquid Deuterium",
J. Phys. A: Math. Gen. 63 (2003) 6159. |
| 17. |
B. Militzer, E. L. Pollock,
"Lowering
of the Kinetic Energy in Interacting Quantum Systems",
Phys. Rev. Lett. 89 (2002) 280401. |
| 16. |
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. |
| 15. |
B. Militzer, D. M. Ceperley,
"Path Integral Monte Carlo Simulation
of the Low-Density Hydrogen Plasma",
Phys. Rev. E 63 (2001) 066404. |
| 14. |
B. Militzer, D. M. Ceperley,
"Path Integral
Monte Carlo Calculation of the Deuterium Hugoniot",
Phys. Rev. Lett. 85 (2000) 1890. |
| 13. |
B. Militzer,
"Path Integral Monte Carlo
Simulations of Hot Dense Hydrogen",
Ph.D. thesis, University of Illinois at Urbana-Champaign (2000). |
| 12. |
B. Militzer, E. L. Pollock,
"Variational Density Matrix
Method for Warm Condensed Matter and Application to Dense Hydrogen",
Phys. Rev. E 61 (2000) 3470. |
| 11. |
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. |
| 10. |
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. |
| 9. |
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. |
| 8. |
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. |
| 7. |
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. |
| 6. |
B. Militzer, M. Zamparelli, and D. Beule,
"Evolutionary Search for
Low Autocorrelated Binary Sequences",
IEEE Trans. Evol. Comput. 2 (1998) 34-39. |
| 5. |
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. |
| 4. |
W. Ebeling and B. Militzer,
"Quantum Molecular Dynamics
of Partially Ionized Plasmas",
Phys. Lett. A 226 (1997) 298 |
| 3. |
B. Militzer,
"Quanten-Molekular-Dynamik mit reaktiven Freiheitsgraden",
in Dynamik, Evolution, Strukturen,
ed. J. Freund, Dr. Köster
publishing company, Berlin, 1996. |
| 2. |
B. Militzer,
"Quanten-Molekular-Dynamik
von Coulomb-Systemen",
Logos publishing company, Berlin, 1996, ISBN 3-931216-08-X |
| 1. |
B.-D. Dörfel and B. Militzer,
"Test of Modular Invariance for Finite XXZ Chains",
J. Phys. A: Math. Gen. 26 (1993) 4875.
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