institut für theoretische physik und astrophysik christian …bonitz/si08/talks/august_6... ·...

IntroductionIntroduction to to densedensequantumquantum plasmasplasmas

Michael BonitzInstitut für Theoretische Physik und Astrophysik

Christian-Albrechts-Universität zu Kiel

Summer Institute „Complex Plasmas“, Stevens Tech, 6 August 2008

PlasmaPlasma

I. Langmuir/L. Tonks (1929): ionized gas - „plasma“„4th state of matter“: solid fluid gas plasma

ideal hot classical gasmade of electrons and ions

BUT: there exist unusual plasmas which- are „non-ideal“, - are non-classical- may contain other particles

= System of many charged particles, dominated by Coulomb interaction

ContentsContents

1. Introduction: Examples of nonideal quantum plasmas

2. Theoretical approaches to quantum plasmas

3. Computer simulations of quantum plasmas

- low temperature: partially ionized plasmas

- high density: quantum plasmas in the universe and lab

- thermodynamics, ionization equilibrium- kinetic theory

- brief overview on existing approaches

• low temperature: partially ionized quantum plasmas

• high density: quantum plasmas in the universe and lab

1. Introduction: Examplesof nonideal quantum plasmas

• nonideal plasmas

Electron density, 1/ccm

Tem

pera

ture

, eV

KeV 4101 ≅

Lightning

Lightning

Magnetic FusionMagnet fusion

SunSun core

Inertial FusionICF

Dusty Dusty PlasmasPlasmas

Plasmas Plasmas in trapsin traps

Plasmas in Plasmas in thethe UniverseUniverse and Laband Lab

300K

Further exist strongly correlated plasmas

WhenWhen isis a a plasmaplasma nonidealnonideal ??Ideal gas behavior of electrons and ions: when Coulomb interaction energy U

is much smaller than kinetic energy K

TNkK B23

=In thermodynamic equilibrium:

∑≠ −

=ji ji

ji

rree

U||2

1

Estimate mean interaction energy: 3/1−∝≡− nrrr ji3/1

2

nreU ∝∝

Degree of nonideality:T

nkTre

KU 3/12||

∝∝=Γ

Nonideal behavior at low temperature or high density

CorrelationCorrelation effectseffects in in classicalclassical plasmasplasmasCoulomb Interaction: rqrU /)( 2=

StrongCoulomb

correlations

1=Γ

TknqTkU

BB

dim/12/ ∝⟩⟨≡ΓCoupling parameter:

175=Γ

Coulomb crystal (OCP)

LetLet usus cool a cool a plasmaplasma

TknqTkU

BB

dim/12/ ∝⟩⟨≡Γ1. Effect: increase of nonideality

2. Effect: electrons become trapped by ionsformation of atoms: e+i Apartially ionized plasma

Further cooling: atoms recombineformation of molecules(neutral gas)

Further cooling: atomic/molecular gas undergoesphase transition(s) to liquid or solid matter

Plasma? Yes: electrons form a plasma (metals) or e-hole plasma (dielectrics)electrons, holes behave as quantum plasma

Atoms, molecules,Condensed matter

require quantummechanics

CorrelationCorrelation effectseffects in in classicalclassical plasmasplasmasCoulomb Interaction: rqrU /)( 2=

1=Γ

TknqTkU

BB

dim/12/ ∝⟩⟨≡ΓCoupling parameter:

175=Γ

Coulomb crystal (OCP)

Two-CP: bound stateformation

Binding energy

SummarySummary: Quantum matter at : Quantum matter at lowlow TT

A. Partially ionized plasma: e-i plasma (classical or quantum) with additional neutral particles (atoms, molecules)

B. Neutral gas or liquid of atoms, molecules(no plasma) requires quantum mechanics

electron-orbitals in hydrogen [Wikipedia]

C. Crystals- lattice of ions plus quantum plasmaof electrons („electron gas“)

- insulators, semiconductors: upon exci-tation formation of quantum plasma of electrons and holes

D. Additional quantum plasma states occurat densities around/above solid density(see below)


Tem

pera

ture

, eV

KeV 4101 ≅

JupiterPlanet coresLightning

Lightning

Magnetic FusionMagnet fusion Inertial FusionICF

Metals

Semiconductors



QuantumQuantum Plasmas in Plasmas in thethe UniverseUniverse and Laband Lab

300K

Partially ionizedquantum plasmas

Quantum Fermi plasmas

WhenWhen isis a a plasmaplasma „„quantumquantum““ ??Quantum wave length („extension“) of a particle (De Broglie):

mvh

=λ

depends on particle mass and velocity

Is an electron a quantum or classical particle ???

Depends on the neighboring „particle“!i.e. on ratio of size to distance

r λIs a person a quantum or classical „particle“ ???

Tmkh Bπλ 2/=Example: thermodynamic equilibrium:

Electron

][10 7

KTm−∝λ

Person (100kg)

mKT ][

1010 167 −−∝λ

Particles are quantum if r≥λ

Quantum Quantum degeneracydegeneracy

transform ratio of length scales to densities: 31

rn ∝

Define quantum degeneracy parameter (dimensionless) :12

3

+=

snλχ

Classical plasma: 1<χ

Quantum plasma: 1>χ

CorrelationCorrelation andand Quantum Quantum effectseffectsCoulomb Interaction: reerU baab /)( =

TkU B/⟩⟨≡Γ

StrongCoulomb

correlations

1=ΓDeBroglie

wave length

Tmkh Bπλ 2/=

r

Overlapof wave functions,

Spin effects

Quantum effects

λ

r=λ

13 == λχ n

175=Γ crystal

LetLet usus compresscompress a a plasmaplasma

1. High-density plasmas in nature: - Interior of Jovian planets (Jupiter, Saturn):

Hydrogen/helium plasma with density up to 32410 −= cmn

- Core of white and brown dwarf stars: 32826 10...10 −= cmn

- neutron stars: (?)10...10 33428 −= cmn

2. High-density plasmas in the laboratory: compression byA. „Traditional methods“: reach

- diamond envils- pinch effect: high current through wires, explosion- launch of shock waves into matter - explosions in chambers

32410 −∝ cmn

Example: plasma compression for inertial confinement fusion

B. Use novel high energy sources: laser or ion beams

WhatWhat happenshappens to to atomsatoms at high at high compressioncompression? ?

Effect 1: Destabilization of atom in plasma environment:- Screening of e-i attraction, reduced binding energy(ionization potential) many-body effect

Effect 2: Overlap of two atoms (electron wave functions) at high n:Tunneling of electrons from one atom to anotherquantum destabilization, tunnel (pressure) ionization of atoms, „Mott effect“at densities corresponding to 32410.., −≅∝ cmneiar B

Both effects occur simultaneously, are equally important

Both effects occur even at zero temperature,Temperature increase helps to destroy atoms, molecules

Mott Effect. Pressure IonizationIdeal Plasma Non ideal Plasma

Lowering of the continuum edge due to screening and quantum effectsreduces number and ionization energy of bound statesvanishing distinction between free and bound states

PartiallyPartially ionizedionized HydrogenHydrogen

1st principle direct path integral Monte Carlo Simulations (V. Filinov, MB)Spontaneous arrangement of electrons and protons (hydrogen)

Hydrogen conductivity(experiment) in cm/ohm

410

210

010

210 −

210 − 010410 − 410

Nellis et al.

+ Fortov et al.

T=3,000-10,000KExistence of an insulator-metal transition?

Nonideal plasma: reduced current conduction: due to Coulomb interaction and bound state formation

Ideal plasma: conductivity increases with density of charged particles


Tem

pera

ture

, eV

KeV 4101 ≅

JupiterPlanet coresLightning

Lightning

Magnetic FusionMagnet fusion Inertial FusionICF

Metals

SemiconductorsDwarfstars



QuantumQuantum Plasmas in Plasmas in thethe UniverseUniverse and Laband Lab

300K

Neutronstars

WhatWhat happenshappens to to thethe plasmaplasma uponuponfurtherfurther compressioncompression? ?

At densities above the Mott point: fully ionized e-i plasma

32410 −≥ cmn

? Question1: Is this plasma classical or quantum?

? Question 2: Is this plasma ideal or nonideal?

Answer 1: Electrons are degenerate (at not too high temperature) 1>>χ

Answer 2: Recall coupling parameterT

nkTre

KU 3/12||

∝∝=Γ

But: this is a classical result, not applicable here!

KineticKinetic energyenergy and and couplingcoupling parameterparameter of of quantumquantum particlesparticles

Coupling (Brueckner) parameter:

3/1

2 1||nEr

erKU

Fs ∝∝=

Electrons are fermions (spin s=1/2) and obey the Pauli principleeach quantum state occupied only by single electronAdding electrons increases (kinetic) energy even when interaction is neglected

Highest occupied level: Fermi energy

( ) 3/23/2

126

222 nE smF += πh

Kinetic energy/N and pressure:

F

F

EpVnEK

52

3/253

=

∝=

Fermions Bosons

With increasing density quantum plasmabecomes ideal!

CorrelationCorrelation andand Quantum Quantum effectseffectsCoulomb Interaction: reerU baab /)( =

TkU B/⟩⟨≡Γ

StrongCoulomb

correlations

1=ΓDeBroglie

wave length

Tmkh Bπλ 2/=

r

r=λOverlap

of wave functions, Spin effects

Quantum effects

λ

13 == λχ n

175=Γ1=sr

100≅sr

crystal

BFs arEUr // ∝⟩⟨≡

- Fermi Energy Ba - Bohr RadiusFE

DenseDense twotwo--componentcomponent plasmasplasmas (1)(1)

Ion-electron mass and charge ratio: M=2,000, … 100,000; Z=1, … 92 (Uranium)

Brueckner parameter of component „a“:3/1

222||

a

a

Faa

asa

a

aa

nZ

EreZr

KU

∝∝=

2/1

3

)2(,

12 aaa

a

aaa kTm

hs

nπ

λλχ =+

=Degeneracy of component „a“:a=e, i

a

aa

aa

a

a

aaa T

nZkTr

eZKU 3/1222||

∝∝=ΓClassical coupling of component „a“:

Charge neutrality leads to restriction: Znn ie =

DenseDense twotwo--componentcomponent plasmasplasmas (2)(2)

1. Low density/high T: electrons and ions classical

2. High density/low T: electrons and ions quantum

2/3

112

2,1MZsi

eie ⋅×

+=⋅=>>>> βχβχχ

3. Intermediate density range: electrons quantum, ions classical

for an electron-proton plasma000,801

==e

i

χχβ

in this density range electrons may be weakly coupled (small r_s) and ions may be strongly coupled (large Gamma)

for example: formation of an ion crystal embedded into an electron Fermi gas

Crystals Crystals in in massmass--asymmetricasymmetric plasmasplasmasCoulomb Interaction: reerU baab /)( =

aaaa TkU B/⟩⟨≡Γ BaFaaasa arEUr // ∝⟩⟨≡Ba - Bohr Radius

StrongCoulomb

correlations

1=Γ

Tkmh Baa πλ 2/=

r

ee r=λ

13 == eee n λχ

175=Γ1=sr

100≅sr

crystal

Atoms, molecules(neutrals)

pp r=λ

100≅spr proton crystal&

Electron gas

Pressure ionization

neutronstar

White dwarf

T = 10,000 K, n = 3⋅1025 сm-3, ρ = 50.2 g/сm3

Filinov, Bonitz, Fortov, JETP Letters 72, 245 (2000)

- proton- electron

- electron

Proton crystallization in dense HydrogenProton crystallization in dense Hydrogen

1st-principlePath integralMonte Carlo simulation

White White dwarfdwarf starstar

D. Schneider, LLNL

classicalfluid and crystalin „quantum sea“of electrons

Size~our EarthMass~our Sun

density:

ERDEρρ 610≅

Crystals in Crystals in twotwo--componentcomponent (TCP) (TCP) CoulombCoulomb systemssystems

Known examples: 1. Ion crystals in white dwarf stars, crust of neutron stars2. Dusty plasmas, expanding neutral plasmas

WhatWhat isis thethe relationrelation betweenbetween 1. and 2.? Do 1. and 2.? Do otherother TCP TCP crystalscrystals existexist ??

Re E

kTT23

=3/11 n

ra

r e

B

se

∝=

Hole crystal in semiconductors

proton crystal

Bonitz, Filinov, Levashov, Fortov, Fehske, Phys. Rev. Lett. 95, 235006 (2005)

ConditionsConditions forfor TCPTCP CoulombCoulomb crystalscrystals

Necessary: I. strong hole coupling (OCP): )3(100,175 Drr crssh

crh ≅≥≅Γ≥Γ

II. no Coulomb bound states:

h

e

e

h

e

h

TT

qqZ

mmM =Θ== ,,

Relevant parameters: mass, charge and temperature asymmetry

Typical for ion/dust crystal: 100...1,10...1,10...10 5124 =Θ== ZM

2.1/ ≅≤= MottsBese rarr

1)(

)(3/4

−=≥e

Motts

crs

ecr

TrZrTMM

ResultsResults

• Critical mass ratio:

• Maximum temperature:cr

scr

R

eB

rMZ

ETk

Γ+Θ

=)1(4

2

PredictedPredicted parametersparameters of of TCP TCP CoulombCoulomb crystalscrystals

][ 3max

−cmn ][max KT

Protons(hydrogen)

Semiconductors(M=100)

][ 3min

−cmn

+6O Ions(white dwarfs)

91026102 ⋅ 33106.3 ⋅

24105⋅ 28100.1 ⋅ 000,66

0.920102.1 ⋅ 20101.2 ⋅

Phase Phase diagramdiagram of of TCP TCP CoulombCoulomb crystalcrystal

Bonitz, Filinov, Fortov, Levashov, and Fehske, Phys. Rev. Lett. 95, 235006 (2005)

Re E

kTT23

=3/11 n

ra

r e

B

se

∝=

11++

= crMMK

Θ= 2

1Z

α

)2(60),3(83 DDM cr =• for Z=1 (e.g. electron-hole plasma):

WhatWhat happenshappens to to thethe ee--i i plasmaplasma uponuponeveneven furtherfurther compressioncompression? ?

At densities above the ion crystal melting: Quantum plasma of electrons and bare nuclei

32810 −≥ cmn

Further compression: overlap of neighboring nucleiPressure dissociation: disintegration of nuclei into free protons, neutrons

Overall Coulomb potential energy U (e-e, p-p, e-p) negativeFavors collaps of all particlesavoided because kinetic (Fermi) energy increases more rapidly than U(r_s decreases with n) Pauli principle stabilizes matter!

Neutron Neutron starstarcrystal and quantum fluidof nuclei

in „quantum sea“of electrons

radius~10kmmass~ our Sun

31510 −≅ cmgρ

Source: Coleman, UMD

WhatWhat happenshappens to to thethe ee--pp--n n plasmaplasma uponuponeveneven furtherfurther compressioncompression? ?

Further compression of quantum electron-proton-neutron plasma:Overlap of neighboring baryons: pressure dissociation into 3 quarks plus quanta of their (strong) interaction – gluonsQuark-gluon plasma

Source: RHIC web site

Does this have any practicalrelevance??

Astrophysicists believe: the whole Universe was in the state of quark-gluon

plasma shortly after the „Big Bang“!

The evolution of the whole Universe since then depends on the properties of the quark-gluon plasma

After After thethe bigbig bangbang

Source: RHIC web site

Can one verify this experimentally ?

To produce a quark-gluon plasma huge densities and particle energies are needed

Big particle colliders in the U.S. and Europe

It is believed that of the quark-gluon plasma has been seenat RHIC (Relativistic Heavy Ion Collider)

Is this of interest for plasma phyics ?

In the quark-gluon plasma Coulomb interaction plays a crucial role

Experiments at RHIC indicate that theQGP is strongly nonidealExperience from plasma physics, nonideal plasmasbecomes crucial for high-energy physics and astrophysics!

ContentsContents







- brief overview on possible approaches

2.1. 2.1. ThermodynamicThermodynamic theorytheory of of partiallypartially ionizedionized plasmasplasmas

High-density plasmas in the Interior of Jovian planets (Jupiter, Saturn)

Hydrogen/helium plasma with total electrondensity of up to

32410 −= cmn

Thermodynamic and transport properties depend on how many electronsare free and how many are bound in atoms, molecules, i.e. on the degree of ionization, the plasma chemical composition.

There exist two approaches: 1. chemical models and 2. Physical models (mostly computer simulations)

Chemical Models of partially ionizeddense Plasmas (1)

Example: partially ionized and dissociated hydrogen: e, p, H, H_2

2HHHHpe

↔+↔+

22 HH μμ =

Hpe μμμ =+

I. Starting point: Choice of relevant particle species

inta

idaa μμμ +=II. Chemical potentials:

⎥⎦

⎤⎢⎣

⎡

+=

12ln

3

a

aaa s

nkT λμ

Ideal part: a) classical particles:

11)2( )],([

04

122/332

+= −

∞+ ∫ aa

anTE

saa e

EdEmn μβπ h

ii) quantum particles (fermions):

Basis: Quantum statistical theories, integral equations etc.

III. Interaction contributions of chemical potentials- charged particle interactions: e-e, e-p, p-p - neutral particle interactions: H-H, H2-H2, H-H2 - charged-neutral particle interactions:


Chemical composition:

Percentage of ionized, atomicand molcular hydrogen(proton number fraction)

Schlanges, Bonitz, Chjan, Contrib. Plasma Phys. 35, 109 (1995)

Mass action law (Saha equation): )(1 intintint2/1~),( HpeB eeenTK

nn EI

A

i μμμββ −+−−=

For more details see book: Kremp et al., „Quantum Statistics of charged particles“,Springer 2003

Maxwell construction coexistence pressure and 2 stable phases

, phase transitions


Problems:

- inconsistent treatment of chargesand neutrals (serious at Mott point)

- subdivision in free and boundparticles artificial!

- Exclusion of particles other thanchosen in the beginning

These problems are avoided in the „physical picture“, e.g. in quantum Monte Carlo simulations.

2.2 2.2 KineticKinetic TheoryTheory forfor quantumquantum plasmasplasmas (1)(1)

Classical plasma Quantum plasma

N-particle probability density (x=r,p):

∫ =1...),,...( 11 NNNN fdxdxxxf

Liouville equation:

0},{ =−∂∂

NNN fHft

With Poisson brackets:

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−∂∂

∂∂

=N

i N

N

N

N

N

N

N

NNN r

fpH

pf

rHfH

1},{

N-particle density operator

1ˆ,ˆ ...1 =NNN Tr ρρ

Von Neumann equation:

[ ] 0ˆ,ˆ1ˆ =−∂∂

NNN Hit

ρρh

With commutator:

abbaba ˆˆˆˆ]ˆ,ˆ[ −=

KineticKinetic TheoryTheory forfor quantumquantum plasmasplasmas (2)(2)


One-particle probability density (x=r,p):

),...(...)( 1211 NNN xxfdxdxxf ∫=

Kinetic equation:

],[ 12111 VfIpf

rU

rfv

tf

=∂∂

∂∂

−∂∂

+∂∂

Mean field approximation: 2112 fff ≈

Linearization: fff δ+= )0(11

Yields Vlasov dielectric function

One-particle density operator:

NNTrF ρ̂ˆ...21 =

Quantum Kinetic equation:

[ ] ]ˆ,[ˆ,ˆ1ˆ12111 FVIFH

iF

t=−

∂∂

h

Mean field approximation:2112

ˆˆˆ FFF ≈

Linearization: FFF ˆˆˆ )0(11 δ+=

Yields „Random phase approximation“( see Norman Horing‘s talk)

For details see book: M. Bonitz, „Quantum Kinetic Theory“

KineticKinetic TheoryTheory forfor quantumquantum plasmasplasmas (3)(3)


Efficient numerical solution of kineticEquation using PIC-MCC

[particle in cell method (for mean field part) plus Monte Carlo methods for collisions.]

],[ 12111 VfIpf

rU

rfv

tf

=∂∂

∂∂

−∂∂

+∂∂

Quantum Kinetic equation:

[ ] ]ˆ,[ˆ,ˆ1ˆ12111 FVIFH

iF

t=−

∂∂

h

For details see book: M. Bonitz, „Quantum Kinetic Theory“

• So far there are no efficient quantumgeneralizations of PIC, manychallenges remain

• alternative approach: classical PIC plus (small) quantum corrections

• we have devolped direct solutions of QK equations using NonequilibriumGreen‘s functions See talk of Karsten Balzer

ContentsContents







- brief overview on possible approaches

RequirementsRequirements forfor quantumquantum simulationssimulations

A. Single-particle properties

- coordinate and momentum not measurable simultaneously(Heisenberg uncertainty)

- Quantum particle has finite extension- quantum particle may be in many states(superposition principle)

- free quantum particle diffuses with time nn Ex),(Ψ

Fermions Bosons

B. Many-particle properties

- Spin statistics, indistinguishability(symmetry/antisymmetry of wave function)Availability of a quantum state for oneparticle depends on the states of all otherparticles (even without interaction!)

3.1 3.1 TypesTypes of of simulationssimulations

Equilibrium Nonequilibrium

- Monte Carlo- Equilibrium Molecular Dynamics

A. Extension of classical methods

B. Special quantum methods

- Exact diagonlization- Density Functional Theory (DFT)- Hartree-Fock

- Schrödinger equationtalk by Sebastian Bauch

- Time-dependent Hartree-Focktalk by Karsten Balzer

- Time-dependent DFT

- Nonequilibrium Molecular Dynamics- Kinetic equations

3.2 Quantum Monte Carlo 3.2 Quantum Monte Carlo

- Generalization of classical Monte Carlo (several methods)

- In particular: using Richard Feynmans‘s „Path integral“ representation of quantum mechanics PIMC

- Very successful „first-principle“ approach, avoids model assumptions

See Talk of Alexei Filinov

For details, see text book „Introduction to Computational Methods forMany-body Systems“, Rinton Press Princeton 2006

3.3 Quantum 3.3 Quantum MolecularMolecular DynamicsDynamics

- Generalization of classical Molecular Dynamics difficult

1. Wave packet MD (modified classical propagation of extended particle)

2. Wigner function MD, see book below

3. Semiclassical MD: classical MD with quantum and spin effects includedvia effective potentials

For details, see text book „Introduction to Computational Methods forMany-body Systems“, Rinton Press Princeton 2006

SummarySummary (1) (1) Quantum plasmas are omnipresent in nature (astrophysics,Condensed matter systems, quark-gluon plasma etc.)

Theoretical approaches are based on quantum statistics: Thermodynamics, Quantum kinetic theory

Efficient first-principle computer simulations have emerged:such as PIMC, DFT, But: much more efforts (quantum MD or PIC) needed

Quantum plasmas become increasingly important for applications: Laser plasmas, ion beam experiments, laser fusion

SummarySummary (2)(2)

For more details see our text books:

M. Bonitz, „Quantum Kinetic Theory“, Teubner 1998

„Introduction to Computational Methods for Many-Particle Systems“,M. Bonitz and D. Semkat (eds.), Rinton Press, Princeton 2006

http://www.theo-physik.uni-kiel.de/~bonitz

SupportedSupported byby DFG via DFG via TransregioTransregio--SFBSFB Greifswald/Kiel Greifswald/Kiel „„Grundlagen Komplexer PlasmenGrundlagen Komplexer Plasmen““

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