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Relativistic and Strongly-Coupled Plasmas Extreme Matter in Plasma-, Astro-, and Nuclear
Physics
Markus H. Thoma
Max-Planck-Institut für extraterrestrische Physik
1. Introduction
2.
Electron-Positron Plasma
3.
Weakly-Coupled Quark-Gluon Plasma
4. Strongly-Coupled Plasma
5. Complex Plasma
6. Strongly-Coupled Quark-Gluon Plasma
1. Introduction
What is a plasma?
Plasma = (partly) ionized gas (4. state of matter)
99% of the visible matter in universe
Plasmas emit light
Plasmas can be produced by
high temperatures
electric fields
kT  mc 2
Relativistic plasmas:
Quantum plasmas:
B 
Strongly coupled plasmas:
h
d
m v th
Q2
G
1
d kT
radiation
(Supernovae)
(White Dwarfs)
(Quark-Gluon Plasma)
G: Coulomb coupling parameter = Coulomb energy / thermal energy
W. dwarfs
Supernova
Quantum Plasmas
bar
106
Pressure
103
Relativistic Plasmas
Complex
Plasmas
Sun
Strongly coupled
Plasmas
Flames
1
Lightening
“Neon”
Tubes
Fusion
10-3
Discharges
10-6
Aurora
Corona
Comets
100
103
Temperature
106
Kelvin
2. Electron-Positron Plasma
What is an electron-positron plasma?
Strong electric or magnetic fields, high temperatures
 massive pair production (E > 2mec2 = 1.022 MeV)
 electron-positron plasma
Examples:
• Supernovae: Tmax = 3 x 1011 K  kT = 30 MeV >> 2mec2
• Magnetars: Neutron Stars with strong magnetic fields B > 1014 G
• Accretion disks around Black Holes
• High-intensity lasers (I > 1018 W/cm2)
 target electrons heated up to multi-MeV
temperatures
Example: Thin gold foil (~1 mm) hit by two
laser pulses from opposite sides
Habs et al.
Equation of state
Notation: h = c = k =1
Assumptions:
• ultrarelativistic gas: T >> me
• thermal and chemical equilibrium
• electron density = positron density
• ideal gas (no interactions)
• infinite extension, isotropic system
Electron and positron distribution function: nF 
Photon distribution function:
nB 
1
eE / T  1
1
eE / T  1
Ultrarelativistic particles: E = p
Particle number density: 
eq
e
d3p
3
 gF 
n
(
p)

0
.
37
T
, gF  4
F
3
( 2 )
Example: T = 10 MeV 
34
3
eq

4
.
9

10
cm
e
Photon density: Photons in equilibrium with electrons and positrons

eq

d3p
3
 gB 
n
(
p)

0
.
24
T
, gB  2
B
3
( 2 )
Energy density: Stefan-Boltzmann law

eq
d3p
d3p
4
 gF 
p
n
(
p)

g
p
n
(
p)

1
.
81
T
F
B
B
( 2 )3
( 2 )3
T = 10 MeV:
eq  3.8  1023 J cm 3
Photons contribute 36% to energy density
Interactions between electrons and positrons  collective phenomena,
e.g. Debye screening, plasma waves
Non-relativistic plasmas (ion-electron):
classical transport theory with scales: T, me
 Debye screening length
Plasma frequency
kT
D 
4 e2  e
4 e2  e
 pl 
me
Ultrarelativistic plasmas: scales T (hard momenta), eT (soft momenta)
Relativistic interactions between electrons  QED
Perturbation theory: Expansion in a = e2/4 =1/137 (e = 0.3)
using Feynman diagrams, e.g. electron-electron scattering
Evaluation of diagrams by Feynman rules 
scattering cross sections, damping and production rates, life times etc.
Interactions within plasma: QED at finite temperature
Extension of Feynman rules to finite temperature (imaginary or real
time formalism)
Polarization tensor:
Relation to dielectric tensor (high-temperature approximation):
3m
 L ( ,k )
L ( ,k )  1 
 1 2
2
k
k
  k 

 1  2k ln   k 


3m
T ( ,k )
T ( ,k )  1 
 1
2

2k 2
 
k2     k 
ln
1   1  2 


2
k


k

 

Effective photon mass: m  eT
3
T  10 MeV  m  1MeV
2
2
Alternative derivation using transport theory (Vlasov + Maxwell equations)
k2
Maxwell equations  L ( ,k )  0, T ( ,k )  2

 propagation of collective plasma modes
 dispersion relations
Plasma frequency
pl  L,T ( k  0 )  m
Plasmon
 1.5  1021 Hz (T  10 MeV )
Debye screening length
pl
1
D 
3m
 1.1 1013 m (T  10 MeV )
Relativistic plasmas  Fermionic plasma modes:
dispersion relation of electrons and positrons in plasma
Electron self-energy:
electron dispersion relation
 Plasmino branch
Examples for further quantities which can be calculated using perturbative
QED at finite temperature (HTL resummed perturbation theory):
• Electron and photon damping rate
• Electron transport rate
• Electron and photon mean free path
• Electron and photon collision time
• Electron and photon viscosity
• Electron energy loss
M.H. Thoma, arXiv:0801.0956, to be published in Rev. Mod. Phys.
Applications to laser induced electron-positron plasmas
T= 10 MeV  equilibrium electron-positron number density
eq  5  1034 cm3
Prediction:
exp  5  1022 cm3
2 laser pulses of 330 fs and intensity of 7 x 1021 W/cm2 on thin foil
B. Shen, J. Meyer-ter-Vehn, Phys. Rev. E 65 (2001) 016405
exp< eq  non-equilibrium plasma
Assumption: thermal equilibrium but no chemical equilibrium
 electron distribution function fF =  nF with fugacity  < 1
exp
d3p
12
 gF 

n
(
p)






10
F
eq
( 2 )3

Non-equilibrium QED:
2
4
e
m2  2  dp pfF ( p)
3 0
M.E. Carrington, H. Defu, M.H. Thoma, Eur. Phys. C7 (1999) 347
 m   meq  1eV
 pl  1.5  1015 Hz
Debye screening length:
D  0.1mm
Collective effects important if extension of plasma L >> D
Electron density > positron density  finite chemical potential m
Temperature high enough  new particles are produced
Example: Muon production via
Muon production exponentially suppressed at low temperatures
T < mm= 106 MeV
Very high temperatures (T > 100 MeV):
Hadronproduction (pions etc.) and Quark-Gluon Plasma
3. Weakly-Coupled Quark-Gluon Plasma (QGP)
Deconfinement transition similar to Mott transition (insulator/conductor):
Electron concentration low  weak screening of ion potential
 electrons bound in atoms  insulator (nucleus)
Electron concentration high  strong screening of ion potential
 free electrons  conductor (QGP = color conductor)
Example: metallic hydrogen in Jupiter
Critical baryon density:
c  10 0
0  0.125 GeV / fm3  2.2 1017 kg / m3
Critical temperature:
Tc  150  200 MeV  (1.8  2.4) 1012 K
Heavy-ion (nucleus-nucleus)
collisions:
RHIC: Au+Au at 200 GeV/N
hot, dense, expanding
fireball
quark-gluon plasma
for 10-22 s?
Space-time evolution of the fireball
Maximum volume (U-U):
3000 fm3
Quark and gluon number:
~ 10000
Pre-equilibrium time:
~1 fm/c = 3 x 10-24 s
Life time of QGP:
~ 5 – 10 fm/c
 good chances for an equilibrated QGP in relativistic heavy-ion collisions
Problem: QGP cannot be observed directly
 discovery of QGP by comparison of theoretical predictions for signatures
with experimental data (circumstantial evidence)
Theoretical description of QGP:
1. Perturbative QCD (finite temperature):
Valid only for small coupling, i.e. at high temperatures (T>>Tc)
Polarization tensor, quark self-energy, dispersion relations,
damping and production rates, transport coefficients, energy loss, …
Apart from color factors similar calculations and results as in the case
of an electron-positron plasma
2. Lattice QCD: non-perturbative method
Valid also for large coupling
Only static quantities (critical temperature, order of phase transition,
equation of state, …), no signatures
3. Classical methods from electromagnetic plasmas:
Transport theory, strongly coupled plasmas (molecular dynamics etc.)
Example:
Strong quenching of hadron spectra at high momenta (jet quenching)
 large energy loss of quarks in QGP
Collisional energy loss of a quark with energy E in a QGP
dE
4

dx
3
 Nf  2 2
E
1 
a s T ln
6 
a sT

Thoma, Gyulassy, Nucl. Phys. B 351
(1990) 491, Braaten, Thoma, Phys.
Rev. D 44 (1991) 2625
RHIC data (quenching of hadron spectra)
radiative energy loss (gluon
bremsstrahlung) not sufficient
 collisional energy loss important
Mustafa, Thoma, Acta Phys. Hung.
A 22 (2005) 93
Quark Matter and Neutron Stars
1. possibility: central density of neutron star > critical
baryon density  hybrid star
Quark matter?
2. possibility: strange quark stars
Speculation: strange quark matter
containing up, down, and strange
quarks more stable than atomic
nuclei (Fe)
Witten (1984)
Self-bound star
made of strange
quark matter
Stöcker
Quark matter: Fermi gas (free quarks)
High-density approximation to quark self-energy (T=0, m large)
 effective quark mass
m*f 
mf
2

m 2f
2

g 2 m 2f
6 2
Quasiparticle approximation
Quark stars have small radii
Reason: quark matter has a larger compressibility than neutron matter
Strange QS
Hybrid star
Schertler, C. Greiner, Schaffner-Bielich, Thoma (2000)
XMM Newton, Chandra: X-ray observation of RXJ1856  R > 16 km
4. Strongly-Coupled Plasmas
Coulomb coupling parameter
Q2
G
d kT
Q: charge of plasma particles
d: inter particle distance
T: plasma temperature
Ideal plasmas: G << 1 (most plasmas: G < 103)
Strongly coupled plasmas: G > O (1)
Examples: ion component in white dwarfs, high-density plasmas at GSI,
complex plasmas, quark-gluon plasma
Ichimaru, Rev. Mod. Phys. 54 (1982) 1017
Numerical simulations of stongly coupled plasmas, e.g. molecular dynamics
One-component plasma (OCP), pure Coulomb-interaction (repulsive):
G > 172  Coulomb crystal
Debye screening

Yukawa system
Q  r / D
V(r)  e
r
Example: White Dwarf
Ions (C,O) in degenerated electron background  OCP good
approximation
Density: 109 kg/m3 , T=106-108 K  G=5-500
Diamond core?
Asteroseismological observations
approximately 90% of the
mass of BPM 37093 has
crystallized (5×1029 kg).
5. Complex Plasma
Complex plasmas = multi component plasmas containing in addition to
electrons, ions and neutral gas microparticles, e.g. dust
Example: microparticles (1-10 mm) in a low-temperature
discharge plasma
Dust particles get highly charged by electron collection:
Q  103...105 e
(higher mobility of electrons than ions)
strong Coulomb interaction between particles (G > 1)
Fortov et al., Phys. Rep. 421 (2005) 1
RF- or DC-discharge in plasma chamber
Noble gases at 300 K and 0.1 – 1.0 mbar
Injection of monodisperse plastic spheres
Electrostatic field above the lower electrode or the glass wall
levitates particles against gravity
Illumination of microparticles with a laser sheet,
recording of scattered light by a CCD camera
Excitation of plasma waves
Direct observation of microparticle system on the microscopic
and kinetic level in real time
Turbulence in particle flow
How many particles are needed for collectivity?
How do macroscopic quantities (e.g. viscosity) develop?
Nanofluidics:
Applications of complex plasmas:
Microscopic model for structure formation, dynamical processes and
self-organisation in strongly interacting many-body systems in plasma,
solid state, fluid, and nuclear physics
Technology: dust contamination in microchip
production by plasma etching, dust in tokamaks, …
Astrophysics: comets, planetary rings, interstellar
clouds, planet formation, noctilucent clouds, …
Plasma crystal
Strong interaction between microparticles:
G ~ Q2, d ~ 100 mm  1 < G < 105, 1 < k <5
Complex plasmas may exist in gaseous, liquid
or solid phase (new states of „soft matter“)
1986: theoretical prediction of the
crystallization of dust particles
in laboratory plasmas
1994: discovery of the plasma crystal
at MPE, in Taiwan and Japan
Thomas et al., Phys. Rev. Lett. 73
(1994) 652
Melting of the crystal by
pressure reduction
Plasma experiments under microgravity
Disturbing effects of gravity on complex plasmas:
• Electrostatic field for levitation of particles neccessary
• Restriction to plasma sheath (electric field for levitation
strong enough) quasi 3D crystals, complicated plasma conditions
• Gravity comparable to force between particles
 structure and dynamics of complex plasmas changed, weak forces
(attraction, ion drag) are covered
• Some experiments (in particular with larger particles) impossible
Microgravity particles in field free bulk plasma
Laboratory
Microgravity
MPE experiments under microgravity
PK-3 Plus
PlasmaLab
BEC
PK-4
2008
ISS
PKE-Nefedov
2006
2004
2002
Texus
2000
1998
1994
1996
Parabolic flights
Parabolic Flights (PK-4)
ISS
PKE-Nefedov
Experiments on the space station from 2001 to 2005 (supported by DLR)
First scientific experiment on board the ISS
Collaboration with Institute for High Energy Densities (IHED, Moscow)
Agglomeration  starting phase of planet formation?
6. Strongly-coupled Quark-Gluon Plasma
Estimate of interaction parameter
Ca S
G2
dT
Thoma, J. Phys. G 31
(2005) L7 and 539
C = 4/3 (quarks), C = 3 (gluons)
T  200 MeV  aS = 0.3 - 0.5
d = 0.5 fm
Ultrarelativistic plasma: magnetic interaction as important as electric
G  1.5 – 6 
QGP Liquid?
RHIC data (hydrodynamical
description with small viscosity, fast
thermalization) indicate QGP Liquid
Attractive and repulsive interaction
 gas-liquid transition at a
temperature of a few hundred MeV
Thoma, Nucl. Phys. A 774 (2006) 30
Example: Static structure function
 experimental and theoretical analysis of liquids
Hard Thermal Loop approximation (T >> Tc):
S ( p) 
2N f T 3
n
p2
, m D  1 / D
2
2
p  mD
interacting gas
QCD lattice simulations 
Thoma, Phys. Rev. D 72 (2005) 094030
QGP liquid?
Example: String theory prediction (AdS/CFT):
Lower limit for ratio of viscosity to entropy density


 6.08  10 13 Ks
s 4k B
Strongly-coupled OCP: Minimum value at G = 12

 4.89
s
4kB
M.H. Thoma and G.E. Morfill, Europhys. Lett. 82
(2008) 65001
Conclusions
• Laser produced electron-positron plasmas can be described by
perturbative QED at finite temperature  PHELIX
• Perturbative QCD for the QGP  predictions of signatures, e.g. jet
quenching, quark matter in neutron stars (no indication)  FAIR, LHC
• Strongly coupled plasmas  new phenomena, e.g. plasma crystal
 PHELIX, FAIR
• Complex plasmas  model for strongly-interacting many-body systems,
applications in technology and astrophysics, microgravity experiments
 PHELIX, FAIR, LHC
• Properties of strongly coupled QGP (equation of state, transport
phenomena, thermalization, etc. ) by comparison with strongly
coupled electromagnetic plasmas  FAIR, LHC
S. Mrowczynski, M.H. Thoma, Annu. Rev. Nucl. Part. Sci. 57 (2007) 61
Thank you very much for your attention!
35. ESA parabolic flight campaign (Bordeaux, October 2003)
PKE-Nefedov on board of the ISS