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Transcript of PHYSICS OF HOT DENSE PLASMAS - EECS at UC Berkeley · Waves in a Plasma v φ x Position Electron...
Nd (nω) laser- produced plasmas
Pinchplasmas
Plasmafocus
Highpressure
arcs
Magneticfusion
plasmasDis-charge
plasmas
Sola
r int
erio
r
Solar C
enter
0.1 eV 1 eV 10 eV 100 eVElectron temperature* (κTe)
Ele
ctro
n de
nsity
(e/c
m3 )
1 keV 10 keV
Alle
n, P
aul
108
1010
1012
1014
1016
1018
1020
1022
1024
1026
PHYSICS OF HOT DENSE PLASMASChapter 6
Ch06_00.horiz.VG
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Chr
omos
pher
e
Transition Corona
Ch06_ProcessPlasma.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Processes in a Plasma
• Particle-particle interactions (short-range “collisions”)• Kinetic theory (velocity distribution function)• Collective motion (electron and ion waves)• Wave-particle interactions (collisionless damping and growth)• Wave-wave interactions (linear and non-linear)• Continuum emission• Atomic physics of ionized species (multiple charge states)• Density and temperature• Spatial profiles• Time dependence
Ch06_Particle2Interctns.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Particle-Particle Interactions:Short Range “Collisions”
+
–
+Ze
–e v
Ch06_VeloctyDistrib_Oct05.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
The Velocity Distribution Function, f(v)
f(v)
ve
∆v v
• f(v) describes the number of particles (e.g., electrons) per unit velocity interval.• Its width is a measure of temperature, with “thermal velocity”.
• Kinetic theory describes how f(v) varies in space and time.• The Maxwellian velocity distribution corresponds to an equilibrium situation (static in time)
with “thermal velocity”, ve, given above in eq. (6.2).
• The area under the curve is normalized to density. For electrons ∫fe(v) dv = ne
(6.2)
(6.1)
Ch06_WavesPlasma.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Waves in a Plasma
vφ
xPosition
Ele
ctro
n de
nsity ωp
ae
kD = 1/λD k
a*
cElectromagnetic wave
Electron-acousticwave
Freq
uenc
y
Wavenumber
ω
Ion-acousticwave
An electron-acoustic wave,typically oscillating at theplasma frequency, ωp.
Dispersion diagram for natural modes of oscillation in a plasma.
Ch06_WavParticle.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Wave-Particle Interactions
vφ
xPosition
Electron plasma wave Electron velocity distribution
Ele
ctro
n de
nsity
• Wave damping or growth• Equilibrium or non-equilibrium velocity distribution
ve
vø =
More slowelectrons
Fewer fastelectrons
v
f(v)
ωk
Ch06_LinearNonLinScat.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Linear and Non-Linear Processes: Scattering as an Example
Three wave mixing among natural modes of the plasma. In resonant mixing the three satisfy conservation of energy and momentum.
Plasmawave
(ωp, kp)
Incidentwave(ωi, ki)
Scatteredwave
(ωs, ks)
(6.3) • Linear scattering
• Non-linear scattering(6.4)
Ch06_PlasmaTheories.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Plasma Theories Address Physical Phenomenaat Various Levels of “Particle Detail”
detailed positionsand velocities asa function of timefor all particles
• Kinetic theory for the evolution of f(v) in space and time
= vi
m = –e[E + vi B]
• Conservation of particles (mass)• Conservation of momentum (forces acting on a fluid)• Conservation of energy
Microscopic
f(v; r, t) ne, ni, κTe
P, v (r, t)
Kinetic Fluid
dridt
dvidt
(6.9)
Plasma Theory
Ch06_PlasmaThry_05.ai
Fluid Description (averages out the kinetic velocity information; r, t dependance only)
Plus Maxwell’s equations
Plus Maxwell’s equations
Plus Maxwell’s equations
“collision term”
Kinetic Description (does not track individual particles; tracks them as a function of coordinates v, r, t )~ ~ ~
(6.15)
(6.25)
(6.40)
(6.43)
Microscopic Description (microscopically tracks all particles individually)
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Ch06_PlasmaModelng.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Plasma Modeling
• Numerical simulations of a finite number of particles to study non-linear processes with limited space-time variations
• Plasma dynamic simulations studying the space-time evolution of electron density and temperature profiles in a grid system, with magnetic fields, radiation and absorption, etc.
Courtesy of G. Dahlbacka (LLNL), K. Estabrook (LLNL), and D. Forslund (Los Alamos)
10 µm
150µm 75 µm
Laserlight
Symmetryaxis
zL1
rr
Densitycontours
Electron temperature
z
75 µm
ρ = 50ρc
ρ = 2ρc
Te = 0.5 keVTe = 0.3 keVTe = 0.1 keV
B = 0.2 MG
B = 2.0 MGB = 0.7 MG
ρ = ρc12
0Distance
Ele
ctro
n ve
loci
ty
24λ–0.2
0.2
0
t = 1600/ωi
Distance
Ele
ctro
n de
nsity
λt = 0nc
4
nc8
Cold electrons (κTe = 1 keV)
0 20 40Electron energy (keV)
Num
ber o
f ele
ctro
ns
60 80
105
104
103
102
101
100
Raman heatedelectrons (κThot = 13 keV)
Ch06_HotDensePlasmas.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Understanding Hot-Dense Plasmas RequiresTheory, Computations and Experiments
Theory
Experiments
Computations
Soft X-Ray/EUV Emission from a Hot-Dense Plasma
Ch06_F05VG.ai
Distance
Laser-plasmainteraction region
Laser light
Hot dense region ofintense x-ray emission
Ele
ctro
n de
nsity
nc• κTe ~ 50 eV to 1 keV• ne ~ 1020 to 1022 e/cm3
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Ch06_LineContinRad.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Line and Continuum Radiation from a Hot-Dense Plasma
Photon energy (ω)
Near thermalcontinuum
L-shell emission lines
Spe
ctra
l em
issi
on in
tens
ity
K-shell emission lines
Non-thermal radiationdue to hot or supra-thermal electrons
(6.136a)
Blackbody Radiation: The Equilibrium Limit
Ch06_BlackbodyRad.ai
(6.137)
(6.143a)
1.4
1.8
1.0
0.6
0.2
0 2 4 6 8Photon energy (x)
Spe
ctra
l brig
htne
ss
x = 2.822
x =
x3
(ex – 1)
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
ωκT
Ch06_LineContinRad2.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Line and Continuum Radiation
+
–
–
+Ze
–e
–e
bv
ω
A broad continuum results fromBremsstrahlung radiation due todiffering electron velocities anddifferent distances of closest approach.
Photon energies for bound-boundtransitions depend on the ionization state.
+Ze
n = 3
n = 2
n = 1
10e– 9e– 8e–
ω
+Ze
ω
+Ze
ω
Xe+10
10 11 12 13 14 15 16 17 18 19 20
10
20
30
40
50
Spe
ctra
l int
ensi
ty, Ι
λ[m
J/(2
π s
r) ·
nm]
Wavelength (nm)
O+5
O+4
Xe+9 Xe+8Xe+11
0
Emission Spectra from a Xenon Plasma
Ch06_F26.ai
Courtesy of M. Klosner and W. Silfvast, U. Central Florida.
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Ch06-IonzBtlnecks.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Ionization “Bottlenecks” Limit the Numberof Ionization States Present in a Plasma
1 143 eV to remove 11th electron, to form Neon-like AR2 No further ionization with proper plasma temperature3 Strong 3s → 2p and 3d → 2p line emission at 4.873 nm and 4.148 nm (254.4 and 298.9 eV)
Ch06_PlasmaTheories.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Plasma Theories Address Physical Phenomenaat Various Levels of “Particle Detail”
detailed positionsand velocities asa function of timefor all particles
• Kinetic theory for the evolution of f(v) in space and time
= vi
m = –e[E + vi B]
• Conservation of particles (mass)• Conservation of momentum (forces acting on a fluid)• Conservation of energy
Microscopic
f(v; r, t) ne, ni, κTe
P, v (r, t)
Kinetic Fluid
dridt
dvidt
Ch06_TheoDescrpPlas1.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Theoretical Description of a Plasma
Microscopic description of all particles
(6.9)
which is coupled to the electric and magnetic fields through Maxwell’sequations (averaged in the same manner). Obtain a fluid level descriptionby forming “velocity moments” of the collisionless Vlasov equation.
form an average f(v, r; t) over some appropriate spatial dimension. Show that f(v, r; t) satisfies a kinetic equation (“collisionless Vlasov equation”)
(6.39)n
Ch06_TheoDescrpPlas2.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Theoretical Description of a Plasma (continued)
to form the fluid equations
again coupled with Maxwell’s equations for E and B, and where
(6.39)
(6.41)
(6.52)
(6.40)
(6.43)
(6.33)
(6.34)
(6.35)
Ch06_MicroDescPlasm1.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Microscopic Description of a Plasma
A formal description of plasma dynamics, suggested by Klimontovich, involves a microscopic distribution function describing the position and velocity of all particles in a six dimensional velocity-position phase space:
where the detailed motion of the ith point particle is described by ri(t) and vi(t). The distribution function is normalized to the total number of particles, N, by thephase-space integral
where we define the shorthand notation, for example in Cartesian coordinates
and
(6.9)
(6.10)
(6.11)
(6.12)
(6.13)
(6.14)
Ch06_MicroDescPlasm2.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Microscopic Description of a Plasma (continued)
The dynamics of the particle distribution can be determined by taking a partial derivative of f(v, r; t) with respect to time:
with Lorentz force on each particle identifiedwith three-dimensional generalization and dri/dt = vi
These combine to give
Using chain rules for differentiation, ∂f∂g
∂g∂t
∂f(g)∂t =
Ch06_KineticDescrip.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
The Kinetic Description of a Plasma
Write the distribution function in terms of a slowly varying part and a fluctuating part, as
Substitute these into the Klimontovich equation (6.15) and average over a spatial scale sufficiently large to give a smoothed kinetic equation for the velocity distribution function. The product of fluctuations term on the right side gives a “collision term”, formally equivalent to a Boltzmann collision term.This is the Vlasov equation describing the evolution of the kinetic velocity distribution function.
Ch06_CollisnlesMxwl_Oct05.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
The Collisionless Maxwell-Vlasov Equations
Written for both electrons and ions, the collisionless Vlasov equation is
(6.32)
(6.26)
(6.27)
(6.28)
(6.29)
(6.30)
(6.31)
Plus Maxwell’s Equations with summed currents and charges due to both electrons and ions
Ch06_Kinetc_Landau.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
A Kinetic Effect: Landau Damping or Landau Growth
Landau damping Landau growth
• More slow electrons than fast electrons (slope negative) at waves phase velocity• Energy transfer from wave to particles• Wave is damped
• More fast electrons (vb > vφ; slope positive)• Energy transfer to the wave; the wave grows• Injected electron beam loses energy; f(v) changes with time.
ω = ωr + iωi
vφ
xPosition
Ele
ctro
nde
nsity
Ele
ctro
nde
nsity
vφ
xPosition
f(v)
ve
vφ v
f(v)
vφ v
Injected electronbeam (vb)ve
Ch06_FluidDescrpPlas.aii
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Fluid Description of a Plasma – Two Approaches
(1) Velocity weighted integrals of the velocity distribution function, f (v)
(2) Conservation of mass, momentum and energy in fluid dynamical control volumes
Rate of mass(or density)
change
Rate ofmass in
Rate ofmass out= –
Rate ofmomentum
change
Rate ofmomentum
in
Rate ofmomentum
out= –
Sum ofall
forces+
∆z∆x
mnv|z + ∆z
∆y
mnv|z
n
Ch06_ContinEqConsrv_05.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
The Continuity Equation for Conservationof Mass or Particles
(6.39)
(6.40)
Collisionless Vlasov equation: multiply by v0 and integrate over all velocities
Combining all three terms one has the fluid mechanical continuity equation
( )
v
Ch06_ConservMomentm1.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Conservation of Momentum:A Force Equation for a Fluid Plasma
Newton’s Second Law of Motion, F = ma, for a plasma at a fluid level ofdescription. From the kinetic theory, taking the mv velocity moment of thecollionless Vlasov equation.
The vv term introduces a dyadic pressure term, where
where v = v + v
and
~~
~
(6.41)
(6.35)and
(6.38)
Ch06_ConservMomentm2.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
Conservation of Momentum:A Force Equation for a Plasma Fluid (continued)
For an isotropic distribution function the dyadic pressure reduces to a scalar pressure, such that
In this case the conservation of momentum equation, for electrons,becomes
(6.43)
(6.42, in thescalar limit)
an evident expression of F = ma for a fluid volume of plasma,sometimes written as
Ch06_ConsrvEnergy1.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
The Conservation of Energy for a Plasma Fluid
To form a conservation of energy equation take the scalermv2/2 velocity moment of the collisionless Vlasov equation
Using similar techniques, this yields a conservation of energy equation:
where Ue is the (random) thermal energy and Qe is the thermal flux vector
(6.36)
(6.37)
(6.53)
(6.52)
e e e e
e
Ch06_ConsrvEnergy2_05.ai
Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley
The Conservation of Energy for a Plasma Fluid(continued)
For an isotropic, collisionless plasma with a symmetric velocitydistribution function, f (–v) = f (v)
thus one obtains the perfect gas law
and for this adiabatic (Q = 0) case, the energy equation yieldsthe adiabatic condition between pressure and density
where the thermodynamic ratio of specific heats is
γ = 1 + (γ = 5/3 for 3 degrees of translational motion)
(6.53)
(6.58)
e e e e
, Qe = 0 , e = 1 = Pe1
e e e
(6.60)e
e e
e
2Ν
nemv2
3~
e ev