PHYSICS OF HOT DENSE PLASMAS - EECS at UC Berkeley · Waves in a Plasma v φ x Position Electron...

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Nd (nω) laser- produced plasmas Pinch plasmas Plasma focus High pressure arcs Magnetic fusion plasmas Dis- charge plasmas S olar interio r Solar Cente r 0.1 eV 1 eV 10 eV 100 eV Electron temperature* (κT e ) Electron density (e/cm 3 ) 1 keV 10 keV Allen, Paul 10 8 10 10 10 12 10 14 10 16 10 18 10 20 10 22 10 24 10 26 PHYSICS OF HOT DENSE PLASMAS Chapter 6 Ch06_00.horiz.VG Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley C h r o m o s p h e r e T r a n sitio n C o r o n a

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

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

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

xPosition

Ele

ctro

nde

nsity

Ele

ctro

nde

nsity

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

nemv2

3~

e ev

Ch06_SummryEqs_05.ai

Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley

Summary of Fluid Equations for anIsotropic, Collisionless Plasma

(plus the same for ions)

(6.44)

(6.40)v

(6.43) (6.58)

(6.60b)e e

e

e e e

e

(6.45)

(6.46)

(6.47)

(6.48)

(6.49)