Ch06_ElectrnAcoustWv1.ai

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

Electron-Acoustic Wave in a Plasma

For small fluctuations, ne/n0 << 1, etc., no directed velocity v0 = 0, and no background magnetic field (B0 = 0), the equations linearize to

where the non-linear product terms, like nev, v v, and v B are neglected.

(uniform iondistribution)

~

(6.75)

(6.76)

(6.77)

0

0

Ch06_ElectrnAcoustWv2.ai

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

Electron-Acoustic Wave in a Plasma (continued)

with sound speed ae given by

and plasma frequency ωp given by

These can be combined to form a wave equation

or

Taking ∂/∂t of Eq. (6.75), of Eq. (6.76),

(6.78)

(6.79)

(6.80)

This can be rewritten as a longitudinal wave equation for electron density fluctuations

Ch06_ElecAcousWvDispr.ai

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

Electron-Acoustic Wave: Dispersion Relation

(6.78)

(6.84)

(6.83)

For an electron density wave of the form

We have, as in chapter 2,

and

so that the wave equation takes the form

This has solutions for finite electron density ne when when the bracketedoperator is zero, giving the dispersion relation for the electron-acousticwave

For long period plasma waves, where k goes to zero, there is a natural oscillation at the electron plasma frequency, ω ωp. For waves of finite k,in the range of 0 ≤ k ≤ ωp/ae, the frequency increases somewhat, to a valueof 2 ωp at k = ωp/ae, as shown in the dispersion diagram, next page.

Ch06_DispersionDiagrm.ai

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

Electron-Acoustic Wave: Dispersion Diagram

(6.84)

Dispersion relation:

Dispersion diagram:

ωp

ae

kD = 1/λD k

a*

cElectromagnetic wave

Electron-acousticwave

Freq

uenc

y

Wavenumber

ω

Ion-acousticwave

Ch06_TransvrseElectro1.ai

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

Transverse Electromagnetic Waves in a Plasma

For transverse waves in a plasma, we neglect the longitudinal fieldcomponents ( → ik terms). The transverse fields are described by

Take ∂∂t of eq. (6.99) and curl of eq. (6.100) to obtain

(6.99)

(6.100)

(6.103)

(6.106)

Eliminate the ∂H/∂t terms and use the vector identity ( E) = ( E) – 2E to form

(6.104)

(6.105)

and

Ch06_TransvrseElectro2.ai

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

Transverse Electromagnetic Waves in a Plasma(continued)

0 for transverse fields

For a plane wave of the form E(r, t) = E0e–i(ωt – kr) this yields a dispersion relation

Rearranging terms

(6.106)

(6.107)

(6.108)

Recognizing c2 = 1/0µ0 and ω2 = e2n/0m, we have the wave equation for a transverse wave in a plasma,

p

Ch06_TransvrseElectro3.ai

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

Transverse Elecromagnetic Waves in a Plasma

(6.108)

Dispersion relation:

Dispersion diagram:

ωp

ae

kD = 1/λD k

a*

c

Electromagnetic wave

Electron-acousticwave

Freq

uenc

y

Wavenumber

ω

Ion-acousticwave

Ch06_PropagatnOvrdns1.ai

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

Propagation in an Overdense Plasma

(6.108)

(6.109)

(6.110)

(6.111)

2

Solving for k

which corresponds to a penetration depth l into the highlyoverdense plasma of

For ω < ωp, k is imaginary, the wave exponentially decays.In the highly overdense limit ω << ωp

Ch06_PropagatnOvrdns2.ai

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

Propagation in an Overdense Plasma (continued)

(6.108)

(6.112a)

(6.112b)

The frequency for which ω = ωp, is referred to as the cutoff or critical frequency, and corresponding electron density is defined as the critical electron density, nc

or in terms of the wavelength (in microns)

Ch06_RefracIndxPlas.ai

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

Refractive Index of a Plasma

(6.113a)

(6.114a)

(6.114b)

For ω > ωp there is a real propagating wave with phase velocity

The refractive index of the plasma is

or equivalently

Ch06_PhaseGroupVelo1.ai

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

Phase Velocity and Group Velocity

In a dispersive medium different frequency components travel with different phase velocities. If there is some frequency modulation, the various components will interfere to form a modulation envelope, which will travel at a different, slower velocity. The velocity with which the envelope moves is known as the “group velocity”. We associate the group velocity with information or energy transport.

Reference: E.C. Jordon, Electromagnetic Waves and Radiating Systems (Prentice-Hall, NJ, 1950).

Modulationenvelope, vg

Individual frequency components, vφ

Ch06_PhaseGroupVelo2.ai

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

Phase Velocity and Group Velocity (continued)

Dispersion relation:

(6.108)

(6.113a)

(6.113b)

Phase velocity:

Group velocity:

where we associate group velocity with information orenergy transport. For ne/nc << 1 both vφ and vg are near c.However, for ne/nc << 1, vφ → and vg → 0.

Ch06_CollisnalAbsrptnTrans.ai

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

Collisional Absorption of a Transverse Wavein a Plasma

The analysis is readily extended to include the effect of collisions betweenelectrons, oscillating due to the transverse wave, and ions. By including acollision term, the electron momentum transfer equation (6.103) becomes

(6.115)

where the momentum transfer is proportional to the electron momentum mvand where vei is the electron-ion collision frequency. The electron velocitycan be written as

and the dispersion relation (6.108) modified, for vei << ω, to

If we set ω = ωr + iωi and solve separately for the real and imaginary parts,we find

and an imaginary component

where the wave now propagates in the plasma as before, but is attenuated bythe loss of energy due to electron-ion collisions.

(6.116)

(6.117)

(6.118a)

(6.118b)

Ch06_WvsMagntzdPlas.ai

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

Waves in a Magnetized Plasma

k

ω

ωc

kD

c

ωρ

Cyclotronresonance

Plasmaresonance

Freq

uenc

y

Wavenumber

Hybridresonance

(θ = 0)

(θ = 0)

(θ = 0)

(θ = π/2)(π/2)

(π/2)

(0)

ω2 + ω2ρ c

(π/2)

(6.5)

(6.8)

(Courtesy of N. Marcuvitz.)

Ch06_NonLinearProc.ai

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

Non-Linear Processes in a Plasma

(6.123a)

(6.123b)

(6.123c)

(6.123d)

Product terms lead to non-linear growth and frequency mixing

Ch06_LinearNonLinScat2.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.

(6.125a)

(6.124b)

• Linear scattering

• Non-linear scattering(6.125b)

ω1, k1

ω2, k2

ω3, k3

Ch06_StimBrillouin.ai

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

Stimulated Brillouin and Raman Scattering of Intense Laser Light

ω

(ωR, kR)

(ωi, ki)

Electron-acoustic(high frequency)

(ωea, kea)

e– - wave

kk

ωi = ωea + ωR

ki = kea + kR

ωi = ωia + ωB

ki = kia + kB

0

ω

(ωB, kB)

(ωia, kia)

(ωi, ki)

Ion-acoustic(low frequency)

0

Electromagneticwave

EM

(a) Brillouin:

Ion-wave

(b) Raman:

ωPeωPe

By resonant three-wave mixing naturally occurring wavesare driven out of the noise, to large amplitude, by an intenseincident electromagnetic wave.

Ch06_StimRamanBackscat.ai

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

Stimulated Raman Backscattering at ne nc/4

(ωea, kea)

k

ae

ki kea = ki + ∆kkR = ∆k

(ωi, ki)

ωp

ω ccIncident electromagnetic wave

Electron-acoustic waveScattered electromagnetic wave

(ωR, kR)

(following H. Motz)

Frequency and wavenumber matching occurs at the quarter-critical surface.For κTe = 1 keV, matching occurs at ωea = 1.005ωp, ωi = 2.005ωp, andki = 3 ωp/c. The phase velocity of the stimulated electron acoustic wave

Trapped electrons within the waves high potential crests can be acceleratedto velocities of c/ 3 , corresponding to electron energies of order 100 keV.and will radiate x-rays of order 100 keV. (Reference: Turner, Drake, Campbell, LLNL).

(6.130)

Ch06_VeryHardXRs_IntensLasRad.ai

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

Very Hard X-Rays Can be Generatedby Intense Laser Radiation

Photon energy (ω)

Near thermalcontinuum

L-shell emission linesS

pect

ral e

mis

sion

inte

nsity

K-shell emission lines

Non-thermal radiationdue to hot or supra-thermal electrons

Typically strong non-linear mixing occurs for

where and

(6.131)

Ch06_ContRadBlackSpec.ai

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

Continuum Radiation and BlackBody Spectra

Hot-dense plasmas with sharp spatial and temporal gradients,rapid expansion, and a variety of temperatures (Te, Ti, Thot, . . .)are not in equilibrium. Nonetheless a great fraction of the radiated energy is in a near-thermal distribution. Thus it is of value to consider the limiting case of blackbody radiation, that emitted by matter in equilibrium with its’ surroundings, and characterized by a single temperature T. Following Planck (1900), the spectral energy density U∆ω inunits of energy per unit volume, per unit frequency interval∆ω at frequency ω, is

in units of ∆2E/∆V ∆ω. In terms of relative spectral bandwidth, ∆ω/ω,

where κ is the Boltzmann constant

(6.134a)

(6.134b)

Ch06_BlackbdyRad.ai

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

Blackbody Radiation

with peak at

(6.136a)

(6.137)

Ch06_BlkbdyRadEqLmt.ai

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

Blackbody Radiation: The Equilibrium Limit

(6.136b)

(6.137)

(6.138b)

Example: κT = 100 eV, B∆ω/ω = 4.47 1017

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) ωκT

2.82 κT

photons/secmm2 mrad2 (0.1% BW)

Ch06_BlkbdyRadSurface.ai

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

Blackbody Radiation Across a Surface

The power per unit area across a surface, in one direction, per unit spectral bandwidth, is

Integrating over all frequencies

The blackbody intensity at any interface is

where σ is the Stefan-Boltzmann constant. Written with κT expressed in electron volts

(6.143a) ; (6.143b)

Example: For κT = 100 eV, I = 1.027 1013 W/cm2.

where θ is mearsured from the surface normal, dΩ = sin θ dθ dφ = 2π sin θ, for 0 ≤ θ ≤ π/2, and where I∆ω/ω has units of energy per unit area per unit of relative spectral bandwidth ∆ω/ω. Since the spectral brightness is isotropic (no θ-dependence), and 0 2π sin θcosθdθ = π, one has

2π

(6.139)

(6.140)

(6.141)

κT

κT

SEMATECH Sources : 3.3.2002Dr. Lebert, Dr. Juschkin , AIXUV

Planck radiator: the most brilliant radiator in nature except for a laser

Thermal PlasmaEmission:

Line radiation

Rekombinationradiation .

Bremssstrahlung

maximum Brilliancefor a given

temperature

Radiation is inequilibrium with

matter(Kirchhoff)

Thermal PlasmaEmission:

Line radiation

Rekombinationradiation .

Bremssstrahlung

maximum Brilliancefor a given

temperature

Radiation is inequilibrium with

matter(Kirchhoff)

( )BBkTISj

)(ννν

ν

α==

10-1 100 101 102 103 104

Wellenlänge [nm]

Log (

Lλ)

Bremsstrahlung

Rekombination radiation

Line radiation

Planck

ωL=ωpe

Ch06_PlanckRadiatr.ppt

Courtesy of Dr. R. Lebert

Soft X-Ray/EUV Emission from a Laser-Produced Plasma

Ch06_F05VG.modif.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

Ch03_NotchFilter.ai

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

The Notch Filter

• Combines a glancing incidence mirror and a filter• Modest resolution, E/∆E ~ 3-5• Commonly used

Mirrorreflectivity(“low-pass”)

Absorptionedge Filter

transmission(“high-pass”)

Photon energy

1.0

Filter/reflectorwith responseE/∆E 4

Ch06_F.24VG.ai

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

Measuring Continuum Emission from a Hot Dense Plasma

Laserirradiated

target

EUV/Soft X-rayemission

Glancingincidence

mirrors

θ

Soft X-raystreak camera

Opticaltrigger

Filter pack

Thin foil cathode substrate

Time

Courtesy of G. Stradling, R. Kauffman, and H. Medecki, LLNL.

• Cross-calibrate each channel with a fast, calibrated EUV/X-ray diode

Ch06_F.25VG.ai

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

Determining an Equivalent Blackbody Temperature

1011

1010

109

108

1070 1000 2000

Time (ps)

Equ

ival

ent r

adia

tion

tem

pera

ture

(eV

)

3000

300

250

200

150

100

790 psec FWHM

50

Au disk2ω, 0.53 mEL = 355, τL 680 psΙ 1 × 1015 W/cm2

Focal diameter 80 m

200 eV

400 eV

600 eV

Time

Rad

iate

d po

wer

per

cha

nnel

Rad

iate

d so

ft x-

ray

pow

er (W

atts

)

(Courtesy of R. Kauffman, LLNL)

Ch06_MeasurePlasRad.ai

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

Measuring Plasma Radiationin Specific Narrow Spectral Bands

Laserirradiated

target

Soft X-rayemission

CHCFSiO2

Layeredtarget Laser

light

Multilayermirrors (5)

25°

Soft x-raystreak camera

Filters (5)

Time

0

1

10–1

10–2

1,000Time (psec)

Rec

orde

d so

ft x-

ray

sign

al (r

elat

ive)

2,000 3,000

102 eV737 eV943 eV

0.27 µm Be on A1.06 µm89J, 720 ps3 × 1014 W/cm2

(Courtesy of G. Stradling)

Ch06_LineContinRad3.ai

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

Multiple Ionization States Result in Many Emission Lines

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

Ch06_F.23VG.ai

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

Soft X-Ray Emission Spectrafrom a Laser Produced Plasma

0.80

0.75

0.70

0.65

0.60

0.55

500 550Photon energy (eV)

Film

den

sity

600 650 700 800750

Cr+14(3s - 2p)

Cr+15(3s - 2p)

Cr+14(3d - 2p) Fe L-edge(707 eV)

(Courtesy of R. Kauffman and L. Koppel, LLNL)

The dominant “neon-like” Cr+14 has 10 electrons: 1s2 2s2 2p6 (ground state)and a 1012 eV ionization energy

1.06 µm laser2 1014 W/cm2

150 psecCr (Z = 24)

Ch06-IonzBtlnecks2.ai

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

Ionization “Bottlenecks” Limit the Numberof Ionization States Present in a Plasma

Cr target at2 1014 W/cm2

κTeq 200 eV4κTeq 800 eV(too low to efficiently ionize neon-like Cr+14 to Cr+15)

(Courtesy of J. Scofield, LLNL)

Ch06_Atomic_IonicBk.ai

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

Three volumes,bound as books

Ch06_ChromiumChart.ai

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

R. Kelley: Atomic and Ionic Spectral Lines

Neon-like Cr

(670 eV)(660 eV)

(594 eV)

(586 eV)