Electron-Acoustic Wave in a Plasmaattwood/sxr2009/... · Univ. California, Berkeley Transverse...

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Ch06_ElectrnAcoustWv1.ai Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Electron-Acoustic Wave in a Plasma For small fluctuations, n e /n 0 << 1, etc., no directed velocity v 0 = 0, and no background magnetic field (B 0 = 0), the equations linearize to where the non-linear product terms, like n e v, v v, and v B are neglected. (uniform ion distribution) ~ (6.75) (6.76) (6.77) 0 0

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Transcript of Electron-Acoustic Wave in a Plasmaattwood/sxr2009/... · Univ. California, Berkeley Transverse...

  • Ch06_ElectrnAcoustWv1.ai

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

    Electron-Acoustic Wave in a Plasma

    For small fluctuations, ne/n0

  • 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

  • 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

  • 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

  • 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, k1ω2, 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– - w

    ave

    kk

    ωi = ωea + ωRki = kea + kR

    ωi = ωia + ωBki = 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) ω

    κT

    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

    (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/cm2Focal 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/cm2150 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/cm2κTeq 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)