PHYSICS OF HOT DENSE PLASMAS
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1 Chapter 6 PHYSICS OF HOT DENSE PLASMAS Solar Center Electron density (e/cm 3 ) High pressure arcs Chromosphere Discharge plasmas Solar interior Nd (nω) laserproduced Plasma focus plasmas Pinch plasmas Magnetic fusion plasmas Transition Corona Allen, Paul ev 1 ev 10 ev 100 ev 1 kev 10 kev Electron temperature* (κt e ) Ch06_00.horiz.VG
2 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_ProcessPlasma.ai
3 Particle-Particle Interactions: Short Range Collisions e v + +Ze Ch06_Particle2Interctns.ai
4 The Velocity Distribution Function, f(v) f(v) v e 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. (6.2) The area under the curve is normalized to density. For electrons f e (v) dv = n e Kinetic theory describes how f(v) varies in space and time. The Maxwellian velocity distribution corresponds to an equilibrium situation (static in time) (6.1) with thermal velocity, v e, given above in eq. (6.2). Ch06_VeloctyDistrib_Oct05.ai
5 Waves in a Plasma An electron-acoustic wave, typically oscillating at the plasma frequency, ω p. ω Dispersion diagram for natural modes of oscillation in a plasma. c Electromagnetic wave Electron density v φ Frequency ω p Electron-acoustic wave Ion-acoustic wave a e a* x k D = 1/λ D k Position Wavenumber Ch06_WavesPlasma.ai
6 Wave-Particle Interactions Electron plasma wave Electron velocity distribution Electron density v φ f(v) v e More slow electrons Fewer fast electrons Position x v ø = ω k v Wave damping or growth Equilibrium or non-equilibrium velocity distribution Ch06_WavParticle.ai
7 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. Incident wave (ω i, k i ) Plasma wave (ω p, k p ) Scattered wave (ω s, k s ) (6.3) (6.4) Linear scattering Non-linear scattering Ch06_LinearNonLinScat.ai
8 Plasma Theories Address Physical Phenomena at Various Levels of Particle Detail Microscopic detailed positions and velocities as a function of time for all particles Kinetic f(v; r, t) Fluid n e, n i, κt e P, v (r, t) dr i dt = v i dv m i = e[e + v i B] dt Kinetic theory for the evolution of f(v) in space and time Conservation of particles (mass) Conservation of momentum (forces acting on a fluid) Conservation of energy Ch06_PlasmaTheories.ai
9 Plasma Theory Microscopic Description (microscopically tracks all particles individually) (6.9) Plus Maxwell s equations (6.15) Kinetic Description (does not track individual particles; tracks them as a function of coordinates v, r, t) ~ ~ ~ collision term (6.25) Plus Maxwell s equations Fluid Description (averages out the kinetic velocity information; r, t dependance only) (6.40) (6.43) Plus Maxwell s equations Ch06_PlasmaThry_05.ai
10 Plasma Modeling Numerical simulations of a finite number of particles to study non-linear processes with limited space-time variations Electron density n c 4 n c 8 Distance t = 0 λ Electron velocity Distance t = 1600/ω i 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. 24λ Number of electrons Cold electrons (κt e = 1 kev) Raman heated electrons (κt hot = 13 kev) Electron energy (kev) 10 µm r B = 0.7 MG B = 2.0 MG r Laser light 150 µm 75 µm 75 µm B = 0.2 MG ρ = 50ρ c z 1 ρ = 2ρ c ρ = ρ 2 c T e = 0.5 kev Density contours T e = 0.3 kev T e = 0.1 kev L1 z Symmetry axis Electron temperature Courtesy of G. Dahlbacka (LLNL), K. Estabrook (LLNL), and D. Forslund (Los Alamos) Ch06_PlasmaModelng.ai
11 Understanding Hot-Dense Plasmas Requires Theory, Computations and Experiments Theory Computations Experiments Ch06_HotDensePlasmas.ai
12 Soft X-Ray/EUV Emission from a Hot-Dense Plasma Hot dense region of intense x-ray emission Electron density n c Laser light κt e ~ 50 ev to 1 kev n e ~ to e/cm 3 Laser-plasma interaction region Distance Ch06_F05VG.ai
13 Line and Continuum Radiation from a Hot-Dense Plasma L-shell emission lines Spectral emission intensity Near thermal continuum K-shell emission lines Non-thermal radiation due to hot or suprathermal electrons Photon energy ( ω) Ch06_LineContinRad.ai
14 Blackbody Radiation: The Equilibrium Limit Spectral brightness x = x 3 (e x 1) x = ω κt Photon energy (x) (6.136a) (6.137) (6.143a) Ch06_BlackbodyRad.ai
15 Line and Continuum Radiation A broad continuum results from Bremsstrahlung radiation due to differing electron velocities and different distances of closest approach. e e v b + +Ze ω n = 3 n = 2 Photon energies for bound-bound transitions depend on the ionization state. 10e 9e 8e ω ω ω n = 1 +Ze +Ze +Ze Ch06_LineContinRad2.ai
16 Emission Spectra from a Xenon Plasma 50 Xe +11 Xe +10 Xe +9 Xe +8 Spectral intensity, Ιλ [mj/(2π sr) nm] O +5 O Wavelength (nm) Courtesy of M. Klosner and W. Silfvast, U. Central Florida. Ch06_F26.ai
17 Ionization Bottlenecks Limit the Number of Ionization States Present in a Plasma ev to remove 11th electron, to form Neon-like AR 2 No further ionization with proper plasma temperature 3 Strong 3s 2p and 3d 2p line emission at nm and nm (254.4 and ev) Ch06-IonzBtlnecks.ai
18 Plasma Theories Address Physical Phenomena at Various Levels of Particle Detail Microscopic detailed positions and velocities as a function of time for all particles Kinetic f(v; r, t) Fluid n e, n i, κt e P, v (r, t) dr i dt = v i dv m i = e[e + v i B] dt Kinetic theory for the evolution of f(v) in space and time Conservation of particles (mass) Conservation of momentum (forces acting on a fluid) Conservation of energy Ch06_PlasmaTheories.ai
19 Theoretical Description of a Plasma Microscopic description of all particles (6.9) 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 ) which is coupled to the electric and magnetic fields through Maxwell s equations (averaged in the same manner). Obtain a fluid level description by forming velocity moments of the collisionless Vlasov equation. n (6.39) Ch06_TheoDescrpPlas1.ai
20 Theoretical Description of a Plasma (continued) (6.39) (6.41) to form the fluid equations (6.52) (6.40) (6.43) again coupled with Maxwell s equations for E and B, and where (6.33) (6.34) (6.35) Ch06_TheoDescrpPlas2.ai
21 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: (6.9) where the detailed motion of the ith point particle is described by r i (t) and v i (t). The distribution function is normalized to the total number of particles, N, by the phase-space integral (6.10) where we define the shorthand notation, for example in Cartesian coordinates (6.11) and (6.12) (6.13) (6.14) Ch06_MicroDescPlasm1.ai
22 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: Using chain rules for differentiation, f(g) t f = g g t with three-dimensional generalization and dr i /dt = v i with Lorentz force on each particle identified These combine to give Ch06_MicroDescPlasm2.ai
23 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_KineticDescrip.ai
24 The Collisionless Maxwell-Vlasov Equations Written for both electrons and ions, the collisionless Vlasov equation is Plus Maxwell s Equations with summed currents and charges due to both electrons and ions (6.26) (6.32) (6.27) (6.28) (6.29) (6.30) (6.31) Ch06_CollisnlesMxwl_Oct05.ai
25 A Kinetic Effect: Landau Damping or Landau Growth Landau damping f(v) v e f(v) v e Landau growth Injected electron beam (v b ) v φ v v φ v v φ v φ Electron density Electron density ω = ω r + iω i Position More slow electrons than fast electrons (slope negative) at waves phase velocity Energy transfer from wave to particles Wave is damped x Position More fast electrons (v b > v φ ; slope positive) Energy transfer to the wave; the wave grows Injected electron beam loses energy; f(v) changes with time. x Ch06_Kinetc_Landau.ai
26 Fluid Description of a Plasma Two Approaches (1) Velocity weighted integrals of the velocity distribution function, f (v) n (2) Conservation of mass, momentum and energy in fluid dynamical control volumes mnv z mnv z + z Rate of mass (or density) change Rate of = mass in Rate of mass out x z y Rate of momentum change Rate of = momentum in Rate of momentum out + Sum of all forces Ch06_FluidDescrpPlas.aii
27 The Continuity Equation for Conservation of Mass or Particles Collisionless Vlasov equation: multiply by v 0 and integrate over all velocities (6.39) ( ) Combining all three terms one has the fluid mechanical continuity equation v (6.40) Ch06_ContinEqConsrv_05.ai
28 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 of description. From the kinetic theory, taking the mv velocity moment of the collionless Vlasov equation. (6.41) where v = v + ~ v and The ~~ vv term introduces a dyadic pressure term, where (6.35) and (6.38) Ch06_ConservMomentm1.ai
29 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) an evident expression of F = ma for a fluid volume of plasma, sometimes written as (6.42, in the scalar limit) Ch06_ConservMomentm2.ai
30 The Conservation of Energy for a Plasma Fluid To form a conservation of energy equation take the scaler mv 2 /2 velocity moment of the collisionless Vlasov equation e (6.52) Using similar techniques, this yields a conservation of energy equation: e e e e (6.53) where U e is the (random) thermal energy and Q e is the thermal flux vector (6.36) (6.37) Ch06_ConsrvEnergy1.ai
31 The Conservation of Energy for a Plasma Fluid (continued) e e e e (6.53) For an isotropic, collisionless plasma with a symmetric velocity distribution function, f ( v) = f (v) n e mv~, Q e = 0, e = 2 e v e 3 1 = P e 1 thus one obtains the perfect gas law e e e (6.58) and for this adiabatic (Q = 0) case, the energy equation yields the adiabatic condition between pressure and density e e e e (6.60) where the thermodynamic ratio of specific heats is 2 γ = 1 + (γ = 5/3 for 3 degrees of translational motion) Ν Ch06_ConsrvEnergy2_05.ai
32 Summary of Fluid Equations for an Isotropic, Collisionless Plasma v (6.40) e e e e (6.60b) (6.43) (6.58) e e e (plus the same for ions) (6.44) (6.45) (6.46) (6.47) (6.48) (6.49) Ch06_SummryEqs_05.ai
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