Introduction to Plasma Physics

Transcription

1 Mitglied der Helmholtz-Gemeinschaft Introduction to Plasma Physics CERN School on Plasma Wave Acceleration November 2014 Paul Gibbon Outline Lecture 1: Introduction Definitions and Concepts Lecture 2: Wave Propagation in Plasmas 2 57

2 Lecture 1: Introduction Plasma definition Plasma types Debye shielding Plasma oscillations Plasma creation: field ionization Relativistic threshold Further reading Introduction 3 57 What is a plasma? Simple definition: a quasi-neutral gas of charged particles showing collective behaviour. Quasi-neutrality: number densities of electrons, n e, and ions, n i, with charge state Z are locally balanced: n e Zn i. (1) Collective behaviour: long range of Coulomb potential (1/r) leads to nonlocal influence of disturbances in equilibrium. Macroscopic fields usually dominate over microscopic fluctuations, e.g.: ρ = e(zn i n e ).E = ρ/ε 0 Introduction Plasma definition 4 57

3 Where are plasmas found? 1 cosmos (99% of visible universe): interstellar medium (ISM) stars jets 2 ionosphere: 50 km = 10 Earth-radii long-wave radio 3 Earth: fusion devices street lighting plasma torches discharges - lightning plasma accelerators! Introduction Plasma types 5 57 Plasma properties Type Electron density Temperature n e ( cm 3 ) T e (ev ) Stars Laser fusion Magnetic fusion Laser-produced Discharges Ionosphere ISM Table 1: Densities and temperatures of various plasma types 1eV 11600K Introduction Plasma types 6 57

4 Debye shielding What is the potential φ(r) of an ion (or positively charged sphere) immersed in a plasma? Introduction Debye shielding 7 57 Debye shielding (2): ions vs electrons For equal ion and electron temperatures (T e = T i ), we have: Therefore, 1 2 m ev 2 e = 1 2 m iv 2 i = 3 2 k BT e (2) ( ) 1/2 ( ) 1/2 v i me me = = = 1 v e m i Am p 43 (hydrogen, Z=A=1) Ions are almost stationary on electron timescale! To a good approximation, we can often write: n i n 0, where the material (eg gas) number density, n 0 = N A ρ m /A. Introduction Debye shielding 8 57

5 Debye shielding (3) In thermal equilibrium, the electron density follows a Boltzmann distribution : n e = n i exp(eφ/k B T e ) (3) where n i is the ion density and k B is the Boltzmann constant. From Gauss law (Poisson s equation): 2 φ = ρ ε 0 = e ε 0 (n i n e ) (4) See, eg: F. F. Chen, p. 9 Introduction Debye shielding 9 57 Debye shielding (4) Combining (4) with (3) in spherical geometry a and requiring φ 0 at r =, get solution: φ D = 1 e r/λd. (5) 4πε 0 r Exercise with Debye length λ D = ( ) 1/2 ε0 k B T e = e n e ( Te ev ) 1/2 ( ) 1/2 ne cm (6) cm 3 a 2 1 r 2 d dr (r 2 dφ dr ) Introduction Debye shielding 10 57

6 Debye sphere An ideal plasma has many particles per Debye sphere: N D n e 4π 3 λ3 D Prerequisite for collective behaviour. Alternatively, can define plasma parameter: g 1 n e λ 3 D 1. (7) Classical plasma theory based on assumption that g 1, which also implies dominance of collective effects over collisions between particles. Introduction Debye shielding Collisions in plasmas At the other extreme, where N D 1, screening effects are reduced and collisions will dominate the particle dynamics. A good measure of this is the electron-ion collision rate, given by: ν ei = π 3 2 n e Ze 4 ln Λ (4πε 0 ) 2 m 2 e v 3 te s 1 v te k B T e /m e is the electron thermal velocity and ln Λ is a slowly varying term (Coulomb logarithm) O(10 20). Can show that ν ei Z ln Λ ; with ln Λ 9N D /Z ω p 10N D Introduction Debye shielding 12 57

7 Plasma classification Introduction Debye shielding Model hierarchy 1 First principles N-body molecular dynamics 2 Phase-space methods Vlasov-Boltzmann 3 2-fluid equations 4 Magnetohydrodynamics (single, magnetised fluid) Time-scales: s Length-scales: m Number of particles needed for first-principles modelling (1): (tokamak), (laser-heated solid) Introduction Debye shielding 14 57

8 Plasma oscillations: capacitor model Consider electron layer displaced from plasma slab by length δ. This creates two capacitor plates with surface charge σ = ±en e δ, resulting in an electric field: E = σ ε 0 = en eδ ε 0 Introduction Plasma oscillations Capacitor model (2) The electron layer is accelerated back towards the slab by this restoring force according to: m e dv dt = m e d 2 δ dt 2 = ee = e2 n e δ ε 0 Or: where d 2 δ dt 2 + ω2 pδ = 0, Electron plasma frequency ω p ( e 2 ) 1/2 ( n e ne ε 0 m e cm 3 ) 1/2 s 1. (8) Introduction Plasma oscillations 16 57

9 Response time to create Debye sheath For a plasma with temperature T e (and thermal velocity v te k B T e /m e ), one can also define a characteristic reponse time to recover quasi-neutrality: t D λ ( D ε0 k B T e = v te e 2 n e m k B T e ) 1/2 = ω 1 p. Introduction Plasma oscillations External fields: underdense vs. overdense If the plasma response time is shorter than the period of a external electromagnetic field (such as a laser), then this radiation will be shielded out. Figure 1: Underdense, ω > ω p : plasma acts as nonlinear refractive medium Figure 2: Overdense, ω < ω p : plasma acts like mirror Introduction Plasma oscillations 18 57

10 The critical density To make this more quantitative, consider ratio: ω 2 p ω 2 = e2 n e ε 0 m e λ 2 4π 2 c 2. Setting this to unity defines the wavelength for which n e = n c, or the Critical density n c λ 2 µ cm 3 (9) above which radiation with wavelengths λ > λ µ will be reflected. cf: radio waves from ionosphere. Introduction Plasma oscillations Plasma creation: field ionization At the Bohr radius a B = 2 me = cm, the electric field strength is: e E a = 4πε 0 ab Vm 1. (10) This leads to the atomic intensity: I a = ε 0cE 2 a Wcm 2. (11) A laser intensity of I L > I a will guarantee ionization for any target material, though in fact this can occur well below this threshold value (eg: Wcm 2 for hydrogen) via multiphoton effects. Introduction Plasma creation: field ionization 20 57

11 Ionized gases: when is plasma response important? Simultaneous field ionization of many atoms produces a plasma with electron density n e, temperature T e 1 10 ev. Collective effects important if ω p τ interaction > 1 Example (Gas jet) τ int = 100 fs, n e = cm 3 ω p τ int = 1.8 Typical gas jets: P 1bar; n e = cm 3 Recall that from Eq.9, critical density for glass laser n c (1µ) = cm 3. Gas-jet plasmas are therefore underdense, since ω 2 /ω 2 p = n e /n c 1. Exploit plasma effects for: short-wavelength radiation; nonlinear refractive properties; high electric/magnetic fields; particle acceleration! Introduction Plasma creation: field ionization Relativistic field strengths Classical equation of motion for an electron exposed to a linearly polarized laser field E = ŷe 0 sin ωt: dv dt ee 0 m e sin ωt v = ee 0 m e ω cos ωt = v os cos ωt (12) Dimensionless oscillation amplitude, or quiver velocity: a 0 v os c p os m e c ee 0 m e ωc (13) Introduction Relativistic threshold 22 57

12 Relativistic intensity The laser intensity I L and wavelength λ L are related to E 0 and ω by: I L = 1 2 ε 0cE 2 0 ; λ L = 2πc ω Substituting these into (13) we find : a (I 18 λ 2 µ) 1/2, (14) where Exercise I 18 = I L Wcm 2 ; λ µ = λ L µm. Implies that for I L Wcm 2, λ L 1 µm, we will have relativistic electron velocities, or a 0 1. Introduction Relativistic threshold Further reading 1 F. F. Chen, Plasma Physics and Controlled Fusion, 2nd Ed. (Springer, 2006) 2 R.O. Dendy (ed.), Plasma Physics, An Introductory Course, (Cambridge University Press, 1993) 3 J. D. Huba, NRL Plasma Formulary, (NRL, Washington DC, 2007) Introduction Further reading 24 57

13 Lecture 2: Wave propagation in plasmas Plasma oscillations Transverse waves Nonlinear wave propagation Further reading Formulary Wave propagation in plasmas Model hierarchy 1 First principles N-body molecular dynamics 2 Phase-space methods Vlasov-Boltzmann 3 2-fluid equations 4 Magnetohydrodynamics (single, magnetised fluid) Wave propagation in plasmas Plasma oscillations 26 57

14 The 2-fluid model Many plasma phenomena can be analysed by assuming that each charged particle component with density n s and velocity u s behaves in a fluid-like manner, interacting with other species (s) via the electric and magnetic fields. The rigorous way to derive the governing equations in this approximation is via kinetic theory, which is beyond the scope of this lecture. We therefore begin with the 2-fluid equations for a plasma assumed to be: thermal: T e > 0 collisionless: ν ie 0 and non-relativistic: velocities u c. Wave propagation in plasmas Plasma oscillations The 2-fluid model (2) n s + (n s u s ) = 0 (15) du s n s m s dt = n s q s (E + u s B) P s (16) d dt (P sn γ s s ) = 0 (17) P s is the thermal pressure of species s; γ s the specific heat ratio, or (2 + N)/N), where N is the number of degrees of freedom. Wave propagation in plasmas Plasma oscillations 28 57

15 Continuity equation The continuity equation (Eq. 15) tells us that (in the absence of ionization or recombination) the number of particles of each species is conserved. Noting that the charge and current densities can be written ρ s = q s n s and J s = q s n s u s respectively, Eq. (15) can also be written: ρ s + J s = 0, (18) which expresses the conservation of charge. Wave propagation in plasmas Plasma oscillations Momentum equation (Eq. 16) governs the motion of a fluid element of species s in the presence of electric and magnetic fields E and B. Remark: In the absence of fields, and assuming strict quasineutrality (n e = Zn i = n; u e = u i = u), we recover the Navier-Stokes equations. Exercise In the plasma accelerator context we will usually deal with unmagnetised plasmas, and stationary ions u i = 0, in which case the momentum equation reads: n e m e du e dt = en e E P e (19) Note that E can include both external and internal field components (via charge-separation). Wave propagation in plasmas Plasma oscillations 30 57

16 Longitudinal plasma waves A characteristic property of plasmas is their ability to transfer momentum and energy via collective motion. One of the most important examples of this is the oscillation of the electrons against a stationary ion background, or Langmuir wave. Returning to the 2-fluid model, we can simplify Eqs.(15-17) by setting u i = 0, restricting the electron motion to one dimension (x) and taking y = z = 0: n e n e ( ue + x (n eu e ) = 0 ) u e + u e = e x m n ee 1 m ( ) d Pe = 0 dt n γ e e P e x (20) Wave propagation in plasmas Plasma oscillations Longitudinal plasma waves (2) Poisson s equation The above system (20) has 3 equations and 4 unknowns. To close it we need an expression for the electric field, which, since B = 0, can be found from Gauss law (Poisson s equation) with Zn i = n 0 = const: E x = e ε 0 (n 0 n e ) (21) Wave propagation in plasmas Plasma oscillations 32 57

17 Longitudinal plasma waves (3) 1D electron fluid equations n e n e ( ue + x (n eu e ) = 0 + u e u e x ) = e m n ee 1 m P e x (22) d dt ( Pe n γ e e ) = 0 E x = e ε 0 (n 0 n e ) Wave propagation in plasmas Plasma oscillations Longitudinal plasma waves (4) Linearization This system is nonlinear, and apart from a few special cases, cannot be solved exactly. A common technique for analyzing waves in plasmas therefore is to linearize the equations, assuming the perturbed amplitudes are small compared to the equilibrium values: n e = n 0 + n 1, u e = u 1, P e = P 0 + P 1, E = E 1, where n 1 n 0, P 1 P 0. These expressions are substituted into (22) and all products n 1 t u 1, u 1 x u 1 etc. are neglected to get a set of linear equations for the perturbed quantities... Wave propagation in plasmas Plasma oscillations 34 57

18 Linearized equations n 1 + n 0 u 1 x = 0, Exercise n 0 u 1 E 1 x = e m n 0E 1 1 m = e ε 0 n 1, P 1 x, (23) P 1 = 3k B T e n 1. N.B. Expression for P 1 results from specific heat ratio γ e = 3 and assuming isothermal background electrons, P 0 = k B T e n 0 (ideal gas) see Kruer (1988). Wave propagation in plasmas Plasma oscillations Wave equation We can now eliminate E 1, P 1 and u 1 from (23) to get: ( 2 2 3v te 2 2 ) x + 2 ω2 p n 1 = 0, (24) with v 2 te = k BT e /m e and ω p given by (8) as before. Finally, we look for plane wave solutions of the form A = A 0 e i(ωt kx), so that our derivative operators become: iω; x ik. Substitution into (24) yields finally: Bohm-Gross dispersion relation for electron plasma waves ω 2 = ω 2 p + 3k 2 v 2 te (25) Wave propagation in plasmas Plasma oscillations 36 57

19 Electromagnetic waves To describe transverse electromagnetic (EM) waves, we need two more of Maxwells equations: Faraday s law (35) and Ampère s law (36), which we come to in their usual form later. To simplify things, taking our cue from the previous analysis of small-amplitude, longitudinal waves, we linearize and again apply the harmonic approximation iω: E 1 = iωb 1, (26) B 1 = µ 0 J 1 + iε 0 µ 0 ωe 1, (27) where the transverse current density is given by: J 1 = n 0 eu 1. (28) Wave propagation in plasmas Transverse waves Electromagnetic waves (2) Ohm s law We now look for pure EM plane-wave solutions with E 1 k. Also note that the group and phase velocities v p, v g v te, so that we can assume a cold plasma with P e = n 0 k B T e = 0. The linearized electron fluid velocity and corresponding current are then: u 1 = e iωm e E 1, J 1 = n 0e 2 where σ is the AC electrical conductivity. iωm e E 1 σe 1, (29) Wave propagation in plasmas Transverse waves 38 57

20 Electromagnetic waves (3) Dielectric function By analogy with dielectric media (see eg: Jackson), in which Ampere s law is usually written B 1 = µ 0 t D 1, by substituting (29) into (36), can show that D 1 = ε 0 εe 1 with ε = 1 + σ iωε 0 = 1 ω2 p ω 2. (30) Wave propagation in plasmas Transverse waves Electromagnetic waves (4) Refractive index From (30) it follows immediately that: Refractive index η ε = ck ω = ( 1 ω2 p ω 2 ) 1/2 (31) with Dispersion relation ω 2 = ω 2 p + c 2 k 2 (32) Exercise The above expression can also be found directly by elimination of J 1 and B 1 from Eqs. (26)-(29). Wave propagation in plasmas Transverse waves 40 57

21 Propagation characteristics Underdense plasmas From the dispersion relation (32) a number of important features of EM wave propagation in plasmas can be deduced. For underdense plasmas (n e n c ): ( ) Phase velocity v p = ω k c 1 + ω2 p > 2ω 2 c ( ) Group velocity v g = ω k c 1 ω2 p < 2ω 2 c Wave propagation in plasmas Transverse waves Propagation characteristics (2) Overdense plasmas In the opposite case, n e > n c, the refractive index η becomes imaginary, and the wave can no longer propagate, becoming evanescent instead, with a decay length determined by the collisionless skin depth c/ω p. Wave propagation in plasmas Transverse waves 42 57

22 Summary: dispersion curves Wave propagation in plasmas Transverse waves Nonlinear wave propagation The starting point for most analyses of nonlinear wave propagation phenomena is the Lorentz equation of motion for the electrons in a cold (T e = 0), unmagnetized plasma, together with Maxwell s equations. We also make two assumptions: 1 The ions are initially assumed to be singly charged (Z = 1) and are treated as a immobile (v i = 0), homogeneous background with n 0 = Zn i. 2 Thermal motion is neglected justified for underdense plasmas because the temperature remains small compared to the typical oscillation energy in the laser field (v os v te ). Wave propagation in plasmas Nonlinear wave propagation 44 57

23 Lorentz-Maxwell equations Starting equations (SI units) are as follows p + (v )p = e(e + v B), (33) E = e ε 0 (n 0 n e ), (34) E = B, (35) c 2 B = e n e v + E ε 0, (36) B = 0, (37) where p = γm e v and γ = (1 + p 2 /m 2 e c2 ) 1/2. Wave propagation in plasmas Nonlinear wave propagation Electromagnetic waves To simplify matters we first assume a plane-wave geometry like that above, with the transverse electromagnetic fields given by E L = (0, E y, 0); B L = (0, 0, B z ). From Eq. (33) the transverse electron momentum is then simply given by: p y = ea y, (38) Exercise where E y = A y /. This relation expresses conservation of canonical momentum. Wave propagation in plasmas Nonlinear wave propagation 46 57

24 The EM wave equation I Substitute E = φ A/; B = A into Ampère Eq.(36): c 2 ( A) + 2 A 2 = J ε 0 φ, where the current J = en e v. Now we use a bit of vectorial magic, splitting the current into rotational (solenoidal) and irrotational (longitudinal) parts: J = J + J = Π + Ψ from which we can deduce (see Jackson!): J 1 c 2 φ = 0. Wave propagation in plasmas Nonlinear wave propagation The EM wave equation II Now apply Coulomb gauge A = 0 and v y = ea y /γ from (38), to finally get: EM wave 2 A y 2 c2 2 A y = µ 0 J y = e2 n e ε 0 m e γ A y. (39) The nonlinear source term on the RHS contains two important bits of physics: n e = n 0 + δn Coupling to plasma waves γ = 1 + p 2 /me 2c2 Relativistic effects Wave propagation in plasmas Nonlinear wave propagation 48 57

25 Electrostatic (Langmuir) waves I Taking the longitudinal (x)-component of the momentum equation (33) gives: d dt (γm ev x ) = ee x e2 2m e γ A 2 y x We can eliminate v x using Ampère s law (36) x : 0 = e ε 0 n e v x + E x while the electron density can be determined via Poisson s equation (34): n e = n 0 ε 0 E x e x., Wave propagation in plasmas Nonlinear wave propagation Electrostatic (Langmuir) waves II The above (closed) set of equations can in principle be solved numerically for arbitrary pump strengths. For the moment, we simplify things by linearizing the plasma fluid quantities: n e n 0 + n v x v 1 + v and neglect products like n 1 v 1 etc. This finally leads to: Driven plasma wave ( 2 γ ω2 p ) E x = ω2 p e 2m e γ 2 0 x A2 y (40) The driving term on the RHS is the relativistic ponderomotive force, with γ 0 = (1 + a 2 0 /2)1/2. Wave propagation in plasmas Nonlinear wave propagation 50 57

26 Cold plasma fluid equations: summary Electromagnetic wave 2 A y 2 c2 2 A y = µ 0 J y = e2 n e ε 0 m e γ A y Electrostatic (Langmuir) wave ( ) 2 + ω2 p 2 γ 0 E x = ω2 p e 2m e γ 2 0 x A2 y Wave propagation in plasmas Nonlinear wave propagation Cold plasma fluid equations: outlook These coupled fluid equations and their fully non-linear variations describe a vast range of nonlinear laser-plasma interaction phenomena: plasma wake generation: Bingham, Assmann blow-out regime: Silva laser self-focussing and channelling: Najmudin, Cros parametric instabilities harmonic generation,... Plasma-accelerated particle beams, on the other hand, cannot be treated with fluid theory and require a more sophisticated kinetic approach. see Pukhov Wave propagation in plasmas Nonlinear wave propagation 52 57

27 Further reading 1 J. Boyd and J. J. Sanderson, The Physics of Plasmas 2 W. Kruer, The Physics of Laser Plasma Interactions, Addison-Wesley, P. Gibbon, Short Pulse Laser Interactions with Matter: An Introduction, Imperial College Press, J. D. Jackson, Classical Electrodynamics, Wiley 1975/ J. P. Dougherty in Chapter 3 of R. Dendy Plasma Physics, 1993 Wave propagation in plasmas Further reading Constants Name Symbol Value (SI) Value (cgs) Boltzmann constant k B JK erg K 1 Electron charge e C statcoul Electron mass m e kg g Proton mass m p kg g Planck constant h Js erg-s Speed of light c ms cms 1 Dielectric constant ε Fm 1 Permeability constant µ 0 4π 10 7 Proton/electron mass ratio m p /m e Temperature = 1eV e/k B K K Avogadro number N A mol mol 1 Atmospheric pressure 1 atm Pa dyne cm 2 Wave propagation in plasmas Formulary 54 57

28 Formulae Name Symbol Formula (SI) Formula (cgs) ( ) 1 ( ε0 k B T e 2 Debye length λ D m kb T e ) 1 2 cm e 2 n e 4πe 2 n e 4π Particles in Debye sphere N D 3 λ3 D ( ) 1 e 2 ne 2 Plasma frequency (electrons) ω pe s 1 ε 0 m e 4π 3 λ3 D ( ) 1 4πe 2 ne 2 s 1 m e ( ) 1 Z 2 e 2 ni 2 Plasma frequency (ions) ω pi s 1 ε 0 m i ( ) 1 4πZ 2 e 2 ni 2 s 1 m i ( ) 1 kb T e 2 Thermal velocity v te = ω pe λ D ms 1 m e ( ) 1 kb T e 2 cms 1 m e Electron gyrofrequency ω c eb/m e s 1 eb/m e s 1 π 3 2 n e Ze 4 ln Λ Electron-ion collision frequency ν ei (4πε 0 ) 2 me 2v3 te s 1 4(2π) 1 2 n e Ze 4 ln Λ 3m 2 e v3 te s 1 Coulomb-logarithm ln Λ ln 9N D Z ln 9N D Z Wave propagation in plasmas Formulary Useful formulae Plasmafrequency Critical density ω pe = ne n c = λ 2 L cm 3 s 1 Debye length λ D = 743 T 1 2 e n 1 2 e cm Skin depth δ = c/ω p = n 1 2 e cm Elektron-ion collision frequency ν ei = n e T 3 2 e Ion-ion collision frequency Quiver amplitude ν ii = Z 4 ( mp a 0 p osc m e c = Relativistic focussing threshold P c = 17 ( nc T e in ev; n e, n i in cm 3, wavelength λ L in µm n e ( ) m i ln Λ s 1 ) 1 2 ni T 3 2 i ln Λ s 1 Iλ 2 L Wcm 2 µm 2 GW ) 1 2 Wave propagation in plasmas Formulary 56 57

29 Maxwell s Equations Name (SI) (cgs) Gauss law.e = ρ/ε 0.E = 4πρ Gauss magnetism law.b = 0.B = 0 Ampère B = µ 0 J + 1 c 2 E B = 4π c J + 1 c E Faraday E = B E = 1 c B Lorentz force E + v B E + 1 c v B per unit charge Wave propagation in plasmas Formulary 57 57