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Introduction to Plasma Physics

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Introduction to Plasma Physics Mitglied der Helmholtz-Gemeinschaft Introduction to Plasma Physics CERN School on Plasma Wave Acceleration 24-29 November 2014 Paul Gibbon Outline Lecture 1: Introduction – Definitions and Concepts Lecture 2: Wave Propagation in Plasmas 2 57 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, ne, and ions, ni , with charge state Z are locally balanced: ne ' Zni : (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(Zni − ne) ) r:E = ρ/ε0 Introduction Plasma definition4 57 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 types5 57 Plasma properties Type Electron density Temperature −3 ∗ ne ( cm ) Te (eV ) Stars 1026 2 × 103 Laser fusion 1025 3 × 103 Magnetic fusion 1015 103 Laser-produced 1018 − 1024 102 − 103 Discharges 1012 1-10 Ionosphere 106 0.1 ISM 1 10−2 Table 1: Densities and temperatures of various plasma types ∗ 1eV ≡ 11600K Introduction Plasma types6 57 Debye shielding What is the potential φ(r) of an ion (or positively charged sphere) immersed in a plasma? Introduction Debye shielding7 57 Debye shielding (2): ions vs electrons For equal ion and electron temperatures (Te = Ti ), we have: 1 1 3 m v 2 = m v 2 = k T (2) 2 e e 2 i i 2 B e Therefore, v m 1=2 m 1=2 1 i = e = e = (hydrogen, Z=A=1) ve mi Amp 43 Ions are almost stationary on electron timescale! To a good approximation, we can often write: ni ' n0; where the material (eg gas) number density, n0 = NAρm=A. Introduction Debye shielding8 57 Debye shielding (3) In thermal equilibrium, the electron density follows a Boltzmann distribution∗: ne = ni exp(eφ/kBTe) (3) where ni is the ion density and kB is the Boltzmann constant. From Gauss’ law (Poisson’s equation): 2 ρ e r φ = − = − (ni − ne) (4) "0 "0 ∗ See, eg: F. F. Chen, p. 9 Introduction Debye shielding9 57 Debye shielding (4) Combining (4) with (3) in spherical geometrya and requiring φ ! 0 at r = 1, get solution: Exercise 1 e−r/λD φD = : (5) 4π"0 r with Debye length " k T 1=2 T 1=2 n −1=2 λ = 0 B e = e e D 2 743 −3 cm (6) e ne eV cm a 2 1 d 2 dφ r ! r 2 dr (r dr ) Introduction Debye shielding 10 57 Debye sphere An ideal plasma has many particles per Debye sphere: 4π N ≡ n λ3 1: (7) D e 3 D ) Prerequisite for collective behaviour. Alternatively, can define plasma parameter: 1 g ≡ 3 neλD Classical plasma theory based on assumption that g 1, which also implies dominance of collective effects over collisions between particles. Introduction Debye shielding 11 57 Collisions in plasmas At the other extreme, where ND ≤ 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: 3 4 π 2 neZe ln Λ −1 νei = 1 s 2 2 3 2 2 (4π"0) mevte p vte ≡ kBTe=me is the electron thermal velocity and ln Λ is a slowly varying term (Coulomb logarithm) O(10 − 20). Can show that νei Z ln Λ ' ; with ln Λ ' 9ND=Z !p 10ND Introduction Debye shielding 12 57 Plasma classification Introduction Debye shielding 13 57 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: 10−15 – 103 s Length-scales: 10−9 – 10 m Number of particles needed for first-principles modelling (1): 1021 (tokamak), 1020 (laser-heated solid) Introduction Debye shielding 14 57 Plasma oscillations: capacitor model Consider electron layer displaced from plasma slab by length δ. This creates two ’capacitor’ plates with surface charge σ = ±eneδ, resulting in an electric field: σ en δ E = = e "0 "0 Introduction Plasma oscillations 15 57 Capacitor model (2) The electron layer is accelerated back towards the slab by this restoring force according to: dv d 2δ e2n δ = − = − = e me me 2 eE dt dt "0 Or: d 2δ + !2δ = 0; dt2 p where Electron plasma frequency 2 1=2 1=2 e ne 4 ne −1 !p ≡ ' 5:6 × 10 −3 s : (8) "0me cm Introduction Plasma oscillations 16 57 Response time to create Debye sheath For a plasma with temperature Te (and thermal velocity p vte ≡ kBTe=me), one can also define a characteristic reponse time to recover quasi-neutrality: λ " k T m 1=2 ' D = 0 B e · = !−1: tD 2 p vte e ne kBTe Introduction Plasma oscillations 17 57 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, ! > !p: Figure 2: Overdense, ! < !p: plasma acts as nonlinear plasma acts like mirror refractive medium Introduction Plasma oscillations 18 57 The critical density To make this more quantitative, consider ratio: !2 e2n λ2 p = e · : 2 2 2 ! "0me 4π c Setting this to unity defines the wavelength for which ne = nc, or the Critical density 21 −2 −3 nc ' 10 λµ cm (9) above which radiation with wavelengths λ > λµ will be reflected. cf: radio waves from ionosphere. Introduction Plasma oscillations 19 57 Plasma creation: field ionization At the Bohr radius 2 a = ~ = 5:3 × 10−9 cm; B me2 the electric field strength is: e Ea = 2 4π"0aB ' 5:1 × 109 Vm−1: (10) This leads to the atomic intensity: " cE2 I = 0 a a 2 ' 3:51 × 1016 Wcm−2: (11) A laser intensity of IL > Ia will guarantee ionization for any target material, though in fact this can occur well below this threshold value (eg: ∼ 1014 Wcm−2 for hydrogen) via multiphoton effects . Introduction Plasma creation: field ionization 20 57 Ionized gases: when is plasma response important? Simultaneous field ionization of many atoms produces a plasma with electron density ne, temperature Te ∼ 1 − 10 eV. Collective effects important if !pτinteraction > 1 Example (Gas jet) 17 −3 τint = 100 fs, ne = 10 cm ! !pτint = 1:8 18 19 −3 Typical gas jets: P ∼ 1bar; ne = 10 − 10 cm Recall that from Eq.9, critical density for glass laser 21 −3 nc(1µ) = 10 cm . Gas-jet plasmas are therefore 2 2 underdense, since ! =!p = ne=nc 1. Exploit plasma effects for: short-wavelength radiation; nonlinear refractive properties; high electric/magnetic fields; particle acceleration! Introduction Plasma creation: field ionization 21 57 Relativistic field strengths Classical equation of motion for an electron exposed to a linearly polarized laser field E = yE^ 0 sin !t: dv −eE ' 0 sin !t dt me eE0 ! v = cos !t = vos cos !t (12) me! Dimensionless oscillation amplitude, or ’quiver’ velocity: vos pos eE0 a0 ≡ ≡ ≡ (13) c mec me!c Introduction Relativistic threshold 22 57 Relativistic intensity The laser intensity IL and wavelength λL are related to E0 and ! by: 1 2πc I = " cE2; λ = L 2 0 0 L ! Substituting these into (13) we find : 2 1=2 a0 ' 0:85(I18λµ) ; (14) Exercise where IL λL I18 = ; λµ = : 1018 Wcm−2 µm 18 −2 Implies that for IL ≥ 10 Wcm , λL ' 1 µm, we will have relativistic electron velocities, or a0 ∼ 1. Introduction Relativistic threshold 23 57 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) http://www.nrl.navy.mil/ppd/content/nrl-plasma-formulary Introduction Further reading 24 57 Lecture 2: Wave propagation in plasmas Plasma oscillations Transverse waves Nonlinear wave propagation Further reading Formulary Wave propagation in plasmas 25 57 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 The 2-fluid model Many plasma phenomena can be analysed by assuming that each charged particle component with density ns and velocity us 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: Te > 0 collisionless: νie ' 0 and non-relativistic: velocities u c. Wave propagation in plasmas Plasma oscillations 27 57 The 2-fluid model (2) @n s + r · (n u ) = 0 (15) @t s s du n m s = n q (E + u × B) − rP (16) s s dt s s s s d (P n−γs ) = 0 (17) dt s s Ps is the thermal pressure of species s; γs 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 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.
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