Plasma Physics Introduction A. Flacco
Structure • The plasma state 5 • Debye screening 16 • Plasma oscillations 18 • Plasma Parameters 19 • Single particle motions 20 A. Flacco/ENSTA - PA201: Introduction Page 2 of 27
Plasmas? (hot/cold, dense/rarefied, . . . ) (d) (a) (b) (c) (f) (e) (g) (h) A. Flacco/ENSTA - PA201: Introduction Page 3 of 27
Plasmas a Candle flame: T = 1000 K − 1400 K, very low ionization; b Orion nebula (M42): T e = 10 4 K, n e = 10 2 − 10 4 cm − 3 ; c Laser produced plasma: T e ∼ keV, T i = 300 K, n e = 10 19 cm − 3 ; d Glow discharge: n = 10 10 cm − 3 , T e = 2 eV; e Joint European Torus (JET): n = 10 25 cm − 3 , k B T e = 100 eV; f Cyclotron proton beam: r = 1 cm, I = 80 pA, K = 230 MeV; g Van Allen Belts (inner and outer): 10 4 to 10 9 particles / cm 2 s, electrons up to 5 MeV, protons up to 400 MeV; h Thruster exhaust: T ∼ 3500 K, pressure 100 bar. A. Flacco/ENSTA - PA201: Introduction Page 4 of 27
Different kinds of plasmas Coupling parameter Ξ: Ξ ≡ |� E p �| � E c � A. Flacco/ENSTA - PA201: Introduction Page 5 of 27
The Plasma State A plasma is a quasineutral gas of charged and neutral particles which ex- hibits collective behaviour (F. Chen) À très haute temperature, la dissociation puis l’ionisation conduisent à la création de populations d’ions et d’électrons libres et ces charges libres induisent un comportement collectif, non linéaire, chaotique et turbulent. (J.-M. Rax) A. Flacco/ENSTA - PA201: Introduction Page 6 of 27
Temperature & ionization Creation of a plasma First Electron Ionization Energy Alkali metal Alkaline earth metal Transition metal 30 Post-transition metal Metalloid He Nonmetal 25 Ionization Energy [eV] Halogen Ne Noble gas 20 Lanthanide Actinide Ar Kr 15 Xe Rn Hg 10 5 Li Na K Rb Cs Fr 0 0 10 20 30 40 50 60 70 80 90 100 Atomic Number A. Flacco/ENSTA - PA201: Introduction Page 7 of 27
Saha Law Ionization degree in a gas Saha Law: n m +1 n e 2 g m +1 � 2 . 4 × 10 21 � T 3 / 2 e − U m +1 / k B T = n m g m Ionization at thermal equilibrium Atmosphere ( N 2 , n i ∼ 10 25 m − 3 , T = 300K): 10 0 n i ≈ 10 − 245 10 -50 n n 10 -100 Glow discharge tube n i /n n ( Ne , n i ∼ 10 16 m − 3 , 10 -150 T e ∼ 10 4 K): 10 -200 n i Na U i =5.13 eV ≈ 3 . 1 n n N U i =14.5 eV 10 -250 He U i =24.59 eV ( Attention: this is false!) 10 -300 10 2 10 3 10 4 10 5 T [K] A. Flacco/ENSTA - PA201: Introduction Page 8 of 27
Saha Law Oxygen ionization vs. temperature 6 O O1 O2 O3 O4 5 O5 O6 e- 4 Relative abundance 3 2 1 0 0 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000 Temperature K A. Flacco/ENSTA - PA201: Introduction Page 9 of 27
Temperature: a review In a gas at thermal equilibrium, atoms follow the Boltzmann distribution: � 2 π k B T � − 3 / 2 e − m v · v / 2 k B T f ( v ) = N m Average kinetic energy (in 3D) gives: � E � = 1 � 1 2 m ( v · v ) f ( v ) d v 3 = 3 2 k B T N R 3 A. Flacco/ENSTA - PA201: Introduction Page 10 of 27
Boltzmann energy distribution Boltzmann distribution on velocity is formed by three (in 3D) independent distributions, with a void mean velocity; the width of the distribution is defined by the temperature. m � 1 / 2 � e − mv 2 i / 2 k B T f ( v i ) = 2 π k B T It is easily calculated that: σ v i = √ k B T � v i � = 0 i � = k B T � v 2 m � v 2 � = 3 k B T m A. Flacco/ENSTA - PA201: Introduction Page 11 of 27
Maxwell-Boltzmann velocity distribution The speed distribution (Maxwell-Boltzmann) is obtained by integrating over ( θ, φ ) in polar coordinates. � 2 π � π � 2 π k B T � − 3 / 2 v 2 e − mv 2 / 2 k B T f ( v ) d v = d φ sin ( θ ) d θ m 0 0 = 4 π v 2 � 2 π k B T � − 3 / 2 e − mv 2 / 2 k B T m From the M-B distribution the most probable speed v m and the mean speed are calculated: � 2 k B T � 1 / 2 v m = m � 8 k B T � 1 / 2 � v � = π m A. Flacco/ENSTA - PA201: Introduction Page 12 of 27
Temperature in a Plasma A Plasma is ionized matter: interaction between particles happens through Lorentz Force: F = q ( E + v × B ) In a plasma, different species can have different temperatures (eg. T i , T e ), each species in its own thermal equilibrium. Temperatures are often expressed in eV via the Boltzmann constant: k B = 1 . 38 · 10 − 23 J K − 1 = 8 . 6 · 10 − 5 eV K − 1 In particular conditions there can exist different components in temperature (eg. T � , T ⊥ ). A. Flacco/ENSTA - PA201: Introduction Page 13 of 27
Long range interaction Coulomb vs. Lennard-Jones �� σ � σ � 12 � � 6 ϕ LJ ( r ) = − 4 ε 1 q 1 q 2 − ϕ C ( r ) = r r 4 πε 0 r H 2 molecule: ε/ k B = 37 K, σ = 2 . 98 Å 10 25 LJ 12-6 Coulomb 10 20 10 25 10 15 10 20 10 15 10 10 10 10 10 5 10 5 V(r)/J V(r)/J 10 0 10 -5 10 0 10 -10 10 -15 10 -5 10 -20 10 -25 10 -10 0.001 0.01 0.1 1 10 100 1000 10000 r/Å 10 -15 10 -20 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 A. Flacco/ENSTA - PA201: Introduction r/Å Page 14 of 27
The Plasma State Parameters and evolution Ionization Coupling Neutrality parameter Parameter Parameter n e Ξ ≡ |� E p �| ε ≡ n e − Zn i α ≡ n e + n n n e + Zn i � E c � α < 1: weak ε ≪ 1: Ξ ≪ 1: weak ionization quasi-neutrality, coupling, kinetic or ideal plasma α ≈ 1: strong ε ≤ 1: beams, ionization space charge Ξ ≥ 1: strong coupling, fluid, effects cristalline A. Flacco/ENSTA - PA201: Introduction Page 15 of 27
Debye Screening ✰ ✰ ✰ ✰ P❧❛s♠❛ ✰ ✰ ✲ ✰ ✲ ✲ ✰ ✲ ✰ ✲ ✰ ✰ ✲ ✲ ✲ ✲ ✰ ✲ ✲ ✲ ✰ ✲ ✲ ✲ ✲ ✰ ✰ ✲ ✲ ✲ ✲ ✰ ✰ ✲ ✰ ✰ ✰ ✲ ✲ ✲ ✰ ✰ ✲ ✰ ✲ ✲ ✰ ✲ ✲ ✰ ✰ ✕ ✰ ✰ ✰ ✰ ✲ ✰ ✰ ✰ ✲ ✲ ✲ ✲ ✰ ✰ ✰ ✲ ✰ ✲ ✲ ✰ ✰ ✰ ✰ ✲ ✰ ✰ ✲ ✰ ✲ ✲ ✰ ✲ ✰ ✰ ✰ ✲ ✰ ✲ ✲ ✲ ✲ ✲ ✲ ✰ ✲ ✰ A. Flacco/ENSTA - PA201: Introduction Page 16 of 27
Debye Screening A test charge introduced in a plasma at the equilibrium perturbs the speed distribution according to: � 1 1 � 2 mu 2 + q φ � � f ( v ) ∝ exp − k B T e where the potential φ must obey Poisson’s law: ε 0 ∇ 2 φ = − e ( n e − n i ) . Plasma charges re-organize to screen the test charge; the new effective potential decreases with an exponential law with the characteristic length λ D : Debye Length: � ε 0 k B T λ D ≡ ne 2 The plasma parameter is the number of charges in a Debye sphere: N D ≡ n 4 3 πλ 3 D ≫ 1 A. Flacco/ENSTA - PA201: Introduction Page 17 of 27
Plasma Oscillation Gauss Law: ∇ · E = ( e /ε 0 ) ( n i − n e ) + + + + + Electron continuity equation: − − − − ∂ n e + + + + + ∂ t + ∇ · (un e ) = 0 − − − − Lorentz force: + + + + + ∂ t = − e ∂ u m E + + + + + E E Electron plasma frequency: − − − − � 1 / 2 � n 0 e 2 + + + + + rad s − 1 ω pe = ε 0 m e E E − − − − + + + + + A. Flacco/ENSTA - PA201: Introduction Page 18 of 27
Plasma parameters and plasma definition A plasma is a quasineutral gas of charged and neutral particles which ex- hibits collective behaviour λ D ≪ L N D ≫ 1 ωτ > 1 A. Flacco/ENSTA - PA201: Introduction Page 19 of 27
Single Particle Motions B 0 = B 0 ˆ z, E 0 = 0 B = B ^ z × B Cyclotron frequency: ω c ≡ | q | B m ✲ + Larmor radius: r L ≡ v ⊥ = mv ⊥ | q | B ω c Motion around a guiding center: ❣✉✐❞✐♥❣ ❝❡♥t❡r x − x 0 = r L sin ( ω c t ) , y − y 0 = ± r L cos ( ω c t ) This describes a circular orbit around a guiding center ( x 0 , y 0 ) which is fixed . The magnetic field generated by the gyration is opposite to the externally imposed field. Plasma is therefore diamagnetic . Arbitrary component v z along B in unaffected: charged particles in space generally follow helicoidal trajectory. A. Flacco/ENSTA - PA201: Introduction Page 20 of 27
Single Particle Motions B = B 0 ˆ z , E = ( E x , 0 , E z ) On the ˆ z component: d v z = q m E z ⇒ v z = qE z m t + v z 0 d t On the orthogonal components: � v x � d v x q = m E x ± ω c v y = v ⊥ exp i ω c t d t − → ± i v ⊥ exp i ω c t − E x d v y v y = = 0 ∓ ω c v x B d t Drift velocity superimposed on the guiding center: v gc = E × B / B 2 ≡ v E , v E = E / B A. Flacco/ENSTA - PA201: Introduction Page 21 of 27
Single Particle Motions: ∇ B drift B 0 � = 0 , E 0 � = 0 , ∇ B ⊥ B The average is taken on the unperturbed orbit. F y = − qv x B z ( y ) ❡ B 0 ± r L (cos ω c t ) ∂ B � � = − qv ⊥ (cos ω c t ) ∂ y v ∇ B � q � 1 B × ∇ B v ∇ B = − 2 v ⊥ r L e B 2 ∇ B B The gradient | B | causes the Larmor radius to be larger in lower field regions, thus resulting in a drift perpendicular to the gradient. A. Flacco/ENSTA - PA201: Introduction Page 22 of 27
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