Plasma Physics Introduction A. Flacco Structure The plasma state - - PowerPoint PPT Presentation

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

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Plasmas?

(hot/cold, dense/rarefied, . . . )

(a) (b) (c) (d) (e) (f) (g) (h)

• A. Flacco/ENSTA - PA201: Introduction

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Plasmas

a Candle flame: T = 1000 K − 1400 K, very low ionization; b Orion nebula (M42): Te = 104 K, ne = 102 − 104 cm−3; c Laser produced plasma: Te ∼ keV, Ti = 300 K, ne = 1019 cm−3; d Glow discharge: n = 1010 cm−3, Te = 2 eV; e Joint European Torus (JET): n = 1025 cm−3, kBTe = 100 eV; f Cyclotron proton beam: r = 1 cm, I = 80 pA, K = 230 MeV; g Van Allen Belts (inner and outer): 104 to 109 particles/cm2 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

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Different kinds of plasmas

Coupling parameter Ξ: Ξ ≡ |Ep| Ec

• A. Flacco/ENSTA - PA201: Introduction

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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

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Temperature & ionization

Creation of a plasma

First Electron Ionization Energy

Alkali metal Alkaline earth metal Transition metal Post-transition metal Metalloid Nonmetal Halogen Noble gas Lanthanide Actinide 30 25 20 15 10 5 10 20 30 40 50 60 70 80 90 100 Ionization Energy [eV] Atomic Number He Ne Ar Kr Xe Hg Rn Li Na K Rb Cs Fr

• A. Flacco/ENSTA - PA201: Introduction

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Saha Law

Ionization degree in a gas Saha Law: nm+1ne nm =

• 2 gm+1

gm 2.4 × 1021 T3/2e−Um+1/kBT

10-300 10-250 10-200 10-150 10-100 10-50 100 102 103 104 105 ni/nn T [K]

Ionization at thermal equilibrium

Na Ui=5.13 eV N Ui=14.5 eV He Ui=24.59 eV

Atmosphere (N2, ni ∼ 1025 m−3, T = 300K): ni nn ≈ 10−245 Glow discharge tube (Ne, ni ∼ 1016 m−3, Te ∼ 104 K): ni nn ≈ 3.1 (Attention: this is false!)

• A. Flacco/ENSTA - PA201: Introduction

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Saha Law

Oxygen ionization vs. temperature

1 2 3 4 5 6 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000 Relative abundance Temperature K O O1 O2 O3 O4 O5 O6 e-

• A. Flacco/ENSTA - PA201: Introduction

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Temperature: a review

In a gas at thermal equilibrium, atoms follow the Boltzmann distribution: f (v) = N

2πkBT

m

−3/2

e−mv·v/2kBT Average kinetic energy (in 3D) gives: E = 1 N

• R3

1 2 m (v · v) f (v) dv3 = 3 2 kBT

• A. Flacco/ENSTA - PA201: Introduction

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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. f (vi) =

• m

2πkBT

1/2

e−mv2

i /2kBT

It is easily calculated that:

              

σvi = √kBT vi = 0 v2

i = kBT

m v2 = 3kBT m

• A. Flacco/ENSTA - PA201: Introduction

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Maxwell-Boltzmann velocity distribution

The speed distribution (Maxwell-Boltzmann) is obtained by integrating over (θ, φ) in polar coordinates. f (v) dv =

2π

dφ

π

sin (θ) dθ

2πkBT

m

−3/2

v2e−mv2/2kBT = 4πv2 2πkBT m

−3/2

e−mv2/2kBT From the M-B distribution the most probable speed vm and the mean speed are calculated:

    

vm =

2kBT

m

1/2

v =

8kBT

πm

1/2

• A. Flacco/ENSTA - PA201: Introduction

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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. Ti, Te), each species in its own thermal equilibrium. Temperatures are often expressed in eV via the Boltzmann constant: kB = 1.38 · 10−23J K−1 = 8.6 · 10−5eV K−1 In particular conditions there can exist different components in temperature (eg. T, T⊥).

• A. Flacco/ENSTA - PA201: Introduction

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Long range interaction

Coulomb vs. Lennard-Jones ϕLJ (r) = −4ε

σ

r

6

−

σ

r

12

H2 molecule: ε/kB = 37 K, σ = 2.98 Å ϕC (r) = 1 4πε0 q1q2 r

10-20 10-15 10-10 10-5 100 105 1010 1015 1020 1025 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 V(r)/J r/Å LJ 12-6 Coulomb

10-25 10-20 10-15 10-10 10-5 100 105 1010 1015 1020 1025 0.001 0.01 0.1 1 10 100 1000 10000 V(r)/J r/Å

• A. Flacco/ENSTA - PA201: Introduction

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The Plasma State

Parameters and evolution Ionization parameter α ≡ ne ne + nn

α < 1: weak ionization α ≈ 1: strong ionization

Coupling Parameter Ξ ≡ |Ep| Ec

Ξ ≪ 1: weak coupling, kinetic

• r ideal plasma

Ξ ≥ 1: strong coupling, fluid, cristalline

Neutrality Parameter ε ≡ ne − Zni ne + Zni

ε ≪ 1: quasi-neutrality, ε ≤ 1: beams, space charge effects

• A. Flacco/ENSTA - PA201: Introduction

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Debye Screening

✕ ✰

P❧❛s♠❛

✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰ ✲ ✰

• A. Flacco/ENSTA - PA201: Introduction

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Debye Screening

A test charge introduced in a plasma at the equilibrium perturbs the speed distribution according to: f (v) ∝ exp

• −

1

2 mu2 + qφ

• 1

kBTe

• where the potential φ must obey Poisson’s law:

ε0∇2φ = −e (ne − ni) . Plasma charges re-organize to screen the test charge; the new effective potential decreases with an exponential law with the characteristic length λD: Debye Length: λD ≡

• ε0kBT

ne2 The plasma parameter is the number of charges in a Debye sphere: ND ≡ n 4 3 πλ3

D ≫ 1

• A. Flacco/ENSTA - PA201: Introduction

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Plasma Oscillation

+ + + + + + + + + + + + + + + − − − − − − − −

Gauss Law: ∇ · E = (e/ε0) (ni − ne) Electron continuity equation: ∂ne ∂t + ∇ · (une) = 0 Lorentz force: ∂u ∂t = − e m E

+ + + + + + + + + + + + + + + − − − − − − − − E E E E

Electron plasma frequency: ωpe =

• n0e2

ε0me

1/2

rad s−1

• A. Flacco/ENSTA - PA201: Introduction

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Plasma parameters and plasma definition

A plasma is a quasineutral gas of charged and neutral particles which ex- hibits collective behaviour λD ≪ L ND ≫ 1 ωτ > 1

• A. Flacco/ENSTA - PA201: Introduction

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Single Particle Motions

B0 = B0ˆ z, E0 = 0 B = B^ z Cyclotron frequency: ωc ≡ |q|B m Larmor radius: rL ≡ v⊥ ωc = mv⊥ |q|B Motion around a guiding center: x−x0 = rL sin (ωct) , y−y0 = ±rL cos (ωct)

❣✉✐❞✐♥❣ ❝❡♥t❡r

+

✲ ×B

This describes a circular orbit around a guiding center (x0, y0) which is fixed. The magnetic field generated by the gyration is opposite to the externally imposed

• field. Plasma is therefore diamagnetic. Arbitrary component vz along B in

unaffected: charged particles in space generally follow helicoidal trajectory.

• A. Flacco/ENSTA - PA201: Introduction

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Single Particle Motions

B = B0ˆ z, E = (Ex, 0, Ez) On the ˆ z component: dv z dt = q m Ez ⇒ vz = qEz m t + vz0 On the orthogonal components:

• dv x

dt

=

q m Ex ± ωcvy dv y dt

= 0 ∓ ωcvx − →

vx

= v⊥ expiωct vy = ±iv⊥ expiωct − Ex

B

Drift velocity superimposed on the guiding center: vgc = E × B/B2 ≡ vE, vE = E/B

• A. Flacco/ENSTA - PA201: Introduction

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Single Particle Motions: ∇B drift

B0 = 0, E0 = 0, ∇B ⊥ B The average is taken on the unperturbed orbit. Fy = −qvxBz (y) = −qv⊥ (cos ωct)

• B0 ± rL (cos ωct) ∂B

∂y

• v∇B = −

q

e

1

2 v⊥rL B × ∇B B2

❡ ∇B v∇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

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Single Particle Motions: Curvature drift

Curvature Drift: Fcf = mv2

• Rc

ˆ r = mv2

• Rc

R2

c

vR = 1 q Fcf × B B2 = mv2

• qB2

Rc × B R2

c

Gradient Correction: Bθ ∝ 1 r = ⇒ ∇|B| B = − Rc R2

c

v∇B = 1 2 m q v2

⊥

Rc × B R2

cB2

Total drift in a curved vacuum field: vR + v∇B = m q Rc × B R2

cB2

• v2

+ 1

2 v2

⊥

• A. Flacco/ENSTA - PA201: Introduction

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Single Particle Motions

B0 = 0, E0 = 0, ∇B B 1 r ∂ ∂r (rBr) + ∂Bz ∂z = 0 Br ≃ − 1 2 r

∂Bz

∂z

• r=0

❇ ˆ z

For the gyration-averaged longitudinal force it holds: Fz = − 1 2 mv2

⊥

B ∂Bz ∂z = −µ ∂Bz ∂z where it has been defined Magnetic Moment: µ = mv2

⊥

2B The resulting force on the parallel direction is then: F = −µ ∂B ∂s = −µ∇B

• A. Flacco/ENSTA - PA201: Introduction

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Magnetic mirrors

dµ dt = 0 − → mv⊥0 B0 = mv′

⊥

2B′ B0 B′ = v2

⊥0

v′2

⊥

= v2

⊥0

v2 ≡ sin2 θ ❇ ■ ■ ˆ z ❝♦✐❧ ∇B ∇B The smallest θ for a confined particle is then: sin2 θm = B0 Bm ≡ 1 Rm (Note: µ is an adiabatic invariant of the motion)

• A. Flacco/ENSTA - PA201: Introduction

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Summary

(what’s to remember)

• Quasi-neutral, charged particles, collective behaviour
• Screening of electric charges: Debye length
• Self-oscillating frequency: plasma frequency
• Particle reacts to magnetic fields (guiding, drifting, etc.)
• A. Flacco/ENSTA - PA201: Introduction

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Summary on Single Particle Motions

Electric field: vgc = E × B/B2 ≡ vE, vE = E/B Orthogonal gradient: v∇B = −

q

e

1

2 v⊥rL B × ∇B B2 Curvature drift: vR + v∇B = m q Rc × B R2

cB2

• v2

+ 1

2 v2

⊥

• Parallel gradient:

F = −µ ∂B ∂s = −µ∇B

• A. Flacco/ENSTA - PA201: Introduction

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