1 PLASMA: WHAT IT IS, HOW TO MAKE IT AND HOW TO HOLD IT Felix I. Parra Rudolf Peierls Centre for Theoretical Physics, University of Oxford
2 Overview Why do we need plasmas? For fusion, among other things Basic properties of a plasma Magnetized plasmas
3 Burning matter to get energy Burning coal = breaking chemical bonds Much less energetic than nuclear bonds Chemical bonds due to electromagnetic force Nuclear bonds due to strong force Strong force > electromagnetic force because it overcomes Coulomb repulsion of nucleons Burning nuclear bonds seems more productive
4 Igniting nuclear fusion reactions Initial energy to start reactions Smaller energy barrier for fusion reaction Deuterium-Tritium Sufficient average energy = sufficient temperature T Use energy from reaction to overcome barrier to produce more reactions Keep energy for sufficiently long time: τ E [s] Thermal insulation Have sufficient number of reactions Fueling sufficiently dense: n [particles/m -3 ]
5 Conditions for fusion Temperature: T = 150 millions K At this temperature, matter is ionized = plasma Hottest place in solar system: JET, Oxfordshire Density: n = 10 20 particles/m 3 A millionth of the atmospheric density Pressure: p = 10 atm Energy confinement time: τ E = 1 10 s Nuclear fusion is harder than fission, but Fuel is virtually inexhaustible No long-lived radioactive waste
6 What does plasma mean? Langmuir 1929: plasma = Greek for to mold Misnomer! Fusion plasmas at 100 million K do not mold
7 Definition of a plasma Ionized gas Dominated by long range interactions Quasineutral
8 Ionized gas Collisions between energetic electrons and atoms IONIZATION Temperature T > ionization energy ~ ev ~ 10,000 K Sufficient with hot electrons Electrons isolated from ions due to small mass In fusion plasmas T = 100 million K How do we ionize? Usually with electric fields
9 Long range vs. short range Long range interactions: plasma currents and charge give magnetic and electric field Short range interactions: Coulomb collisions Balance potential energy with kinetic energy e 2 4πε 0 b ~ T e b = e 2 4πε 0 T e
10 Long range interactions dominate Very few particles in sphere of radius b 4π 3 b3 n e = 1 3 e 6 n e ( ε 0 T e ) <<1 3 Kinetic energy much larger than Coulomb potential Γ = T e e 2 4πε 0 r ~ T e e 2 4πε 0 n e 1/3 ~ 4πε 0T e e 2 n e 1/3 >>1 Note that 4π 3 b3 n e ~ 1 Γ 3 <<1
11 Continuum approach Distribution function f (r, v,t)d 3 v d 3 r = number of particles with velocity v at r Maxwell s equations E = 1 q s f s d 3 v ε 0 s E = B t B = 0 B = µ 0 q s f s vd 3 v + 1 E s c 2 t
12 Particle conservation Number of particles must be conserved along trajectories Equation with characteristics = particle trajectories f s t + v f s + q s m s ( E + v B) v f s = C ss ' f s, f s ' s ' [ ] Collision operator Weak coupling means only binary collisions matter But Coulomb interaction is long range. Do particles infinitely far away matter? No, because
13 Quasineutrality Natural tendency to neutrality: opposite charges attract Not sufficient energy to separate opposite charges L >> λ D ω << ω p Surprisingly, every volume in an ionized gas is almost perfectly neutral! By definition, plasma is quasineutral
14 Debye length Layers of positive and negative charge in uniform plasma + λ D Sufficient potential difference to attract electrons eδφ ~ T e Poisson s equation 2 φ = e(n i n e ) ε 0 Δφ λ D 2 ~ en e ε 0 λ D = ε 0 T e e 2 n e
15 Validity of quasineutrality Plasma size >> λ D any charge is shielded Electrons have to be sufficiently fast to respond ω << v e λ D = 1 λ D 2T e m e ~ ω p = e 2 n e ε 0 m e = plasma frequency Need sufficient electrons in Debye sphere 4π 3 λ 3 Dn e = 4π 3 ( ε 0 T ) 3/2 e = e 3 1/2 n e 1 3 4π Γ3/2 >>1 Weak coupling = sufficient particles for shielding
16 Coulomb collisions Coulomb interaction is long range, but it does not reach beyond λ D! Equation with binary collisions valid f s t + v f s + q s m s ( E + v B) v f s = C ss ' f s, f s ' s ' [ ]
17 Confining very hot plasmas Do not confine: just get fusion faster than plasma life time H bombs Inertial confinement fusion Within a physical container Lots of plasma energy lost to the wall Very low temperatures (at most 100,000 K) With gravity Only star-size entities can achieve fusion Magnetic confinement
18 Magnetized plasmas For strong magnetic field plasmas show a particular behavior Quasineutrality helps magnetization: makes E small In principle, v B E ~ v c <<1 Need to satisfy L >> ρ ω << Ω
19 Gyromotion Gyromotion or Larmor motion ( B/ t = 0 = B) Free motion parallel to B Perpendicular force balance mv 2 ρ Gyroradius: = ev B ρ = mv eb = v Ω Gyrofrequency: Ω = eb m
20 E B drift Gyromotion + perpendicular electric field? Perpendicular drift E v > v B v E = 1 B 2 E B v < v
21 Gyrokinetic motion Two-scale expansion [Catto, Plasma Phys. 1978] Separate fast and slow times t g = Ωt, t t = ωt ~ ρ * t g to calculate particle position r(t g, t t ) Simplified for approximately circular gyromotion r(t g,t t ) R(t t )+ ρ(t t )( sint g ê 1 + cost g ê ) 2 +... Gyrokinetic equations of motion To lowest order perpendicular average motion = 0 dr dt = v ˆb + v + other d drifts ; dρ dt =...; dv dt =... O(v t ) O(ρ * v t ) O(ρ 2 * v t )+...
22 Gyrokinetic equations Moving rings of charge f s t + dr dt f + dρ f s R s dt ρ + dv dt f s = C ss ' f s, f s ' v s ' [ ]
23 Tokamaks and stellarators Need to account for drifts in design Tokamak Stellarator