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Basic Plasma Parameters F Basic Plasma Parameters F. Robicheaux Auburn University Alabama, USA Tools Pencil & paper, calculator, PC/laptop, workstation, local cluster, national supercomputer center Plasma = ionized gas Properties: Hot? (need to ionize the atoms/molecules) Density (Sun > 1024 cm-3, space 1 cm-3) Good conductor of electricity Magnetic fields (sometimes) Examples: Stars Ionosphere Fusion devices Solar Wind Interstellar Gas Strongly Coupled Plasmas? JPB 36, 499 Strongly Coupled Plasmas If the temperature is low enough, PE > KE. Highly correlated motion between the charged particles. Average volume occupied by one electron: 4πa3/3 = 1/n a = (3/4πn)1/3 9 −3 For ne = 10 cm : a = 6.2 µm << size of plasma Coulomb coupling parameter: Γe I <PE>/<KE> 2 Γe = (e /4πε0a)/kBTe 9 −3 For Te = 100 K & ne = 10 cm : Γe = 0.027 9 −3 For Te = 1 K & ne = 10 cm : Γe = 2.7 Dimensionless number! Dimensional analysis? Summary Coulomb coupling parameter: Γe I <PE>/<KE> 2 1/3 Γe = (e /4πε0a)/kBTe where a = (3/4πn) Screening? Slightly more red (- charges) +Q near a +Q charge due to the thermal distribution n ~ exp[-q V(r)/k T] Plasmas are conductors: no E-fields?! Charges should be screened by the free charges in the plasma, but nonzero T prevents perfect screening. Debye length – the distance that a charge screened by a factor of ~ e = 2.718… 2 Debye length from electrons & ions: λD = ε0 k B Te / 2 ne e Debye Length Theory Need to self consistently solve for the potential. The change in electron density due to a potential is r δρ(r) = e n e {exp[-e V(r)/kB Te ] - exp[e V(r)/kB Te ]} 2 ≅ - 2 n e e V(r)/kB Te Poisson’s equation for the potential is 2 2 1 ∂ δρ(r) ∇ V(r) = 2 [rV(r)] = - r ∂r ε0 2 2 2 1 ∂ 2 n e e 1 2 [rV(r)] = V(r) = V(r) r ∂r ε0 k B Te λD Debye Length Physics The solution to this equation is exp(-r/λ ) V(r) = V D 0 r The Coulomb coupling parameter can also be written as 2 Γe = (1/3) (a/λD) The coupling is small when there are a large number of electrons within a Debye sphere. (Many charges shift by a small amount.) Strong coupling occurs when there are few charges within a Debye sphere. Summary Coulomb coupling parameter: Γe I <PE>/<KE> 2 1/3 Γe = (e /4πε0a)/kBTe where a = (3/4πn) 2 1/2 Debye screening length: λD = (ε0 kB Te/2 ne e ) Langmuir Wave (plasma frequency) Imagine pulling the electrons in a region of the plasma slightly (distance x) to the left. What happens? – neutral + The electrons oscillate with the electron plasma frequency E ω = (e2 n /ε m )1/2 Increasing t p e 0 e 9 −3 For ne = 10 cm : fp = 280 MHz 7 −3 For ne = 10 cm : fp = 28 MHz The ions are essentially stationary. 1 2 KE = me V ne x& 2 E 2 1 2 1 e ne PE = V ε0 E = V ε0 x + neutral – 2 2 ε0 Plasma Frequency Theory, v<<c Solve Maxwell’s equation + Newton’s equation for small changes in density. r r r r ∇ ⋅E(r, t) = ρ(r, t)/ε0 ∂n(r, t) r + ∇ ⋅[n(r, t) vr(r, t)] = 0 ∂t ∂vr(r, t) r r + [vr(r, t)⋅∇]vr(r, t) = - e E(r, t)/m ∂t e Use E = E0 + δE, n = ne + δn, ρ = ρ0 -e δn, v = δv ne is the background electron number density assumes the background electron flow is 0 Plasma Frequency Theory, v<<c Solve Maxwell’s equation + Newton’s equation for small changes in density. r r r r ∇ ⋅δE(r, t) = - e δn(r, t)/ε0 ∂δn(r, t) r + n ∇ ⋅δvr(r, t) = 0 ∂t e ∂δvr(r, t) r = - e δE(r, t)/m ∂t e Use the t derivative of the middle equation and div of last 2 r 2 ∂ δn(r, t) e n e r 2 + δn(r, t) = 0 ∂t me ε0 Summary Coulomb coupling parameter: Γe I <PE>/<KE> 2 1/3 Γe = (e /4πε0a)/kBTe where a = (3/4πn) 2 1/2 Debye screening length: λD = (ε0 kB Te/2 ne e ) 2 1/2 Electron plasma frequency: ωp = (e ne/ε0 me) Ion Motion The large scale fields in the plasma can also give motion to the ions. Regions of slightly higher ion density is not completely screened by electrons. D Ions will be pushed out of region of high ion density and pulled into low. If the modulation in space is sinusoidal D wave 1/2 ω = (kB Te qi/e mi) k (if Te ~ Ti, then v ~ thermal speed) Ion Acoustic Wave (theory) The ion acoustic wave (for low T) can be found by noting that the electron charge density must almost exactly cancel the ion charge density. r r qi r e V(r) n e (r, t) ≅ ni (r, t) ≅ C exp e k B Te k T V(r, t) = B e ln[]n (r, t) + cons e i r r k B Te r r E(r, t) = - r ∇ni (r, t) e ni (r, t) Ion Acoustic Wave (theory) The equations for the ion density and velocity flow: ∂n(r, t) r + ∇ ⋅[n(r, t) vr(r, t)] = 0 ∂t ∂vr(r, t) r r + [vr(r, t)⋅∇]vr(r, t) = q E(r, t)/m ∂t i i r k T r E(r, t) = - B e ∇n(r, t) e n(r, t) Use n = ni + δn, v = δv ni is the background ion number density assumes the background ion flow is 0 Ion Acoustic Wave (theory) The equations for the change in ion density and velocity flow: ∂δn(r, t) r + n ∇ ⋅δvr(r, t) = 0 ∂t i ∂δvr(r, t) r = q E(r, t)/m ∂t i i r k T r E(r, t) = - B e ∇δn(r, t) e ni Use the t derivative of the 1st equation and div of 2nd & 3rd 2 r ∂ δn(r, t) k B Te qi 2 r 2 − ∇ δn(r, t) = 0 ∂t e mi 2 This has ignored terms ~ (k λD) and (Ti/Te) (ion pressure) Summary Coulomb coupling parameter: Γe I <PE>/<KE> 2 1/3 Γe = (e /4πε0a)/kBTe where a = (3/4πn) 2 1/2 Debye screening length: λD = (ε0 kB Te/2 ne e ) 2 1/2 Electron plasma frequency: ωp = (e ne/ε0 me) Ion acoustic wave dispersion relation: 1/2 ω = (kB Te qi/e mi) k Thermalization/Randomization How does the Maxwell-Boltzmann distribution become established? How does the direction of travel of an electron become randomized due to scattering with the ions? Two body collisions are all that is needed. Rate for a collision process is n <v σ> 4 2 4 2 Collision thermalization time: 1/τ =ne v[e ln(Λ)/4πε0 v me ] 2 Λ = 4πε03kBTe λD/e ~ 1/θmin>>1 9 −3 For Te = 100 K & ne = 10 cm : ln(Λ) = 6.0 ; τ = 0.064 µs 9 −3 For Te = 10 K & ne = 10 cm : ln(Λ) = 2.5 ; τ = 0.005 µs Thermalization/Randomization The collision between two charged particles is well studied. The angle through which the particle scatters depends on the charge, reduced mass, relative velocity and impact parameter. 2 2 tan(θ/2) = Q e /(4 π ε0 µ v b) b θ +Q Rutherford scattering cross section (Q=1): 4 2 4 2 dσ/d(cosθ) = 2πe /[(µ 4πε0) v (1 – cosθ) ] Angle Randomization The electron-ion collisions will give a spread in velocity directions for the electron. In a time δt, the direction spreads by an amount: 2 4 2 4 2 <δθ > = n v δt[e ln(Λ)/2πε0 v me ] where 2 2 Λ = 4πε0me v λD/e Why does the Debye length come into this expression? The plasma modifies the Coulomb potential at large distances. This changes the amount that scatters into small angles (large impact parameter). The Debye length gives the distance over which the potential is present. Angle Randomization (Theory) The angle deviation in a time δt diverges if all scattering angles are allowed. Restrict the minimum angle using the Debye length for the maximum impact parameter: 2 2 tan(θmin/2) ~ θmin/2 = e /(4 π ε0 µ v λD) The angle deviation is given by cos(θmin ) dσ δθ2 = n v δt ∫ 2 [1 - cos(θ )] d cos(θ ) -1 d cos(θ ) e4 2 = n v δt ln 2 2 4 2 π ε0 me v θmin Summary Coulomb coupling parameter: Γe I <PE>/<KE> 2 1/3 Γe = (e /4πε0a)/kBTe where a = (3/4πn) 2 1/2 Debye screening length: λD = (ε0 kB Te/2 ne e ) 2 1/2 Electron plasma frequency: ωp = (e ne/ε0 me) Ion acoustic wave dispersion relation: 1/2 ω = (kB Te qi/e mi) k 4 2 4 2 Collision thermalization time: 1/τ =ne v[e ln(Λ)/4πε0 v me ] 2 Λ = 4πε03kBTe λD/e ~ 1/θmin>>1 Three Body Recombination (TBR) Two electrons collide in the field of an ion so that one electron loses so much energy to become bound. +Q +Q Three body recombination rate (e− + e − + A+ J e − + A*): −39 6 −1 9/2 Γ = 2 X 10 m s neni(eV/kBTe) 9 −3 −4 −1 For Te = 50 K & ne = 10 cm : Γ = 10 µs 9 −3 −1 For Te = 10 K & ne = 10 cm : Γ = 0.1 µs 9 −3 −1 For Te = 1 K & ne = 10 cm : Γ = 4000 µs Recombination into states bound by ~4kBTe (size of atom ~ distance between ions at 1 K!) Three Body Recombination (theory) It is not possible to calculate the TBR rate using pencil & paper.
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