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
Coulomb coupling parameter: Γe I
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
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< 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< 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 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 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 +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. Can calculate how the rate scales with different parameters then use Monte Carlo simulation to determine unknown dimensionless parameters. Γ= n v σ P = const. X n2 T-9/2 2 b = size of atom ~ e /(4 πε0 kB T) n = number density v = velocity ~ T1/2 σ = cross section = π b2 ~ T-2 P = probability for finding another electron ~ n b3 ~ n T-3 TBR (qualified) In most plasmas, the time to scatter the resulting Rydberg atom into the ground state is short compared to 1/Γ. The TBR rate is then the rate for generating ground state atoms. For ultra-cold plasmas, the time to scatter to the ground state is much longer than the capture time. Need to follow electron-Rydberg collisions. The capture step itself is the result of MANY collisions. Atoms can re-ionize if BE < 8 kB Te. The many collisions takes time and it could be important if the ion were only briefly exposed to the electrons. Summary Coulomb coupling parameter: Γe I +Q +Q Again, this problem is too complicated for pencil and paper. But don’t need to solve for all possible incident energies and binding energies. Classically, the Coulomb problem scales (only E0/BE important). Numerically, solve for one binding energy and scale to other BE’s. l-mixing is much faster than all other processes. electron-Rydberg collisions (trends) Excitation and ionization requires the incoming electron loses energy; de-excitation more likely when BE >> E0. The ratio of probability for de-excitation by different energy amounts does not change rapidly as E0 t 0. The de-excitation cross section diverges as E0 t 0 because the dipole can pull electrons in from large distances and there is no energy barrier. Atoms bound by roughly 5 kB Te are ~ equally likely to be re-ionized or driven to much larger BE. Classically, the excitation and de-excitation rates are smooth functions that can be fit by simple expressions. Photon Emission Rydberg atoms can lose energy by emitting a photon. +Q +Q Accelerating charged particles emit light. 2 2 3 Classical: P = 2 q a /(3 c 4 πε0) 2 2 2 3 Quantum: P = 4 q ω |pif| /(3 c 4 πε0) Photon Emission (Simplified) The matrix elements for full quantum calculation of all transitions are extremely tedious. In many (most?) plasmas, the collisions of the electrons with the atoms cause the angular momentum to completely mix. Only need to calculate radiation from one n-manifold to another. 16 α4 c Z4 1 A = ni →nf 2 2 2 3 3 π a 0 ni ni n f (ni - n f ) Summary Coulomb coupling parameter: Γe I