Kinetic Theory of Instability-Enhanced Collisions and Its Application to Langmuir s Paradox and the Multi-Species Bohm Criterion

Transcription

1 Kinetic Theory of Instability-Enhanced Collisions and Its Application to Langmuir s Paradox and the Multi-Species Bohm Criterion Scott D. Baalrud in collaboration with Chris C. Hegna and James D. Callen University of Iowa Physics Department Colloquium, February 6, 2012 University of Iowa Physics Department Colloquium February 6, 2012, p 1

2 Introduction Part 1: Langmuir s paradox Explain: What is Langmuir s paradox? Review physics of plasma-boundary regions (sheaths) Show that ion-acoustic instabilities can be excited near boundaries Discuss a new kinetic theory that treats wave-particle scattering Show how wave-particle scattering can resolve Langmuir s paradox Part 2: The multi-species Bohm criterion Explain: What is the multi-species Bohm criterion? Describe a long-standing discrepancy between experiments and theory for what the speed of each ion species is when it leaves a plasma Show how ion-ion streaming instabilities can be excited near boundaries Show how wave-particle scattering can resolve this discrepancy as well University of Iowa Physics Department Colloquium February 6, 2012, p 2

3 figure S.D. Baalrud Irving Langmuir: A founder of plasma physics February 4, 2012 Studied low pressure mercury plasmas energized by a biased hot filament Spherical glass chamber 3 cm in diameter Early incandescent light bulb Plasma Sources Sci. Technol. 18 (2009) N Hershkowitz and Y-C Ghim (Kim) Nobel Prize: Chemistry (1932) for his discoveries and investigations in surface chemistry Langmuir, Phys. Rev. 26, 585 (1925) Figure 1. Experimental tube used by Tonks and Langmuir to study plasma oscillations. Named plasma and sheath Used electrostatic probes (Langmuir probes) to measure plasma properties Figure 2. Experimental tube used by Revans to study standing waves. Probe P was movable by a magnet. Substituting equation (5)gives (using SI units and measuring temperatures in electron volts) follows. 2 ( 2 ) ( ξ Consider an initially uniform ion distribution and a + ne2 2 ξ x 2 t 2 m i ε 0 x m i 2 ) ξ = 0. (8) University of Iowa Physics Department Colloquium 2 T e February t 2 6, 2012, p 3 uniform electron background with a charge density n e located 1

4 Sheaths are a plasma-boundary effect Electrons diffuse much faster than ions in these plasmas v T e = 2T e /m e and v T i = 2T i /m i Here m i m e and T e 1 ev T i ev (room temperature) Sheaths are thin regions where strong electric fields form to balance electron and ion currents lost to boundaries Sheath length scale is the Debye length: λ De = T e /(4πe 2 n e ) n i = n e n i n e n o n i n o 0 s plasma presheath sheath n e z ~ i» De ~ De p o z w sheath edge s University of Iowa Physics Department Colloquium February 6, 2012, p 4

5 figure S.D. Baalrud Bohm discovered important properties of presheaths February 4, 2012 Langmuir showed plasma is quasineutral, with nonneutral sheaths near boundaries In 1949, Bohm showed that ions must satisfy the condition V c s = T e /m i at the plasma-sheath interface It has later been shown that equality typically holds The Bohm criterion is useful for calculating the ion flux to a boundary Implies the existence of a presheath Typical presheaths have a potential drop of approximately T e /2 over an ion-neutral collision length (much longer than the sheath) Fellow of the Royal Society Expert in theoretical physics, philosophy and neuropsychology (photo credit: An American-born British physicist, philosopher and neuropsychologist 1 University of Iowa Physics Department Colloquium February 6, 2012, p 5

6 Langmuir expected a truncated e distribution The electrostatic potential drop of the sheath near a boundary is φ s = T ( ) e e ln πmi 8m e (Calculated from equating electron and ion fluxes to a boundary) Only electrons with velocity greater than V,c = 2 φ s /m e will be lost plasma presheath sheath ~ i» De ~ De ln f e Measurement locations p z wall Not missing in Langmuir s measurement sheath edge s v, v c University of Iowa Physics Department Colloquium February 6, 2012, p 6

7 Langmuir did not find the expected truncation! The electron-electron collision length was estimated to be 28 cm in Langmuir s plasma Chamber diameter was only 3 cm Appears to be no mechanism to thermalize electrons Langmuir measured the electron distribution with an electrostatic probe (a Langmuir probe) Straight line indicates Maxwellian e φ s 10 ev Why do electrons come into equilibrium anomalously quickly? Important: Most ionization is due to tail electrons that should be missing! Langmuir, Phys. Rev. 26, 585 (1925) University of Iowa Physics Department Colloquium February 6, 2012, p 7

8 figure Gabor named and studied Langmuir s paradox S.D. Baalrud February 4, 2012 In the mid 1950 s, Dennis Gabor named this problem Langmuir s paradox He repeated and extended Langmuir s experiments Called this one of the worst discrepancies known in science Used an oscillogram and found waves in the MHz frequency range near the plasma boundaries Did not know the origin of these waves, but suggested that they might contribute to electron scattering No theory for the wave dispersion or scattering existed at that time Gabor, Ash and Dracott, Nature 176, 916 (1955) Nobel Prize: Physics (1971) for his invention and development of the holographic method (photo credit: Hungarian-born British engineer and physicist most famous for invention of the holographic method 1 University of Iowa Physics Department Colloquium February 6, 2012, p 8

9 figure Landau discovered collisionless wave damping In 1946, Landau showed that waves can damp in a plasma even without collisions Electrostatic waves are described by the roots of ˆε( k, ω) = 1 + 4πq 2 s d 3 v k f s ( v)/ v k s 2 m s ω k v He was the first to properly resolve the singular integral by realizing ω = ω R + iω I S.D. Baalrud February 4, 2012 Landau damping also works in reverse, where waves can grow by tapping an energy source (flows, gradients, etc.) Landau s prediction was decades before its time First measured by Malmberg and Wharton in 1966 One instability is the ion-acoustic instability, excited by relative ion-electron flow Nobel Prize: Physics (1962) for his pioneering theories for condensed matter (photo credit: Leading Russian physicist who made many seminal contributions to quantum, plasma and condensed 1 matter physics University of Iowa Physics Department Colloquium February 6, 2012, p 9

10 Ion-acoustic instabilities can be excited in the presheath Ion flow provides an energy sources for ion-acoustic instabilities Require that T e /T i 1 (T e /T i 50 in Langmuir s experiment) ω ± = ( k V ± )( kc s 1 i 1 + k2 λ 2 De ) πm e /(8M i ) 1 + k 2 λ 2 De Unstable for V c s / 1 + k 2 λ 2 De (very short wavelength) ω R ω pi V i /c s 1/4 1/2 1 Mi γ m e ω pi kλ De kλ De University of Iowa Physics Department Colloquium February 6, 2012, p 10

11 A kinetic theory for wave-particle scattering We used the dressed particle method and the BBGKY hierarchy to derive a kinetic theory that accounts for wave-particle interactions Get 2 terms to the collision operator C(f s, f s ) = C LB (f s, f s )+C IE (f s, f s ) (Here LB stands for Lenard-Balescu and IE for instability-enhanced) Mathematically, instabilities are poles of ˆε = 0 in the upper-half of the complex frequency (ω) plane: Unstable Stable University of Iowa Physics Department Colloquium February 6, 2012, p 11

12 The theory works best for convective instabilities Convect out of the plasma (or region of interest) before reaching nonlinear levels, or... Modify f to reduce wave amplitudes before nonlinear regime Instability-Enhanced regime Coulomb collision dominated Nonlinear effects Total Coulomb level I-E level Distance (or time) Fluctuation-induced collisions δ/ ln Λ 10 (2 3) less frequent than Coulomb collisions in stable plasmas Fluctuations must grow for 2γt 5 for instability amplification University of Iowa Physics Department Colloquium February 6, 2012, p 12

13 One slide for the experts: The theory is similar to quasilinear theory with one important difference: it accounts for the discrete particle origin of fluctuations The component C IE (f s, f s ) have Landau form: diffusion and drag terms C IE (f s, f s ) = ( v D IE,diff f ) s v v [ ] D IE,drag f s where and D IE,diff = Q IE = 2q2 s q2 s πm s d 3 v Q IE f s ( v ) m s, D IE,drag = d 3 k k k k 4 j ˆε( k,ω) ω d 3 v Q IE 1 f s ( v ) m s v γ j e 2γ jt 2 [ (ωr,j ][ k v) ω 2 + γ 2 j j (ωr,j ] k v ) 2 + γj 2 The total collision operator, C(f s ) = s C(f s, f s ), is a diffusion equation since s D IE,drag = 0 However, the component C IE (f s, f s ) are also an important concept This allows one to derive the spectral energy density (initial condition of quasilinear theory) University of Iowa Physics Department Colloquium February 6, 2012, p 13

14 Properties of kinetic and quasilinear theories differ Density is conserved for collisions between individual species (kinetic) d 3 v C(f s, f s ) = 0 Total density is conserved in quasilinear theory: d 3 v s C(f s, f s ) = 0 Momentum is conserved for collisions between individual species (kinetic) d 3 v m s vc(f s, f s ) + d 3 v m s vc(f s, f s ) = 0 Total momentum is conserved in quasilinear theory: d 3 v s m s vc(f s, f s ) = 0 The sum of particle and wave energy is conserved (in both theories) Kinetic theory obeys an H-theorem (entropy production) Maxwellian is the unique equilibrium in the kinetic theory (not unique in quasilinear theory) Like species s = s equilibrate on a faster timescale than s s Component-equilibria (e.g., for individual species) are uniquely Maxwellian Approach to Maxwellian is hastened by instabilities University of Iowa Physics Department Colloquium February 6, 2012, p 14

15 IA instabilities explain Langmuir s measurements Recall that Langmuir s plasma was 3 cm long, and the estimated e-e collision length was about 30 cm Need instabilities to enhance the collision frequency at least 10 times For thermal electrons (v = v T e ), and Langmuir s discharge parameters, the IE collision theory predicts 100 times enhancement Also important to know that these collisions generate a Maxwellian φ T e 10 2 ν ν o ν o ν IE ν total V/c s z/l S. D. Baalrud, J. D. Callen and C. C. Hegna, Phys. Rev. Lett. 102, (2009). z/l University of Iowa Physics Department Colloquium February 6, 2012, p 15

16 The theory is valid in the presheath Kinetic theory neglects nonlinear effects δ E f v δ E δf v Green line shows z max below which the linear theory is valid Red line shows z min for which wave-particle interactions dominate nonlinear m M e i Z De linear, wave-particle presheath range particle-particle 3 n De S. D. Baalrud, J. D. Callen and C. C. Hegna, Phys. Plasmas 15, (2008) University of Iowa Physics Department Colloquium February 6, 2012, p 16

17 Part 2: The Bohm criterion with multiple ion species For plasmas with one ion species, the Bohm criterion determines the speed that they leave the plasma V = c s = T e /m i What happens if more than one ion species is present? Species can denote different mass, charge, etc. Generalizing the Bohm criterion gives i n i n e c 2 s,i V 2 i v 2 T i /2 = 1 Here c s,i = T e /m i is the individual sound speed of species i This only provides 1 constraint in as many unknowns as there are ion species What determines the Bohm criterion? (What additional constraints are required to determine the speed of each ion species at the sheath edge?) University of Iowa Physics Department Colloquium February 6, 2012, p 17

18 Theories predict individual sound speeds If ions are collisionless, energy conservation over the presheath is V i = 2e φ ps /m i Putting this constraint into the multi-species Bohm criterion gives e φ ps = T e /2 Thus, each ion species obtains its individual sound speed V i = c s,i = T e /M i However, the presheath length is comparable to the ion-neutral collision length (so collisions may play a role) Franklin 1 has accounted for ion-neutral collisions Shows that ion-neutral collisions usually do not substantially change the individual sound speed prediction Cause a larger φ ps, but speeds are the same Substantial changes only if one species is much more collisional than the other (a property of the ion-neutral collision cross sections) 1R.N. Franklin, J. Phys. D: Appl. Phys. 33, 3186 (2000). University of Iowa Physics Department Colloquium February 6, 2012, p 18

19 Experiments do not agree with these theories Experiments measure speeds close to a system sound speed: n1 V i = c s = c 2 s,1 n + n 2 c 2 s,2 e n e Suggests ion-ion friction is important, but theory predicts it shouldn t Appl. Phys. Lett. 91, (Ar-Xe plasma, equal densities, T e = 0.7 ev, T Ar = T Xe = ev) D. Lee, N. Hershkowitz and G. D. Severn, Appl. Phys. Lett. 91, FIG. 4. Color online Spatial profiles of the plasma potential and Ar Xe + (2007). velocities in the Ar 0.5+Xe 0.2 mtorr plasma. University of Iowa Physics Department Colloquium February 6, 2012, p 19

20 Two-stream instabilities can be excited in the presheath Use fluid theory (assumes T i = 0) to calculate the dispersion relation Numerical solution is shown below (blue solid line) An analytic solution is required for the kinetic theory (red dotted line): ( ω n2 c 2 s2 k V n e c n 1 c 2 ) s1 V s n e c i k V α ( k V ) 2 1 (1 + k 2 λ 2 s 1 + α k 2 Vup 2 De ) in which α = n 1 M 2 /(n 2 M 1 ) and V 2 up = c2 s [ α/(1 + α) 2 ] University of Iowa Physics Department Colloquium February 6, 2012, p 20

21 Two-stream instabilities have been measured Log of power spectra from fluctuations in ion saturation current (a) Adding Ar to a 1 mtorr He plasma (b) As a function of distance from the boundary (1 mtorr Ar, 0.1 mtorr Xe) Noah Hershkowitz Phys. Plasmas 12, N. Hershkowitz, Phys. Plasmas 12, (2005) FIG. 21. Log of power spectra measured from fluctuations in ion saturation current as a function of a the addition of Ar to a 1 mtorr He plasma and b the distance from the wall with 0.1 mtorr Ar and 1 mtorr He plasma. Curves are displaced vertically for clarity. University of Iowa Physics Department Colloquium February 6, 2012, p 21 FIG. 22. a The peak frequency observed in Fig. 21 vs A

22 Two-stream instabilities rapidly enhance friction Use our new kinetic theory to calculate the ion-ion friction R 1 2 = d 3 v m 1 vc(f 1, f 2 ) I-E friction dominates for z/λ De 15 (stiff system) Presheath length: l 10 3 λ De length for I-E friction to dominate Cold ion limit predicts common sound speed V 1 = V 2 = c s S. D. Baalrud, C. C. Hegna and J. D. Callen, Phys. Rev. Lett. 103, (2009) University of Iowa Physics Department Colloquium February 6, 2012, p 22

23 V c is a critical parameter Accounting for finite T i gives stabilization for V V c = O(v T i ) Need to determine V c from dispersion relation For v T 1 /v T 2 4, or v T 1 /v T 2 1/4, an approximation from fluid theory V fl c where α = n 1 m 2 /(n 2 m 1 ) 1 + α = vt 2 1 2α + αv2 T 2 For 1/4 v T 1 /v T 2 4, an approximation from kinetic theory V kin c 3 ( 2 v 1 T 2 v T n )( 2 T 1 vt n 1 T + n ) 1 T 2 vt n 2 T 1 First constraint is V c from instabilities or energy conservation V 1 V 2 = V c { V fl c, or V kin c if c s1 c s2 c s1 c s2 if > c s1 c s2 Second constraint is the 2-species Bohm criterion n 1 n e c 2 s1 V n 2 c 2 s2 n e (V 1 V c ) = 1 2 University of Iowa Physics Department Colloquium February 6, 2012, p 23

24 Speeds become locked with V = V c A handy formula for V c c s,1 c s,2 is V 1 c s + n 2 c 2 s2 V c and V 2 c s n 1 c 2 s1 V c n e n e Speeds differ from c s by an amount of order O(v T i ) V c s1 c s c s2 c 2 s Collisional friction n c V! V 2 2 s2 1 = cs + 2 ne cs n c V = c! " V!V c 2 1 s1 2 s 2 ne cs c c c 2 s No collisional friction sheath Distance Bulk plasma University of Iowa Physics Department Colloquium February 6, 2012, p 24

25 Experiments validate the theory: Ar + and Xe + Measurements of ion flow speeds using laser induced fluorescence (LIF) For v T 1 /v T 2 = 1.8, use V = V kin c Handy formula: V 1 c s + n c 2 s2 n e c 2 s in the Bohm criterion (solid lines) V c and V 2 c s n 1 n e c 2 s1 c 2 s V c (dashed) 1=Ar +, 2=Xe +, T e = 0.7 ev, T 1 = T 2 = 0.05 ev 1400 c s, V1 and V2 [m/s] c s c s, n 1 /n e C.-S. Yip, N. Hershkowitz and G. Severn, Phys. Rev. Lett. 104, (2010). University of Iowa Physics Department Colloquium February 6, 2012, p 25

26 Experiments validate the theory: He + and Xe + Measurements of ion flow speeds using laser induced fluorescence (LIF) For v T 1 /v T 2 8, use V = V fl c Handy formula: V 1 c s + n c 2 s2 n e c 2 s in the Bohm criterion (solid lines) V c and V 2 c s n 1 n e c 2 s1 c 2 s V c (dashed) 1=He +, 2=Xe +, T e = 0.7 ev, T 1 = T 2 = 0.05 ev 5000 c s,1 V1 and V2 [m/s] c s 1000 c s, n 1 /n e N. Hershkowitz, C.-S. Yip and G. D. Severn, Phys. Plasmas 18, (2011) University of Iowa Physics Department Colloquium February 6, 2012, p 26

27 Summary Derived a kinetic theory for wave-particle interactions The theory works best for convective instabilities that either: (1) Leave the plasma before reaching nonlinear levels (2) Modify the plasma to reduce the instability amplitude before nonlinear levels Applied the theory to resolve Langmuir s paradox Instability-enhanced scattering can dominate Can shrink e/e mean free path to the presheath length scale Predicts Maxwellian within the presheath Applied the theory to calculate instability-enhanced collisional friction in plasmas with two ion species Considered ion-ion two-stream instabilities in the presheath Instability-enhanced friction quickly becomes large after onset (stiff system) This provides a condition V = V c, which determines the Bohm criterion University of Iowa Physics Department Colloquium February 6, 2012, p 27