kinetic theory of instability-enhanced collisions and its...

Kinetic Theory of Instability-Enhanced Collisionsand 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

IntroductionPart 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


Irving Langmuir: A founder of plasma physics

• Studied low pressure mercury plasmas en-

ergized by a biased hot filament

• Spherical glass chamber 3 cm in diameter

• Early incandescent light bulbPlasma Sources Sci. Technol. 18 (2009) 014018 N Hershkowitz and Y-C Ghim (Kim)

Figure 1. Experimental tube used by Tonks and Langmuir to studyplasma oscillations.

(using SI units and measuring temperatures in electron volts)follows.

Consider an initially uniform ion distribution and auniform electron background with a charge density ne locatedbetween two planes perpendicular to the x-axis. If each ion isdisplaced a small distance ξ that depends on x, the change inion density is

δni = −ni∂ξ

∂x. (1)

Assuming the electron density satisfies the Boltzmann relation

ne = ne0 exp

(eφ

Te

)≈ ne0

(1 +

eφ

Te

)(2)

for small displacements, then

δne ≈ ne0eφ

Te. (3)

Poisson’s equation gives

∂2φ

∂x2= − e

ε0(δni − δne) ≈ e

ε0

(ni

∂ξ

∂x+ ne0

eφ

Te

). (4)

The ion equation of motion is

e∂φ

∂x= −mi

∂2ξ

∂t2. (5)

Equating equations (4) and (5) gives

e∂2φ

∂x2= −mi

∂

∂x

(∂2ξ

∂t2

)≈ ne2

ε0

(∂ξ

∂x+

eφ

Te

). (6)

Taking the derivative of equation (6) with respect to x

∂2

∂x2

(∂2ξ

∂t2

)+

ne2

miε0

(∂2ξ

∂x2+

e

Te

∂φ

∂x

)= 0. (7)

Figure 2. Experimental tube used by Revans to study standingwaves. Probe P was movable by a magnet.

Substituting equation (5) gives

∂2

∂x2

(∂2ξ

∂t2

)+

ne2

miε0

(∂2ξ

∂x2− mi

Te

∂2ξ

∂t2

)= 0. (8)

This can be rewritten as

∂2

∂x2

(∂2ξ

∂t2+

ne2

miε0ξ

)−

(1

λ2De

∂2ξ

∂t2

)= 0,

∂2

∂x2

(∂2ξ

∂t2+ ω2

piξ

)−

(1

λ2De

∂2ξ

∂t2

)= 0, (9)

where the electron Debye length λDe ≡√

ε0Te/ne2 and theion plasma frequency ωpi ≡

√ne2/ε0mi.

They looked for a solution of the form

ξ = exp[i2π

(f t − x

λ

)]. (10)

This gave

f =(

ω2pi

(2π)2 + (λ2/λ2De)

)1/2

. (11)

For low frequency long wavelength modes, λ � λDe, 2πf �ωpi, they pointed out that the phase velocity was

vφ = f λ ≈ ωpiλDe =√

Te

mi(12)

for λ ≈ λDe, ω ≈ ωpi.

In 1933, five years after Langmuir’s prediction of ionacoustic waves, J J Thomson provided a derivation of ionacoustic waves [5] from ion fluid equations and the Boltzmann

2

Langmuir, Phys. Rev. 26, 585 (1925)

figure

S.D. Baalrud

February 4, 2012

Nobel Prize: Chemistry (1932)“for his discoveries and investiga-tions in surface chemistry”

1

– Named plasma and sheath

– Used electrostatic probes(Langmuir probes) to mea-sure plasma properties


Sheaths are a plasma-boundary effect• Electrons diffuse much faster than ions in these plasmas

vTe =√

2Te/me and vT i =√

2Ti/mi

• Here mi � me and Te ' 1 eV � Ti ' 0.023 eV (room temperature)

• Sheaths are thin regions where strong electric fields form to balance elec-

tron and ion currents lost to boundaries

• Sheath length scale is the Debye length: λDe =√Te/(4πe2ne)

z

z

no

ne

ni

ni ≈ neni = ne

s0

no

ow

s

p

plasma presheath sheath

~ i » De ~ De

sheathedge


Bohm discovered important properties of presheaths

• Langmuir showed plasma is quasineutral,

with nonneutral sheaths near boundaries

• In 1949, Bohm showed that ions must sat-

isfy the condition

V ≥ cs =√Te/mi

at the plasma-sheath interface

• It has later been shown that equality typi-

cally 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 Te/2 over an ion-neutral col-

lision length (much longer than the sheath)

figure

S.D. Baalrud

February 4, 2012

Fellow of the Royal SocietyExpert in theoretical physics, phi-losophy and neuropsychology

(photo credit: www.wikipedia.org)

1

– An American-born Britishphysicist, philosopher andneuropsychologist


Langmuir expected a truncated e− distribution

• The electrostatic potential drop of the sheath near a boundary is

∆φs =Te

eln

(√πmi

8me

)

(Calculated from equating electron and ion fluxes to a boundary)

• Only electrons with velocity greater than V‖,c =√

2∆φs/me will be lost

z

s

p

plasma presheath sheath

~ i » De ~ De

sheathedge

wall

cv||, ||v

efln

Not missing in

Langmuir’s measurement

Measurement locations


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∆φs ' 10 eV

•Why do electrons come into equi-

librium anomalously quickly?

• Important: Most ionization is due

to tail electrons that should be

missing! Langmuir, Phys. Rev. 26, 585 (1925)


Gabor named and studied Langmuir’s paradox

• In the mid 1950’s, Dennis Gabor named this

problem Langmuir’s paradox

• He repeated and extended Langmuir’s ex-

periments

• 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 scat-

tering existed at that time

Gabor, Ash and Dracott, Nature 176, 916 (1955)

figure

S.D. Baalrud

February 4, 2012

Nobel Prize: Physics (1971)“for his invention and develop-ment of the holographic method”

(photo credit: www.nobelprize.org)

1

– Hungarian-born British en-gineer and physicist most fa-mous for invention of theholographic method


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

ε̂(~k, ω) = 1 +∑

s

4πq2s

k2ms

∫d3v

~k · ∂fs(~v)/∂~v

ω − ~k · ~v

• He was the first to properly resolve the sin-

gular integral by realizing ω = ωR + iωI

• 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 instabil-

ity, excited by relative ion-electron flow

figure

S.D. Baalrud

February 4, 2012

Nobel Prize: Physics (1962)“for his pioneering theories forcondensed matter”

(photo credit: www.nobelprize.org)

1

– Leading Russian physi-cist who made many sem-inal contributions to quan-tum, plasma and condensedmatter physics


Ion-acoustic instabilities can be excited in the presheath

• Ion flow provides an energy sources for ion-acoustic instabilities

• Require that Te/Ti � 1 (Te/Ti ' 50 in Langmuir’s experiment)

ω± =

(~k · ~V ±

kcs√1 + k2λ2

De

)(1∓ i

√πme/(8Mi)

1 + k2λ2De

)

• Unstable for V & cs/√

1 + k2λ2De (very short wavelength)

0 2 4 6 8 10−2

0

2

4

6

8

10

kλDe

ωR

ωpi

0 2 4 6 8 10−0.15

−0.1

−0.05

0

0.05

0.1

kλDe

√Mi

me

γ

ωpi

1/41/21

Vi/cs


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(fs, fs′) = CLB(fs, fs′)+CIE(fs, fs′)

(Here LB stands for Lenard-Balescu and IE for instability-enhanced)

• Mathematically, instabilities are poles of ε̂ = 0 in the upper-half of the

complex frequency (ω) plane:

Unstable

Stable


The theory works best for convective instabilities• Convect out of the plasma (or region of interest) before reaching nonlin-

ear levels, or . . .

• Modify f to reduce wave amplitudes before nonlinear regime

Distance (or time)

Coulomb level

Nonlinear effectsCoulomb collisiondominated

Instability-Enhancedregime

Total

I-E level

• 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


One slide for the experts:

• The theory is similar to quasilinear theory – with one important differ-

ence: it accounts for the discrete particle origin of fluctuations

• The component CIE(fs, fs′) have Landau form: diffusion and drag terms

CIE(fs, fs′) =∂

∂~v·(DIE,diff ·

∂fs

∂~v

)−

∂

∂~v·[~DIE,dragfs

]

where

DIE,diff =

∫d3v′

↔QIE

fs′(~v′)

ms

, ~DIE,drag =

∫d3v′

↔QIE ·

1

ms′

∂fs′(~v′)

∂~v′

and

↔QIE=

2q2sq

2s′

πms

∫d3k

~k~k

k4

∑

j

γj e2γjt

∣∣∂ε̂(~k,ω)∂ω

∣∣2ωj

[(ωR,j − ~k · ~v)2 + γ2

j

][(ωR,j − ~k · ~v′)2 + γ2

j

]

• The total collision operator, C(fs) =∑

s′ C(fs, fs′), is a diffusion equation

since∑

s′~DIE,drag = 0

• However, the component CIE(fs, fs′) are also an important concept

• This allows one to derive the spectral energy density (initial condition of

quasilinear theory)


Properties of kinetic and quasilinear theories differ

• Density is conserved for collisions between individual species (kinetic)∫d3v C(fs, fs′) = 0

– Total density is conserved in quasilinear theory:∫d3v

∑s′ C(fs, fs′) = 0

• Momentum is conserved for collisions between individual species (kinetic)∫d3vms~vC(fs, fs′) +

∫d3vms′~vC(fs′, fs) = 0

– Total momentum is conserved in quasilinear theory:∫d3v

∑s′ ms′~vC(fs, fs′) = 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 6= s′

– Component-equilibria (e.g., for individual species) are uniquely Maxwellian

– Approach to Maxwellian is hastened by instabilities


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 = vTe), and Langmuir’s discharge parameters,

the IE collision theory predicts 100 times enhancement

• Also important to know that these collisions generate a Maxwellian

−2 −1.5 −1 −0.5 00

0.2

0.4

0.6

0.8

1

1.2

z/l−2 −1.5 −1 −0.5 0

100

101

102

z/l

V/cs

φ

Teν

νoνIEνo

νtotal

S. D. Baalrud, J. D. Callen and C. C. Hegna, Phys. Rev. Lett. 102, 245005 (2009).


The theory is valid in the presheath

• Kinetic theory neglects nonlinear effects

δ ~E ·∂f

∂~v� δ ~E ·

∂δf

∂~v

• Green line shows zmax below which the linear theory is valid

• Red line shows zmin for which wave-particle interactions dominate

3

Den

linear,

nonlinear

wave-particle

particle-particle

Dei

e Z

M

m

presheathrange

S. D. Baalrud, J. D. Callen and C. C. Hegna, Phys. Plasmas 15, 092111 (2008)


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 = cs =√Te/mi

•What happens if more than one ion species is present?

• “Species” can denote different mass, charge, etc.

• Generalizing the Bohm criterion gives

∑

i

ni

ne

c2s,i

V 2i − v2

T i/2= 1

• Here cs,i =√Te/mi 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?)


Theories predict individual sound speeds

• If ions are collisionless, energy conservation over the presheath is

Vi =√

2e|∆φps|/mi

• Putting this constraint into the multi-species Bohm criterion gives

e|∆φps| = Te/2

• Thus, each ion species obtains its individual sound speed

Vi = cs,i =√Te/Mi

• However, the presheath length is comparable to the ion-neutral collision

length (so collisions may play a role)

• Franklin1 has accounted for ion-neutral collisions

– Shows that ion-neutral collisions usually do not substantially change the individualsound speed prediction

– Cause a larger |∆φps|, 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).


Experiments do not agree with these theories• Experiments measure speeds close to a “system” sound speed:

Vi = cs =

√n1

nec2s,1 +

n2

nec2s,2

• Suggests ion-ion friction is important, but theory predicts it shouldn’t

(Ar-Xe plasma, equal densities, Te = 0.7 eV, TAr = TXe = 0.023 eV)

1100±60 m/s, which is close to the vph and is not consistentwith its own Bohm velocity. On the other hand, the xenonion velocity is 940±50 m/s at the sheath edge. This is muchfaster than CXe and is just barely in agreement with the IAWvelocity within experimental uncertainties. From the two ve-locity measurements, it is evident that the results exclude oneof the simple solutions, i.e., the ions do not have their ownBohm velocities near the sheath edge. The data appear tosupport the other simple solution that the ions approach theIAW velocity near the sheath edge. Substituting the mea-sured values into the left hand side of Eq. �2� gives 0.97,which satisfies the generalized Bohm criterion in two-ionspecies plasmas.

This work was supported by DOE Grant No. DE-FG02-97ER54437. One of us �G.D.S.� expresses thanks for thesupport by DOE �DE-FG02-03ER54728� and NSF�CHEM0321326�.

1D. Bohm, in The Characteristics of Electrical Discharges in MagneticField, edited by A. Guthrie and R. K. Wakerling �McGraw-Hill, NewYork, 1949�, Chap. 3, p. 77.

2L. Oksuz and N. Hershkowitz, Phys. Rev. Lett. 89, 145001 �2002�.3L. Oksuz and N. Hershkowitz, Plasma Sources Sci. Technol. 14, 201�2005�.

4K.-U. Riemann, IEEE Trans. Plasma Sci. 23, 709 �1995�.5A. M. A. Hala, Ph.D. dissertation, College of Engineering, University ofWisconsin-Madison, 2000.

6G. D. Severn, X. Wang, E. Ko, and N. Hershkowitz, Phys. Rev. Lett. 90,145001 �2003�.

7G. D. Severn, X. Wang, E. Ko, N. Hershkowitz, M. M. Turner, and R.McWilliams, Thin Solid Films 506-507, 674 �2006�.

8X. Wang and N. Hershkowitz, Phys. Plasmas 13, 053503 �2006�.9D. Lee, G. Severn, L. Oksuz, and N. Hershkowitz, J. Phys. D 39, 5230�2006�.

10G. D. Severn, D. A. Edrich, and R. McWilliams, Rev. Sci. Instrum. 69, 10�1998�.

11G. D. Severn, D. Lee, and N. Hershkowitz �unpublished�.12H. Salami and A. J. Ross, J. Mol. Spectrosc. 233, 157 �2005�.13A. M. Keesee, E. E. Scime, and R. F. Boivin, Rev. Sci. Instrum. 75, 4091

�2004�.14H.-J. Woo, K.-S. Chung, T. Lho, and R. McWilliams, J. Korean Phys. Soc.

48, 260 �2006�.15J. R. Smith, N. Hershkowitz, and P. Coakley, Rev. Sci. Instrum. 50, 210

�1979�.16X. Wang and N. Hershkowitz, Rev. Sci. Instrum. 77, 043507 �2006�.17S. B. Song, C. S. Chang, and D.-I. Choi, Phys. Rev. E 55, 1213 �1997�.18A. M. A. Hala and N. Hershkowitz, Rev. Sci. Instrum. 72, 2279 �2001�.

FIG. 3. �Color online� �a� Ar and �b� Xe IVDFs as a function of distance zfrom the plate in the Ar 0.5+Xe 0.2 mTorr plasma.

FIG. 4. �Color online� Spatial profiles of the plasma potential and Ar+–Xe+

velocities in the Ar 0.5+Xe 0.2 mTorr plasma.

041505-3 Lee, Hershkowitz, and Severn Appl. Phys. Lett. 91, 041505 �2007�

Downloaded 05 Apr 2009 to 72.33.11.94. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

D. Lee, N. Hershkowitz and G. D. Severn, Appl. Phys. Lett. 91, 041505 (2007).


Two-stream instabilities can be excited in the presheath• Use fluid theory (assumes Ti = 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 ' ~k ·(n2

ne

c2s2

c2s

~V1 +n1

ne

c2s1

c2s

~V2

)+ i

~k ·∆~V√α

1 + α

√√√√1−(~k ·∆~V )2

k2∆V 2up

(1 + k2λ2De)

in which α =√n1M2/(n2M1) and ∆V 2

up = c2s[1 +

√1 + 32α/(1 + α)2]


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)

VII. CONCLUSIONS

Overall, ion sheaths in single-ion species weakly colli-sional plasma witheV/Te@1 are better understood thanelectron sheaths. Both ion and electron sheaths can exist andthe Child–Langmuir law provides good fits to potential vsposition wheneV/Te@1. Electron sheaths are much lesscommon than ion sheaths because they require that a suffi-cient area be available in the plasma chamber for the ion losscurrent and many chambers are not big enough. The charac-teristics of potential dips preceding many electron sheathsare not well understood.

In weakly collisional plasmas, most of the ion accelera-tion to the sheath boundary takes place in a presheath withina collision lengthl of the boundary. Emissive probe datashow the plasma potential in the presheath varies asef /Te

=Îx/l, measured from the sheath/presheath boundary inagreement with Riemann’s predictions. WheneV/Te@1, theChild–Langmuir sheath provides a good fit. A transition re-gion, the Debye sheath, in which the ion density becomesnegligible is found between the presheath and Child–Langmuir sheath, so sheaths are normally thicker than theChild–Langmuir sheaths. The electric field in the transitionregion is found to beE<Te/elD in agreement with thequalitative predictions of Godyak. LIF data indicatev<cs atthe presheath/sheath boundary for single species plasma.Both electron and ion sheaths are seen in rf plasmas. Double

layers provide a way for a “sheath” to form away from theboundary.

More than one ion species are present in most plasmas ofinterest, but sheaths and presheaths of multiple speciesplasma are not well understood. LIF data indicatevAr .cs atthe presheath/sheath boundary for an Ar+–He+ plasma and itis not clear how the individual ion velocities are determined.Furthermore, the presheath was shown to be ion-ion un-stable. The multiple species presheath/sheath problem stillneeds lots of work.

ACKNOWLEDGMENTS

The author is grateful to Eunsuk Ko for assistance inpreparing this paper.

This work was supported by the Department of EnergyGrant No. DE-FG02-97ER5447.

1F. F. Chen,Introduction to Plasma Physics and Controlled Fusion, 2nd ed.sPlenum, New York, 1974d, Chap. 8.

2Irving Langmuir, Phys. Rev.33, 954 s1929d.3D. Bohm,The Characteristics of Electrical Discharges in Magnetic Field,edited by A. Guthrie and R. K. WakerlingsMcGraw-Hill, New York,1949d, Chap. 3.

4P. C. Stangeby and A. V. Chankin, Phys. Plasmas2, 707 s1995d.5D. J. Koch and W. N. G. Hitchon, Phys. Fluids B1, 2239s1989d.6G. A. Emert, R. M. Wieland, A. T. Mense, and J. N. Davidson, Phys.Fluids 23, 803 s1980d.

FIG. 21. Log of power spectra measured from fluctuations in ion saturationcurrent as a function ofsad the addition of Ar to a 1 mTorr He plasma andsbd the distance from the wall with 0.1 mTorr Ar and 1 mTorr He plasma.Curves are displaced vertically for clarity.

FIG. 22. sad The peak frequency observed in Fig. 21 vs Ar+ concentration.sbd Frequencies and growth rates predicted by fluid and kinetic equations.

055502-10 Noah Hershkowitz Phys. Plasmas 12, 055502 ~2005!

Downloaded 05 Feb 2012 to 132.177.40.110. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions

N. Hershkowitz, Phys. Plasmas 12, 055502 (2005)


Two-stream instabilities rapidly enhance friction• Use our new kinetic theory to calculate the ion-ion friction

~R1−2 =

∫d3vm1~vC(f1, f2)

• I-E friction dominates for z/λDe ' 15 (stiff system)

• Presheath length: l ∼ 103λDe � length for I-E friction to dominate

• Cold ion limit predicts common sound speed V1 = V2 = cs

S. D. Baalrud, C. C. Hegna and J. D. Callen, Phys. Rev. Lett. 103, 205002 (2009)


∆Vc is a critical parameter

• Accounting for finite Ti gives stabilization for ∆V ≤ ∆Vc = O(vT i)

• Need to determine ∆Vc from dispersion relation

• For vT1/vT2 & 4, or vT1/vT2 . 1/4, an approximation from fluid theory

∆V flc =

√1 + α

2α

√v2T1 + αv2

T2

where α = n1m2/(n2m1)

• For 1/4 . vT1/vT2 . 4, an approximation from kinetic theory

∆V kinc ≈ −

3

2|vT2 − vT1|+

√1

2

(1 +

n2

n1

T1

T2

)(v2T1 +

n1

n2

T2

T1

v2T2

)

• First constraint is ∆Vc from instabilities or energy conservation

V1 − V2 = ∆Vc ≡{

∆V flc , or ∆V kin

c if ≤ |cs1 − cs2|cs1 − cs2 if > |cs1 − cs2|

• Second constraint is the 2-species Bohm criterion

n1

ne

c2s1

V 21

+n2

ne

c2s2

(V1 −∆Vc)2= 1


Speeds become “locked” with ∆V = ∆Vc• A handy formula for ∆Vc � |cs,1 − cs,2| is

V1 ' cs +n2

ne

c2s2

c2s

∆Vc and V2 ' cs −n1

ne

c2s1

c2s

∆Vc

• Speeds differ from cs by an amount of order O(vT i)

sc

2sc

1sc

c

s

s

e

sV

c

c

n

ncV !+=

2

2

22

1

2

11

2 2

s

s c

e s

cnV c V

n c= ! "

cV!

sheath

V

Distance Bulk plasma

No collisional frictionCollisional friction


Experiments validate the theory: Ar+ and Xe+

• Measurements of ion flow speeds using laser induced fluorescence (LIF)

• For vT1/vT2 = 1.8, use ∆V = ∆V kinc in the Bohm criterion (solid lines)

• Handy formula: V1 ' cs + n2ne

c2s2c2s

∆Vc and V2 ' cs − n1ne

c2s1c2s

∆Vc (dashed)

1=Ar+, 2=Xe+, Te= 0.7 eV, T1 = T2 = 0.05 eV

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

1200

1400

1600

n1/ne

V1andV2[m

/s]

cs,2

cs,1

cs

C.-S. Yip, N. Hershkowitz and G. Severn, Phys. Rev. Lett. 104, 225003 (2010).


Experiments validate the theory: He+ and Xe+

• Measurements of ion flow speeds using laser induced fluorescence (LIF)

• For vT1/vT2 ' 8, use ∆V = ∆V flc in the Bohm criterion (solid lines)

• Handy formula: V1 ' cs + n2ne

c2s2c2s

∆Vc and V2 ' cs − n1ne

c2s1c2s

∆Vc (dashed)

1=He+, 2=Xe+, Te= 0.7 eV, T1 = T2 = 0.05 eV

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

6000

n1/ne

V1andV2[m

/s]

cs

cs,1

cs,2

N. Hershkowitz, C.-S. Yip and G. D. Severn, Phys. Plasmas 18, 057102 (2011)


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 = ∆Vc, which determines the Bohm criterion


kinetic theory of instability-enhanced collisions and its...

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