Plasma Physics Lecture 3 Ian Hutchinson

8/3/2019 Plasma Physics Lecture 3 Ian Hutchinson

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Chapter 3Collisions in Plasmas3.1 Binary collisions between charged particlesReduced-mass for binary collisions:Two particles interacting with each other have forcesF12 orce on 1 from 2.FZ1orce on 2 from 1.By Newton's 3rd law, F12= F Z 1 .Equ ations of m otion:

mlrl = F12 ; m2r2= FZ1Combine to get

r l - 2= F12 (3.2)which may be written

mlm2 d2ml + (3.3)m2 - 1 - 2 ) =dt2 F12

If F l 2 depends only on the difference vector rl - r2 , hen this equation is identical to theequ atio n of a particle of "Reduce d Mass" m, = moving a t position r = rl - 2 withrespect to a fixed center of force:m,r = F12(r) . (3.4)

This is the e quatio n we analyse, but actually particle 2 does move. An d we need t o recognizethat when interpre t ing mathematics .If FZ1 n d r l - 2 are always parallel, the n a general form of th e trajecto ry can be writtenas an integral. To save time we specialize immediately to the Coulomb force

Solution of this s tan da rd (New ton's) problem:


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1: projectile- ,111 , ,' '. . . . . . . . . . . . . . . . . . . . . . . .A- - - - - - - - - - - - - - - -Ib Impact Parameter, 2: target

Figure 3.1: Geometry of the collision orbitAngular momentum is conserved:

m, r28 = const . = m,bvl (0 clockwise from symmetry)Subs t i tu te u =; t h e n 8 =3 u2 bv,Also

Then radial acceleration is

This orbi t equat ion has the e lementary solut ion

Th e s in 0 term is absent by symmetry. T he other cons tant of integrat ion, C, must be deter-mined by initial conditio n. A t initial (far di sta nt ) angle, 01, u l oo= = 0. o

There:

Hence sin O1 1 / C b bt a n & = -= --cosO1 --V'42- / C bgo4neo rn,(b~~)~


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Notice tha t tan81 = 1 when b = b g ~ .This is when O1 = -45" and x = 90". So particleemerges at 90" to initial direction whenb = bg O "90" imp act param eter" (3.16)

Finally:

3.1.1 Frames of ReferenceKey quantity we want is the scattering angle but we need to be careful about referenceframes.Most "na tura l" fram e of ref is "Cen ter-of-Mass" fra m e, in which C of M is sta tio na ry . C ofM has position:

and velocitv (in lab frame)

Now

So motion of either particle in C of M fr ame is a factor tim es difference vector, r .Velocity in lab frame is obtained by adding V to th e C of M velocity, e.g. + VAng l es of position vectors and velocity dz f f erences are s a m e in all frames.Ang l es (i.e. directions) of velocities are n o t s a m e .

3.1.2 Scattering AngleIn C of M f r am e is just the final angle of r .

- 201 -+x = r(81 is negative)


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Figure 3.2: Relation between O1 and x

X bcot - =2 -90x b90t a n - =2 -But scattering angle (defined as exit velocity angle relative to initial velocity) in lab frameis different.Final velocity in CM frame

vLM = V ~ C MC O S x,, sin xc) m2= vl (cos XC , sin x,) (3.27)m l+ m2[ xC - x and vl is initial relative velocity]. Final velocity in Lab frameSo angle is given by

v+GG732211 cos Xc V ml+m2cot XL = -7n2211. cosecx, + (3.29)cot X,m 1 t m 2 si n xz ~1 m2

For the specific case when m2 is initially a stationary target in lab frame, thenv m1v1= and hencem l+ m2

mlcot XL = c o s e c x , + cot X,m2

This is exact.Small angle approximation (cot x i , osecx i 1 ivesX X

So small angles are proportional, with ratio set by the mass-ratio of particles


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Center-of-Mass Frame / y tParticle 1

Laboratory Fram e,Stationary Target

,I..- ..Particle- 1 !. /f X,.> ,.'Jo

Particle 2

Figure 3.3: Collisions viewed in Center of Mass and Laboratory frame

3.2 Differential Cross-Section for Scattering by AngleRutherford Cross-Section

By definition the cross-section, 0, for any specified collision process when a particle is passingthrough a density n2 of targets is such tha t the number of such collisions per unit path lengthis n20.Sometimes a continuum of types of collision is considered, e.g. we consider collisions atdifferent angles (x) to be distinct. In that case we usually discuss differential cross-sections(e.g $) defined such that number of collisions in an (angle) element dx per unit path lengthis n2$dx. [Note th at 4~ is just notation for a number. Some authors just write ~ ( x ) ,ut Idxfind that less clear.]Normally, for scattering-angle discrimination we discuss the differential cross-section per unitsolid angle: dff-

dflsThis is related to scat tering angle integrated over all azimuthal directions of scattering by:


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Figure 3.4: Scattering angle and solid angle relationship.

dCl, = 2.rrsinxdxSo that since

we have

Now, since x is a function (only) of the impact parameter, b, we just have to determine thenumber of collisions per unit length at impact parameter b.

+d l -

. ... 'Figure 3.5: Annular volume corresponding to db.

Think of the projectile as dragging along an annulus of radius b and thickness db for anelementary distance along its path, d It thereby drags through a volume:

Therefore in this distance it has encountered a total number of targetsd!2.rrbdb . n2


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at imp act para m ete r b(db). By definitio n this is equ al t o d!$dbn2. Hence th e differentialcross-section for scattering (encounter) at impact parameter b is

Again by definition, since x is a function of b

[d bld x is n egative but different,ial cross-sect,ions are posit,ive.]Substi tute and we get

[T his is a gen eral re sult for classical collisions.]For Coulomb collisions, in C of M f rame,

bcot (- =-b90+ db ,d X= b b9090 c o t 2x- = -- cosec - .d x d x 2 2 2

Hencedff b90 cot $ b90 2x- cosecd fl , s i n x 2 -2- bgo cos $ /s in 5 12 2sin $ cos $ sin25

This is the Rutherford Cross-Section.

for scatterin g by Coulom b forces through an angle x measured in C of M frameNotice that & i i s x i .Th is is because of the long-range na tur e of the Co ulom b force. Distan t collisions ten d todominate . (X i H b i a).


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3.3 Relaxation ProcessesThere are 2 (main) different types of collisional relaxation process we need to discuss for atest particle moving through a background of scatterers:

1. Energy Loss (or equilibrium)2. Momentum Loss (or angular scattering)

The distinction may be illustrated by a large angle (90") scatter from a heavy (stationary)targetIf the target is fixed, no energy is transferred to it. So the energy loss is zero (or small ifscat terer is just 'heavy'). However, the m o m e n t u m in the x direction is comple te ly ' los t ' inthis 90" scatter.This shows that the timescales for Energy loss and momentum loss may be very different.

3.3.1 Energy LossFor an initially stationary target, the final velocity in lab frame of the projectile is

So the final kinetic energy is

1 2mlm2= -m1v; {1+ 2 sin2 (m1+ m2) -

Hence the kinetic energy lost is AK = K - K t1 4mlm2 2 X C

= m1V2 sin - (3.53)2 (m 1+ m2) 21 4mlm2 1 xc b

= m 1 v 1 [using cot- 1 (3.54)2 ( m l + m2)2 (&))+ 1 2 b90(exact). For small angles x > 1 this energy lost in a single collision isapproximately


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If wh at we are asking is: how fast does the project i le lose energy? Th en we need ad d up th eeffects of all collisions in an elemental length d! at al l relevant impact parameters.The cont r ibut ion f rom impact parameter range db a t b will equal the number of targetsencountered t imes AK :

-1 4m1m2n2d!2rbdb m l v 12 ( F ) '( m l +m2)'encounters .Loss per encounter ( A K )This must be integrated over al l b to g et to ta l energy loss

dK m1m2- = K n 2 8nb;, [In b]::d! (m1+m2I2We see there is a problem both limits of the integral ( bi 0, i o)diverge logarithmically.That is because the formulas we are integrat ing are approximate.

1. We are using small-angle approx for AK .2. We are assum ing the Co ulomb force applies but th is is a plasm a so there is screening.

3.3.2 Cut-offs Estimates1. Small-angle approx breaks down around b = bgO. Jus t t runcate the integra l there ;ignore contr ibut ion s from b < bgO.2 . Debye Shielding says really the potential varies as

""P (2) 14~ instead ofr K -rso approximate this by cut t ing off integral at b = AD equivalent to

b = b90 . b , = AD


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So Coulomb Logarithm is '1nA'

Because these cut-offs are in in term result is not sensitive to their exact values.One commonly uses Collision Frequency. Energy Loss Collision Frequency is

Sub st i tu te for bgO a nd m, (in bgO)

Collision t ime TK TK 1/uKEffective (Energy Loss) Cross-section [+% = ffKn2]

3.3.3 Momentum LossLoss of x-momentum in 1 collision is

(small angle appr ox). Hence rate of mom entum loss can be obtaine d using an integralidentical to the energy loss but with the above parameters:

Note for the fut ure reference:


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Therefore M o m e n t u m L o ssCollision Frequency

Col l i s ion Time T = l / v pCross-Section (effective) 0 = vp/n2vlNotice ratio Energy Loss v~ m l + m2 2ml (3.78)This is

Third case, e.g. electrons i hows that most ly the angle of velocity scatters. ThereforeMomentum 'Scatter ing' t ime is of ten called '90" scatter ing' t ime to 'diffuse ' through 90" inangle.

3.3.4 'Random Walk' in angleW h e n m l


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So can think of a kind of 'cross-section' for 'ugO'90" scattering as such thatn2L 'ogo ' = 1 when L n2 8~ b 1nA = 1 (3.85)i.e. ' ffgo' = 8~ b 1n A (= 2oP) (3.86)

Th is is 8 in,\ larger tha n cross-section for 90" sca tterin g i n single collision.Be Careful! 'gg0' s not a usual type of cross-section because the whole process is reallydiffusive in angle.Actually all collisio~lprocesses due t ,o coulomb force are best t re ated (in a M athem aticalway) as a diffusion in velocity space

i okker-Plmck equation.

3.3.5 Summary of different types of collisionT h e Energy Loss collision frequency is to do with slowing down to rest and exchangingenergy. It is required for calculatingEquil ibrat ion Times (of Temperatures)

Energy Transfer between species.T h e M o m e n t u m L o s s frequency is to do with loss of directed velocity. It is require d forcalculat ing

Mobil i ty: Conductivi ty/Resist ivi tyViscosityParticle DiffusionEnergy (The rma l) Diffusion

Usually we distinguish between electrons and ions because of their very different mass:Energy Loss [S tat ionary T argets] M o m e n t u m L o ss

Sometimes one distinguishes between 'transverse diffusion' of velocity and 'momentum loss'.The ratio of these two is


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= 1 like particles

Hence

1 . . = P v . . -Z Z - Kvii (= vii!!) (Like Ions)

[B ut note: ions are slowed down by electrons long before being angle sca ttere d.]

3.4 Thermal Distribution CollisionsSo far we have calculated collision frequencies with stationary targets and single-velocityprojectiles but generally we shall care about thermal (Maxwellian) distributions (or nearlythermal) of both species. This is harder to calculate and we shall resort to some heuristiccalculations.

Very rare for thermal ion velocity to be - electron. So ignore ion motion.Average over electron distribution.Mo mentum loss to ions from (assume d) drif t ing Maxwellian electron d istr ibution :

Each electron in this distr ibution is losing momentum to the ions at a rate given by thecollision frequency 424: 4 ~ ( m evp = ni +m i) In A (3.98)( 4 7 ~ ~ ) ~imzv3so total rate of loss of momentum is given by (per unit volume)

To evaluate this integral approximately we adopt the following simplifications


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1. Ignore variations of 1nA with v and just replace a typical the rm al value in A =X D / ~ S O ( ~ I ) .

2. Supposerthat drift velocity vd is small relative'o th e typical therm al velocity, writtenv, = t , /m, an d express f, in terms of u = to first order in ud = -:Zle Zletaking x-axis along ud and denoting by f, the unshifted MaxwellianThen momentum loss rate per unit volume

To evaluate this integral, use the spherical symm etry of f , to see th at :

Th us th e Maxwell-averaged m omentum-loss frequency is

(where p = m,vdn, is the m om entum per unit volume attr ib utab le to drif t)

(sub stitu ting for therm al electron velocity, v,, and dropping order term ) , where Ze = q i .This is the standard form of electron collision frequency.


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Ion momentum loss to electrons can be treated by a simple Galilean transformation of thee i case because it is s ti l l the electron thermal motions that matter.

Ions + Electrons Ions + Electrons

Figure 3.6: Ion-electron collisions are equivalent to electron-ion collisions in a moving refer-ence frame.Rate of mom entum transfer , 2 , s same in both cases:

(since drift velocities a re the sam e).Ion momentum loss to electrons is much lower collision frequency than e i because ionspossess so much m ore mo men tum for th e sam e velocity.

Ion-ion collisions can be treated somewhat like e i collisions except that we have toaccount for moving targets i .e . their thermal motion.Consider two different ion species moving relative to each other with drift velocity vd; th etargets ' thermal motion affects the momentum transfer cross-section.Using our previous expression for momentum transfer, we can write the average rate oftransfe r per un it volume as: [see 3.74 "n ote for futu re reference"]

where v, is the relative velocity (vl - v 2 ) a n d bgO s expressed


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and m, is the reduced mass m 1 t m 2Since everything in the integral apart from f 1 f 2 depends only on the relative velocity, weproceed by transforming the velocity coordinates from v l , v 2 o being expressed in terms ofrelative (v,) and ave rage ( V say)

Take f l and f 2 to be shifted Maxwellians in the overall C of M frame:

where mlvdl + m2Vd2 = 0.T h e n

to first order in vd. Convert CM coordinates and find (after algebra)

where M = ml +m2. Note also that (i t can be shown) d3vld3v2= d3v,d3V. Hencevrm,v,47r b:, in Anln2dt

M V ~ mrv,2 m,exp exp (1 + T ~ d . ~ r )3v.d3v (3.115)and since*nothing except the exponent ia l depends' (-1on V , hat integral can be done:- -m,v,2 m,dt = / v r mrvr47r 1nA nln2 (-) ,27rT exp 27r (1 + ,vd.vr) d3vr (3.116)This integral is of just the same type as for e - collisions, i.e


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where vrt = E , gO(vrt) s the n inety degree impa ct p aram eter ev aluate d a t velocity vt,,and f o is the normalized Maxwellian.

This is the general result for momentum exchange rate between two Maxwellians drifting atsmall relative velocity vd.To get a collision frequency is a m att er of deciding which species is statio na ry an d so what themo men tum density of the moving species is . Suppose we regard 2 as targets the n m om entumdensity is nlmlvd so

Th is expression works immediately for electron-ion collisions sub sti tut ing m, -. m e, recov-ering previous.For equal-mass ions m, m i 1=- ,mi and vr t =m i t m i m y =p.iSub st i tut ing, we get 1 47rv ..7%-- i (z) n h ( 3 . 1 2 0 )37rZ m? Ti"tha t i s , 5 t imes the e - expression but w ith ion paramete rs sub stitu ted. [Note , however,that we have considered the ion species to be different.]

Electron-electron collisions are covered by the same formalism, so

However, the physical case under discussion is not so obvious; since electrons are indistigu-ishable how do we define two different "drifting maxwellian" electron populations? A morespecific discussion- would be needed to make this rigorous.Ge ner ally v,, ve il& : electron-electron collision frequency - lectron-ion (for mom entumloss)3.4.5 Summary of Thermal Collision FrequenciesFor momentum loss:


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-v,, " 1 -- ei . (electron param eters) (3.123)2/2Energy loss K v related to th e above (pv) by

Transverse 'dzffuszon'f mo men tum 'v, related to th e above by:

3.5 Applications of Collision Analysis3.5.1 Energetic ('Runaway') ElectronsConsider a n energetic (;met$ >> T) electron travelling through a plasma. It is slowed down(loses momentum) by collisions with electrons and ions (Z ) , with collision frequen cy:

Hence (in the absence of other forces)

This is equivalent to saying that the electron experiences an effective 'Frictional' force


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Notice1. for Z = 1 slowing dow n is on electrons ions2. Ff decreases with v increasing.

Suppose now there is an electric field, E. The electron experiences an accelerating ForceTota l force

d e4 8.rrln.hF = - ( m v ) = e E Ff = e E (3.134)d tTwo Cases ( Wh e n E is accelerating)

1. e E < I F f : Electron Slows Down2. e E > IF f : E lect ron Speeds Up!

Once the electron energy exceeds a certain value its velocity increases continuously and thefriction force becomes less an d less effective. Th e electron is th en s aid to ahve become a'runaway.

3.5.2 Plasma Resistivity (DC)Consider a bulk distribution of electrons in an electric field. They tend to be accelerated byE and decelerated by collisions.In this case, considering the electrons as a whole, no loss of total electron momentum bye - e collisions. Hence the friction force we need is just that due to Gi.If the electrons have a mean drift velocity vd(


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Therefore, for a plasma, 1 nee2g = = -7 meGi

Substitute the value of vei and we get

1- Ze2m2 8~ inA- (for a single ion species).( ~ T C , ) ~ 6 2

Notice1. Density cancels out because more electrons means (a) more carriers but (b) more

collisions.2. Main dependence is 7 K T;~/~ . High electron temperatu re implies low resistivity (highconductivity).3. This expression is only approximate because the current tends to be carried by the

more energetic electrons, which have smaller vei; thus if we had done a proper averageover f (v,) we expect a lower numerical value. Detailed calculations give

for Z = 1 (vs. -. in our expression). This is 'Spitzer' resistivity. The detailedcalculation value is roughly a factor of two smaller than our calculation, which is nota negligible correction!

3.5.3 DiffusionFor motion parallel to a magnetic field if we take a typical electron, with velocity v -. vt, itwill travel a distance approximately

before being pitch-angle scattered enough to have its velocity randomised. [This is an order-of-magnitude calculation so we ignore u,,.] !s the mean free path.Roughly speaking, any electron does a random walk along the field with step size ! nd stepfrequency vei.Thus th e diffusion coefficient of this process is

Similarly for ions


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(if T, 2 Ti)U t e

Hence !, 2 tiMean free paths for electrons and ions are - ame.The diffusion coefficients are in the ratio

Di 1 : Ions diffuse slower in parallel direction. (3.148)

Diff usio n Pe rp en di cu la r to Mag. Field is different

Figure 3.7: Cross-field diffusion by collisions causing a jump in the gyrocenter (GC) position.Roughly speaking, if electron direction is changed by -- 90" the Guiding Centre - oves bya distance r ~ .ence we may think of this as a random walk with s tep size r~ andfrequency vei. ence

2Del U t e-2 Zevei2 (3.149)Ion transport is similar but requires a discussion of the effects of l ike and unl ike collisions.Particle transport occurs only via unl ike collisions. To show this we consider in more detailthe change in guiding center position at a collision. Recall mv = qv A B which leads to

4v l = r~ A B (perp. velocity only). (3.150)mThis gives

B A mv lr~ = (3.151)4B2At a collision the particle position does not change (instantaneously) but the guiding centerposition (ro)does.


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Change in r~ is due to the momentum change caused by the collision:

The total momentum conservation means that A (mv) for the two particles colliding is equaland opposite. Hence, from our equation , for like particles, A r o is equal and opposite. Themean position of guiding centers of two colliding like particles (rol+ rO2)/2does not change.No net cross field particle (guiding center) shift.Unlike collisions (between particles of different charge q) do produce net tr anspor t of particlesof either type. And indeed may move rol and ro2 n same direction if they have oppositecharge.

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Dil 2 - ' t i -rr rLiPvie . (3.155)Notice that ri ,/ri , -.mi/me ; Wie/vei .%miSo Dil/De1 -. 1 (for equal temperatures). Collisional diffusion rates of particles are samefor ions and electrons.However energy transport is different because it can occur by like-like collisinsThermal Diffusivity:

1r i i ~ i i mi m2xiIxe - 2 -- .- =(3 (equal T)r~~ Vei me ,?Collisional Thermal transport by Ions is greater than by electrons [factor - milme)' ]3.5.4 Energy EquilibrationIf T, # Ti then there is an exchange of enegy between electrons and ions tending to makeT, =Ti . As we saw earlier

K,, me 1ez - 2me pYei =- vei (3.159)mi miSo applying this to averages


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Thermal energy exchange occurs - m,/mi slower than momentum exchange. (Allows T, #. SoFrom this one can obtain the heat exchange rate (per unit volume), Hei, ay:

Important point:

'Electrons and Ions equilibrate among themselves much faster than with each other'

3.6 Some Orders of Magnitude1. 1nA is very slowly varying. Typically has value - 12 to 16 for laboratory plasmas2. vei-- 6 x 10-"(ni/m3) / (T , / ~V ) ; (1nA = 15,Z = 1) .

e.g. = 2 x 105s-' (when n = 1020m-3 and T, = IkeV.) For phenomena which happenmuch faster than this, i.e. T


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Result for most Fusion schemes it looks as if Ohmic heating does not quite yet get us to therequired ignit ion tem per atu re. We need auxil l iary heating, e .g. Neu tral Beam s. (These slowdown by collisions.)

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Plasma Physics Lecture 3 Ian Hutchinson