Coulomb Collisions - CAE Usershomepages.cae.wisc.edu/~callen/chap2.pdf · CHAPTER 2. COULOMB COLLISIONS 3 and consequently deceleration of the test particle’s initial, directed

CHAPTER 2. COULOMB COLLISIONS 1

Chapter 2

Coulomb Collisions

The characteristics and effects of Coulomb collisions between charged particlesin a plasma are very different from those of the more commonly understoodcollisions of neutral particles. The fundamental differences can be illustrated byexamining trajectories of neutral and charged particles as they move through apartially ionized gas. As shown in Fig. 2.1, neutral particles move along straight-line trajectories between distinct collision events. Collisions occur when neutralatoms or molecules come within about an atomic radius (of order 1 A= 10−10 m— see Section A.7) of another particle (a neutral or a charged particle) and theelectric field force associated with the atomic potential (of order eV) is operative;the resultant “strong,” typically inelastic, collision causes the initial neutral tobe scattered in an approximately random direction.

In contrast, as a charged “test” particle moves through an ionized gas itsimultaneously experiences the weak Coulomb electric field forces surroundingall the nearby charged particles, and its direction of motion is deflected as itpasses by each of them, with the closest encounters producing the largest de-flections — see Fig. 2.2. As was discussed in Section 1.1, the Coulomb potential(and hence electric field) around any particular background charged particle ina plasma is collectively shielded out at distances beyond a Debye length. Thus,the only background particles that exert a significant force on the test parti-cle’s motion are those within about a Debye length of its trajectory. However,since plasmas usually have a very large number of particles within a Debyesphere [ (4π/3) nλ3

D >> 1 ], even in traversing only a Debye length the testparticle’s motion is influenced by a very large number of background particles.The Coulomb electric field forces produced by individual background particlesare small and can be assumed to be experienced randomly by the test parti-cle as it passes close to individual background particles — as indicated in theelectron trajectory shown in Fig. 2.2. The effect of many successive, elasticCoulomb “collisions” of a test particle with background charged particles leadsto a random walk (Brownian motion) process. Thus, the effects of the manycumulative small-angle, elastic Coulomb collisions are diffusion of the test parti-cle’s direction of motion (at constant energy in the center-of-momentum frame)

DRAFT 20:46July 22, 2006 c©J.D Callen, Fundamentals of Plasma Physics


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Figure 2.1: The trajectory of a neutral particle in a partially ionized gas exhibits“straight-line” motion between abrupt atomic collisions. In this and the nextfigure, the (assumed stationary) random positions of “background” particles inthe partially ionized plasma are indicated as follows: neutral particles (circles),electrons (minus signs) and ions (plus signs). The typical distance betweenneutral particle collisions is called the “collision mean free path.”

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Figure 2.2: The trajectory of a “test” charged particle (electron) in a partiallyionized gas exhibits continuous small-angle deflections or scatterings of its di-rection of motion. The largest deflections occur when it passes close to anothercharged particle. The “collision length” of a charged particle in a plasma is de-fined to be the average distance it moves in being deflected through one radian.



and consequently deceleration of the test particle’s initial, directed velocity. Ex-ploration of these Coulomb collision effects is the main subject of this chapter.

Because electrons have less inertia and typically have larger speeds thanions, their collision rates are usually the largest in plasmas. Thus, we firstconsider the momentum loss and velocity-space diffusion of a test electron as itmoves through a plasma. Electron collisions are initially investigated using theLorentz (simplest) collision model in which their collisions are assumed to occuronly with a background of stationary ions. Next, since the collisional effectsdecrease as electron speed increases, we determine the energy (usually on thehigh energy tail of a Maxwellian distribution) at which electrons “run away” inresponse to an electric field; also, the plasma electrical resistivity is determinedby balancing the average collisional deceleration of an entire flowing electronspecies against the electron acceleration induced by an electric field. Then,we discuss the various Coulomb collisional processes (momentum loss, velocityspace diffusion and energy exchange, and their time scales) that occur betweenelectrons and ions in a plasma. The chapter concludes with sections that developa more complete model of Coulomb collision effects, both on test particles andon an entire plasma species, that takes into account collisions with all types ofbackground charged particles that are also in motion. Finally, applications ofthis more complete model to the evolution of the velocity of any type of testparticle and to the thermalization of a fast ion in a plasma are discussed.

2.1 Lorentz Collision Model

To illustrate Coulomb collision effects, we first consider the momentum loss andvelocity diffusion of a test electron moving through a randomly distributed back-ground of plasma ions that have charge Zie and are stationary. (The particles inthe background that are being collided with are sometimes called field particles.)The background plasma electrons, which must be present for quasineutrality,will be neglected except insofar as they provide Debye shielding of the Coulombpotentials around the background ions. However, the “test” electron can bethought of as being just one particular electron in the plasma. This simplest andmost fundamental model of collisional processes in a plasma is called the Lorentzcollision model . It provides a reasonably accurate description of electron-ion col-lisional processes and, in the limit Zi >> 1 where electron-electron collisionaleffects become negligible (see Table 2.1 in Section 2.9), for electron Coulombcollision processes as a whole.

The electron test particle velocity v will be assumed to be large compared tothe change ∆v due to any individual Coulomb interaction with an ion. Hence,the test electron will be only slightly deflected from its straight-line trajectoryduring a single collision. Figure 2.3 shows a convenient geometry for describingthe Coulomb collision process.1 In the rest frame of the electron, the background

1The geometry shown in Fig. 2.3 and the pedagogical approach we use for exploringCoulomb collision processes follows that developed in Chapter 13 of Jackson, Classical Elec-trodynamics, 1st and 2nd Editions (1962, 1975).



electron

ion

bcosφ

bsinφ

b

y

z

x

vt

φ

v

Figure 2.3: Geometry for considering the Coulomb collision of an electron havingcharge qe = −e with an ion of charge qi = Zie. The ion is placed at the originof the coordinate system, which in the electron rest frame is moving in the −ez

direction at the electron speed v. The electron passes the ion at an “impactparameter” distance b at the closest point, which occurs at t = 0.

ion, which we place at the origin of the coordinate system, is seen to be movingwith a velocity −v ez along a straight-line trajectory x(t) = −vt ez. The electronis instantaneously at the position

x = b (ex cos ϕ + ey sinϕ) + vt ez, |x| =(b2 + v2t2

)1/2, (2.1)

in which b is known as the impact parameter . It is the distance of closestapproach, which by assumption will occur at time t = 0. The electrostaticpotential around the ion is the Coulomb potential φ(x) = Zie/(4πε0|x|).Thus, the electric field force experienced by the test electron with charge qe = −eat its position x is

F = qeE = − (−e)∇(

Zie

4πε0|x|

)= − Zie

2x4πε0|x|3

. (2.2)

Next, we calculate the momentum impulse me∆v on the test electron as itpasses the background ion. Integrating Newton’s second law (m dv/dt = F)over time from long before (t → −∞) to long after (t → +∞) the Coulomb“collision” that takes place during the time t where |t| ∼ ∆t ∼ b/v, we see that



a single electron-ion Coulomb collision induces:

me∆v =∫ ∞

−∞dt qeE = −

∫ ∞

−∞dt

Zie2x

4πε0|x|3. (2.3)

Using the specification of x in (2.1), we find

∆v⊥ = − Zie2b

4πε0me(ex cos ϕ + ey sinϕ)

∫ ∞

−∞

dt

(b2 + v2t2)3/2

= − 2Zie2

4πε0meb v(ex cos ϕ + ey sinϕ) . (2.4)

(This expression is relativistically correct if me is replaced by the relativisticmass γ me = me/

√1 − v2/c2.) Note that the perturbation of the electron ve-

locity is in a direction perpendicular to its direction of motion. There is nocomponent along the direction of particle motion (ez direction), at least in thisfirst order where the particle trajectory is the unperturbed one — because the zcomponent of the Coulomb force is an odd function of z or t. Hence, to this firstor lowest order there is no momentum loss by the particle. Rather, a typicalelectron is only deflected by a small angle ∆ϑ ∼ ∆v⊥/v << 1 in velocity space.Using a typical impact parameter b ∼ n

−1/3e , the average inter-particle spacing

in the plasma, and a typical electron speed v ∼ vTe, the typical deflection angleis ∆ϑ ∼ 1/[4π(neλ

3D)2/3] << 1.

Since the background ion is at rest in the Lorentz collision model, electronenergy is conserved during the elastic Coulomb collision process. Thus, we haveme|v|2/2 = me|v + ∆v|2/2 = m(|v|2 + 2v · ∆v + ∆v · ∆v), from which we findthat the component of ∆v parallel to v can be determined from

v · ∆v = − 12∆v · ∆v − 1

2∆v⊥· ∆v⊥, (2.5)

as indicated in Fig. 2.4. That is, because of electron energy conservation, thereduction in electron velocity along its direction of motion is given by half ofthe negative of the square of the perpendicular (⊥) deflection. The net velocitychange along the ez or parallel (‖) direction of electron motion induced by asingle Coulomb collision with a background ion is thus (v · ∆v ≡ v ∆v‖)

∆v‖ − 12v

∆v⊥· ∆v⊥ = − 2Z2i e4

4πε02m2eb

2v3. (2.6)

Note that while ∆v⊥ is a first order quantity in terms of the weak Coulombelectric field between the two particles given in (2.2), ∆v‖ is a second orderquantity, as evidenced by the square of the 4πε0 factor in the denominator.The result in (2.6) can also be obtained directly by integrating the Coulombelectric field force along a perturbed (by the Coulomb collision) trajectory ratherthan the straight-line electron trajectory that was assumed in the precedinganalysis — see Problem 2.5.



constantenergysurface

v+∆v

v

∆v ∆v⊥

∆v‖

Figure 2.4: Change in electron velocity vector from v before the Coulomb colli-sion to v + ∆v afterward. The change takes place at constant electron energy,which means constant radius in this diagram, and hence results in ∆v‖ < 0.

Next, we take account of the entire background distribution of ions, assum-ing that electron collisions with individual ions are statistically random andthus that their effects can be summed independently. For a density ni of ions,adopting a cylindrical geometry in which the radius is b and the azimuthal an-gle is ϕ, the number of ions passed by the electron per unit time is ni

∫d3x/dt

= ni (dz/dt)∫

dA = ni v∫

dϕ∫

b db (cf., Fig. 2.2). Hence, the net or ensembleaverage2 Coulomb collisional force in the direction of electron motion is

〈F‖〉 ≡ me

〈∆v‖〉∆t

= niv

∫ 2π

0

dϕ

∫ ∞

0

b db me∆v‖ = − 4π niZ2i e4

4πε02mev2

∫db

b. (2.7)

Here, ∆t is a typical interaction time for individual Coloumb collisions (∆t ∼b/v ∼ 1/[ωpe(neλ

3D)1/3]), which is short compared to the time for the test

electron to traverse a Debye sphere (∼ λDe/v ∼ λDe/vTe ∼ 1/ωpe). It is alsocertainly short compared to the time scale on which the test particle velocityv changes significantly due to Coulomb collisions [∆t << 1/ν, where ν is thecollision frequency defined in (2.14) below].

The integral over the impact parameter b in (2.7) is divergent at both itsupper and lower limits:

∫ ∞0

d b/b =⇒ ln(∞/0) ?! We restrict its range of in-

2In an ensemble average one averages over an infinite number of similar plasmas (“realiza-tions”) that have the same number of particles and macroscopic parameters (e.g., density n,temperature T ) but whose particle positions vary randomly from one realization to the next.



tegration through physical considerations that can be more rigorously justifiedby detailed analyses. The maximum impact parameter will be taken to bethe Debye length since the Coulomb electric field force decays exponentially inspace from the value given in (2.2) for distances larger than the Debye length(cf., Fig. ??):

bmax = λD. (2.8)

To estimate the minimum impact parameter bmin, we note that when theCoulomb potential energy qeqi/(4πε0|x|) becomes as large as the electronkinetic energy mev

2/2: ∆v‖ becomes comparable to |∆v⊥|, the scattering anglebecomes 90o [see (??) in Appendix A.1], and our weak interaction approximationbreaks down. Hence, we determine a classical minimum impact parameter by|∆v‖| = |∆v⊥|, which yields

bclmin =

Zie2

4πε0(mev2) Zie

2

4πε0(3Te)=

Zi

12πneλ2De

4.8×10−10 Zi

Te(eV)m.

(2.9)

Here, we have approximated mev2/2 by 3Te/2, which is appropriate for a ther-

mal electron in a Maxwellian distribution [cf., (??) in Appendix A.4].Quantum mechanical effects become important when they could induce scat-

tering through an angle ϑ of 90o, which occurs [for wave scattering processes— see (??) in Appendix A.7]) when the distance of closest approach b is lessthan half the radian de Broglie wavelength λh/2π ≡ h/mv = h/(2πmv). Thisphysical process yields a quantum-mechanical minimum impact parameter3 (forv vTe ≡

√2Te/me )

bqmmin ≡ h

2mev h

4πmevTe 1.1×10−10 1

T1/2e (eV)

m. (2.10)

The relevant minimum impact parameter bmin is the maximum of classicaland quantum-mechanical minimum impact parameters. Quantum-mechanicaleffects dominate for Te

>∼ 20 Z2i eV. With these specifications of the limits of

integration, the impact parameter integral in (2.7) can be written as

ln Λ ≡∫ bmax

bmin

db

b= ln

(λD

bmin

), bmin = max

bclmin, bqm

min

,

Coulomb logarithm. (2.11)

It is called the Coulomb logarithm because it represents the sum or cumulativeeffects of all Coulomb collisions within a Debye sphere for impact parametersranging from bmin to λD.

3In Chapter 13 of Jackson’s Classical Electrodynamics the factor of 2 is omitted in thedefinition of the quantum-mechanical minimum impact parameter, but then the argumentof the Coulomb logarithm in (2.11) is multiplied by a factor of 2 when quantum-mechanicaleffects dominate.



To determine the relative magnitude and scaling of Coulomb collision effects,it is convenient to assume classical effects determine the minimum impact pa-rameter. When classical effects dominate (bmin = bcl

min), the Coulomb logarithmbecomes

ln Λcl ≡ ln(

λD

bclmin

) ln

(12π neλ

3De

Zi

). (2.12)

Since the definition of a plasma (cf., Section 1.8) requires that neλ3De >>>> 1,

plasmas have ln Λcl >> 1. For example, typical magnetic fusion experiments inlaboratory plasmas have nλ3

D ∼ 106, and hence ln Λ ∼ 17.Having defined the impact parameter integral in (2.7), the total Coulomb

collisional force on a test electron along its direction of motion thus becomes

me

dv‖dt

= 〈F‖〉 = me

〈∆v‖〉∆t

= −[

4πniZ2i e4

4πε02m2ev

3ln Λ

]mev‖ = − ν mev‖. (2.13)

The Coulomb collisional drag force in the last form of this equation is called thedynamical friction force — because it is proportional to the test particle velocity.Here, we have defined a net momentum loss or slowing down4 Coulomb collisionfrequency for a particle of speed v in the Lorentz collision model:

ν(v) ≡ 4π neZie4 ln Λ

4πε02m2ev

3 ωpe

ln (12π neλ3De/Zi)

4π neλ3De/Zi

(Te

mev2

)3/2

,

Lorentz collision frequency. (2.14)

In this definition we have taken into account the condition for quasineutrality ina plasma: ne = Zini. Note from the last form in (2.14) that the electron collisionfrequency is smaller than the electron plasma frequency by a very large factor[∝ 1/(neλ

3De), which is by definition a small number in a plasma]. The Lorentz

collision frequency can also be shown to be given by ν(v) = niσmv in whichσm = 4π (bcl

min)2 ln Λ is a momentum transfer cross-section — see Problems 2.6,2.7. It can also be deduced from the Langevin equation in which the stochasticforce is due to Coulomb collisions — see Problem 2.8.

For classical “hard” collisions with b < bclmin, the maximum parallel mo-

mentum transfer is given by max(∆v‖) = 2v. The collision frequency for hardcollisions can be estimated using a cross section of σhard π(bcl

min)2: νhard =niσhard max(∆v‖) 2πniv(bcl

min)2, which is smaller than the collision frequencyin (2.14) by a factor of 1/(2 ln Λ) << 1. Thus, the net Coulomb collision fric-tional force is dominated by the cumulative small angle collisions with impactparameters b ranging between bmin and λD that are embodied in the ln Λ integralin (2.11). That is, the Coulomb logarithm represents the degree to which cumu-lative small-angle collisions dominate over hard collisions for Coulomb collisionprocesses in plasmas.

4Note that in the Lorentz collision model there is no energy transfer and only loss ofdirected momentum — see Problem 2.4. It is thus unfortunate and rather misleading thatthe Lorentz collision frequency is often called a ”slowing down” frequency in plasma physics.



Detailed treatments of the physical phenomena of hard collisions for b ≤ bmin

(see Problems 2.7, 2.24) and of the Debye shielding process (see Chapter 13) forb >∼ bmax = λD yield order unity corrections to the ln Λ ≡ ln (bmax/bmin) factorin (2.14). However, because these corrections are small and quite complicated,it is customary to neglect them in most plasma physics calculations. Thus,the Coulomb collision momentum loss frequency given in (2.14) and the otherCoulomb collision processes calculated in this chapter should be assumed to beaccurate to within factors of order 1/(ln Λ) ∼ 5 − 10%; evaluation of Coulombcollision processes and their effects to greater accuracy is unwarranted.

Finally, we use our result for the Coulomb collision frictional force 〈F‖〉 on asingle electron to calculate the net frictional force on a “drifting” Maxwellian dis-tribution of electrons flowing slowly (compared to their thermal speed) througha background of fixed, immobile ions. For a small net flow speed V‖ in the ez

direction, the appropriate flow-shifted Maxwellian distribution for electrons is5

fMe(v) = ne

(me

2πTe

)3/2

exp(− me|v − V‖ez|2

2Te

) ne

(me

2πTe

)3/2

e−mev2/2Te

[1 +

mev‖V‖Te

+ · · ·]

=ne e−v2/v2

T e

π3/2v3Te

[1 +

2 v‖V‖v2

Te

+ · · ·]

, (2.15)

in which in the last form we have used the convenient definition of the electronthermal speed vTe ≡

√2Te/me. Multiplying (2.13) by this distribution and

integrating over the relevant spherical velocity space (v‖ ≡ vζ = v cos ϑ), theMaxwellian-average (indicated by a bar over F‖) of the Coulomb collisionalfrictional force density on the drifting electron fluid becomes

ne〈F‖〉 ≡∫

d3v fMe(v) 〈F‖〉

≡ −∫ 2π

0

dϕ

∫ 1

−1

dζ

∫ ∞

0

v2dv ν(v) mevζ2vζV‖v2

Te

nee−v2/v2

T e

π3/2v3Te

= − νemeneV‖. (2.16)

Here, we have defined the Maxwellian-averaged electron-ion collision frequency

νe ≡ 43√

πν(vTe) =

4√

2π niZ2i e4 ln Λ

4πε02 3 m1/2e T

3/2e

5×10−11neZi

[Te(eV)]3/2

(ln Λ17

)s−1,

fundamental electron collision frequency. (2.17)

This is the average momentum relaxation rate for the slowly flowing Maxwelliandistribution of electrons. Since many transport processes arise from collisional

5Here, and throughout this text, a capital letter V (V) will indicate the average flow speed(velocity) of an entire species of particles while a small letter v (v) will indicate the speed(velocity) of a particular particle, or a particular position in velocity space.



relaxations of flows in a plasma, this average or reference electron collisionfrequency is often the fundamental collision frequency that arises — in theplasma electrical conductivity (see Section 2.3 below) and plasma transportstudies (cf., Problem 2.10).

Since a typical, thermal electron moves at the thermal speed vTe, it is con-venient to define the characteristic length scale over which the momentum in aflowing distribution of electrons is damped away by

λe ≡ vTe

νe 1.2×1016 [Te(eV)]2

ne Zi

(17

ln Λ

), electron collision length. (2.18)

Note that (in contrast to neutral particle collisions) it is not appropriate tocall this length a collision “mean free path” — because a very large numberof random small-angle Coulomb collisions deflect particles’ velocities and causethe net momentum loss over this length scale. [The total number of collisionsinvolved is of order nλ3

D as a test electron traverses a Debye length times afactor of λe/λD nλ3

D/ ln Λ, or of order (nλ3D)2/ ln Λ >>>>>> 1.] For the

relevant length and time scales in some typical plasmas, see Problems 2.1–2.3

2.2 Diffusive Properties of Coulomb Collisions

The Coulomb collision process causes more than just momentum loss by the elec-trons. As indicated in Fig. 2.4, the dominant collisional process in individualcollisions is deflection of the test particle velocity in a random direction per-pendicular to the original direction of motion. The net perpendicular Coulombcollision force defined analogously to the net parallel force in (2.7) vanishes:

〈F⊥〉 ≡ m〈∆v⊥〉

∆t= niv

∫ 2π

0

dϕ

∫ bmax

bmin

b db me∆v⊥ = 0.

While the ensemble average perpendicular force vanishes, because of the ran-domness of the impact angle ϕ, velocity-space deflections caused by Coulombcollisions do have an effect in the perpendicular direction. Namely, they lead todiffusion of the test particle velocity v in directions perpendicular to v. For ageneral discussion of diffusive processes see Appendix A.5.

The temporal evolution of the velocity of a test particle as it undergoes ran-dom Coulomb collisions with background ions is illustrated in Fig. 2.5. While forlong times (many Coulomb collisions) the average of the perpendicular velocitycomponent vx vanishes (〈vx〉 = 0), its square and the reduction of the velocitycomponent in the original direction of motion increase approximately linearlywith time — 〈v2

x〉 (〈∆v2x〉/∆t) t [see (??) in Appendix A.5] and v0 − vz ∼

(〈∆v‖〉/∆t) t. The fact that the average of v2x increases linearly with time while

the average of vx vanishes indicates a diffusive process for the x (perpendicular)component of the test particle velocity — see Appendix A.5. Because thereis no preferred direction in the plane perpendicular to the original direction of



Figure 2.5: Temporal evolution of vx, v2x and vz components of the test particle

velocity as it undergoes random Coulomb collisions with background ions. Notethat for times long compared to an individual Coulomb collision time the averageof vx vanishes, but v2

x and v0 − vz increase approximately linearly with time t.



motion, we obtain 〈v2x〉 = 〈v2

y〉 = 〈v2⊥〉/2 = (1/2)(〈∆v2

⊥〉/∆t) t; hence there isvelocity diffusion equally in both the x and y directions.

To mathematically describe the diffusion in velocity space, we calculate themean square deflection of the test electron as it moves through the backgroundions by the same ensemble-averaging procedure as that used in obtaining theaverage parallel force in (2.7). We obtain

〈∆v2⊥〉

∆t≡ niv

∫ 2π

0

dϕ

∫ bmax

bmin

b db ∆v⊥· ∆v⊥ =8π niZ

2i e4

4πε02m2vln Λ = 2 ν v2.

(2.19)

Thus, as can be inferred from (2.5), and from Figs. 2.4 and 2.5, in the Lorentzscattering model the rate of velocity diffusion (〈∆v2

⊥〉/v2∆t) for the test electronis twice the rate of momentum loss (〈∆v‖〉/v∆t). Note that for the collisionalprocess being considered the velocity diffusion takes place at constant energyand in directions perpendicular to the test particle velocity v; there is no speed(energy) diffusion in the Lorentz collision model because the background parti-cles (ions here) are assumed to be immobile and hence to not exchange energywith the test electron.

In the spherical velocity space we are using, the “pitch-angle” through whichthe random scattering, deflections and diffusion take place is defined by sinϑ ≡v⊥/v =

√v2

x + v2y /v. Since the Coulomb collision process is a random walk or

diffusion process (in pitch-angle), the time required to diffuse the test particlevelocity vector through a small angle ϑ v⊥/v << 1 is much less than theLorentz collision model (momentum loss) time 1/ν, which is effectively the timescale for scattering through 90 — see Problem 2.12 for a specific example.From (2.19) we can infer that collisional scattering through an angle ϑ << 1(but ϑ must be greater than the ∆ϑ for any individual Coulomb interaction soa diffusive description applies) occurs in a time [see Fig. 2.5 and (??)]

t ∼ (v⊥/v)2/ν ∼ ϑ2/ν << 1/ν, time to diffuse through ϑ << 1. (2.20)

As time progresses, a test particle’s “pitch-angle” ϑ in velocity space is randomlydeflected or scattered. Thus, over time the pitch-angle of a test particle assumesa probability distribution whose width is given by

√〈ϑ2〉 ∼

√νt.

For the Lorentz collision model the probability distribution of a test particlewith an initial velocity v0 [i.e., f(v, t = 0) ≡ δ(v − v0)] can be shown (seeSection 11.2) to be given for short times by

ft(v, ϑ, t) δ(v − v0)2πv2

0

(e−ϑ2/2νt

νt

)=

δ(v − v0)2πv2

0

(e− v2

⊥/(2v2νt)

νt

)for νt << 1.

(2.21)

This distribution function is normalized so it represents one test particle: i.e.,∫d3vft = 1. The delta function in speed, δ(v − v0), represents the fact that



the test particle speed stays constant at the initial speed |v0| ≡ v0 — becausethe test particle energy (speed) is constant in the Lorentz collision model. Thefactor e−ϑ2/2νt/(νt) represents the diffusion in pitch-angle ϑ that takes place ina time t; it indicates that ft is reduced by a factor of e−1/2 0.61 for diffusionover a pitch-angle of ϑ << 1 in the short time t ∼ ϑ2/ν indicated in (2.20). Thevelocity-space diffusion properties of Coulomb collision processes are exploredin greater detail in Chapter 11.

The dynamical friction and diffusion coefficients for the Lorentz collisionmodel can be written in a coordinate-independent, vectorial form as follows.First, note that the parallel or z direction here is defined to be in the initialelectron velocity direction: ez ≡ v/v. Thus, we can write the dynamical frictionforce coefficient due to Coulomb collisions in the form

〈∆v〉∆t

=〈∆v‖〉

∆tez = − ν(v)v. (2.22)

Similarly, because velocity diffusion occurs equally in all directions perpendicu-lar to v, we have 〈∆v2

x〉/∆t = 〈∆v2y〉/∆t = (1/2)〈∆v2

⊥〉/∆t; hence the (secondrank tensor) diffusion coefficient can be written as

〈∆v ∆v〉∆t

=12〈∆v2

⊥〉∆t

(exex + eyey) = ν(v) (v2I − vv), (2.23)

in which I is the identity tensor [see (??) in Appendix D.7]. These forms for〈∆v〉/∆t and 〈∆v ∆v〉/∆t will be useful in Section 11.1 where we will developa Lorentz Coulomb collision operator for use in plasma kinetic theory.

2.3 Runaway Electrons and Plasma Resistivity

Next, we consider the combined effects of a macroscopic electric field E and thedynamical friction due to Coulomb collisions on test electrons in a plasma. Usingthe dynamical friction force given in (2.13) using the vectorial form indicatedin (2.22), Newton’s second law for this situation can be written in the form

medvdt

= qeE − ν mev. (2.24)

The electric field may be externally imposed, or arise from a collective responsein the plasma. The electric field E, which we take to be in the −ez direc-tion accelerates electrons (qe = −e) in the −E or +ez direction; Coulombcollisions exert a dynamical friction force that opposes this acceleration. Ina more complete Coulomb collision model that includes electron-electron col-lisions (see Section 2.7 below), the Lorentz collision frequency ν gets replacedby a “slowing down” (subscript S) electron (momentum relaxation) collisionfrequency νe

S = νe/eS + ν

e/iS , in which ν

e/eS and ν

e/iS are the momentum loss rates

for electron-electron and electron-ion collisions, which will be derived explictly



below, in Section 2.9. For electron-ion collisions, since electron speeds are typ-ically much greater than the ion thermal speed and little energy is transferredduring the collisions because of the large disparity in masses, the ions are es-sentially immobile during the Coulomb collision process. Thus, the Lorentzcollision model is applicable and the relevant electron-ion collision frequency issimply the Lorentz collision frequency: ν

e/iS = ν(v), as given in (2.14). Electron-

electron collisions are in general more complicated — because during collisionsboth particles are in motion and energy is transferred. With these simplifica-tions and adaptations, the equation governing the velocity of a single electronin the ez direction, (2.24) can be rewritten in the more precise one-dimensionalform

me

dv‖dt

= (−e)(−E) − (νe/eS + ν

e/iS ) mev‖ = eE − νe

S mev‖. (2.25)

We first consider the combined electric field and Coulomb collision effectson energetic test electrons in the high energy tail of a Maxwellian distribution.For these energetic test electrons the background electrons can be consideredat rest and the electron-electron momentum loss collision frequency is simplyν

e/eS = 2 ν(v)/Zi (see Table 2.1) — the factor of two comes from the inverse

dependence on the reduced electron rest mass [see (2.55) below] and the 1/Zi

factor eliminates the dependence on the ion Zi in the Lorentz model collisionfrequency. The total momentum loss collision frequency for these energeticelectrons can thus be written as

νeS = (1 + 2/Zi) ν(v) = (1 + 2/Zi) ν(vTe) v3

Te/v3, for v >> vTe. (2.26)

Here, the unity multiplicative factor (on ν) represents electron-ion collisions andthe 2/Zi factor represents electron-electron collisions. In the limit Zi >> 2 thisoverall electron momentum relaxation rate becomes simply the Lorentz modelcollision frequency and electron-electron collision effects are negligible.

The dynamical friction force νeS(v)mev‖ in (2.25) with the νe

S given in (2.26)decreases as v−2 for electrons in the high energy tail of a Maxwellian distribu-tion. The dependence of the electric field and dynamical friction forces on thespeed v of a tail electron are illustrated in Fig. 2.6. As indicated, when theelectric field force exceeds the dynamical friction force, electrons are freely ac-celerated by the electric field. Such electrons are called runaway electrons. Theenergy range for which runaway electrons occur is determined by eE > νe

Smev:

mev2

2 Te> (2 + Zi)

ED

|E| , (2.27)

where

ED ≡ 2πnee3 ln Λ

4πε02Te=

e ( 12 ln Λ)

4πε0λ2De

=mevTeν(vTe)

eZi, Dreicer field (2.28)



Force

v

e|E|

runawayelectrons

Figure 2.6: Relative strengths of the electric field e|E| and dynamical frictionνe

S(v)mev forces on an electron as a function of the electron speed v. Runawayelectrons occur when the electric field force exceeds the dynamical friction force.

is a critical electric field strength, called the Dreicer field.6 For weak electricfields (|E| << ED), the energy at which electron runaways occur is far outon the high energy tail of the Maxwellian electron distribution and only anexponentially small fraction of electrons run away — see Problem 2.13. [Forrelativistic electron energies the dynamical friction decreases less rapidly than1/v2 and no runaways are produced for a weak electric field satisfying |E|/ED <2 Te/(mec

2) — see Problem 2.14.] High Zi ions increase the energy for electronrunaway relative to that for protons — because they increase the frictional dragdue to Coulomb collisions. Note also from the middle form of the critical electricfield defined in (2.28) that its magnitude is roughly (to within a factor of 1

2 ln Λ ∼10) what is required to substantially distort the Coulomb electric field around agiven ion [cf., (2.2)] at distances of order the Debye length. Alternatively, it canbe seen from the last form in (2.28) that the Dreicer field is approximately theelectric field strength at which typical, thermal energy electrons with v ∼ vTe

in a Maxwellian distribution become runaways — see Problem 2.15 for a moreprecise estimate. Thus, when the electric field is larger than the Dreicer field,the entire distribution of electrons responds primarily to the electric field andcollisional effects are small.

For weak electric fields |E| << ED, most plasma electrons will be onlyslightly accelerated by the E field before Coulomb collisions relax the momentumthey gain. However, the velocity distribution of electrons will acquire a net flowvelocity Ve in response to the E field. Since the more massive ions have muchmore inertia and are accelerated less by the electric field, they acquire a much

6H. Dreicer, Proceedings of the Second United Nations International Conference on thePeaceful Use of Atomic Energy (United Nations, Geneva, 1958), Vol. 31, p. 57. See also,Phys. Rev. 115, 238 (1959).



smaller [by a factor ∼ (me/mi)1/2 <∼ 1/43 << 1] flow, which can be neglected.Thus, the electron flow in response to the electric field will correspond to anelectric current flowing in the plasma. The proportionality constant betweenthe current and electric field is the plasma electrical conductivity, which we willnow determine.

For electrons with a flow-shifted Maxwellian distribution as in (2.15) thathave a flow velocity Ve relative to the ions (V‖ez → Ve −Vi), the average (overthe Maxwellian distribution) frictional force is given in (2.16). Adding electricfield force and electron inertia effects yields the electron momentum densityequation

menedVe

dt= − eneE − meneνe(Ve − Vi), (2.29)

in which νe is the fundamental electron collision frequency defined in (2.17). Inequilibrium (t >> 1/νe, d/dt → 0) we obtain the current induced by an electricfield:

J = −nee(Ve − Vi) = σ0E, Ohm’s law (2.30)

in which

σ0 =nee

2

meνe≡ 1

η, reference (subscript 0) plasma electrical conductivity,

(2.31)

where η is the plasma resistivity. The electron collision frequency that entersthis formula is νe, which is the (electron-ion) Lorentz collision frequency (2.14)averaged over a flowing Maxwellian distribution of electrons given in (2.17). (Inthis analysis the electron Coulomb collision frequency is assumed to be muchgreater than the electron-neutral collision frequency. See Problems 2.19, 2.20for situations where this assumption is not valid and the electrical conductivityis modified.) Note also that since ne/νe ∝ T

3/2e , the electrical conductivity in a

plasma increases as T3/2e — an inverse dependence compared to solid conductors

whose electrical conductivity decreases with temperature. The conductivity inplasmas increases with electron temperature because the noise level [see (??)]and collision frequency [see (2.17)] decrease with increasing electron temperatureand Debye length. For some perspectives on the magnitude and effects of theelectrical conductivity in plasmas, see Problems 2.16–2.18.

In a more complete, kinetic analysis with the Lorentz collision model (seeSection 11.4), the electric field distorts the electron distribution function morethan indicated by the simple flow effect in (2.15). Specifically, we can inferfrom (2.25) and (2.26) that higher energy electrons receive larger momentuminput from the electric field because the Coulomb collision dynamical frictionforce decreases as v−2. Thus, the current is carried mainly by higher energy(v ∼ 2 vTe), lower collisionality electrons than is embodied in the simple flow-shifted Maxwellian distribution. Since the collision frequency decreases as 1/v3,



the Maxwellian-averaged collision frequency is reduced (see Section 11.4), bya factor of 3π/32 0.2945 ≡ αe; thus the electrical conductivity in a kineticLorentz model is increased relative to that given in (2.31) by the factor 1/αe.

Electron-electron collisions are momentum conserving for the electron dis-tribution function as a whole. Thus, they do not contribute directly to themomentum loss process or plasma electrical conductivity. However, in a kineticdescription the electric field distorts the electron distribution function awayfrom a flow-shifted Maxwellian. Then, electron-electron collisions have an indi-rect effect of reducing the net flow (and electrical conductivity) in response toan electric field — as they try to force the electron distribution to be close to aMaxwellian. Details of this process will be discussed in Section 12.3.

The net result of these kinetic and electron-electron effects, which is obtainedfrom a complete, kinetic analysis that was first solved numerically by Spitzerand Harm,7 is that the effective electron collision frequency is reduced by ageneralized factor αe. Thus, the electrical conductivity becomes

σSp =nee

2

meαeνe=

σ0

αe, Spitzer electrical conductivity. (2.32)

The generalized factor αe ranges from 0.5129 for Zi = 1 to 3π/32 0.2945for Zi → ∞ (Lorentz kinetic model). A later analytic fluid moment analysis8

has shown that this factor can be approximated to three significant figures (seeSection 12.3), which is much more accuracy than warranted by the intrinsicaccuracy (∼ 1/ ln Λ <∼ 10%) of the Coulomb collision operator, by

αe 1 + 1.198Zi + 0.222Z2i

1 + 2.966Zi + 0.753Z2i

. (2.33)

2.4 Effects of Coulomb Collisions

So far we have concentrated on the electron momentum relaxation effects ofCoulomb collisions using a Lorentz collision model. In this section we discussphenomenologically more general Coulomb collision effects on electrons as wellas the collisonal effects on ions, and between ions and electrons. A complete,rigorous treatment of Coulomb collision effects begins in Section 2.6.

The Lorentz collision model takes into account electron-ion collisions butneglects electron-electron collisions. However, these two collisional processesoccur on approximately the same time scale, at least for ions with a Zi that isnot too large. As indicated in the preceding section, electron-electron collisionstend to relax the electron velocity distribution toward a Maxwellian distributionfunction. They do so on approximately the fundamental electron collision timescale 1/νe. However, as indicated in (2.25) and (2.26), the collisional relaxationof electrons in the high energy tail of the distribution is slower. The character-istic time τ for tail electrons to equilibrate toward a Maxwellian distribution is

7L. Spitzer and R. Harm, Phys. Rev. 89, 977 (1953).8S.P. Hirshman, Phys. Fluids 20, 589 (1977).



τ ∼ (v/vTe)3/νe for v >> vTe. (For an application where this effect is impor-tant see Problem 2.21.) In contrast, all electrons with v <∼ vTe relax toward aMaxwellian distribution on the same time scale as the bulk (see Section 2.7 andProblem 2.32): τ ∼ 1/νe.

As indicated in (2.16) and (2.29), the net Coulomb collisional force densityon a Maxwellian distribution of electrons flowing relative to the ions is

Re ≡ −meneνe(Ve − Vi) =neeJ

σ, collisional friction force density.

(2.34)

This is the electron force density that was introduced in the electron fluid mo-mentum balance given in (2.29). Note also from a temporal solution of (2.29)that the electron flow (momentum) will relax exponentially to its equilibriumvalue at the rate νe, i.e., on the electron time scale τe = 1/νe. Because Coulombcollisions are momentum conserving, any momentum lost from the electronsmust be gained by the ions. Thus, the Coulomb collisional force density on ionsis given by

Ri = −Re. (2.35)

Ion-ion collisions are analogous to electron-electron collisions and compli-cated — during Coulomb collisions both particles are in motion and energy isexchanged between them. Nonetheless, considering a Lorentz-type model forion-ion collisions using the framework developed in Section 2.1, it is easy tosee that the appropriate ion collision frequency should scale inversely with thesquare of the ion mass and the cube of the ion speed. A detailed analysis (seeSections 2.6–2.10) of the effects of ion-ion collisions yields a flowing-Maxwellian-averaged ion collision frequency given by

νi =4√

π niZ4i e4 ln Λ

4πε02 3 m1/2i T

3/2i

=(

me

mi

)1/2 (Te

Ti

)3/2Z2

i√2

νe,

fundamental ion collision frequency. (2.36)

The√

2 factor (in the denominator at the end of the second formula) en-ters because of the combined effects of the reduced mass [see (2.55) below]and the motion of both particles during ion-ion collisions. Note that for anelectron-proton (Zi = 1) plasma with Te ∼ Ti the ion collision frequency issmaller than the electron collision frequency by a square root of the mass ratio:νi/νe ∼ (me/2mi)1/2 <∼ 1/60 << 1. Because of their very disparate masses,ion-electron collisional effects are typically smaller than ion-ion collisional ef-fects by a factor of (me/mi)1/2 <∼ 1/43 << 1; hence they are negligible for ioncollisional effects. As for electrons, ion collisions drive the velocity distribu-tion of ions toward a Maxwellian distribution on the ion collisional time scaleτi = 1/νi ∼ (2mi/me)1/2/νe >> 1/νe. In addition, like electrons, ions in thehigh energy tail of the distribution relax toward a Maxwellian distribution moreslowly: τ ∼ (v/vTi)3/νi for v >> vTi.



We now determine the small energy transfer from electrons to ions duringCoulomb collisions, which we have heretofore neglected. Momentum is con-served during a Coulomb collision. Thus, if an electron acquires an impulseme∆ve during a electron-ion collision, the ion acquires an impulse determinedfrom momentum conservation:

me∆ve + mi∆vi = 0 =⇒ ∆vi = − (me/mi) ∆ve.

The energy exchange from electrons to ions initially at rest during a Coulombcollision will thus be

mi

2∆vi · ∆vi =

mi

2

(me

mi

)2

∆ve · ∆ve (

me

mi

)me

2∆v2

⊥.

The net energy exchange from a test electron moving through the backgoundstationary ions can thus be evaluated using (2.19):

mi

2〈∆vi · ∆vi〉

∆t=

(me

mi

)mev

2ν(v). (2.37)

Note that this energy exchange rate is smaller than the basic Lorentz collisionfrequency ν by a factor of me/mi

<∼ 10−3 << 1 — because lightweight electronstransfer very little energy to the massive ions in Coulomb collisions.

Integrating this last result over a Maxwellian distribution of the electrons,the Maxwellian-averaged rate of energy (ε) density transfer from electrons toinitially stationary background ions (≡ ν

e/iε in Section 2.10) becomes∫

d3v fMemi

2〈∆vi · ∆vi〉

∆t≡ ν

e/iε neTe = 3

me

miνe neTe.

A more complete analysis (see Section 2.10) shows that if the background ionshave a Maxwellian velocity distribution (instead of being stationary and immo-bile as they are in the Lorentz model) Te → Te −Ti in this formula, as would beexpected physically. Thus, the rate of ion energy density increase from Coulombcollisions with electrons is

Qi ≡ νe/iε ne(Te − Ti) = 3

me

miνe ne(Te − Ti), ion collisional heating density.

(2.38)

In the absence of other effects, the equation governing ion temperature evolutionbecomes

32

nidTi

dt= Qi = 3

me

miνe ne(Te − Ti). (2.39)

Here, (3/2)(nidTi/dt) represents the rate of increase of ion internal energy in theplasma. From (2.39) we see that for a constant electron temperature the charac-teristic time scale on which Coulomb collisions equilibrate the ion temperature



to the electron temperature is τi−e = 3/(2 νe/iε ) = (mi/2me)/νe

>∼ 103/νe >>1/νe. (For a more precise determination of the temporal evolution of the colli-sional equilibration of the electron and ion temperatures in a plasma, see Prob-lem 2.23.)

Because energy is conserved in the elastic Coulomb collisions, energy gainedby the ions is lost from the electrons. In addition, the electrons are heated by thework they do per unit time in flowing relative to the ions against the collisionalfriction force density Re given in (2.34). Thus, the total electron heating dueto Coulomb collisions is given by

Qe = − (Ve − Vi) · Re − Qi = J2/σ − Qi. (2.40)

In the absence of other effects and using 1/σ = η, the electron temperatureevolution equation becomes

32

nedTe

dt= Qe = ηJ2 − Qi. (2.41)

In these equations ηJ2 is the joule or ohmic heating induced by a current densityJ flowing in a plasma with resistivity η. Because the plasma resistivity scalesas T

−3/2e , for a constant current density the joule heating rate of a plasma

decreases as it is heated. Thus, joule heating becomes less effective as theelectron temperature increases. Note also that despite the complexity of thedependence of νe on Te, the characteristic time scale for Coulomb collisionalrelaxation of the electron temperature to its equilibrium value is roughly thesame as the ion temperature equilibration time scale τi−e.

2.5 Numerical Example of Collisional Effects*

In order to illustrate the evaluation of and numerical values for these variouscollisional processes, we will work them out for a particular plasma example.The plasma example will be chosen to be typical of laboratory experimentsfor magnetic fusion studies, but the plasma will be assumed to be infinite anduniform, and in equilibrium — so there will not be any spatial or temporalinhomogeneity effects. For the plasma electrons we assume an electron densityne = 2×1019 m−3 and electron temperature Te = 1 keV. For these parametersthe electron plasma period [inverse of electron plasma frequency from (??)] is1/ωpe = 1/[56(2×1019)1/2] 4×10−12 s−1, the electron Debye length from (??)is λDe = 7434 [103/(2×1019)]1/2 5.3×10−5 m, and the number of electrons inan electron Debye cube is neλ

3De 3×106. These parameters clearly satisfy the

criterion neλ3De >> 1 for the plasma state.

The ions in laboratory plasmas often include impurities in addition to thedesired hydrogenic species. We will take into account an impurity species toshow how the various plasma collision rates presented in the preceding sectionsneed to be modified to take into account multiple species of ions, and in partic-ular impurities. For our example laboratory plasma we will assume a dominant



deuterium (atomic weight AD = 2, charge ZD = 1) ion species with relativedensity nD/ne = 0.64 and fully ionized carbon (AC = 12, ZC = 6) impuritieswith a relative density of nC/ne = 0.06. Note that even though the carbon iondensity is only 6% of the electron density the carbon ions supply 36% of the ioncharge needed for charge neutrality:

∑i niZi = [0.64 + (0.06)(6)]ne = ne. Both

the deuterium and carbon ion temperatures will be assumed to be 0.5 keV.In order to calculate the ln Λ factor for the fundamental electron collision rate

we first need to determine the maximum and minimum collisional impact pa-rameters bmax and bmin. The maximum impact parameter is the overall plasmaDebye length in the plasma which is defined in (??). For our multi-speciesplasma the Debye length can be calculated from the electron Debye length bytaking out common factors in the ratio of λD to λDe:

λD ≡ λDe

[∑s

ns

ne

Te

TsZ2

s

]−1/2

, (2.42)

which for our plasma yields λD = λDe/[1 + (0.64)(2) + (0.06)(2)(62)]1/2 λDe/2.6 2×10−5 m. Classical and quantum mechanical minimum impactparameters for electron-deuteron collisions in this plasma are estimated from(2.9) and (2.10): bcl

min = 4.8×10−10/103 = 1.4×10−12 m and bqmmin = 1.1×

10−10/(103)1/2 = 3.5×10−12 m. Since the quantum mechanical impact pa-rameter is larger, we use it for bmin and thus have ln Λ ≡ ln (λD/bqm

min) ln [(2×10−5)/(3.5×10−12)] ln (5.7×106) 16. Since the Coulomb collision fre-quency is only accurate to order 1/ ln Λ 1/16 0.06, in the following we willgive numerical values to only about 6% accuracy; more accuracy is unwarrantedand misleading.

In calculating the electron collision frequency we need to take account of allthe ion species. From (2.17) we see that the electron-ion collision frequency isproportional to niZ

2i . Thus, for impure plasmas it is convenient to define

Zeff ≡∑

i niZ2i∑

i niZi=

∑i niZ

2i

ne, effective ion charge, (2.43)

in which the sum is over all ion species in the plasma. Hereafter in this sectionwe will designate the main ions with a subscript i and the impurities with asubscript Z. For our example plasma we obtain Zeff ≡ (niZ

2i + nZZ2

Z)/ne =0.64(12) + (0.06)(62) = 2.8. The overall electron collision frequency νe definedin (2.17) for an electron-ion plasma can be written for an impure plasma interms of the electron-deuterium (dominant ion with Zi = 1) collision frequencyν

e/Zi=1S (in the notation used in Section 2.10) as νe = Zeff ν

e/Zi=1S , in which

νe/Zi=1S ≡ νe

Zeff=

4√

2π nee4 ln Λ

4πε02 3 m1/2e T

3/2e

5×10−11 ne

[Te(eV)]3/2

(ln Λ17

)s−1. (2.44)

For our example plasma νe/Zi=1S (5×10−11)(2×1019)(16/17)/(103)3/2 3×

104 s−1, which gives νe = (2.8)(3×104) 8.4×104. Hence, for our example



plasma the time scale on which the electron distribution becomes a Maxwellianand electron flows come into equilibrium is τe ≡ 1/νe 12 µs. The distancetypical electrons travel in this time is the electron collision length (2.18) λe ≡vTe/νe, which is about 230 m for our plasma. Finally, the reference electricalresistivity calculated from (2.31) is about 1.5×10−7 Ω ·m. For impure plasmasit is appropriate to replace the Zi in (2.33) by Zeff , which then yields αe 0.4for Zeff 2.8. Thus, the Spitzer electrical resistivity for our example plasma is6×10−8 Ω ·m. For reference, the resistivity of copper at room temperature isabout 1.7×10−8 Ω ·m, a factor of about 3.5 smaller.

To calculate the ion collision frequency for the dominant ions (subscript i)in an impure plasma we need to include both their self-collisions and their colli-sions with impurities (subscript Z). Since the masses of impurity ions are ratherdisparate from the dominant ions (mi << mZ → AD << AC for our exam-ple plasma), the

√2 rest mass factor is not appropriate for collisions between

dominant ions and impurities. Thus, the appropriate collision frequency for thedominant ions in an impure plasma becomes

νi = fi νi/iS (2.45)

with

νi/iS ≡ 4

√π niZ

4i e4 ln Λ

4πε02 3 m1/2i T

3/2i

=(

niZ4i

ne

) (me

mi

)1/2 (Te

Ti

)3/2ν

e/Zi=1S√

2, (2.46)

fi ≡ 1 +√

2(

nZZ2Z

niZ2i

) (mi

mZ

)1/2

, ion collisions impurity factor. (2.47)

For multiple impurity species (Z) one just sums the second term in fi overthem. For our example plasma fi = 1 +

√2 [(0.06)(62)/0.64](2/12)1/2 3, and

νi 3(0.64)(1/3672)1/223/2(3×104)/√

2 1.9×103 s−1. Thus, the ions willrelax toward a Maxwellian distribution and their equilibrium flow on the ioncollisional time scale τi = 1/νi 530 µs. The ion collision length defined byλi = vTi/νi is about 120 m for our plasma, which is about a factor of two lessthan the electron collision length λe.

Finally, we calculate the longest time scale process — ion-electron energyexchange. We must again take account of impurities in the calculation. Here,since an electron-ion mass ratio is involved, we obtain

νe/iε = fi−e

(3

me

mi

) (niZ

2i

ne

)ν

e/Zi=1S (2.48)

in which the relevant factor to include impurity effects is

fi−e = 1 +(

nZZ2Z

niZ2i

) (mi

mZ

), ion-electron energy exchange impurity factor.

(2.49)

Again, for multiple impurity species (Z) one just sums the second term in fi−e

over them. For our example plasma fi−e = 1 + [(0.06)(62)/(0.64)](2/12)



1.6. In the presence of impurities the time scale for ion-electron temperatureequilibration becomes [see discussion after (2.39)] τi−e ≡ 3/(2 ν

e/iε ), which for

our plasma is τi−e = (3672/2)/[(1.6)(0.64)(3×104)] 60 ms.In summary, the electron, ion and ion-electron collision times in our example

plasma are τe : τi : τi−e 12 : 530 : 60 000 µs. Their ratios are in roughaccord with their anticipated mass ratio scalings of 1 : (mi/me)1/2 : mi/me =1 : 61 : 3672. Note also that even the electron (shortest) of these collisional timescales are much much longer (by a factor ∼ neλ

3De 3×106 >> 1) than the

plasma oscillation period 1/ωpe 4×10−6 µs.Implicit in the preceding analysis is the assumption that no other physical

processes operate on the charged particles in the plasma on these character-istic collision time ( τe − τi−e ∼ 10 − 104 µs) or length ( λe, λi

>∼ 100 m)scales. In practice, in most plasmas many other processes (for example, tem-poral variations, gyromotion in magnetic fields, and spatial inhomogeneities)vary more rapidly than one or more of these collisional effects and modify orimpede the collisional processes. Such combined collision and geometric effectswill be discussed later, particularly in Part IV: Transport. Note, however, thateven in the limit of very short time scales (compared to τe) Coulomb collisioneffects are not insignificant; as indicated by (2.20), in a time t they diffusivelyspread the velocity vectors of charged particles in a plasma through a pitch-angleϑ v⊥/v (νt)1/2. This velocity diffusion effect is important in smoothing outsharp gradients in velocity space and leads to collisional boundary layers inotherwise “collisionless” plasmas. Thus, Coulomb collisions will often play asignificant role even in “collisionless” plasmas. In fact, as we will see in laterchapters, Coulomb collisions provide the fundamental irreversibility (entropy-producing dissipative mechanisms) in plasmas.

2.6 Collisions with a Moving Background+

The most general Coulomb collision processes are those where a test particlespecies (s) collides with an arbitrary background species (s′) of plasma par-ticles that are in motion, which we now consider. The test particle charge,mass, position and velocity vectors will be taken to be qs, ms,x and v while thecorresponding quantities for the background particles will be indicated by thecorresponding primed quantities: qs′ , ms′ ,x′ and v′. The background particleswill be assumed to have an arbitrary velocity distribution given by fs′(v′).

The procedure we follow to determine the Coulomb collision processes forthis general case follows that used in the Lorentz collision model except thatnow the basic interaction is most conveniently calculated in a center-of-mass (orreally -momentum) frame. To develop the equations of motion in a center-of-momentum frame, we first note that the equations of motion of the interactingtest and background particles are given by

msdvdt

= qsE(x) =qsqs′

4πε0x − x′

|x − x′|3, (2.50)



ms′dv′

dt= qs′E(x′) =

qsqs′

4πε0x′ − x

|x′ − x|3. (2.51)

Note that the forces in these equations are equal and opposite — because of theconservative nature of the Coulomb force. Defining the center-of-momentumposition R and velocity U vectors as

R =msx + ms′x′

ms + ms′, U =

msv + ms′v′

ms + ms′, (2.52)

and the corresponding relative position r and velocity u vectors

r = x − x′, u = v − v′, (2.53)

we find the equations of motion in (2.50), (2.51) become

dUdt

= 0, mss′dudt

=qsqs′r

4πε0|r|3, (2.54)

in which mss′ is defined by

mss′ ≡ msms′

ms + ms′, reduced mass. (2.55)

From the first relation in (2.54) we see that the center-of-momentum velocityU is constant throughout the collisional interaction of the particles.

The equation describing the force on the relative velocity u ≡ v − v′ in(2.54) is analogous to that in (2.2) for the Lorentz collision model. Adoptinga coordinate system analogous to that in Fig. 2.3 in which v is replaced byu ≡ |v − v′|, we readily find that the change ∆u in a single Coulomb collisioninteraction between a test particle (s) and background particle (s′) is

∆u⊥ =1

mss′

∫ ∞

−∞dt

qsqs′r4πε0r3

=2qsqs′

4πε0mss′bu(ex cos ϕ + ey sinϕ) . (2.56)

Since the total energy is constant in the center-of-momentum frame for an elas-tic Coulomb collision, using a geometry analogous to that in Fig. 2.4, with vreplaced by the relative velocity u, and relations (2.5), (2.6), we obtain

u · ∆u = − 12∆u · ∆u − 1

2∆u⊥ · ∆u⊥ =⇒ ∆u‖ = − 2q2

sq2s′

4πε02m2ss′b2u3

.

(2.57)

Next, we want to determine the dynamical friction and diffusion coefficients〈∆v〉s/s′

and 〈∆v ∆v〉s/s′for test particles s colliding with background particles

s′. To do so we must relate ∆v to the relative ∆u determined above. Utilizingthe momentum conservation relations arising from U = constant in (2.52) withv → v + ∆v, v′ → v′ + ∆v′ and u → u+ ∆u from before to after the collision,we find

∆v′ = − ms

ms′∆v, ∆v =

mss′

ms∆u. (2.58)



Then, taking account of the velocity distribution fs′(v′) of the background par-ticles, we define the average vectorial dynamical friction and tensorial velocitydiffusion coefficients to be

〈∆v〉s/s′

∆t≡

∫d3v′ fs′(v′) u

∫dϕ

∫b db

mss′

ms∆u, (2.59)

〈∆v ∆v〉s/s′

∆t≡

∫d3v′ fs′(v′) u

∫dϕ

∫b db

m2ss′

m2s

∆u∆u. (2.60)

Using (2.56) and (2.57), the integrations in (2.59) and (2.60) can be per-formed with a specification of the impact parameter integral in (2.11) general-ized to a test particle (s) colliding with a moving background (s′) as follows:

ln Λss′ ≡∫ bmax

bmin

db

b= ln

(λD

bmin

), bmin = max

bclmin, bqm

min

(2.61)

in whichbclmin ≡ qsqs′

4πε0mss′u2, bqm

min =h

4πmss′

√u2

. (2.62)

The u2 indicates an average of u2 over the distribution of background particles;an appropriate typical value for this quantity is given in (2.113) below. In whatfollows we will implicitly assume that ln Λss′ is independent of v′ so that it can bebrought outside the v′ integration in equations (2.59) and (2.60); retaining theln Λ inside the v′ integration would only yield negligible (additional) correctionsof order 1/ ln Λ to the results we obtain below.

Thus, performing the integrations in (2.59) and (2.60) utilizing the impactparameter integral in (2.61) and the facts that

∂u

∂v=

uu

,∂

∂v1u

= − uu3

,∂2u

∂v∂v=

u2I − uuu3

=1u

(exex + eyey) , (2.63)

for our present velocity space coordinate system we obtain (for an alternatederivation using the Rutherford differential scattering cross section see Prob-lem 2.24):

〈∆v〉s/s′

∆t= − ms

mss′Γss′

∫d3v′fs′(v′)

uu3

≡ Γss′∂Hs′(v)

∂v, (2.64)

〈∆v ∆v〉s/s′

∆t= Γss′

∫d3v′fs′(v′)

u2I − uuu3

≡ Γss′∂2Gs′(v)

∂v∂v, (2.65)

in whichΓss′ ≡ 4πq2

sq2s′ ln Λss′

4πε02m2s

, (2.66)

Gs′(v) ≡∫

d3v′ fs′(v′) |v − v′|, (2.67)

Hs′(v) ≡ ms

mss′

∫d3v′

fs′(v′)|v − v′| =

(1 +

ms

ms′

) ∫d3v′

fs′(v′)|v − v′| . (2.68)



The G and H functions are formally similar to the electrostatic potential due toa distributed charge density for which Poisson’s equation −∇2φ = ρq(x)/ε0 hasthe solution φ(x) =

∫d3x′ ρq(x′)/(4πε0|x − x′|). They are called Rosenbluth

potentials9. Using the facts that

∇2v

1u≡

(∂

∂v· ∂

∂v

)1u

= − 4π δ(u) = − 4π δ(v − v′), (2.69)

∇2vu =

∂

∂v· ∂u

∂v=

∂

∂v·(u

u

)=

2u

, (2.70)

the Rosenbluth potentials can be shown to satisfy the relations

∇2vHs′(v) = − 4π (1 + ms/ms′)fs′(v),

∇2vGs′(v) = 2Hs′(v)/(1 + ms/ms′),

∇2v∇2

vGs′(v) = − 8πfs′(v).

(2.71)

Note that since the second of these equations shows that Hs′ is proportional toa Laplacian velocity space derivative of Gs′ , the Rosenbluth potential Gs′ is thefundamental one from which all needed quantities can be derived.

From the analogy of the first of the forms in (2.71) to electrostatics andthe definition of 〈∆v〉/∆t in (2.64) in terms of the Rosenbluth potential Hs′ ,we see that the dynamical friction 〈∆v〉/∆t tries to relax the test particle ve-locity to the centroid of the velocity distribution of the background particlesfs′(v) — see Problems 2.25 and 2.26. However, the velocity space diffusion〈∆v ∆v〉/∆t causes the velocity distribution of the test particles to maintain athermal spread comparable to that of the background particles. The dynamicalbalance between these two collisional processes on an entire distribution of testparticles determines their collisional distribution function — see Chapter 11.

Finally, using (2.63) and vector identities from Appendix D.3, we note that

∂

∂v·(

u2I − uuu3

)=

(∂

∂v1u

)· I −

(∂

∂v1u3

)· uu − 1

u3

∂

∂v· uu = − 2

uu

. (2.72)

Thus, we find that for Coulomb collisions the dynamical friction and velocitydiffusion coefficients are related by the important relation

〈∆v〉s/s′

∆t=

ms

2mss′

∂

∂v· 〈∆v ∆v〉s/s′

∆t=

(1 + ms/ms′

2

)∂

∂v· 〈∆v ∆v〉s/s′

∆t.

(2.73)

The total collisional effects on a test particle due to Coulomb collisions withall types of background particles are obtained by simply adding the contributionsfrom each species of background particles:

〈∆v〉s∆t

=∑s′

〈∆v〉s/s′

∆t,

〈∆v ∆v〉s∆t

=∑s′

〈∆v ∆v〉s/s′

∆t. (2.74)

9M.N. Rosenbluth, W. MacDonald and D. Judd, Phys. Rev. 107, 1 (1957).



Note also that the combination of this summation of species effects and, moreimportantly, of the fact that the Rosenbluth potentials are integrals over thebackground distribution functions, means that the dynamical friction and ve-locity diffusion coefficients are not sensitively dependent on detailed features offs′(v). (Recall the analogous weak dependence of an electrostatic potential tothe distribution of charges inside a surface.) Thus, evaluation of the Rosen-bluth potentials for Maxwellian background distributions will be useful bothin describing test particle collisional processes in Maxwellian plasmas and inother plasmas of interest where the distribution functions are reasonably closeto Maxwellians.

2.7 Collisions with a Maxwellian Background+

Specific test particle collisional effects due to dynamical friction and velocitydiffusion can be worked out in the rest frame of the background particles for anisotropic Maxwellian velocity distribution of the background particles:

fMs′(v) = ns′

(ms′

2πTs′

)3/2

e−ms′v2/2Ts′ =

ns′e−v2/v2T s′

π3/2v3Ts′

. (2.75)

Here, we have defined a “typical” thermal speed

vTs′ ≡ (2Ts′/ms′)1/2. (2.76)

Note that this speed is not the average speed [see (??) in Appendix A.4] fora Maxwellian distribution, which is (8Ts′/πms′)1/2; however, it is the mostprobable speed [see (??) in Appendix A.4] and it is mathematically convenient.

For a Maxwellian velocity distribution the Rosenbluth potential Gs′(v) de-fined in (2.67) can be evaluated in a spherical coordinate system in the relativevelocity space u = v − v′ as follows:

Gs′(v) ≡∫

d3v′ fs′(v′) |v − v′| =∫

d3u fs′(u + v)u

=ns′

π3/2v3Ts′

∫ ∞

0

2πu2du

∫ 1

−1

d(cos ϑ)u e−(v2+u2+2uv cos ϑ)/v2T s′

= − ns′v2Ts′√

π v

∫ ∞

0

u2du

v3Ts′

[e−(v+u)2/v2

T s′ − e−(v−u)2/v2T s′

]= − ns′vTs′√

x

1√π

[−4

√x

∫ ∞

√x

dy ye−y2 − 2∫ √

x

0

dy (y2 + x)e−y2

]

=ns′vTs′√

x

2√π

[2√

x

∫ ∞

√x

dy ye−y2+

∫ √x

0

dy y2e−y2+ x

∫ √x

0

dy e−y2

](2.77)

in which√

x ≡ v/vTs′ . The integrals in the last forms of (2.77) are relatedto the error function or probability integral (cf., Problem 2.27), but are most



conveniently written in terms of

ψ(x) ≡ 2√π

∫ x

0

dt√

t e−t, Maxwell integral, (2.78)

which has the properties

ψ′ ≡ dψ

dx=

2√π

√x e−x, ψ + ψ′ =

2√π

∫ √x

0

dy e−y2 ≡ erf (√

x ). (2.79)

Physically, the Maxwell integral is the normalized integral of a Maxwellian ve-locity distribution out to a sphere of radius v. Utilizing these definitions, wefind that the Rosenbluth potential Gs′(v) for a Maxwellian distribution of back-ground particles can be written as

Gs′(v) = ns′vTs′1√x

[(x + 1)ψ′(x) + (x + 1/2)ψ(x)] , (2.80)

in which

x ≡ xs/s′=

ms′v2

2 Ts′=

v2

v2Ts′

, relative speed parameter. (2.81)

The important parameter xs/s′is the square of the ratio of the test particle

speed to the thermal speed of the background particles of species s′.Thus, for an isotropic Maxwellian velocity distribution of background parti-

cles the Rosenbluth potential Gs′(v)=Gs′(v); that is, it depends only on the testparticle speed v, not its velocity v. Then, as can be shown from (2.65), 〈∆v ∆v〉is a diagonal tensor with elements 〈∆v2

x〉 = 〈∆v2y〉 = 〈∆v2

⊥〉/2 and 〈∆v2‖〉. Fur-

ther, it can be shown that 〈∆v〉 is in the ez or v direction. [These propertiesare valid for any distribution function for which the Rosenbluth potential Gs′

depends only on the test particle speed v.] Substituting the Rosenbluth poten-tial in (2.80) into (2.64) and (2.65), and utilizing (2.71) or (2.73), we find thatthe relevant dynamical friction and velocity diffusion coefficients are given by

〈∆v‖〉s/s′

∆t= Γss′

ms

2mss′

∂

∂v

[1v2

∂

∂v

(v2 ∂Gs′

∂v

)]= −

[ms

mss′ψ (x)

]ν

s/s′

0 v, (2.82)

〈∆v2⊥〉s/s′

∆t= Γss′

2v

∂Gs′

∂v= 2

[ψ(x)

(1 − 1

2x

)+ ψ′(x)

]ν

s/s′

0 v2, (2.83)

〈∆v2‖〉s/s′

∆t= Γss′

∂2Gs′

∂v2=

[ψ(x)

x

]ν

s/s′

0 v2. (2.84)

Note that in contrast to the Lorentz collision model, we now find 〈∆v2‖〉 = 0

— because the background particles are of finite mass and in motion, and hencecan exchange energy with the test particle during a Coulomb collision. The netrate of change of the test particle energy, which is given by (m/2)〈∆v2〉/∆t ≡



(m/2)〈(v + ∆v) · (v + ∆v)− v2〉/∆t, can be determined from these coefficientsas well:

〈∆v2〉s/s′

∆t= 2v

〈∆v‖〉s/s′

∆t+

〈∆v2⊥〉s/s′

∆t+

〈∆v2‖〉s/s′

∆t

= − 2 [(ms/ms′)ψ(x) − ψ′(x)] νs/s′

0 v2. (2.85)

The fundamental collision frequency for all these processes is

νs/s′

0 (v) ≡ ns′Γss′

v3=

4π ns′q2sq2

s′

4πε02m2sv

3ln Λss′ , reference collision frequency,

(6.6×10−11s−1)ns′Z2

s Z2s′

(ms/me)1/2 (Es/eV)3/2

(ln Λss′

17

), (2.86)

which is a straightforward generalization of the collision frequency ν(v) derivedfor the Lorentz collision model in (2.14): ν

e/i0 = ν(v) = νe(3

√π/4)v3

Te/v3.These dynamical friction and velocity diffusion coefficients can be used to

elucidate the rates at which the various Coulomb collision processes affect thetest particle velocity. Thus, we define the rates for momentum loss or slowingdown (νS), perpendicular diffusion (ν⊥), parallel or speed diffusion (ν‖) andenergy loss (νε) resulting from collisions of a test particle s on a Maxwellianvelocity distribution of background particles s′ as follows:10

d

dt(msv) = − ν

s/s′

S msv ≡ ms〈∆v〉s/s′

∆t, momentum loss,

d

dt|v − v|2⊥ = ν

s/s′

⊥ v2 ≡ 〈∆v2⊥〉s/s′

∆t, perpendicular diffusion,

d

dt|v − v|2‖ = ν

s/s′

‖ v2 ≡〈∆v2

‖〉s/s′

∆t, parallel diffusion,

d

dtεs = − ν

s/s′

ε εs ≡ ms

2〈∆v2〉s/s′

∆t, energy loss.

(2.87)

Here, msv is the test particle momentum, v is its average velocity [see (2.97) and(2.104) below for a detailed specification of v], |v−v|2⊥ and |v−v|2‖ indicate thediffusional spread of the test particle velocity in directions perpendicular andparallel to its direction of motion, and εs ≡ msv

2/2 is the test particle energy.From the definitions in (2.85) through (2.87) we see that νε is not an inde-

pendent quantity:

νε = 2 νS − ν⊥ − ν‖. (2.88)

10For an alternative representation of these various collisional processes using the notationand functions Chandrasekhar introduced for stellar collisions see Problem 2.27.



Figure 2.7: Coulomb collisional effects on the velocity of a test particle: mo-mentum loss or slowing down (νS), angular or perpendicular (ν⊥) and speedor parallel (ν‖) diffusion of the original test particle velocity. The contoursshown are lines of e−1 0.37 probability (see Section 2.10) at the short times(t1, t2, t3) = (0.002, 0.02, 0.2)/ν⊥ for an energetic electron (mev

2/2Te = 10) inan electron-proton plasma for which ν‖/ν⊥ 1/40, νε/ν⊥ 1/2.

From (2.82)–(2.88), we find the relevant frequencies for collisions of a test par-ticle (species s) with a Maxwellian background (species s′) are

νs/s′

S = −[(

1 +ms

ms′

)ψ

]ν

s/s′

0 , momentum loss, slowing down (2.89)

νs/s′

⊥ = 2[ψ + ψ′ − ψ

2x

]ν

s/s′

0 , perpendicular, pitch-angle diffusion, (2.90)

νs/s′

‖ =[ψ

x

]ν

s/s′

0 , parallel, speed diffusion, (2.91)

νs/s′

ε = 2[

ms

ms′ψ − ψ′

]ν

s/s′

0 = 2 νs/s′

S − νs/s′

⊥ − νs/s′

‖ , energy loss. (2.92)

The total effects due to a test particle’s collisions will all species of backgroundparticles are obtained by summing over s′ as indicated in (2.74); for example,νs

S ≡∑

s′ νs/s′

S . The overall effects of Coulomb collisions in slowing down anddiffusing the test particle velocity are indicated schematically in Fig. 2.7.

Equations (2.81)–(2.92) provide a very complete and useful description ofthe evolution of the velocity of a test particle of species s suffering Coulombcollisions with Maxwellian background particles s′ — see Problems 2.28–2.33



for some illustrative applications of them. In addition, as discussed in the nextsection, they can be used to develop a Monte Carlo scattering operator fornumerical studies of the effects of Coulomb collisions on the velocity of a testparticle. Finally, as we did for the Lorentz collision model [cf., (2.22), (2.23)],we write the dynamical friction and velocity diffusion coefficients for Coulombcollisions of test particles of species s with a species s′ of Maxwellian backgroundparticles in coordinate-independent vectorial forms:

〈∆v〉s/s′

∆t= − ν

s/s′

S v, dynamical friction, (2.93)

〈∆v ∆v〉s/s′

∆t=

12

νs/s′

⊥ (v2I − vv) + νs/s′

‖ vv, velocity diffusion. (2.94)

2.8 Evolution of Test Particle Velocity+

To further illustrate the Coulomb collision effects, we examine the collisionalevolution of the velocity of a test particle for short times where the velocitychanges are small. The test particle will be assumed to be colliding with aplasma whose components have Maxwellian distributions. Thus, the results ofthe previous section will be applicable.

A test particle of species s will be taken to have an initial velocity v0 in theez or parallel direction (cf., Figs. 2.4 and 2.7). Integrating the first equationof (2.87) over a short time t >> ∆t (for validity of the dynamical friction andvelocity diffusion coefficients), we find that the mean parallel or ez componentof the test particle velocity after a time t is

v‖ = v0(1 − νSt), νS = νsS ≡

∑s′

νs/s′

S (v0). (2.95)

This result is valid for νSt << 1 and indicates the monotonic decrease in testparticle momentum due to Coulomb collisons. Similarly, the test particle energyafter a short time t can be obtained directly by integrating the last equation of(2.87) over time:

12mv2 =

12mv2

0(1 − νεt), νε = νsε ≡

∑s′

νs/s′

ε (v0). (2.96)

Thus, the average test particle speed v, which will be used below, is defined by

v ≡√

v2 v0(1 − νεt/2). (2.97)

Similar to (2.95), these formulas are only valid for νεt << 1.The angular (perpendicular) velocity and speed (parallel) diffusion processes

in velocity space indicated by ν⊥ and ν‖ have to be treated differently. Becausethese Coulomb collision effects are random in character and diffusive, they leadto a Gaussian probability distribution P (v−v) of the velocity about the average



(slowing down) test particle speed v. Since the diffusion results from purelyrandom processes, we can anticipate (and will derive in Chapter 11) that thisprobability distribution will be Gaussian and of the form:

P (v − v) =1

2π v2

(e−ϑ2/2σ2

⊥

σ2⊥

) (e−(v−v)2/2σ2

‖√

2π σ‖

). (2.98)

Here, ϑ2 ≡ [arcsin−1(v⊥/v0)]2 (v2x + v2

y)/v20 and v is defined in (2.97). Note

that since in the limit σ‖ → 0 the last term in braces becomes δ(v− v) [see (??)and (??) in Appendix B.2], this probability distribution reduces to the shorttime Lorentz model test particle distribution given in (2.21) for σ⊥ =

√νt.

Taking velocity-space averages of various quantities A(v) over this probabil-ity distribution [A ≡

∫ ∞0

2πv2 dv∫ π

0sinϑ dϑ P (v− v) A(v)], we find that while

the average of the diffusive deflections vanish [vx = vy = 0, v‖ − v‖ = 0], thediffusive spreads in the perpendicular and parallel (to v0) directions are

v2 − v2 v2⊥ = v2

x + v2y = 2σ2

⊥v20 , (v‖ − v‖)2 = σ2

‖. (2.99)

To determine the probability variances σ⊥ and σ‖ for the diffusive Coulombcollisional processes, we integrate the middle two equations in (2.87) over a shorttime t, and obtain (keeping only first order terms in νt << 1)

v2⊥ = (ν⊥t) v2

0 , ν⊥ = νs⊥ ≡

∑s′

νs/s′

⊥ (v0), (2.100)

(v‖ − v‖)2 = (ν‖t) v20 , ν‖ = νs

‖ ≡∑s′

νs/s′

‖ (v0). (2.101)

Comparing (2.99), (2.100) and (2.101), we see that for Coulomb collisions

σ⊥ =√

ν⊥t/2, σ‖ =√

ν‖t v0. (2.102)

The relative collisional spreads (half-widths in velocity space to points wherethe probability distribution drops to e−1/2 0.61 of its peak value) of the testparticle velocity in the directions perpendicular and parallel (i.e., for speed orenergy diffusion) relative to its initial velocity v0 are given by

δϑ δv⊥/v0 ≡ σ⊥ =√

ν⊥t/2, δv/v0 ≡ σ‖/v0 =√

ν‖t. (2.103)

Note that in comparing the perpendicular diffusion factor in (2.98) for whichσ⊥ =

√ν⊥t/2 with the perpendicular diffusion in the Lorentz model as given in

(2.21), we need to realize that ν⊥ = 2 ν and hence that σ⊥ =√

νt for the Lorentzcollision model. These formulas indicate that, even for very high temperatureplasmas with nλ3

D >> 1 where the Coulomb collision rates are very slow, onlya short time is required to diffuse the test particle velocity through a small|δv| << |v0|. For example, as indicated in (2.20), the time required to diffuse aparticle’s velocity through a small angle ϑ δv⊥/v0 << 1 is only t 2 ϑ2/ν⊥ <<1/ν⊥ [or, t ϑ2/ν << 1/ν for the Lorentz collision model]. Thus, because of the



diffusive nature of Coulomb collisions, it takes much less time to scatter throughan angle ϑ << 1 in velocity space than it does to scatter through 90 (ϑ ∼ 1).The various diffusive collisional effects are illustrated in Fig. 2.7. There, thecontours shown indicate where the probability distribution P in (2.98) is equalto e−1 0.37 of its peak value for ν⊥t = 0.002, 0.02, 0.2, for a typical set of testparticle parameters.

The change in the average energy mv2/2 can also be obtained using (2.95),(2.99) and (2.102). This procedure yields, correct to first order in νt << 1,

12mv2 = 1

2m[v2‖ + v2

⊥

]= 1

2m[v‖

2 + 2(v‖ − v‖)v‖ + (v‖ − v‖)2 + v2⊥

] 1

2m[v20(1 − 2νSt) + σ2

‖ + 2σ2⊥] = 1

2mv20 [1 − (2νS − ν⊥ − ν‖)t].

This result is the same as (2.96) because of the relation between the variouscollisional processes given in (2.88).

The formulas developed in this section also provide a basis for a probablistic(Monte Carlo) numerical approach for inclusion of Coulomb collision effects inother plasma processes such as single particle trajectories. Thus far we havefound that after a short time t a test particle’s velocity and speed decreaseaccording to (2.95) and (2.97). However, the test particle also acquires a diffusivespread in the perpendicular and parallel directions as given by (2.98) with thespreads (variances) defined in (2.102). Further, the velocity space latitudinalangle ϕ [cf., (2.1)] is completely randomized by successive individual Coulombcollisions — for time scales t >> ∆t. Hence, defining a random variable ξto be evenly distributed between 0 and 1, and independent random variablesη1, η2 sampled from a normal probability distribution [i.e., Gaussian such asindicated in the σ‖ part of (2.98)] with zero mean and a mean square of unity(i.e., η1 = η2 = 0 but η2

1 = η22 = 1), we find that the total velocity vector v after

a short time t (νt << 1) can be written as

v = v0 (1 − νεt/2)[ez ( 1 + η1

√ν‖t ) + |η2|

√ν⊥t/2 (ex cos 2πξ + ey sin 2πξ)

].

(2.104)

In the Lorentz collision model where νε = 0 and ν‖ = 0, this result simpifies to

v = v0

[ez + |η2|

√ν⊥t/2 (ex cos 2πξ + ey sin 2πξ)

], Lorentz collision model.

(2.105)

Either of these forms can be used to develop a Monte Carlo algorithm for ad-vancing the test particle velocity v taking into account the Coulomb collisiondynamical friction and velocity space diffusion effects.

Since (2.104) implies a change in the velocity of the test particle, in or-der to preserve the momentum and energy conserving properties of the elasticCoulomb collision process, the velocity of the background particles must alsochange, at least on average. Hence, in order to develop a complete Monte Carlo-based Coulomb collision operator we should consider simultaneously both a test



and a background particle. Then, the change in velocity δv ≡ v − v0 for thetest particle is determined from (2.104), and that for the background particle isgiven by δv′ = − (ms/ms′)δv — see (2.58).

2.9 Test Particle Collision Rates+

We now consider the various Coulomb collision effects on typical electrons andions in a plasma. For simplicity the plasma will be assumed to be composed ofelectrons and only one species of ions with charge qi = Zie, and to have equalelectron and ion temperatures, with both species of particles having Maxwellianvelocity distributions. Thus, the formulas derived in the Section 2.7 will apply.

For illustrative purposes we consider collisional effects on a test electronand a test ion in the plasma, each having speeds equal to the thermal or mostprobable speeds for their respective species:

ve = vTe, vi = vTi. (2.106)

Then, the reference collision frequencies νs/s′

0 for electron-ion (e/i), electron-electron (e/e), ion-ion (i/i) and ion-electron (i/e) collisions are simply related:

νe/i0 = Zi ν

e/e0 , ν

i/i0 = Z2

i

√me/mi ν

e/i0 , ν

i/e0 = Zi

√me/mi ν

e/i0 , (2.107)

in which we have neglected the small differences in ln Λss′ for differing s ands′ and made use of the quasineutrality condition ne = niZi. Further, sincethe ratio of ion to electron mass is very large (1836 for protons), we find thatthe relative speed parameters xs/s′

defined in (2.81) for the various collisionalprocesses are given by

xe/i =mi

me>> 1, xe/e = 1, xi/i = 1, xi/e =

me

mi<< 1. (2.108)

Thus, we will need both small and large argument expansions of the Maxwellintegral ψ(x) and its derivative ψ′(x), as well as evaluation of them at theparticular value of unity.

The behavior of ψ(x) and other functions of interest are sketched in Fig. 2.8.For x = 1, we have ψ = 0.4276, ψ′ = 0.4151 and ψ +ψ′−ψ/2x = 0.6289. Smalland large argument expansions of interest in evaluating νS , ν⊥, ν‖, and νε are:

x << 1

ψ(x) (4x3/2/ 3√

π )(1 − 3x/5 + 3x2/14 − · · · ),ψ′ = (2

√x e−x/

√π ) (2 x1/2/

√π)(1 − x + x2/2 − · · · ),

ψ + ψ′ − ψ/2x (4x1/2/ 3√

π )(1 − x/5 + 3x2/70 − · · · ),

(2.109)



Figure 2.8: Maxwell integral ψ(x) and related functions.

x >> 1

ψ(x) 1 − (2√

x e−x/√

π )(1 + 1/2x − 1/4x2 + · · · ),ψ′(x) = 2

√x e−x/

√π,

ψ + ψ′ − ψ/2x 1 − 1/2x + (e−x/√

π x3/2)(1 − 1/x + · · · ).

(2.110)

Using only the lowest order of these approximations in (2.89)–(2.92), we findthe relationships between various collisional processes listed in Table 2.1. Therates are all referred to the electron-ion collision frequency ν

e/i0 , which is the

same as the Lorentz collision frequency in (2.14). The νe/iS and ν

e/i⊥ components

of the first (e/i) column are the same as those given by the Lorentz collisionmodel [cf., (2.13) and (2.19)]. All the other electron processes indicated in thetable arise from the finite mass ratio between the electrons and ions, and the factthat the background particles are in motion. Note that in this general collisionmodel the electron-ion parallel (speed) diffusion (νe/i

‖ ) and energy loss (νe/iε )

are of order me/mi << 1 compared to the Lorentz collision model processes —because of the inefficiency of energy transfer in collisions of particles with verydisparate masses.

From Table 2.1 we see that the various collisional processes naturally splitinto three groups of rates: ν

e/i0 , Z2

i

√me/mi ν

e/i0 , and Zi(me/mi) ν

e/i0 . The

fastest of these rates is the Lorentz collision rate; however, all the electron-electron collisional processes also occur at roughly the same rate and so shouldalso be taken into account in investigations of electron collisional processes. (Theelectron-electron collision processes are small in a plasma where the ions all havehigh charge states Zi >> 1 since then ν

e/i0 = Zi ν

e/e0 >> ν

e/e0 .) Physically, on

this fastest time scale of 1/νe/i0 , electron momentum is relaxed by collisions



Table 2.1: Relative Coulomb collision rates for thermal speed test electrons andions with charge Zi in a Maxwellian plasma with Te = Ti.

e/i e/e i/i i/e

slowingdown ν

s/s′

S / νe/i0 1

0.86Zi

0.86Z2i

√me

mi0.75Zi

me

mi

perpendiculardiffusion ν

s/s′

⊥ / νe/i0 2

1.26Zi

1.26Z2i

√me

mi1.50Zi

me

mi

speeddiffusion ν

s/s′

‖ / νe/i0

me

mi

0.43Zi

0.43Z2i

√me

mi0.75Zi

me

mi

energyloss ν

s/s′

ε / νe/i0 2

me

mi

0.03Zi

0.03Z2i

√me

mi− 0.75Zi

me

mi

on both electrons and ions, and the electrons relax within themselves throughall the processes. The electron-electron collisions relax the electrons toward aMaxwellian distribution (see Chapter 11). On the next lower rate or longertime scale — by a factor of order

√mi/me

>∼ 43 >> 1 — ion-ion collisionsrelax the ions toward a Maxwellian distribution. Finally, on the longest timescale, which is a factor of about mi/me

>∼ 1836 >> 1 slower than 1/νe/i0 , there

is energy transfer between the electrons and ions, and ion momentum loss to theelectrons. [The energy loss rate ν

i/eε is negative here because we are evaluating

it for a test particle whose energy mv2/2 = T is less than the average particleenergy in the plasma, mv2/2 = 3T/2 — see (??) in Appendix A.4].

2.10 Plasma Collision Rates+

Next, we consider the overall collisional relaxtion rates for the entire electron andion species of charged particles in a plasma. First, we consider the temperatureequilibration rate for a Maxwellian distribution of test particles of species scolliding with a Maxwellian distribution of background particles s′. Multiplyingthe test particle energy loss equation defined in the last line of (2.87) by anisotropic Maxwellian velocity distribution of test particles s in the form givenin (2.75) and using the first property of ψ given in (2.79) to integrate the ψ

contribution to the νs/s′

ε defined in (2.92) by parts once, we find

32ns

dTs

dt= − ν

s/s′

ε ns(Ts − Ts′), (2.111)

where



nsνs/s′

ε = nsms

ms′

[4√π

νs/s′

0 (vTss′)]

= ns′ νs′/sε =

4√π

4π nsns′q2sq2

s′ ln Λss′

4πε02msms′v3Tss′

,

(2.112)

is the average energy density exchange rate between the species. Here,

vTss′ ≡ [2(Ts/ms + Ts′/ms′)]1/2 =√

v2Ts + v2

Ts′ (2.113)

is the appropriate mean thermal velocity for a combination of test and back-ground particles, both with Maxwellian distributions. From the equality ofnsν

s/s′

ε and ns′ νs′/sε , it is obvious that

nsdTs

dt= −ns′

dTs′

dt, (2.114)

as required by energy conservation — energy lost from the test particle species isgained by the (dissimilar) background species with which it suffers Coulomb col-lisions. For a couple of applications of these temperature equilibration formulassee Problems 2.35 and 2.36.

For typical electron-ion plasmas where me << mi and Te is not too differentfrom Ti so that Te/me >> Ti/mi (vTe >> vTi), (2.111) becomes [cf., (2.38)]

32ne

dTe

dt= − ν

e/iε ne(Te − Ti) = − 3

me

mineνe(Te − Ti) ≡ −Qi. (2.115)

In the next to last expression we have used the fundamental νe defined in (2.17).The relevant formula for the electron-ion energy transfer rate ν

e/iε in a plasma

with impurities (see Problem 2.39) was given previously in (2.48) and (2.49).Finally, we calculate the momentum relaxation rate for two Maxwellian dis-

tributions of particles that are drifting (flowing) slowly relative to each otherwith velocity V ≡ Vs − Vs′ , assuming |V| << vTss′ . In the rest frame of thebackground particles (s′), the drifting test particle (s) distribution function canbe written as in (2.15):

fs(v) = ns

(ms

2πTs

)3/2

exp(− ms|v − V|2

2Ts

) nse

−v2/v2T s

π3/2v3Ts

[1 +

2v ·Vv2

Ts

+ · · ·]

. (2.116)

Multiplying the momentum loss rate formula in the first line of (2.87) by thisdistribution function and integrating over velocity space, again integrating onceby parts and using the first relation in (2.79), we find

msnsdVs

dt= − ν

s/s′

S msns(Vs − Vs′), (2.117)

where



msnsνs/s′

S = msns

[4

3√

π

ms

mss′ν

s/s′

0 (vTss′)]

=4

3√

π

4π nsns′q2sq2

s′ ln Λss′

4πε02mss′v3Tss′

(2.118)

is the average momentum density exchange rate between the s and s′ species ofparticles, and vTss′ is the average thermal velocity defined in (2.113). From thesymmetric form of msnsν

s/s′

S in terms of the species labels s and s′, it is clearthat the momentum lost from the s species is gained by the s′ species and thusmomentum is conserved in the Coulomb collisional interactions between the twospecies of particles: msns dVs/dt = −ms′ns′ dVs′/dt.

Specializing again to an electron-ion plasma and assuming as usual thatvTe >> vTi, we find that (2.117) and (2.118) reduce to [cf., (2.34)]

menedVe

dt= −meneνe(Ve − Vi) ≡ Re, (2.119)

where

νe = νe/iS =

43√

πν

e/i0 (vTe) =

4√

2π neZie4 ln Λ

4πε02 3 m1/2e T

3/2e

≡ 1τe

. (2.120)

This electron momentum relaxation rate is the same as that obtained in (2.17)for the Lorentz collision model and shows that the fundamental Maxwellian-averaged electron-ion collision frequency νe is in fact ν

e/iS . Electron-electron

collisions do not contribute to the momentum relaxation process because theyare momentum conserving for the electron species as a whole. Note also thatthe collisional momentum relaxation process acts on the difference between theelectron and ion flow velocities. Thus, the net effect of Coulomb collisions is torelax the electron flow to the ion flow velocity. Finally, the relevant formula forthe electron-ion collisional “slowing down” rate ν

e/iS in a plasma with impurities

is just νe = Zeff νe/Zi=1S [see (2.44) and Problem 2.37].

For the slightly fictitious case of two ion species with charge qi = Zie thathave equal temperatures but are drifting relative to each other with velocity V,the ion momentum relaxation rate is given by

minidVdt

= −miniνiV, (2.121)

where [cf., (2.36)]

νi = νi/iS =

4√

π niZ4i e4 ln Λ

4πε02 3 m1/2i T

3/2i

≡ 1τi

. (2.122)

As can be anticipated from the νi/iS entry in Table 2.1, this momentum relaxation

rate is a factor of order Z2i

√me/mi slower than that for electrons. Also, the

ion-electron collisional effects due to νi/eS have been neglected in the average ion

momentum loss rate because they are a factor of order√

me/mi smaller thanthe ion-ion collisional effects. The numerical factor in (2.122) is

√2 smaller than



that in (2.120) because of the rest mass and average thermal velocity factorsfor this equal mass case. Finally, the relevant formula for the ion-ion collisional“slowing down” rate ν

i/iS in a plasma with impurities (see Problem 2.38) was

given in (2.45).

2.11 Fast Ion Thermalization+

In attempting to heat plasmas one often introduces “fast” ions (through ab-sorption of energetic neutrals, from radiofrequency wave heating, or directly asenergetic charged fusion products such as α particles), which have speeds in-termediate between the ion and electron thermal speeds. These fast ions heatthe plasma by transferring their energy to the background plasma electrons andions during the Coulomb collision slowing down process. This collisional fastion slowing down and energy transfer process will now be considered in detail.

For simplicity we consider an electron-hydrogenic (proton, deuteron or triton— mi = 1, 2 or 3 but Zi = 1) background plasma in which both species havea Maxwellian velocity distribution. The electron and ion temperatures will beassumed to be unequal, but comparable in magnitude. The fast or test ionwill be allowed to have a mass (mf ) and charge (qf = Zfe) different fromthe background ions. Because the fast ion speed is intermediate between theelectron and ion thermal speeds, the relative speed parameters in (2.81) for thefast ion-ion (f/i) and fast ion-electron (f/e) collisions are given by

xf/i =miv

2

2Ti=

v2

v2Ti

>> 1, xf/e =mev

2

2Te=

v2

v2Te

<< 1, (2.123)

in which v is the fast ion speed. From (2.86) we see that the reference collisionfrequencies ν

f/s′

0 are equal for the electron-hydrogenic ion background plasma:

νf/i0 = ν

f/e0 . (2.124)

Using the approximations (2.123) in (2.109) and (2.110), we find that thefast ion transfers energy to the plasma electrons and ions at the rates definedin (2.92), which are given to lowest significant order by

νf/iε 2

mf

miν

f/i0 , (2.125)

νf/eε 2

mf

me

4 (xf/e)3/2

3√

πν

f/e0 = 2

mf

me

43√

π

v3

v3Te

νf/e0 . (2.126)

From the definition of νf/s′

0 in (2.86) we see that it depends on v−3. Thus, νf/iε

also depends on v−3. However, νf/eε is independent of the fast ion speed v —

because the appropriate relative speed for fast ion-electron collisions when thefast ion speed v is slower than vTe is the electron thermal speed.

Adding together the fast ion energy losses via collisions with backgroundplasma ions and electrons, the total fast ion energy loss rate becomes



dεdt

= −(ν

f/eε + ν

f/iε

)ε, (2.127)

in which

ε = mfv2/2 (2.128)

is the instantaneous fast ion energy. Since νf/eε is independent of the fast ion

energy, it is convenient to define a characteristic fast ion slowing down time interms of it:

τS ≡ 2

νf/eε

1

νf/eS

=mf

me

4πε02 3 m1/2e T

3/2e

(4√

2π) neZ2fe4 ln Λ

=(

mf

me

)1νe

, (2.129)

fast ion slowing down time. (2.130)

Here, the approximate equality to 1/νf/eS follows because for mf/me >> 1 the

fast ions are not significantly scattered by the electrons; thus, they lose energyto the plasma electrons at twice the rate they lose momentum to them.

The rate of transfer of fast ion energy to plasma ions can be referenced tothe transfer rate to the electrons in terms of a critical energy εc ≡ mfv2

c/2 asfollows:

νf/iε

νf/eε

=(εc

ε

)3/2

=v3

c

v3, (2.131)

where

εc ≡ mfv2c

2= Te

[3√

π

4

√mf

me

mf

mi

]2/3

15 Te

(mf

m1/3p m

2/3i

)(2.132)

in which mp is the proton mass. (For the appropriate modifications when mul-tiple species of ions are present, see Problems 2.40 and 2.50.) In terms of thiscritical energy, (2.127) can be written as

dεdt

= − 2 ετS

[1 +

(εc

ε

)3/2]

. (2.133)

The fast ion energy transfer rates as a function of energy are illustrated in Fig.2.9. For fast ion energies greater than εc the energy transfer is primarily toelectrons, while for ε < εc it is primarily to ions.

Since (2.133) applies for all fast ion speeds between the electron and ionthermal speeds, it will be valid for all fast ion energies during the thermalizationprocess. Thus, its solution will give the fast ion energy as a function of time as ittransfers its energy to the background plasma. To solve (2.133) it is convenientto convert it to an equation for the fast ion speed v ≡

√2 ε/mf , for which it

becomes

dv

dt= − v

τS

[1 +

v3c

v3

], (2.134)



Figure 2.9: Fast ion energy transfer rate versus energy ε. The energy transferis primarily to electrons for ε > εc, but to ions for ε < εc.

vc ≡√

2 εc

mf=

[3√

π

4me

mi

]1/3

vTe. (2.135)

Multiplying (2.134) by v2 and integrating over time from t = 0 where the initialfast ion speed will be taken to be v0 to the current time t where it has speed v(assumed > vTi

), we obtain

t =τS

3ln

(v30 + v3

c

v3 + v3c

), (2.136)

orv3(t) = (v3

0 + v3c ) e−3t/τS − v3

c . (2.137)

The fast ion energy ε(t) = mfv2(t)/2 during the slowing down process can bereadily obtained from this last result.

The decay of the fast ion energy with time is illustrated in Fig. 2.10. Notethat for initial energies much greater than the critical energy εc the fast ionenergy decays exponentially in time at a rate 2/τS = ν

f/eε due to collisions with

electrons, as is apparent from (2.133). However, when the fast ion energy dropsbelow εc the energy transfer is predominantly to the ions and the fast ion energydecays much faster than exponentially. The total lifetime for thermalization(i.e., to v vTi << vc) of the fast ion into the background plasma ions is

τf (τS/3) ln [1 + (ε0/εc)3/2] = (τS/3) ln [1 + v30/v3

c ]. (2.138)

A couple of applications of these fast ion slowing down effects and formulas aredeveloped in Problems 2.41 and 2.42.



E/Ec

t/τs

0 0.5 1.1

.3

1

3

10

Figure 2.10: Decay of fast ion energy ε versus time during thermalization intoa background plasma for various initial ratios of ε to the critical energy εc.

Next, we calculate the fraction of the fast ion energy transferred to thebackground plasma electrons and ions over the entire fast ion slowing downprocess. Since in many plasma situations the fast ions are also susceptible toother, direct loss processes such as charge-exchange, we introduce a probabilityexp(−t/τcx) that the fast ion will remain in the plasma for a time t againstcharge-exchange losses at rate 1/τcx. Then, the fraction Ge of the total fastion energy ε0 ≡ mfv2

0/2 transferred to the electrons during the thermalizationprocess is given by

Ge ≡ 1ε0

∫ τf

0

dt

(− dε

dt

)ν

f/eε e−t/τcx

νf/eε + ν

f/iε

=2v20

∫ v0

0

v3 v dv

v3 + v3c

[v3 + v3

c

v30 + v3

c

]τS/3τcx

. (2.139)

Similarly, the fraction Gi of fast ion energy transferred to the ions is (for thesimpler case where τcx → ∞, see also the form given in Problem 2.43)

Gi ≡1ε0

∫ τf

0

dt

(− dε

dt

)ν

f/iε e−t/τcx

νf/eε + ν

f/iε

=2v20

∫ v0

0

v3c v dv

v3 + v3c

[v3 + v3

c

v30 + v3

c

]τS/3τcx

. (2.140)

The fraction of fast ion energy lost due to charge-exchange is 1 − Gi − Ge.However, a portion of this energy may be absorbed in the plasma if some of thefast neutrals produced by charge-exchange are reabsorbed before they leave theplasma.

The fractions Ge, Gi of fast ion energy transferred to plasma electrons andions during the thermalization process as a function of ε0/εc is illustrated inFig. 2.11. Note that the integrated fractions Gi, Ge become equal for ε0

<∼ 2 εc,which is significantly larger than the value of ε0 εc where the instantaneousenergy transfer rates are equal — recall Fig. 2.9. Also, charge-exchange lossesbecome significant for τS/τcx

>∼ 1, and can greatly diminish the fast ion energy



0.1 0.2 0.5 1 2 5 100

0.2

0.4

0.6

0.8

1

01

3

10

0

1

3

10

τsτcx

____ =

electrons

ions

Eo/Ec

Gi,Ge

Figure 2.11: Fraction of the fast ion energy ε0 transferred to background plasmaelectrons (Ge) and ions (Gi) as a function of the ratio of the initial energy ε0

to the critical energy εc. The variation with τS/τcx indicates the influence ofdirect fast ion losses (at rate 1/τcx) during the thermalization process.

transfer to the plasma for τS/τcx >> 1. For some typical applications of fastion slowing down and energy transfer processes and their effects on plasmas,see Problems 2.44–2.47. One key result for fusion experiments is that for 3.52keV alpha particles slowing down in a Te ∼ 10 keV deuterium-tritium plasmafor which εc ∼ 330 keV we obtain ε0/εc ∼ 10 and hence (from Fig. 2.11) thealpha particle deposits over 80% of its energy in the plasma electrons.

In addition to energy loss, the fast ions experience perpendicular and paralleldiffusion in velocity space during their thermalization. The relative importanceof the various Coulomb collision processes on the fast ion for the conditionsgiven in (2.123) are indicated in Table 2.2. From this table we see that forε >> εc the momentum and energy losses by the fast ions to the electronsare the dominant processes because then the velocity space diffusion effectsindicated by ν⊥, ν‖ are small. However, for ε < εc the fast ions lose energyprimarily to the background ions and their perpendicular or angular diffusionrate in velocity space becomes equal to their energy loss rate. For some typicalapplications of fast ion scattering processes and their effects on plasmas, seeProblems 2.48–2.50.

The energy or speed diffusion process indicated by ν‖ is negligible until thefast ion energy is reduced to approximately the ion temperature in the back-ground plasma. Since the energy diffusion process is thus negligible during thefast ion thermalization process, and the perpendicular diffusion has no effecton the energy transfer rates, our characterization of the fast ion slowing downprocess as one of a monotonic decrease in the fast ion energy is a reasonablyaccurate one. A kinetic description that allows for pitch-angle (ϑ) scatteringalong with the fast ion energy loss process is developed in Section 11.4.



Table 2.2: Relative Coulomb collision rates for fast ions with vTi << v << vTe

slowing down in a plasma with Maxwellian electrons and ions (εc ∼ 15 Te).

f/i f/e

slowingdown ν

f/s′

S / νf/e0

(1 +

mf

mi

)mf

me

43√

π

(v

vTe

)3

=(

εεc

)3/2

perpendiculardiffusion ν

f/s′

⊥ / νf/e0 2

83√

π

(v

vTe

)<< 1

speeddiffusion ν

f/s′

‖ / νf/e0

Ti

mfv2/2<< 1

43√

π

(v

vTe

)<< 1

energyloss ν

f/s′

ε / νf/e0 2

mf

mi

mf

me

83√

π

(v

vTe

)3

= 2(

εεc

)3/2

REFERENCES AND SUGGESTED READING

The basic Coulomb collision processes were first worked out in the analogous con-text (see Problem 2.9) of the gravitational interaction of stars:

Chandrasekhar, Principles of Stellar Dynamics (1942).

S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).

A comprehensive application to Coulomb collisions in a plasma was first presented in

Spitzer, Physics of Fully Ionized Gases (1962).

The most general development of the dynamical friction and velocity diffusion coeffi-cients for a plasma in terms of the Rosenbluth potentials originated in the paper

M.N. Rosenbluth, W. MacDonald and D. Judd, Phys. Rev. 107, 1 (1957).

The most comprehensive treatments of the Coulomb collision effects on test particlesin a plasma are found in

B.A. Trubnikov, “Particle Interactions in a Fully Ionized Plasma,” in Reviews ofPlasma Physics, M.A. Leontovich, ed. (Consultants Bureau, New York, 1965),Vol. I, p. 105.

D.V. Sivukhin, “Coulomb Collisions in a Fully Ionized Plasma,” in Reviews ofPlasma Physics, M.A. Leontovich, ed. (Consultants Bureau, New York, 1966),Vol. IV, p. 93.

A brief, but very useful summary of the important Coulomb collision formulas in thischapter is given in

Book, NRL Plasma Formulary (1990), p. 31.

A book devoted almost entirely to the subject of Coulomb collision effects in a plasmain which numerous examples are worked out is

Shkarofsky, Johnston and Bachynski, The Particle Kinetics of Plasmas (1966).



Also, most books on plasma physics have chapters devoted to discussions of Coulombcollision effects. Among the most descriptive and useful are those in

Spitzer, Physics of Fully Ionized Gases (1962), Chapter 5

Rose and Clark, Plasmas and Controlled Fusion (1961), Chapter 8.

Schmidt, Physics of High Temperature Plasmas (1979), Chapter 11.

Krall and Trivelpiece, Principles of Plasma Physics (1973), Chapter 6.

Golant, Zhilinsky and Sakharov, Fundamentals of Plasma Physics (1980), Chap-ter 2.

The original theory of runaway electrons was developed in

H. Dreicer, Proceedings of the Second United Nations International Conferenceon the Peaceful Use of Atomic Energy (United Nations, Geneva, 1958), Vol. 31,p. 57. See also, Phys. Rev. 115, 238 (1959).

The thermalization of a fast ion in a Maxwellian plasma was first developed in

D.J. Sigmar and G. Joyce, Nuclear Fusion 11, 447 (1971).

T.H. Stix, Plasma Physics 14, 367 (1972).

Inclusion of charge-exchange loss and geometry effects on neutral-beam-injected fastions are discussed in

J.D. Callen, R.J. Colchin, R.H. Fowler, D.G. McAlees and J.A. Rome, “NeutralBeam Injection into Tokamaks,” Plasma Physics and Controlled Nuclear FusionResearch 1974 (IAEA, Vienna, 1975), Vol. I, p. 645.

The Monte Carlo computational approach to including Coulomb collisional effects hasbeen developed primarily in the context of investigating transport processes in

R. Shanny, J.M. Dawson and J.M. Greene, Phys. Fluids 10, 1281 (1967).

K.T. Tsang, Y. Matsuda and H. Okuda, Phys. Fluids 18, 1282 (1975).

T. Takizuka and H. Abe, J. Comput. Phys. 25, 205 (1977).

A.H. Boozer and G. Kuo-Petravic, Phys. Fluids 24, 851 (1981).

PROBLEMS

2.1 Consider the length scales relevant for electron Coulomb collision processes in atypical university-scale magnetic fusion plasma experiment that has ne = 2×1013

cm−3 and Te = Ti = 1 keV. Calculate: a) the distance of closest approach bmin;b) the average interparticle spacing; c) the maximum interaction distance bmax;and d) the average collision length λe = vTe/νe for electrons in this plasma.What is the ratio of each of these lengths to the mean interparticle spacing? /

2.2 Consider the length scales relevant for electron Coulomb collision processes ina laser-produced electron-proton plasma that has ne = 1029 m−3 and Te = Ti

= 1 keV. Calculate: a) the distance of closest approach bmin; b) the averageinterparticle spacing; c) the maximum interaction distance bmax; and d) theaverage collision length λe = vTe/νe for electrons in this plasma. What is theratio of each of these lengths to the mean interparticle spacing? /

2.3 Estimate the time scales relevant for electron Coulomb collision processes in theearth’s ionosphere at a point where ne = 1012 m−3, Te = 1 eV. For simplicity,use a Lorentz collision model and assume the ions have Zi = 1 and Ti Te.Calculate the times for: a) a typical Coulomb interaction at the average inter-particle spacing; b) an electron to traverse the Debye shielding cloud; and c) theaverage electron collision time τe = 1/νe. How long (or short) are each of thesetimes compared to the “plasma period” ω−1

pe ? /



2.4 Consider the “slowing down” of an electron using the Lorentz collison model. a)Show that an electron with an initial velocity v0 loses momentum exponentiallyin time at a decay rate given by ν(v0). b) What is the electron energy after itsmomentum is totally depleted? c) Calculate the distance the electron travelsin its original direction of motion while losing its momentum. d) Evaluatethe momentum decay rate and distance the electron travels for a plasma withZi = 5, ne = 1019m−3, Te = Ti = 100 eV, and an initial electron “test particle”energy of 1 keV. [Hint: Be careful to distinguish between an electron’s velocity(a directional, vector quantity) and its speed (a scalar quantity).] //

2.5 In Section 2.1 we derived the momentum impulse ∆v for a single Coulombcollision in the Lorentz collision model by integrating me(dv/dt) = qeE over anunperturbed “straight-line” trajectory to determine ∆v⊥ and then used ∆v‖ −∆v⊥ · ∆v⊥/2v. Show that the result for ∆v‖ given in (2.6) can be obtaineddirectly by integrating E[x(t)] along a perturbed electron trajectory x(t) = x+xthat includes the first order effects x due to the E field of the ion on the electrontrajectory. [Hint: First calculate the perturbed velocity v, and then make useof the fact that

∆v‖ = ez · qe

me

∫ ∞

−∞dt E(x + x) =

qe

me

∫ ∞

−∞dt ez ·

[E(x) + x · ∂E

∂x

∣∣∣∣x

+ · · ·]

=e

me

∫ ∞

−∞dt v ·∇

∫ t

−∞dt′ Ez(t

′) =Zie

2

4πε0mev

∫ ∞

−∞dt v‖

t

(b2 + v2t2)3/2.] ///

2.6 Show that the Lorentz collision frequency ν can be derived from the Rutherforddifferential scattering cross-section dσ/dΩ given in (??) in Appendix A.1 andProblem 2.24 as follows. a) First, show that for the Lorentz collision modelthe scattering angle ϑ is given for typical small-angle Coulomb collisions byϑ 2Zie

2/(4πε0mv2b) = 2bclmin/b, and that the differential scattering cross-

section is dσ/dΩ = |(b db dϕ)/(dϕ d cos ϑ)| (b/ϑ)|db/dϑ| = 4(bclmin)2/ϑ4. b)

Then, determine the effective cross-section for momentum transfer σm, which isdefined by

σm ≡∫

dΩ (dσ/dΩ) (1 − cos ϑ).

In performing this integral discuss the maximum, minimum scattering anglesϑmax, ϑmin in terms of the bmin, bmax interaction distances. c) Finally, showthat ν = ni σmv yields the Lorentz model collision frequency given in (2.14).///

2.7 Use the full Rutherford differential scattering cross-section and the procedureoutlined in the preceding problem to give an alternative derivation of the Lorenzcollision frequency that takes into account classical “hard,” or large angle colli-sions; i.e., do not initially assume ϑ << 1. ///

2.8 The Lorentz collision frequency ν can also be determined from the Langevinequation

medv

dt= −meν v + ∆F(t)

in which −meν v is the dynamical friction force, and ∆F(t) is a stochastic force,which for Coulomb collisions is that given by (2.2). a) Assuming ν is constant in



time, use an integrating factor eνt in solving the Langevin equation to determinethe particle velocity v(t) after its initialization to v0 at t = 0. b) Next, calculatethe ensemble average of the electron kinetic energy as a function of time, andshow that it yields

〈v2(t)〉 v20 e−2νt +

1 − e−2νt

2ν

∫ ∞

−∞dτ

〈∆F(0) · ∆F(τ)〉m2

e

.

c) Show that for times long compared to the duration of individual Coulombcollisions but short compared to the momentum loss collision time (b/v <<t << 1/ν), the electron kinetic energy is constant through terms of order νtwhen

ν ≡ 1

2m2ev2

∫ ∞

−∞dτ 〈∆F(0) · ∆F(τ)〉.

Thus, ν is proportional to the autocorrelation function of the Coulomb colli-sion force ∆F. d) Next, show that for Coulomb collisions between electronsand a stationary background of randomly distributed ions this formula yieldsthe Lorentz collision frequency given by (2.14). e) Finally, making use of theequilibrium (νt → ∞) statistical mechanics (thermodynamics) property that forrandom (Brownian) motion due to a stochastic force ∆F the ensemble-averagekinetic energy me〈v2〉/2 of a particle is Te/2, note that this last result yields

ν =me

2 Te

∫ ∞

−∞dτ

〈∆F(0) · ∆F(τ)〉m2

e

, fluctuation-dissipation theorem,

which is also related to Nyquist’s theorem for noise in electrical circuits. [Hints:a) Here, 〈f〉 ≡ ni

∫d3x f = ni

∫ ∞−∞ dz

∫b db

∫ 2π

0dϕ f ; b) The electron position

for an ion at x = zez at time t = 0 is x = b(ex cos ϕ+ ey sin ϕ)− (z− vt)ez.] ///

2.9 Consider cumulative small-angle collisional interactions of a test star of mass Mt

and velocity v in a galaxy for which the gravitational force between it and groupsof field stars with density nf and mass Mf is given by [cf., (??) in Appendix A.6]FG = −GMtMf (xt − xf )/|xt − xf |3 — an attractive inverse square law forcelike that for Coulomb collisions of oppositely charged particles. a) Develop amodel for collisions of this test star with other, background stars and show thatthe reference gravitational collision frequency analogous to (2.14) is

νG =4πG2nfM2

f

v3ln

[D0u2

G(Mt + Mf )

],

where bmax is taken to be D0, the mean distance between stars (∼ n−1/3f ) and

bmin has been taken to be the minimum interaction distance given by an expres-sion analogous to that implied by (2.9). b) Estimate the time in (years) for thevelocity of our sun to scatter through 90, assuming that our sun is a typicalstar in our galaxy which has a mass of 2×1030 kg, a velocity of 20 km/s and amean separation from other stars in our galaxy of 1 parsec ( 3×1013 km). c)Will you be concerned about this scattering process in your lifetime? ///

2.10 Estimate the diffusion coefficient D for a Fick’s law representation (Γ = −D∇n)of the particle flux Γ due to electron-ion Coulomb collision effects in an inho-mogeneous plasma as follows. For simplicity, use a Lorentz collision model and



assume the electrons have a density gradient but no temperature gradient. a)Show by balancing the pressure gradient force density −∇pe = −Te∇ne againstthe frictional drag induced by Coulomb collisions that D = Te/meνe = νeλ

2e/2.

b) Estimate the magnitude of this diffusion coefficient for the plasma describedin Problem 2.3. c) Compare the result to the viscous diffusion coefficient formolecules of air at the earth’s surface. d) Why is the diffusivity of chargedparticles in a plasma so much larger? //

2.11 a) Estimate the D-D fusion reaction rate (use σfv 10−17 cm3/s) for a plasmawith Te = Ti = 40 keV, and ne = 1020 m−3. b) Compare this rate to typicalelectron and ion Coulomb collision rates in this fusion plasma. c) How manytimes do electrons and ions scatter through 90 during a typical D-D fusion inthis plasma? d) How far do typical electrons and ions travel in a characteristicfusion reaction time? /

2.12 Consider the angular scattering of a beam of 100 eV electrons introduced intoan Argon laboratory plasma that has Te = 3 eV, Ti = 1 eV, Zi = 3 and anelectron density ne = 1019 m−3. Using the Lorentz collision model, estimatethe distance over which the beam electrons are: a) scattered through an angleof 6; b) scattered via small angle collisions through an angle of about 90; andc) deflected 90 via hard collisions. Finally, d) estimate the angle ∆ϑ throughwhich a beam electron is scattered in a typical Coulomb interaction when theimpact parameter b is given by the mean interparticle spacing of ions. //

2.13 Determine the energy at which electrons “run away” in response to an electricfield in an impure plasma as follows. Assume that a nearly Maxwellian plasma(Ti ∼ Te) is composed of electrons and various species of ions with charge Zi,for which charge neutrality requires ne =

∑i niZi. Calculate the frictional

drag force on electrons in the high energy tail (mev2/2 >> Te) of the electron

distribution. Show that the energy at which electrons run away is given by(2.27) with Zi now replaced by the Zeff defined in (2.43). Also, estimate thefraction of electrons that are runaways for |E|/ED = 0.1 and Zeff = 2. //

2.14 As electrons become relativistic (v → c) the dynamical friction force decreasesless rapidly than the 1/v2 indicated in (2.25) and Fig. 2.6. In fact, it becomesnearly constant for γ ≡ (1− v2/c2)−1/2 >> 1. Then, if the electric field is weakenough, there are no runaway electrons. Determine the dynamical friction forceon relativistic electrons in a nonrelativistic plasma composed of electrons andions of charge Zi as follows. a) First, show that the change in perpendicularmomentum (p ≡ γ mev) in a single Coulomb collision is given by

∆p⊥ = − 2Zie2

4πε0bv(ex cos ϕ + ey sin ϕ).

b) Next, use the relativistic form of the total particle energy (ε =√

m2ec4 + p2c2)

to show that for Coulomb scattering (constant energy) collisions between highenergy electrons and background electrons or ions of mass mi the change inparallel momentum is

∆p‖ − ∆p⊥ · ∆p⊥2p

[1 + γ

me

mi

].

c) Show that the frictional force induced by Coulomb collisions of the high energy



electron with the background plasma is thus

〈F‖〉 − 4π nee4 lnΛ

4πε02mev2

(1 +

1 + Zi

γ

).

d) Finally, show that for a weak electric field satisfying

|E|/ED < 2Te/(mec2),

no runaway electrons will be produced in the plasma. ///+

2.15 Estimate the electric field strength at which the entire electron distributionfunction runs away as follows. a) First, assume the ions are at rest and theelectrons are described by a flow-shifted Maxwellian as defined in (2.116). b)Then, transform to the electron rest frame where V = 0. In this frame the ionsall have a velocity −V. c) Show that the frictional force on a test ion is given by

(mi/me) mi νi/e0 (v) ψi/e(x)V in which xi/e = V 2/v2

Te. d) Find the maximum ofthis frictional force as a function of V/vTe (cf., Fig. 2.8). e) Then, use the factthat this frictional force must be equal and opposite to the maximum force onthe electron distribution to estimate the critical electric field strength (in termsof the Dreicer field) for total electron runaway. f) Also, show that at this electricfield strength an average electron is accelerated to roughly its thermal speed inan appropriate electron collision time. //+

2.16 At what electron temperature is the electrical resistivity of an electron-protonplasma with lnΛ ∼ 17 the same as that of copper at room temperature for whichη 1.7×10−8 Ω ·m? /

2.17 In a typical university-scale tokamak experiment an electron-proton plasma withlnΛ ∼ 17 is heated to a temperature of about 300 eV by the joule or “ohmic”heating induced by an electric field of about 0.5 V/m. a) What current density(in A/cm2) does this electric field induce in such a plasma? b) What is the jouleheating rate (in W/cm3)? /

2.18 Determine the plasma electrical impedance to an oscillating electric field asfollows. a) First, assume a sinusoidal electric field oscillating at a (radian)frequency ω: E(t) = E e−iωt. b) Then, solve an appropriate electron fluidmomentum density equation and show that the frequency-dependent electricalconductivity can be written as

σ(ω) =nee

2

me(νe − iω).

c) Over what frequency range is the plasma resistive (dissipative, real) and overwhat range is it reactive (imaginary)? d) What frequency ranges (in Hz) arethese in the earth’s ionosphere for the parameters of Problem 2.3? //

2.19 The plasma electrical conductivity is modified in a plasma with neutral par-ticles. Add a neutral friction force −meneνenVe, where νen = nnσenv is theMaxwellian-averaged electron-neutral collision frequency, to the right of (2.29)and show that in equilibrium the modified electrical resistivity is given by

η =me(νe + νen)

nee2. //



2.20 Determine the neutral density range over which the effects of neutral particleson the electrical conductivity can be neglected using the result given in thepreceding problem as follows. The reaction rate σenv for ionization of atomichydrogen by electrons is approximately (to within about a factor of two)

σenv 1.5×10−8 cm3/s for 10 eV ≤ Te ≤ 104 eV.

a) How small must the ratio of the neutral to electron density (nn/ne) beto neglect electron-neutral collision effects in an electron-proton plasma forTe = 10, 102, 103, and 104 eV? b) Explain why this density ratio varies so dra-matically with electron temperature. /

2.21 In high neutral pressure, low temperature, partially ionized plasmas (e.g., in the“glow discharge” in fluorescent light bulbs), electron-neutral collisions competewith Coulomb collisions. In particular, they can become dominant in the highenergy tail of the electron distribution function, thereby causing it to effectivelyvanish for energies above a “cut-off” energy. Estimate the cut-off energy fora Te = 3 eV, ne = 1010 cm−3 electron-proton plasma that has a hydrogenneutral density determined by a 3 mm Hg filling pressure, assuming an electron-ionization rate coefficient σenv = 10−10 cm3/s for this Te = 3 eV plasma. /

2.22 a) Sketch the variation of the energy transfer rate Qi in (2.38) from electronsto ions in a Maxwellian electron-proton plasma as a function of Te/Ti. b) Findthe value of Te/Ti at which the maximum energy transfer occurs. c) Explainphysically why the energy transfer rate decreases for increasing Te/Ti >> 1. /

2.23 Consider the thermal equilibration of an electron-ion plasma with Te > Ti. a)Eliminating Ti in favor of the final temperature T∞ = (Te + Ti)/2, show that inthe absence of joule heating, Eq. (2.41), which governs the electron temperatureevolution, can be reduced to

dz

dt= − z − 1

τ∞ z3/2, τ∞ =

mi

me

τe

4

1

z3/2=

mi

me

4πε02 3 m1/2e T

3/2∞

16√

2π neZie4 lnΛ

in which z ≡ Te/T∞. b) Integrate this equation to obtain in general

− t

τ∞= ln

∣∣∣∣z1/2 − 1

z1/2 + 1

∣∣∣∣ +2

3z3/2 + 2 z1/2 + C,

where C is a constant to be determined from the initial conditions. c) Estimatethe temporal range over which Te decays exponentially in time toward T∞ andindicate the decay rate. d) Discuss the relationship of this decay rate to a simpleone derivable from (2.41) with Te fixed and τe = constant. //

2.24 a) Utilize the Rutherford differential scattering cross-section

dσ

dΩ=

q2sq2

s′

4u4m2ss′

1

sin4 ϑ/2, tan

ϑ

2=

qsqs′

mss′u2b=

bclmin

b,

in which ϑ is the scattering angle, to give an alternate derivation of the frictionaldrag and velocity diffusion coefficients 〈∆v〉s/s′/∆t and 〈∆v ∆v〉s/s′/∆t definedin (2.59), (2.60) that takes into account classical “hard,” or large angle collisions.b) Show that the results reduce to those given in (2.64) and (2.65) in the limitlnΛ >> 1. [Hint: bdb dϕ = dσ = (dσ/dΩ) dΩ = (dσ/dΩ) sin ϑ dϑ dϕ and afterthe collision the test particle velocity in the center-of-momentum frame is givenby u + ∆u = (ex sin ϑ cos ϕ + ey sin ϑ sin ϕ + ez cos ϑ) u.] ///+



2.25 a) Show that the rate of momentum and energy loss of a test particle of species sby collisions with background particles having an arbitrary velocity distributionfs′(v) can be written, in analogy with electrostatics, as

msdv

dt= −Qs

∂Φ

∂v,

d

dt

(msv

2

2

)= Qs

(−v · ∂Φ

∂v− mss′

msΦ

),

where the analogous potential Φ and charge Qs are defined by

Φ(v) ≡ −Hs′(v) = − ms

mss′

∫d3v′ fs′(v

′)

|v − v′| , Qs ≡ msΓss′ .

b) Show that for an infinitely massive, immobile background (Lorentz collisionmodel) these formulas reduce, in analogy with an electrostatic point charge Qs

at the origin of velocity space, to

msdv

dt= −Qs

ns′

v3v = −msν(v)v,

d

dt

(msv

2

2

)= 0. ///+

2.26 Use the formulas derived in the preceding problem to consider collisions of atest particle s with a spherically symmetric velocity distribution of backgroundparticles that all have the same speed V : fs′(v) = (ns/4πV 2)δ(v−V ). a) Showthat for these collisional processes the analogous potential Φ is given by

Φ =ms

mss′ns′

1/V, v < V,1/v, v > V.

b) Calculate the momentum and energy loss rates for test particle (s) speedsv < V , and v > V . c) Discuss the results obtained in analogy with electrostatics,and in particular explain by analogy with electrostatics why the test particle sexchanges no momentum with the background when v < V. ///+

2.27 In the original work on stellar collisions Chandrasekhar introduced the function

G(z) ≡ Φ(z) − zΦ′(z)

2z2, where Φ(z) ≡ 2√

π

∫ z

0

dy e−y2= erf (z).

Show that this “Chandrasekhar function” G is related to the Maxwell integralψ by

G(√

x ) =ψ(x) v2

Ts′

2v2=

ψ

2x,

and hence that the various collision frequencies for Coulomb collisions of a testparticle of species s with a Maxwellian distribution of background particles ofspecies s′ can be written as

νs/s′

S = νss′

(2Ts

Ts′

) (1 +

ms′

ms

)G(v/vTs′)

(v/vTs),

νs/s′

⊥ = 2 νss′ [Φ(v/vTs′) − G(v/vTs′)] (v3Ts/v3),

νs/s′

‖ = 2 νss′G(v/vTs′)(v3Ts/v3)

in which

νss′ ≡ νs/s′

0 (v)v3

v3Ts

=4π ns′q

2sq2

s′ lnΛss′

m2sv

3Ts

is a reference collision frequency which has the advantage of being independentof the particle speed v. //+



2.28 Discuss the changes that occur in Problem 2.4 when general Coulomb collisionsare allowed for instead of the Lorentz collision model. In particular, indicate theapproximate magnitude and direction of changes in the momentum decay rate,the distance traveled and the rate of energy transfer to the ions for the plasmaparameters indicated. //+

2.29 Consider the Coulomb collision scattering processes on a D−T fusion-producedα particle (ε0 = 3.52 MeV) in a thermonuclear plasma (50% D, 50% T , Te = Ti

= 10 keV, ne = 1020 m−3). a) Calculate the collision rates for slowing down(νS), perpendicular diffusion (ν⊥), parallel diffusion (ν‖) and energy loss (νε) ofthe α particle in the plasma. b) Discuss which collisional processes (α/e, α/D,or α/T ) dominate each of these rates and why. c) How long will it take such afusion-produced α particle to deposit half of its energy in the plasma? /+

2.30 The direction of energy transfer in Coulomb collisions of test particles witha background plasma depends on the test particle energy and other parame-ters. Estimate the particular test particle energies at which there is no energyexchange between test electrons, test protons and a Maxwellian backgroundelectron-proton plasma that has Te = Ti, but Te of the same order of magnitudeas Ti. /+

2.31 Evaluate the ratio of energy diffusion to energy loss for electrons on the highenergy tail (mev

2/2 > Te) of a Maxwellian electron distribution function. Usethis result to: a) find the probability that a tail electron will gain rather thanlose energy; and b) discuss phenomenologically how the energy dependence ofthe Maxwellian tail of the electron distribution function is determined. //+

2.32 Consider a test electron with an energy of 10% of the electron temperature ina plasma. a) Show that the electron gains energy approximately linearly withtime from a Maxwellian background of electrons. b) Estimate the time required(in terms of τe) for the test electron to acquire an energy approximately equalto the plasma electron temperature. //+

2.33 It is of interest to drive the current in a tokamak plasma by means other thanvia the usual inductive electric field. Thus, one often seeks [see N.J. Fisch,Rev. Mod. Phys. 59, 175 (1987)] to drive currents by radiofrequency wavesthat impart momentum to a selected group of suprathermal (v >> vTe) elec-trons. Coulomb collision effects relax these suprathermal electrons back intothe background distribution and thus limit the current produced. Estimatethe steady-state “efficiency” J/Pd for such a process as follows. Consider asuprathermal electron that has a large velocity v0 relative to the thermal speedof background electrons which will be assumed to have a Maxwellian velocitydistribution. Assume the ions in the plasma have charge Zi, a Maxwellian dis-tribution, and a comparable temperature to the background electrons. a) Showthat the z-directed velocity component and speed of the suprathermal electronare governed by dvz/dt = − (2 + Zi)ν

e/e0 (v) vz and dv/dt = − ν

e/e0 (v) v, respec-

tively. b) Combine these equations and show that for vTe < v < v0 their solutioncan be written as vz = vz0[v(t)/v0]

2+Zi . c) Then, show that the current inducedin the plasma by one suprathermal electron over the time it takes for it to slowdown to the thermal energy of the background electrons is

Jz = qe

∫dt vz qevz0

νe/e0 (v0)

1

5 + Zi.



d) However, show that the sum over an isotropic distribution of such suprather-mal electrons yields no net current in the plasma. Next, consider the effect ofa small momentum input via radiofrequency waves at v0 = v0ez that increasesthe electron velocity to (v0 + δv) ez where δv << v0. e) Calculate the ratio ofthe perturbed current δJ to the power (energy) input δPd from the wave neededto produce this change and show it is given by

δJ

δPd=

4

5 + Zi

qe

me νe/e0 (v0) v0

∝ v20 .

f) Also consider wave momentum input in directions perpendicular to the initialsuprathermal velocity direction z with vz0 = 0; show that it too can inducecurrent in the z direction (with reduced efficiency) and explain physically howthis is possible. g) Finally, show that when the current and power dissipated

are normalized to neqevTe and nemev2Teν

e/e0 (vTe), respectively, the normalized

steady-state “current-drive efficiency” for z-directed momentum input is givenby

J

Pd=

4

5 + Zi

(v0

vTe

)2

.

h) In what range of speeds is this type of current drive most efficient? i) Howdoes it compare to the normalized efficiency J/Pd for the usual ohmic current-drive by an electric field with |E| << ED? ///+

2.34 a) Show that for test electrons with energies much larger than the electrontemperature in an electron-ion plasma with Ti ∼ Te that νe

ε [2/(2 + Zi)]νeS

and that the velocity friction and diffusion coefficients can be written to lowestorder as

〈∆v〉e∆t

= −(2 + Zi) νe/e0 v,

〈∆v ∆v〉e∆t

= νe/e0 [(1 + Zi)(v

2I − vv) + (v2Te/v2)vv].

b) How large are the most significant terms that have been neglected for thebeam electrons in Problem 2.12 /+

2.35 A hydrogen ice pellet is injected into a hot Maxwellian electron-proton plasmawith Te = Ti = 2 keV and ne = 5×1019 m−3. Assume the pellet doubles theplasma density. a) Neglecting the energy expended in the ionization processes(∼ 30–100 eV), what is the temperature the hot plasma and the cold, pelletproduced plasma (at say 10 eV) will equilibrate to? b) Estimate the time scaleson which the electron and ion plasma components become Maxwellians at theirnew temperatures, and equilibrate to a common temperature. //+

2.36 It is usually difficult to measure directly the ion temperature of hydrogenic ions(protons, deuterons, tritons) in a hot plasma. However, it is often possible todetermine the temperature of trace amounts of impurity ions in hot plasmasby measuring the Doppler broadening of the line radiation produced by the de-excitation of excited, highly ionized states of the impurity ions. Show that theimpurity temperature is close to the ion temperature in Maxwellian plasmaswith comparable electron (e), ion (i), and impurity (Z) temperatures as follows.Assume for simplicity that the impurities are heated only through Coulomb



collisions with the hot plasma electrons and the dominant, hydrogenic ions. a)First, write down an energy balance equation for the impurity species. b) Next,show that in equilibrium since the impurity mass is much closer to that of thehydrogenic ions than to that of the electrons, the impurity temperature can bewritten as

TZ Ti −(ν

Z/eε /ν

Z/iε

)(Ti − Te)

= Ti −(

ne

niZ2i

) (me

mi

)1/2 (Ti

Te

)3/2

(Ti − Te).

c) Finally, estimate the difference between the impurity and ion temperaturesfor Ti = 10 keV, Te = 5 keV in a predominantly electron-deuteron plasma. //+

2.37 Using the formulas in Section 2.10, show that the total electron collision fre-quency νe for a muliple ion species, impure plasma is as implied in (2.44). /+

2.38 Using the formulas in Section 2.10, show that the impurity factor fi given in(2.47) is correct. /+

2.39 Using the formulas in Section 2.10, show that the impurity factor fi−e given in(2.49) is correct. /+

2.40 a) Show that for a fast ion of mass mf and charge Zfe with vTi << vf << vTe

that is slowing down in a plasma composed of a mixture of ions of mass mi andcharge Zie the slowing down is governed by (2.133), with the fast ion slowingdown time τS unchanged, but that the critical speed vc and energy εc are nowgiven by

v3c ≡ 3

√π

4

me

mf[Z]mv3

Te, εc 15 Te

(√mf

mp[Z]m

)2/3

, [Z]m ≡∑

i

niZ2i /ne

(mi/mf ).

Here, [Z]m is a mass-weighted effective Zi for energy transfer processes in animpure plasma. b) For the parameters of Problem 2.29, what fraction of thealpha particle energy is transferred to the plasma ions and electrons? //+

2.41 Estimate the distance traveled by a fusion-produced 3.52 MeV α particle inslowing down in an infinite, homogeneous thermonuclear plasma as follows. a)First, calculate the alpha particle energy loss rate per unit distance traveled(dε/dz) in terms of quantities derived in Section 2.11. b) Then, integrate toobtain the total distance z the alpha particle travels in slowing down from itsinitial velocity to the thermal velocity of the background plasma. c) Finally,estimate the distance traveled for the parameters of Problem 2.29 using theformulas developed in Problem 2.40. //+

2.42 Determine the current driven in a tokamak plasma by the fast ions introducedby energetic netral beam injection [T. Ohkawa, Nuclear Fusion 10, 185 (1970)]as follows. Consider introducing a beam of fast ions of density nf , and chargeZfe with velocity Vf such that vc << Vf << vTe. a) Calculate the relative flowVe − Vi induced by the beam ions for a plasma having ions of charge Zi anddensity ni. Assume nf << ne and that the beam ions transfer their momentumonly to plasma electrons for simplicity. b) Show that the net current in theplasma due to the three plasma components is given by J = nfZfeVf (1−Zf/Zi).c) Explain physically why there is no current when the fast beam ions havethe same charge as the background ions, which is sometimes called a plasma



shielding effect. [Hint: The beam momentum input to the electrons is the sameas the loss of fast ion momentum by collisions with the Maxwellian electronbackground, namely mfnf ν

f/eS Vf .] //+

2.43 For the case where there are no direct particle losses during fast ion slowingdown (τcx → ∞), show that the fraction of fast ion energy transferred to thebackground plasma ions can be written as

Gi =2

x2

[−1

6ln

(1 + x)2

1 − x + x2+

1√3

arctan2x − 1√

3+

1√3

arctan1√3

]in which x ≡ v0/vc. //+

2.44 For the parameters of Problem 2.29, calculate the fast ion slowing down andenergy transfer characteristics for a D-T fusion-produced alpha particle: a) thecritical energy εc, and b) the total lifetime from birth to thermalization in thebackground D-T plasma. and c) the fraction of the alpha particle energy thatwill be transferred to the background plasma electrons and ions. /+

2.45 In the early, 1970s experiments that injected energetic neutral beams into toka-mak plasmas there was concern that charge-exchange of the injected fast ionswith neutrals in the plasma would cause the fast ions to be lost from the plasmabefore they could deposit their energy in the background electrons and ions.Consider 40 keV deuterium beam injection into a ne = 2×1013 cm−3, Te = 1.3keV, Ti = 0.5 keV electron-deuteron plasma. a) What are the critical energyεc, slowing down time τS and fast ion lifetime τf for fast ions in this plasma?b) In the absence of charge-exchange losses, what fraction of the fast ion en-ergy is transferred to the backgound plasma electrons, and ions? c) Assuminga charge-exchange cross section of σcx = 7×10−16 cm−2 and a neutral densityof nn = 2×108 cm−3, what is τcx at the initial fast ion energy? (Since σcxvis approximately constant below 20 keV per nucleon, τcx is nearly independentof energy.) d) How much does charge-exchange reduce the fractions of fast ionenergy transferrred to the background plasma electrons and ions? e) Whichtransfer fraction is affected the most? f) Why? /+

2.46 Energetic neutral atoms from neutral beams are absorbed in plasmas via theatomic collision processes of electron ionization, proton ionization and chargeexchange. The ionization processes are ∼ σion/(σcx + σion) probable (∼ 30% forthe parameters of the preceding problem). They produce an electron whose ini-tial speed is approximately the same as the injected fast neutral atom but whosekinetic energy is much lower. a) For the parameters of the preceding problem,what is the energy of such electrons? Since such electrons are born with lowenergies and take energy from the backgound plasma as they are heated to theplasma electron temperature, they represent an initial heat sink. b) Approxi-mately how long does it take for these electrons to be heated by the plasma tothe background electron temperature of 1.3 keV? (Hint: See Problem 2.32.) c)How does this time compare to the fast ion slowing down time τS? d) Abouthow long does it take for the injected energetic neutral beam to add net energyto the plasma? //+

2.47 In the “wet wood burner” approach to controlled thermonuclear fusion [J.M.Dawson, H.P. Furth and F.H. Tenney, Phys. Rev. Lett. 26, 1156 (1971)] it isproposed to obtain energy multiplication through fusion reactions of energeticdeuterons as they slow down in a background triton plasma. a) Show that the



energy multiplication factor F , which is defined as the ratio of fusion energyproduced to the initial deuteron energy ε0, can be written as

F =

(εf

εo

)(nτS)

∫ v0

0

v3 dv

v3 + v3c

σf (v) ≡(

εf

ε0

)(nτS) σfv

in which εf is the energy produced per fusion and σf (v) is the speed- (energy-)dependent fusion cross section. b) For D-T fusion with εf = 22.4 MeV (17.6MeV from the reaction products and 4.8 MeV from assuming energy multi-plication by neutrons absorbed in a surrounding lithium blanket) and σfv 2.8×10−22 m3/s for 120 keV deuterons, find the minimum electron temperatureat which energy multiplication is possible. c) For this “critical” temperature,what is the probability that a deuteron will undergo a fusion reaction during itsslowing down? //+

2.48 Consider the angular or perpendicular diffusion of a fast proton with energy 40keV injected into a Maxwellian electron-proton plasma that has Te = Ti = 1 keVand ne = 3×1019 m−3. a) Estimate the time at which Coulomb collisions scatterthe velocity space angle of the fast proton through δϑ = δv⊥/v = 0.1 radian(∼ 6). b) Compare this time to the fast ion slowing down time τS defined in(2.130). c) Discuss the physical reason why one of these times is much shorterthan the other. /+

2.49 Consider the proposition that an electric field E is applied to keep fast ionswith vTi << v << vTe from slowing down in a plasma. a) First, calculatethe momentum loss rate of the fast ion and find its minimum as a function ofthe fast ion speed v. b) Next, calculate the minimum electric field [in termsof the Dreicer field defined in (2.28)] required to prevent fast ions from slowingdown. c) Then, discuss the degree to which such an electric field would causerunaway electrons. d) Estimate the rates of perpendicular and parallel diffusionat the fast ion speed at which the minimum momentum loss rate occurs. e)Finally, discuss the effects these processes might have on the proposed scheme(cf., Fig. 2.7). //+

2.50 a) Show that for a fast ion slowing down in the plasma described in Problem 2.40the dynamical friction and diffusion coefficients can be written to lowest orderas

〈∆v〉f∆t

= −(

[Z]m + (Zeff + [Z]m)v3

c

v3

)v

τS

〈∆v ∆v〉f∆t

=1

τS

(Zeff

v3c

v3(v2I − vv) +

2[Te + Teff(v3c/v3)]

mf

vv

v2

)in which the angular scattering Zeff is defined in (2.43), the energy transfer ormass-weighted [Z]m is defined in Problem 2.40 and the effective ion temperatureis defined by

Teff ≡ 1

[Z]m

∑i

niZ2i /ne

(mi/mf )Ti.

b) How large is the most significant term that has been neglected in theseapproximate results and where does it contribute for fast ions slowing downfor the situation described in Problem 2.48? //+



2.51 Write a Monte Carlo type computer code for exploring the Coulomb scatteringof energetic test electrons in an electron-proton plasma. Use it to determinenumerically the answers to parts a) and b) of Problem 2.12. //+

2.52 Write a Monte Carlo type computer code for exploring the Coulomb collisionprocesses for fast ions slowing down in an electron-ion plasma. Use it to deter-mine numerically the answers to the questions in Problem 2.48. ///+


Coulomb Collisions - CAE Usershomepages.cae.wisc.edu/~callen/chap2.pdf · CHAPTER 2. COULOMB COLLISIONS 3 and consequently deceleration of the test particle’s initial, directed

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