Chapter 2

KINETIC THEORY

2.1 Distribution Functions

A plasma is an ensemble of particles electrons e, ions i and neutrals n withdierent positions r and velocities v which move under the inuence of externalforces (electromagnetic elds, gravity) and internal collision processes (ionization,Coulomb, charge exchange etc.)

However, what we observe is some average macroscopic plasma parameterssuch as j - current density, ne - electron density, P - pressure, Ti - ion temperatureetc.

These parameters are macrsocopic averages over the distribution of particlevelocities and/or positions.

In this lecture we

Introduce the concept of the distribution function f(r,v, t) for a givenplasma species;

Derive the force balance equation (Boltzmann equation) that drives thetemporal evolution of f(r,v, t);

Show that low order velocity moments of f(r,v, t) give various importantmacroscopic parameters;

Consider the role of collision processes in coupling the charged and neutralspeicies dynamics in a plasma and

Show that low order velocity moments of the Boltzmann equation giveuid equations for the evolution of the macroscopic quantities.

30

2.2 Phase Space

Consider a single particle of species . It can be described by a position vector

r = xi+ yj + zk

in configuration space and a velocity vector

v = vxi+ vy j + vzk

in velocity space. The coordinates (r,v) dene the particle position in phasespace.

For multi-particle systems, we introduce the distribution function f(r,v, t)for species dened such that

f(r,v, t) dr dv = dN(r,v, t) (2.1)

is the number of particles in the element of volume dV = dv dr in phase space.Here, dr d3r dxdydz and dv d3v dvxdvydvz f(r,v, t) is a positivenite function that decreases to zero as |v | becomes large.

y

x

z

rdx

dy

dz

v

v

v

v

dv

x

y

z

z

dvxdvy

o o

Figure 2.1: Left: A conguration space volume element dr = dxdydz at spatial

position r. Right: The equivalent velocity space element. Together these two

elements constitute a volume element dV = drdv at position (r,v) in phase

space.

The element dr must not be so small that it doesnt contain a statisticallysignicant number of particles. This allows f(r,v, t) to be approximated by

2.3 The Boltzmann Equation 31

a continuous function. For example, for typical densities in H-1NF 1012 m3,dr 1012 m3 = dr f(r,v, t) dv 106 particles.

Some defnitions:

If f depends on r, the distribution is inhomogeneous If f is independent of r, the distribution is homogeneous If f depends on the direction of v, the distribution is anisotropic If f is independent of the direction of v, the distribution is isotropic A plasma in thermal equilibrium is characterized by a homogeneous, isotropic

and time-independent distribution function

2.3 The Boltzmann Equation

As we have seen, the distribution of particles is a function of both time andthe phase space coordinates (r,v). The Boltzmann equation describes the timeevolution of f under the action of external forces and internal collisions. Theremainder of this section draws on the derivation given in [2].

f(r,v, t) changes because of the ux of particles across the surface boundingthe elemental volume drdv in phase space. This can arise continuously due toparticle velocity and external forces (accelerations) or discontinuously throughcollisions. The collisional contribution to the rate of change f/t of the distri-bution function is written as (f/t)coll.

To account for continuous phase space ow we use the divergence theoremSE.ds =

V.E dV (2.2)

applied to the six dimensional phase space surface S bounding the phase spacevolume V enclosing the six dimensional vector eld E.

Now note that conservation of particles requires that the rate of particle owover the surface ds bounding the element V plus those generated by collisionsbe equal to the rate at which particle phase space density changes with time.If we let V = (v,a) be the generalized velocity vector for our mathematicalphase space (r,v), then the rate of ow over S into the volume element is

Sds. [V f ]

Compare V f with the denition of particle ux (r, t) - a conguration spacequantity [Eq. (2.14)]. Using the divergence theorem, this contribution can bewritten

V

drdv [r.(vf) + v.(af)]

32

where r. is the divergence with respect to r and v. is the divergence operatorwith respect to v. For example

r. i x

+ j

y+ k

z

Since V is an elemental volume, the integrand changes negligibly and canbe removed from the integral. The inclusion of the continuous phase space owterm and the discontinuous collision term then gives the result

f

t= r.(vf)v.(af) +

(f

t

)coll

. (2.3)

The acceleration can be written in terms of the force F = ma applied or actingon the particle species. For plasmas, the dominant force is electromagnetic, theLorentz force

F = q(E + vB) ma. (2.4)Using the vector identity (product rule)

.(fV ) = f .V + f.Vwe can write

v.(vB)f = (vB).vf (why?)Moreover, since r and v are independent coordinates, we nally obtain the Boltz-mann equation

f

t+ v.rf + q

m(E + vB).vf =

(f

t

)coll

(2.5)

Since f is a function of seven variables r, v, t its total derivative with respectto time is

df

dt=

f

t

+f

x

dx

dt+

f

y

dy

dt+

f

z

dz

dt

+f

vx

dvxdt

+f

vy

dvydt

+f

vz

dvzdt

(2.6)

The second line is just the expansion of the term v.rf and the third, the terma.vf . Thus Eq. (2.3) is simply saying that df/dt = 0 unless there are collisions!When there are no collisions, (f/t)coll = 0 and Eq. (2.5) is called the Vlasovequation.

The quantity df/dt is called the convective derivative. It contains terms dueto the explicit variation of f (or any other conguration or phase space quantity)

2.3 The Boltzmann Equation 33

with time (f/t) and an additional term v.f that accounts for variations inf due to the change in location in time (convection) of f to a position where fhas a dierent value. As an example, consider water ow at the outlet of a tap.The water is moving, but the ow pattern is not changing explicitly with time(f/t) = 0. However, the velocity of a uid element changes on leaving the tapdue to, for example, the force of gravity. The ow velocity changes in time as thewater falls because the term (v.)v is nonzero. When the tap is turned o, theow slows and stops due to (f/t) = 0.

What does df/dt = 0 mean? With reference to Fig 2.2, consider a group ofparticles at point A with 2-D phase space density f(x, v). As time passes, theparticles will move to B as a result of their velocity at A and their velocity willchange as a result of forces acting. Since F = F (x, v), all particles at A will beaccelerated the same amount. Since the particles move together, the density atB will be the same as at A, f can only be changed by collisions.

It is important to understand that x and v are independent variables or co-ordinates, even though v = dx/dt for a given particle. The reason is that theparticle velocity is not a function of position - it can have any velocity at anyposiiton.

Collisions

A

B

x

dx

vx

dvxCollisions

Figure 2.2: Evolution of phase space volume element under collisions

34

2.4 Plasma Macroscopic Variables

Macroscopic variables (measurable quantities) are obtained by taking appropriatevelocity moments of the distribution function. It is perhaps not surprising that thedynamical evolution of these quantities can be described by equations obtainedby taking the corresponding velocity moments of the Boltzmann equation. Theseequations (the fluid equations) are derived later in this chapter. For simplicity,and where possible, we drop the subscript denoting the particle species for theremainder of this analysis.

2.4.1 Number density

Integration over all velocities gives the particle number density

n =

f(r,v, t) dv (2.7)

This is the zeroth order velocity moment.Multiply by mass or charge to obtain

mass density : m = m

f(r,v, t) dv (2.8)

charge density : q = q

f(r,v, t) dv (2.9)

2.4.2 Taking averages using the distribution function

The quantity

f(r,v, t) =f(r,v, t)f(r,v, t)dv

=f(r,v, t)

n(r, t)(2.10)

is the probability of nding one particle in a unit volume of ordinary (congura-tion) space centred at r and in a unit volume of velocity space centred at v attime t. The distribution average of some quantity g(r,v, t) is then the value of gat (r,v, t) times the probability that there is a particle with these coodinates

g(r, t) =

dv g(r,v, t) f(r,v, t) (2.11)

or

g(r, t) =1

n(r, t)

dv g(r,v, t) f(r,v, t) (2.12)

2.4.3 First velocity moment

The average velocity is obtained from the rst velocity moment of f .

v(r, t) =1

n(r, t)

dv v f(r,v, t) (2.13)

2.4 Plasma Macroscopic Variables 35

Note that v(r, t) can be a function of r and t though v (a coordinate) is not.This is because f varies with r and t.

Multiplying the average velocity by the particle number density gives theparticle flux

= nv =

dv v f(r,v, t) (2.14)

In a similar fashion, multiplying the ux by the charge density gives the chargeux density or current density for a given species

j = qnv = q (2.15)

2.4.4 Second velocity moment

Taking the second central moment with respect to average velocity v gives thepressure tensor (or kinetic stress tensor)

P (r, t) = m

dv (v v)(v v) f(r,v, t) (2.16)

We dene vr = v v, the random thermal velocity of the particles where v isthe mean drift velocity. Then vrvr is the tensor or dyadic

vrvr =

vxvx vxvy vxvzvyvx vyvy vyvz

vzvx vzvy vzvz

(2.17)

and clearly (vrvr) = (vrvr) so thatP has only six independent components

which we write as

P = m

dv vrvr f(r,v, t)

= mnvrvr (2.18)

Now consider the distribution function f to be isotropic in the system driftingwith velocity v(r, t) (see Fig 2.3). Because f is isotropic in this system, vr = 0and

vrvr = 0 = (2.19)= v2r = v

2r = v

2r/3 (2.20)

(show this result) andP p = nmv2r/3. (2.21)

p is the rate of transfer of particle momentum in any direction. It has dimensionsforce/area or pressure. v2r (or simply v

2 when the plasma is not drifting) is the

36

vy

Beam

Drifting Maxwellianvy

vx

vx

Anisotropicdistribution

Figure 2.3: Examples of velocity distribution functions

mean square thermal speed and vrms =

v2 is the rms thermal speed. Note thatthe average kinetic energy is given from Eq. (2.12) by

Ur 1n

dvr

1

2mv2r f(r,v, t) (2.22)

and that this is therefore related to the pressure by

p =2

3nUr. (2.23)

The average particle kinetic energy is also related to the second velocity momentof the distribution function.

2.5 Maxwell-Boltzmann Distribution 37

2.5 Maxwell-Boltzmann Distribution

So far we have shown how low order velocity moments of the particle distri-bution function are related to macroscopic parameters such as current density,pressure etc. without looking at the details of the distribution function itself. Aparticularly important velocity distribution function is the Maxwell-Boltzmanndistribution, or Maxwellian. It describes the spread of velocities for a gas whichis in thermal equilibrium. Such a system can be described by a simple Gaussianspread of velocities, the half-width being related to the gas temperature. As al-ways, the zeroth moment of such a distribution is the particle number density.Because a Gaussian has even symmetry, the odd moment will vanish unless thedistribution happens to be drifting with respect to the laboratory frame. Thesecond velocity moment is related to the temperature and also, not surprisingly,reveals a link between the gas pressure and temperature. In this section, we intro-duce this important distribution and its variants and discuss some of its physicalramications.

2.5.1 The concept of temperature

For a gas in thermal equilibrium the most probable distribution of velocities canbe calculated using statistical mechanics to be the Maxwellian distribution. In1-D

fM(v) = A exp

(mv

2/2

kBT

)(2.24)

This is homogeneous and isotropic. Using Eq. (2.7), this can be written in termsof the density as

fM(v) = n(

m

2kBT

)1/2exp

(mv

2/2

kBT

)(2.25)

If we dene

vth =

(2kBT

m

)1/2(2.26)

then

fM(v) =nvth

exp

( v

2

v2th

). (2.27)

If we follow this analysis through in 3-D we nd

fM(v) =n

(

vth)3exp

( v

2

v2th

)(2.28)

The function fM describes a 3-D distribution of velocities. It is useful to alsocalculate the distribution of particle speeds gM(v) by averaging the Maxwellian

38

over the velocity directions. The result obtained is

gM(v) = 4n(

m

2kBT

)1/2v2 exp (v2/v2th) (2.29)

It can be shown by dierentiation that the peak of this distribution occurs atv = vth.

Several other important characteristic velocities can be related to temperaturein the case of a Maxwellian distribution. For example, the mean particle speedis calculated in Sec. 2.5.2. Perhaps most important is the rms thermal speed

vrms =

3kBT

m. (2.30)

Relation to kinetic energy

The average particle kinetic energy can be calculated from Eq. (2.22) as

Ur Eav = 1vth

1

2mv2 exp (v2/v2th) dv. (2.31)

It is a standard result that

v2 exp (av2) dv = 12

a3(2.32)

so that we may simplify to obtain

Eav =1

4mv2th =

1

2kBT (2.33)

In three dimensions, it can be shown that

Eav =3

2kBT, (2.34)

implying that the plasma particles possess 12kBT of energy per degree of freedom.

In plasma physics it is customary to give temperatures in units of energy kBT .This energy is expressed in terms of the potential V through which an electronmust fall to acquire that energy. Thus, eV = kBT and the temperature equivalentto one electron volt is

T = e/k = 11600K (2.35)

Thus, for a 2eV plasma, kBT = 2 eV and the average particle energy Eav =32kBT = 3 eV For a 100 eV plasma (H-1NF conditions), T 106K and

vthi 1.6 105m/svthe 6.7 106m/s.

These velocities are in the ratio of the square root of the particle masses.

2.5 Maxwell-Boltzmann Distribution 39

Relation to pressure

The pressure is also a second moment quantity, so that it wil not be surprisingto nd it related to temperature in the case of a Maxwellian distribution. It isinstructive to review the concept of plasma pressure using a simple minded modeland show how the result relates to that obtained using distribution functions.

Consider the imaginary plasma container shown in Fig. 2.4. The number ofparticles hitting the wall from a given direction per second is nAv/2 (only halfcontribute). The momentum imparted to the wall per collision is mv (mv) =2mv so that the force (rate of change of momentum) is F = (2mv)(nAv/2) andthe pressure is p = F/A = nmv2 per velocity component. The energy per degreeof freedom is 1

2mv2 = 1

2kBT so that p = nkBT

A

Figure 2.4: Imaginary box containing plasma at temperature T

If we sum over all plasma species we obtain

p =

nkT (2.36)

That is, the electrons and ions contribute equally to the plasma pressure!Explanation:

By equipartition, 12miv

2thi =

12mev

2the so that vthe = vthi

mi/me whereupon the

momenta imparted per collision are related by mevthe = mivthi

me/mi. Note

however, that the collison rate for electrons is

mi/me faster.If we use distribution functions, we obtain for a Maxwellian

Ur = Eav =3

2kBT [Eq. (2.34)]

p =2

3nUr [Eq (2.22)]

40

Combining these two recovers Eq. (2.36)

2.5.2 Thermal particle ux to a wall

Ignoring the eects of electric elds in the plasma sheath, it is useful to calculatethe random particle flux to a surface immersed in a plasma in thermal equilibrium.This ux is dened as

s = nvs =

dv fM(r,v, t) v.s (2.37)

where s is the normal to the surface under consideration. We assume f isMaxwellian and consider the ux in only one direction, since in both directionsthe nett ux crossing a plane vanishes (why?). In spherical polar coordinates,dv = v2 sin dddv and v.s = v cos and we have

s = 0

dv v3fM(v) /20

d sin cos 20

d

= 0

dv v3fM(v)

= n(

m

2kBT

)3/2 0

dv v3 exp

( mv

2

2kBT

)

= n

(kBT

2m

)1/2

nv/4 (2.38)

where

v 1n

0

dv |v | fM(v)

=

(8kBT

m

)1/2(Maxwellian) (2.39)

is the mean speed of the particles. The ux in a given direction is thus non-zero.

Question: Why doesnt the plasma vessel melt?

1. Low energy ux (E = Eavs)

2. Magnetic bottle (thermal insulation)

3. Pulsed experiments (short duration)

But the heat ux is a problem for fusion reactors.

2.5 Maxwell-Boltzmann Distribution 41

2.5.3 Local Maxwellian distribution

We dene the local Maxwellian velocity distribution as

fLM(v) = n(

m

2kBT

)3/2exp

( mv

2

2kBT

)(2.40)

where now

n n(r, t)T T (r, t)

depend on the spatial coordinates and time. fLM can be shown to satisfy theBoltzmann equation. Though each species in a plasma is characterised by itsown distribution function, the time evolution of these distributions are coupledby collisions between the species and the self-consistent electric eld that arisesthrough particle diusion.

2.5.4 The eect of an electric eld

Till now we have implicitly ignored the inuence of electric elds upon the dis-tribution function. When more than a single particle species (for example ionsand electrons) are considered, however, such elds can arise spontaneously in theplasma due to dierences in the mobility or transport of the two species. Theelectric eld in turn aects the spatial distribution of particle velocities throughthe Lorentz force (we here ignore complications due to magnetic elds). Thiseect is important for the electrons owing to their much greater mobility. Thespatial variation of the electric potential (E = ) suggests the use of theconcept of a local Maxwellian.

In equilibrium (/t = 0) and for B = 0, the Boltzmann equation for elec-trons becomes

v.rfe emE.vfe = 0. (2.41)

It can be veried by substitution that, in thermal equilibrium, the electron dis-tribution function diers from Maxwellian by a factor related to the electricpotential:

fe = fLM exp

(e

kBTe

). (2.42)

The exponent is independent of v so that integration over v gives the Boltzmannrelation

ne = ne0 exp

(e

kBTe

)(2.43)

42

where ne0 is the electron density in the zero potential region and the exponentialterm is known as the Boltzmann factor. Substituting from Eq. (2.28) we have

fe(r,v) = A exp

(mv2/2 + ekBTe

)= A exp (E/kBTe) (2.44)

where E is the sum of the kinetic and potential components of the electronenergy. The expression indicates that only the fastest electrons can overcomethe potential barrier and enter regions of negative potential. In other words,the electron density (zeroth moment of fLM) is depleted in regions where theelectric potential is negative. We shall visit this concept again in relation to theelectric sheath that is established at the boundary between a plasma and a wallin Chapter 3.

2.6 Quasi-neutrality

In Chapter 1 we introduced a number of fundamental concepts and criteria thatcharacterise the plasma state. In this and the following two sections we apply thetools of kinetic theory to obtain a more rigorous treatment of Debye shielding,plasma oscillations and collision phenomena.

The potential at a distance r from an isolated positive test charge Q is

=Q

40r. (2.45)

When immersed in a plasma Q attracts negative charge and repels positive ions.As a result, the number density of ions and electrons will vary slightly around Qbut will be in charge balance for large r (i.e. ne = ni for Z = 1).

The eect of this screening will be to exponentially damp over a distancecomparable to the Debye length

= exp (r/D)

To show this we start with Poissons equation

.E = /0, E = 2 = /0.

is composed of e and i as given by Eq. (2.9) plus the contribution from +Qat r = 0. Over time scales comparable with the plasma frequency, the ions areimmobile and we have i = eni = en0 where n0 is the charge density in thepotential free (unperturbed) region, while for the electrons (c.f. Eq. (2.43))

e(r) = en0 exp [e(r)/kBTe]

2.6 Quasi-neutrality 43

and Poissons equation gives

2(r) en00

[exp (e/kBTe) 1] = Q0

(r).

If the electrostatic potential energy e kBTe then exp (e/kBTe) 1+e/kBTeand

2(r) 12D

= Q0

(r).

For spherical symmetry, only the r variation remains and the equation becomes

1

r2d

dr

(r2

d

dr

) (r)

2D= 0 (r = 0).

It can be veried by substitution that the solution is

=Q

40rexp (r/D). (2.46)

Noting that

(r) n0e2(r)/kBTe + Q(r)= Q

4r2Dexp (r/D) + Q(r),

the total charge is obtained by integration

QT =

(r) dr = 4

dr r2(r)

= Q2D

0

dr r exp (r/D) + Q

(r)dr

= 0. (2.47)

Thus neutralization of the test particle takes place on account of the imbalanceof charged particles in the Debye sphere.

Check: e/kBTe 1 (range of validity)

e

kBTe=

e2

40rkBTeexp (r/D)

=D3

exp (r/D)r

1 for r > D/ (2.48)

This condition is valid for most particles in the Debye sphere.

44

Figure 2.5: Electrostatic Coulomb potential and Debye potential as a function of

distance from a test charge Q.

2.7 Kinetic Theory of Electron Waves

As another example of the importance of a kinetic theory, we consider the de-scription of electron plasma oscillations (waves) in a warm plasma. When thermaleects (a distribution of velocities) are included in the treatment of plasma oscil-lations, it is found that the disturbances, which were found to be pure oscillationsin Sec 1.1.2, do indeed propagate and are strongly damped by a linear collisionlessmechanism known as Landau damping after the Soviet physicist who predicted itmathematically in 1946. The analysis presented here closely follows those givenin [3, 4].

To describe these phenomena, we consider a simple one-dimensional system,ignore collisions and assume no magnetic eld. Because of their much greaterinertia, we regard the ions as providing a xed uniform background at frequenciesnear pe. The Vlasov equation can be written

f

t+ v

f

x+

qE

m

f

v= 0. (2.49)

In the presence of a wave, we allow the equilibrium distribution function f0 to beperturbed an amount f1 with an associated small electric eld E = E1 due to thedisturbance. Substituting f = f0 + f1 into the 1-D Valsov equation, expanding

2.7 Kinetic Theory of Electron Waves 45

and retaining terms up to rst order in small quantities gives

f1t

+ vf1x

+qE

m

f0v

= 0. (2.50)

We assume a plane harmonic form for the response f1 exp (ikx it) to theinitial one-dimensional perturbation, recognizing that may be complex in orderto allow for wave damping. This allows the substitutions

t i (2.51)

x ik (2.52)

to obtain

if1 + ikvf1 + qEm

f0v

= 0. (2.53)

(We have implicitly integrated f over the velocity variables orthogonal to v, sothat f is one-dimensional.) Solving for the perturbation gives

f1 = qEim

f0/v

kv . (2.54)

By analogy with Eq. (2.7) the perturbed density is given by

n1 = qEim

dvf0/v

kv . (2.55)

This in turn gives rise to an electric potential uctuation 1 that can be obtainedusing E1 = 1/x and Poissons equation E/x = qn1/0. Substituting inEq. (2.55) gives the dispersion relation (a relation linking the phase velocity ofthe wave to its frequency and wavenumber)

1 = 2pe

k2

dvf0/v

(v /k) . (2.56)

We have normalized the unperturbed distribution function to the electron numberdensity f0 = f0/n (see Eq. (2.10)) and used Eq. (1.14) for the plasma frequency.Note that the integrand contains a singularity at vph = /k and the integralrequires to be evaluated using contour integration in the complex plane. However,the correct result can also be obtained using the less rigorous approach describedbelow.

Fig. 2.6 shows the distribution f0(v) and the integrand. Assuming vph >>vthe, the latter can be conveniently separated into two parts

(i) the central region dominated by f0/v

(ii) the region near v = vph dominated by the singularity

46

Figure 2.6: The contributions (i) and (ii) noted in the text [3]

Contribution of the central region

In this region, v vph, allowing the integrand to be expanded in powers of v/vphto give for the integral:

1vph

dvf0/v

1 v/vph = 1

vph

dvf0v

1 + v

vph+

(v

vph

)2+

(v

vph

)3+ . . .

1v2ph

dv vf0v

1v4ph

dv v3f0v

(2.57)

where we have truncated the series after the two most signicant terms (the termscontaining even powers of v are zero by the antisymmetry of the integrand). Theseintegrals can be evaluated by parts and substituted into Eq. (2.56) to obtain

1 =2pe2

(1 +

3k2

2v2)

where we have made use of the denition of the mean square particle speed andignored the contribution from the singularity. For f0 Maxwellian, we have

1

2mv2 =

1

2kBT

and the above can be solved for 2 in terms of k2 to obtain (show this)

2 2pe +3kBT

mk2. (2.58)

This is the so-called Bohm-Gross dispersion relation for electron plasma waves(or Langmuir waves). The inclusion of nite electron temperature has contributed

2.7 Kinetic Theory of Electron Waves 47

a propagating component (k-dependent) to the dispersion relation. This resultcan also be obtained using the uid equations (Chapter ??) which describe theevolution of distribution averaged quantities. Some important non-equilibriumphysical eects (such as Landau damping) are lost in this averaging process. Itis the remaining integral over the singularity that reveals this.

Contribution of the singularity

Within a narrow range of velocity around the singularity, we may regard f0/vas constant and remove it from the integral. With u = v /k, the integral canbe written

2pek2

(f0v

)/k

du

u(2.59)

Now aa

du

u= ln a lna = ln a ln [a exp (i + i2n)]

= ln a ln a (i + i2n)= i (2.60)

where we have taken n = 0. Thus the singularity component becomes

i2pe

k2

(f0v

)/k

(2.61)

and the full dispersion relation for electron plasma waves is

1 =2pe2

(1 +

3k2

2v2) i

2pe

k2

(f0v

)/k

(2.62)

Treating the imaginary part (damping term) as small, and neglecting the thermalcorrection, we can use Taylors theorem and rearrange to obtain (Prove thisresult)

= pe

1 + i

2

2pek2

(f0v

)/k

(2.63)

It is left as an exercise to show that if f0 is a one-dimensional Maxwellian, thedamping is nally given by

Im

(

pe

)= 0.22

(pekvth

)2exp

( 12k22D

)(2.64)

where D is given by Eq. (1.15). Since Im() is negative, the wave is damped, theeect becoming important for waves of wavelength smaller or comparable withthe Debye length.

48

2.7.1 The physics of Landau damping

We have noted the appearance of a small electric eld associated with the elec-tron wave and an accompanying disturbance of the distribution from thermalequilibrium. The Boltzmann relation requires that the density varies with theelectric potential according to Eq. (2.43). This is shown schematically in Fig.2.7 where is plotted the potential energy q = e of an electron in an electronplasma wave. Thus, at the wave potential energy peaks, the electron density islower than in the troughs.

Figure 2.7: Diagram showing the electron potential energy e in an electronplasma wave. Regions labelled A accelerate the electron while in B, the electron

is decelerated [3].

Now consider electrons at the potential energy peak moving with velocityslightly faster than the phase velocity of the wave. These electrons will be accel-erate, sacricing potential for kinetic energy. Those particles in the trough willbe decelerated. However, there are more particles in the trough than at the peak,so the net eect is deceleration and the electrons give energy to the wave.

This argument can be inverted, however, by considering electrons movingslightly slower than the wave. The net eect is opposite: the wave gives energyto the electrons. There is remaining, however, a fundamental asymmetry in thatthere are more particles moving slower than the wave velocity vph than goingfaster due to the slope of the distribution function. Thus, overall, there is a netloss of energy by the wave which is damped and a corresponding modication ofthe distribution function around v = vph. There is no dissipation inherent in theprocess. However, collisions will cause the perturbed distribution to relax backto Maxwellian in the absence of a source for the wave excitation which couldmaintain the perturbation.

2.8 Collision Processes 49

2.8 Collision Processes

Collisions mediate the transfer of energy and momentum between various speciesin a plasma, and as we shall see later, allow a treatment of highly ionized plasmaas a single conducting uid with resistivity determined by electron-ion collisions.

Collisions which conserve total kinetic energy are called elastic. Some exam-ples are atom-atom, electron-atom, ion-atom (charge exchange) etc. In inelas-tic collisions, there is some exchange between potential and kinetic energies ofthe system. Examples are electron-impact ionization/excitation, collisions withsurfaces etc. In this section, we consider both types of collision process, withparticular emphasis on Coulomb collisions between charged particles, this beingthe dominant process in a plasma.

A simple expression for the collision term on the right of Eq. (2.5) is the Krookcollision term (

f

t

)coll

= (f fM

)(2.65)

where fM is the Maxwellian towards which the system is tending and is themean (constant) collision time. The model integrates to give

f(t) = fM + (f(0) fM) exp (t/). (2.66)The model is not very good if the masses of the species are very dierent.

2.8.1 Fokker-Planck equation

2.8.2 Mean free path and cross-section

To properly treat the physics of collisions we need to introduce the concept ofmean free path - a measure of the likelihood of a collision event. Imagine electronsimpinging on a box of neutral gas of cross-sectional area A (see Fig 2.8).

If there are nn atoms m3 in volume element Adx the total area of atoms in

the volume (viewed along the x-axis) is nnAdx where is the cross-section andnnAdx A so that there is no shadowing. The fraction of particles makinga collision is thus nnAdx/A - the fraction of the cross-section blocked by atoms.If is the incident particle ux, the emerging ux is = (1 nndx) and thechange of with distance is described by

d

dx= nn. (2.67)

The solution is

= 0 exp (x/mfp)mfp =

1

nn(2.68)

50

dx

Area A n atoms/cm3

x

Incoming particle

Figure 2.8: Particles moving from the left and impinging on a gas undergo colli-

sions.

where mfp is the mean free path for collisions characterised by the cross-section. The physics of the interaction is carried by , the rest is geometry.

The mean free time between collisions, or collision time for particles of ve-locity v is = mfp/v and the collision frequency is =

1 = v/mfp = nnv.Averaging over all of the velocities in the distribution gives the average collisionfrequency

= nnv (2.69)

where we have allowed for the fact that can be energy dependent as we shallsee below.

2.8.3 Coulomb collisions

Coulomb collisions between free particles in a plasma is an elastic process. Letus consider the Coulomb force between two test charges q and Q

F =qQ

40r2 C

r2(2.70)

This is a long range force and the cross-section for inetraction of isolated chargesis innity! It is quite dierent from elastic hard-sphere encounters such asthat which can occur between electrons and neutrals for example. In a plasma,however, the Debye shielding limits the range of the force so that an eectivecross-section can be found. Nevertheless, because of the nature of the force, themost frequent Coulomb deections result in only a small deviation of the particlepath before it encounters another free charge. To produce an eective 90 scat-tering of the particle (and hence momentum transfer) requires an accumulationof many such glancing collisions. The collision cross-section is then calculated bythe statistical analysis of many such small-angle encounters.

2.8 Collision Processes 51

Consider the force Eq. (2.70) on an electron as it follows the unperturbed pathshown in Fig. 2.9 In this picture, only the perpendicular component mattersbecause the parallel compoent of the force reverses direction after q passes Q.Thus F/F = b/r or

F = Cb/r3 =cb

(x2 + b2)3/2

where b is the impact parameter for the interaction. Since x (the parallel coordi-nate) changes with time, we integrate along the path to obtain the net perpen-dicular impulse delivered to q

(mv) =

Fdt = cb

dx

(x2 + b2)3/2dt

dx

F

xq

Q

bImpactparameter

Unperturbed path

Figure 2.9: The trajectory taken by an electron as it makes a glancing impact

with a massive test charge Q

Now dx

(x2 + b2)3/2=

x

b2(x2 + b2)1/2

1/b2 x 1/b2 x

so that

v =2c

mvb. (2.71)

We consider a statistical average over a random distribution of such smallangle collisions. For a random walk with step length s, the total displacementafter N steps is s =

Ns (i.e. (s)2 =

(s)2) where N is the number of

steps. The total change in velocity is thus

(v)2 = N(v)2. (2.72)

52

Now integrate over the range of impact parameters b to estimate the number ofglancing collisions in time t. Small angle colisions are much less likely because ofthe geometrical eect

N(b) = n(2bdb)vt. (2.73)

We combine equations (2.71), (2.72) and (2.73) and integrate over impact param-eter:

(v)2 =8nC2t

m2v

bmaxbmin

db

b. (2.74)

bmin is the closest approach that satises the small deection hypothesis. Weobtain this be setting v = v

bmin =2c

mv2=

2qQ

40mv2. (2.75)

Outside D the charge Q is not felt. We thus take bmax = D and

bmaxbmin

db

b= ln

(Dbmin

)= ln.

ln is called the Coulomb logarithm and is a slowly varying function of electrondensity and temperature. For fusion plasma ln 6 16. One usually setsln = 10 in quantitative estimations.

We are nally in a position to nd the elapsed time necessary for a net 90

deection i.e. (v)2 = v2. Using Eq. (2.74)

(v)2 = v2 =8nc2t

m2vln

and

90 = 1/t =8nq2Q2 ln

16220m2v3

.

For electron-ion encounters, q = e and Q = Ze so

90ei ei = niZ2e4 ln

220m2v3

(2.76)

and ei = nieiv implies that

ei =Z2e4 ln

220m2v4

(2.77)

Lets review this derivation

Perpendicular impulse 1/vb Angular deection v/v 1/v2

2.8 Collision Processes 53

Random walk to scatter one radian (v/v = 1) 1/()2 1/v4

Integrate over b ln .The dependence 1/v4 has very important ramications. A high temperatureplasma is essentially collisionless. This means that plasma resistance decreasesas temperature increases. In some circumstances populations of particles canbe continually accelerated, losing energy only through synchrotron radiation (forexample runaway electrons in a tokamak).

Energy transfer in electron-ion collisions

A pervading theme in plasma physics is me mi. This has consequences forcollisions between the species in that we expect very little energy transfer betweenthe species. To illustrate this, consider such a collision in the centre-of-mass frame(a direct hit).

m M

After

Before

Vv

v0 V0

Figure 2.10: Collison between ion and electron in centre of mass frame.

mv0 + MV0 = 0 = mv + MV momentum

mv20 + MV20 = mv

2 + MV 2 energy

mv20

(1 +

m

M

)= mv2

(1 +

m

M

)eliminate V

v = v0 and V = V0. (2.78)Now translate to frame in which ion M is at rest (initially). In this frame theinitial electron energy is

Ee0 =1

2m(v0 V0)2 = 1

2mv20

(1 +

m

M

)2and the nal ion energy is

Ei =1

2M(2V0)

2 = 2M

(m2v20M2

).

54

Their ratio is given by

ion (nal)

electron (initial)=

2m2v2012mv20M

(1 +

m

M

)2

4mM

(2.79)

and thus

e i E 4meEemi

(2.80)

e e E Ee (2.81)i i E Ei (2.82)

For glancing collisions the energy transfer between ions and electrons is even less.Coulomb collisions result in very poor energy transfer between electrons andions. The rate of energy transfer is roughly (me/mi) slower than the e-i collisionfrequency . On the other hand, energy transfer rate and collision frequency are

the same for collisions between ions ii ei

me/mi:

iiei

=m2ev

3e

m2i v3i

=(memi

)1/2 (mev2emiv2i

)3/2=(memi

)1/2for Te = Ti

2.8.4 Light scattering from charged particles

2.8.5 Nuclear fusion

This is an inelastic process where the nal kinetic energy of the system is muchgreater than the initial kinetic energy. This excess energy can be harnessedto produce power. For example, Fig. 1.7 shows schematically the Deuterium-Tritium reaction upon which present-day fusion reactors are based. To overcomethe Coulomb repulsion requires 400 keV of kinetic energy. Upon fusing, the twoisotopes transmute to a fast neutron and a helium nucleus (-particle). Evenin a plasma whose temperature is only 10 keV, there are sucient particles inthe wings of the distribution to sustain a fusion reactor. The reason is that ina fusion reaction, the energy release is a huge 17.6 MeV - more than enough toaccount for the relatively small number of fusing ions.

Using energy and momentum conservation, we can estimate the relative en-ergies of the reaction by-products. In the centre-of-mass frame we have

m1v1 = m2v21

2m1v

21 +

1

2m1v

21 = 17.6 MeV

2.8 Collision Processes 55

where subscripts 1 and 2 refer respectively to the -particle and the neutron.Eliminating v1 gives

1

2m2v

22

(1 +

m2m1

)= 17.6 MeV.

Substituting m2/m1 = 1/4 shows that 14.1 MeV is carried by the neutron whichescapes the conning magnetic eld to deliver useful energy elsewhere. The fast (3.5 MeV) is conned by the magnetic eld where it delivers its energy to thedeuterium and tritium ions.

For the D-T reaction in a high temperature plasma with n 1 1020 m3and 1029 m2 at 100 keV where v 5 106 m/s, the mean free path for afusion collision is

mfp =1

n= 109 m

and = mfp/v 200 s. In other words, D and T have to be conned for 200s and travel a million km without hitting the container walls! The situation ismitigated by the fact that 17.6 MeV 10 keV so that not all particles need tofuse in order to achieve a net energy gain.

2.8.6 Photoionization and excitation

To understand these processes, we must rst look at the atomic physics of the H-atom. This is governed by the Coulomb force. Taking a semi-classical approach,we may equate the centripetal force with the Coulomb attraction

F =mv2

r=

e2

40r2.

The kinetic energy of the electron is 12mv2 = e2/(80r) and the potential en-

ergy is e2/(40r) so that the total energy of the atomic system is KE+PE =e2/(80r) (it is a bound system).

Given that electrons are bound in states having de Broglie wavelength satis-fying n = 2r, we may obtain for the energies of the bound states

En = 13.6 eV/n2 n = 1, 2, 3, . . .

(To see this, we write nh = hkr = mvr, or r = nh/mv and substitute into theexpression for the kinetic energy to obtain r = (n2/Z)a0 where a0 is the Bohrradius. This is nally subsituted into the expression for the total energy of theatom.) It is clear that the energy required to ionize the H atom is 13.6 eV andto excite from the ground state is E2 E1 = 10.2 eV. Photons of these energiesreside in the deep ultarviolet of the spectrum. Since the photon energy is usedto disrupt a bound system, this is an inelastic process.

56

2.8.7 Electron impact ionization

The area of the hydrogen atom is a20 = 0.88 1020m2. If an electron comeswithin a radius a0 of the atom and has energy in excess of 13.6 eV, ionizationwill probably result. The behaviour of the electron impact ioization cross-sectionas a function of electron energy is shown in Fig. 2.11.

Figure 2.11: The electron impact ionization cross-section for hydrogen as a func-

tion of electron energy. Note the turn-on at 13.6 eV. [2]

The cross-section falls for energies greater than 100 eV since the electron doesnot spend much time in the vicinity of the atom. The cross section, averagedover the distribution of electron velocities, can be used to estimate the electronmean free path for a given neutral density. Again this is an inelastic process.

2.8.8 Collisions with surfaces

Reactions with surfaces are complex. Gases are readily adsorbed on a cleansurface, sticking with probability 0.3 to 0.5 to form a monolayer surface. Thebinding energy is a strong function of the type of gas however, being smallestfor the noble gases. Heating at temperatures > 2000 will usually produce anatomically clean surface in

2.9 Fluid Equations 57

At low energies, the released electron energy is less than the work function and itcannot escape. At intermediate energies we can have > 1 (gain). The shape ofthe plot of secondary emission coecient (number of electrons emitted for eachincident electron) versus incident energy is similar for all materials. Because ofthe mass dierence, electrons have low probability of ejecting atoms.

Figure 2.12: Electron secondary emission for normal incidence on a typical metal

surface [2]

Secondary electron emission by ions

The number of secondary electrons emitted per positive ion is denoted i. Typi-cally i < 0.3 and is roughly independent of the ion speed.

2.9 Fluid Equations

Just as the velocity moments of the distribution function give important macro-scopic variables, so the velocity moments of the plasma kinetic equation (Boltz-mann equation Eq. (2.5)) give the equations that describe the time evolution ofthese macroscopic parameters. Because these equations (apart from the Lorentzforce) are identical with the continuum hydrodynamic equations, the theoriesusing the low-order moments are called fluid theories. Remember that the uidequations can be applied separately to each of the species that constitute theplasma.

58

2.9.1 Continuityparticle conservation

The equation of continuity is derived from the zeroth velocity moment of theBoltzmann equation. Its analogue for the particle distribution function is justthe number density. It is not surprising that continuity describes conservation ofparticle number. We shall derive the moment equations only for the zeroth mo-ment and simply state the result for the higher order moments of the Boltzmannequation.

Since v0 = 1, the zeroth order moment implies a simple integration overvelocity space to obtain

f

tdv +

v.rfdv + q

m

(E + vB).vfdv =

(f

t

)coll

dv (2.83)

The rst term gives f

tdv =

t

fdv =

n

t. (2.84)

Since v is independent of r (independent variables) the second term givesv.rfdv = r.

vfdv = .(nv) .(nu) (2.85)

where we have used Eq. (2.14) and dened u as the uid velocity. The terminvolving E in the third integral vanishes:

E.

f

vdv =

v.(fE)dv =

S

fE.dS = 0 (2.86)

where the surface S is the v-space surface at v . The rst step is validbecause E is independent of v while the second is an application of Gauss vectorintegral theorem. This last integral vanishes because f vanishes faster than v2

(area varies as 4v2) as v (i.e. nite energy system). The vB componentcan be written

(vB).fv

dv =

v.(fvB)dv

f

v(vB)dv = 0 (2.87)

The second term can be transformed to a surface integral which again vaishes atinnity. Because /v and vB are mutually perpendicular, the second righthand term also vanishes. Finally, the 4th term of Eq. (2.83) vanishes becausecollisions cannot change the number of particles (at least for warm plasmas,where recombination can be ignored). The nal result is

n

t+.(nu) = 0 (2.88)

which expresses conservation of particles . Multiplying by m or q yields conser-vation of mass or charge.

2.9 Fluid Equations 59

Note that taking the zeroth moment of the Boltzmann equation has intro-duced a higher order moment u the rst velocity moment of f . Its descriptionwill require taking the rst order moment of the Boltzmann equation. As we shallsee, this will in turn generate second order moments and so on. This is knownas the closure problem. At some stage, the sequence must be terminated bysome reasonable procedure. Usually this is eected by setting the third velocitymoment of f (describing the thermal conductivity) to zero.

2.9.2 Fluid equation of motion

The rst moment equation (multiply by mv the particle momentum and inte-grate) expresses conservation of momentum. The resulting equation of motion isgiven by

mn

[u

t+ (u.)u

]= qn(E + uB). P +Pcoll (2.89)

where the left hand side will be recognized as proportional to the convectivederivative of the average velocity u. The right side includes the Lorentz force,the divergence of the pressure tensor (a second order quantity) and the exchangeof momentum due to collisions with other species in the plasma:

Pcoll =

mv

(f

t

)coll

dv =

[

t

mvfdv

]coll

(2.90)

Collisions between like particles cannot produce a net change of momentum ofthat species.

Pressure tensor

The elements of the pressure tensor are dened by Eq. (2.16) such that Pij =mnvivj . For an isotropic Maxwellian distribution,

P=

p 0 00 p 00 0 p

(2.91)

and . P= P where P = mnv2r/3 = nkT is the scalar pressure. In thepresence of a magnetic eld, isotropy can be lost and the species may assumedistict temperatures along and across the magnetic eld so that p = nkT andp = nkT with

P=

p 0 00 p 0

0 0 p

where it is customary to take the z direction to conincide with the direction ofB. Note that the pressure is still isotropic in the plane normal to B.

60

The collision term

If a species (for example electrons) is interacting with another via collisions,the momentum lost per collision will be proportional to the relative uid velocityuu where denotes the background uid (for example ions or neutrals). Ifthe mean free time between collisions is approximately constant, the resultingfrictional drag term can be written as

Pei = mene(ue ui)ei

(2.92)

where Pei is the force felt by the electron uid owing with velocity ueui relativeto the background (in this case ion) uid. By conservation of momentum, themomentum lost (gained) through collisions by the electrons on ions must equalthe momentum gained (lost) by the ions. Thus

Pei = Pie. (2.93)What does this say about ie compared with ei?

2.9.3 Equation of state

The second moment equation describes conservation of energy. It is derived bymultiplying the Boltzmann equation by mv2/2 and integrating over velocity. Theresult is

t

(nm

2v2)+ r.

(nm

2vv2

)= j.E +

dv

nm

v2

(f

t

)coll

. (2.94)

The left hand side terms represent the change of energy density with time and theenergy ow across the surface bounding the volume element of interest. The sumof these two terms equals the electric power dissipated in the plasma plus anyenergy that might be generated per unit time through collisions. The magneticeld does no work because it always acts in a direction perpendicular to theparticle velocity.

The equation can be dramatically simplied under the assumption that thedistribution function is a drifting Maxwellian (v = u+vr where vr is the randomor thermal velocity component) and provided collisions do not create new par-ticles (ionization) or change the temperature (thermal conductivity = 0). Thiscorresponds to a system undergoing adiabatic changes (no heat ow). The secondmoment equation then becomes the equation of state:

dp

dn=

5p

3n(2.95)

On integration this givesp = p0n

5/3. (2.96)

2.9 Fluid Equations 61

More generally, we can writep = C (2.97)

where C is a constant, = (2 + N)/N (2.98)

is the ratio of specic heats (Cp/CV ) and N is the number of degrees of freedom.In 3-D, N = 3, = 5/3 and we recover Eq. (2.96). Equation (2.97) can also bewritten as

pp

= nn

(2.99)

or p = kBT n (2.100)For the isothermal case (T=constant, thermal conductivity innite) we have

p = (nkBT ) = kBTn (2.101)and = 1

2.9.4 The complete set of uid equations

Let us dene

= niqi + neqe (2.102)

j = niqiui + neqeue (2.103)

as net charge and current densities respecticely. Ignoring collisions and viscosity(i.e. the pressure tensor scalar pressure p) we have uid equations for speciesj = i, e

njt

+.(njuj) = 0 (2.104)

mjnj

[ujt

+ (uj.)uj]

= qjnj(E + ujB)pj (2.105)pj = Cjn

jj (2.106)

supplemented by Maxwells equations for the eld

.E = 0

(2.107)

E = Bt

(2.108)

.B = 0 (2.109)B = 0j + 00E

t. (2.110)

These two sets of equations provide 16 independent equations for the 16 unknownsni, ne, pi, pe, ui, ue, E andB. Two of Maxwells equations are redundant. In thiscourse we shall examine and employ these equations to solve a range of simpleplasma physics problems.

62

2.9.5 The plasma approximation

In many plasma applications it is usually possible to assume ni = ne and.E = 0at the same time. On time scales slow enough for the ions and electrons to move,Poissons equation can be replaced by ni = ne (i.e. quasi-neutrality). Poissonsequation holds for electrostatic applications:

B

t= 0 E = 0 E = 2 = /0.

The electric eld can be found from the equations of motion. If necessary, thecharge density can then be found from Poissons equation. but it is usually verysmall. This, so-called plasma approximation is only valid for low frequencyapplications. For high frequency electron waves, for example, where charges canseparate, it is not a valid approximation.

Problems

Problem 2.1 A chamber of volume 0.5 m3 is filled with hydrogen at a pressure of

10 Pa at room temperature (20. Calculate

(a) The number of H2 molecules in the chamber

(b) the mass of the gas in the chamber (proton mass = 1.67 1027 kg)

(c) the average kinetic energy of the molecules

(d) the RMS speed of the molecules

(e) the energy in Joules required to heat the gas to 100,000 K, given that it takes 2

eV to dissociate a H2 molecule into H atoms and 13.6 eV to ionize each atom.

Assume that Te = Ti and that the plasma is fully ionized.

(f) the total plasma pressure when Te = Ti = 105K

(g) the RMS speeds of the ions and electrons when Te = Ti = 105K

(h) The electron plasma frequency pe, Debye length D, number of particles in the

Debye sphere ND and the ion-electron collision frequency ei.

Problem 2.2 In the two dimensional phase space (x, v), show the trajectory of

a particle at rest, moving with constant velocity, accelerating at a constant rate,

executing simple harmonic motion and damped simple harmonic motion.

2.9 Fluid Equations 63

Problem 2.3 For any distribution f which is isotropic about its drift velocity v,

prove that

vrvr = v2r/3

where is the Kronecker delta

Problem 2.4 If a particle of mass m, velocity v, charge q approaches a massive

particle (M ) of like charge q0, show by energy considerations that the distanceof closest approach is 2qq0/(40mv

2).

Problem 2.5 Calculate the cross-section 90 for an electron Coulomb collsion with

an ion which results in at least a 90 deflection of the electron from its initial trajectory.

(HINT: This cross-section is b2 where b is the maximum impact parameter for such

a collision). Show that the ratio ei/90 = 2loge where ei is the collsion cross

section calculated in the notes for multiple small angle Coulomb collisions. Comment

on the significance of this result.

Problem 2.6 Consider the motion of charged particles, in one dimension only, in

an electric potential (x). Show, by direct substitution, that a function of the form

f = f(mv2/2 + q) is a solution of the Boltzmann equation under steady state

conditions.

Problem 2.7 Since the Maxwellian velocity distribution is isotropic (independent

of velocity direction) it is of interest to define a distribution of speeds v =| v |. Thatis, we want a new function g(v) which describes the numer of particles with speeds

between v and v + dv. Show that the desired distribution of speeds (in 3-d) is

g(v) = 4v2n(m

2kT)3/2 exp (mv2/2kT )

Plot this distribution and calculate the most probable particle speed, or in other words,

the speed for which g(v) is a maximum.

Problem 2.8 Consider the following two-dimensional Maxwellian distribution func-

tion:

f(vx, vy) = n0(m

2kT) exp [m(v2x + v2y)/2kT ]

(a) Verify that n0 represents correctly the particle number density, that is, the number

of particles per unit area. (b) Draw, in 3-d, the surface for this distribution function,

plotting f in terms of vx and vy. Sketch on this surface curves of constant vx,

constant vy and constant f .

64

Problem 2.9 The entropy of a system can be expressed in terms of the distribution

function as

S = k

dr dv f f

Show that the total time derivative of the entropy for a system which obeys the

collisionless Boltzmann equtaion is zero.