Chapter 3

GASEOUS ELECTRONICS

In this chapter we study the properties of a plasma in an electric field. Ourtreatment of magnetized plasmas will await consideration of individual chargedparticle orbits in spatially and time varying electric and magnetic fields presentedin Chapter 4. Thus in this chapter, the Lorentz force is simple F = qE. Welook at basic phenomena such as plasma breakdown, equilibrium, diffusion andplasma-wall interactions, including sheath physics and Langmuir probes. Tocommence, let us look at plasma equilibrium in the presence of an E-field.

3.1 Plasma in an Electric Field

Force balance – no collisions

In this section we look at the force balance between plasma pressure and electricfield, ignoring the effects of collisions. Under these conditions, we retrieve theBoltzmann relation for a plasma immersed in a spatially varying electric potential(electric field). To show this, we take E = Ek. The z-component of the plasmaequation of motion Eq. (2.89) then reduces to

mn

[∂u

∂t+ (u.∇)u

]z

= qnE − ∂p

∂z

where we have ignored collisions (i.e. we ignore diffusion processes). We simplifyby further assuming that the system is in steady state (∂/∂t = 0) and thatvelocity gradients can be ignored, in which case the left side of the equationvanishes. Substituting for ∇p from Eq. (2.101) (the electrons have high thermalconductivity) gives

qnE = kBT∂n

∂z.

66

Taking q = −e and E = −∂φ/∂z = 0 gives

e∂φ

∂z=

kBTe

ne

∂ne

∂z

whose solution is the previously stated Boltzmann relation

ne = n0 exp

(eφ

kBTe

)(3.1)

where n0 is the density in the potential free region. This expresses the balancebetween electrostatic and pressure forces that must hold in a plasma (electronsare mobile and respond to pressure forces). Thus, there is just enough chargeimbalance to compensate the pressure force felt by the electrons (see Fig 3.1)

Figure 3.1: Illustrating the Boltzmann relation. Because of the pressure gradient,

fast mobile electrons move away, leaving ions behind. The nett positive charge

generates an electric field. The force F e opposes the pressure gardient force F p.[4]

Force balance – including collisions

In a real plasma, diffusion processes (collisions) will eventually flatten or smoothout density gradients unless they are supported by an external power source. Toshow this we retain collisions with a suitably defined collision frequency ν andassume that a fluid element does not move into a different region of E or p inless than a collision time so that the convective derivative can be ignored. Thenthe force balance can be written

0 = ±enE − kBT∇n − mnνu

3.1 Plasma in an Electric Field 67

where the ± accounts for both ions and electrons. We can solve for the speciesdrift velocity

u = ±(

e

mν

)E −

(kBT

mν

) ∇n

n(3.2)

≡ ±µE − D∇n

n(3.3)

where

µ =|q |mν

(3.4)

is the particle mobility and

D =kBT

mν(3.5)

is the diffusion coefficient. Note that

D ∼ v2th/ν = (v2

th/ν2)ν = λ2

mfp/τ (3.6)

where λmfp is the distance between collisions and τ = 1/ν is the collision time.The diffusion coefficient therefore is proporitonal to the square of the step lengthdivided by the time between collisions. The step length is therefore very impor-tant for diffusion processes.

The diffusion coefficient and mobility are related by the Einstein relation:

µj =|q | Dj

kBTj

. (3.7)

Using ν = nσvth ∼ m−1/2 we find that µ ∼ m−1/2 so that µe � µi and theelectrons are much more mobile than ions. This has significant consequences forplasma diffusion as shown below. The species particle flux defined by Eq. (2.14)can now be written as

Γj = nuj = ±µjnE − Dj∇n. (3.8)

When either E = 0 or the particles are uncharged, we recover “Fick’s Law ofDiffusion”

Γ = −D∇n (3.9)

which shows that a net flux of particles from a more dense to less dense regionoccurs simply because there are more randomly moving particles in the denseregion.

In highly ionized magnetized plasma, Fick’s law needs to be reappraised.Moreover, collective wave effects and microturbulent convection can significantlyenhance the rate of diffusion.

68

3.2 Resistivity

To obtain an estimate for the plasma resistivty, we start with some simple defi-nitions

Ohm′s law E = V/L = IR/L (3.10)

Resistivity η R = ηL/A (3.11)

=⇒ E = Iη/A = jη (3.12)

where L is the length of conductor, A is its cross-sectional area, V is the appliedvoltage and and j is the current density.

Inside a plasma, electrons are being accelerated by the E field and deceleratedby collisions. They acquire a net drift velocity given by Eq. (3.3) (and taking∇n = 0).

Γei= nue

i= ±µe

inE (3.13)

Using the definition for j Eq. (2.103) we have

j = niqiui + neeue

= e(Γi − Γe)

= ne(µi − µe)E. (3.14)

With µe � µi, and in 1-D, we obtain

j =ne2

meνE (3.15)

which gives for the plasma resistivity

η =meν

ne2(3.16)

For a fully ionized plasma, ν = ν90ei [Eq. (2.76)], so

η =me

Zne2

niZ2e4 ln Λ

2πε20m

2ev

3e

.

Now in 3-D (three degrees of freedom) mev2e/2 = 3kBTe/2 and the Coulomb

plasma resistivity can be expressed

η =Ze2m1/2

e ln Λ

6√

3πε20(kBTe)3/2

. (3.17)

ln Λ is only weakly dependent on plasma parameters and for the prupose ofstudying the scaling of Eq. (3.17) can be regarded as constant. Thus

η ∼ T−3/2e

3.3 Plasma Decay by Diffusion 69

with almost no density dependence.Reason: j increases with n (more charge carriers) but the frictional drag (colli-sions) also increases with n i.e. νei = nσeiv and the two effects cancel.

For a weakly-ionized plasma where ν is dominated by collisions with neutrals,

j = −neeue ue = −µeE ⇒ j = neeµeE

µe depends on the density of neutrals (not electrons) through the collision fre-quency νen so that now the current is proportional to the density ne of chargecarriers (electrons).

3.2.1 Ohmic dissipation

An easy way to heat a plasma is to pass a current through it. The power dissipatedis I2R (or power density = j2η) and this appears as an increase in electrontemperature through frictional drag on the ion fluid. This is known as Jouleor Ohmic heating. However, η ∼ T−3/2

e implies that the plasma is such a goodconductor at thermonuclear temperatures (i.e. > 1 keV) that ohmic heating istoo slow - the plasma is effectively collisionless.

Numerically

η = 5.2 × 10−5 Z ln Λ

T3/2e (eV)

Ohm − m (3.18)

The table below compares resistivity for a typical high-temperature plasma andsome well known metals.

η Ohm-m

H-1NF (100eV) 5 × 10−7

Cu 2 × 10−8

St. Steel 7 × 10−7

Hg 1 × 10−6

3.3 Plasma Decay by Diffusion

Consider the plasma container shown schematically in Fig. 3.2. As a boundarycondition we take n(±L) = 0 and use the fluid equations to study the plasmadecay as a function of time due to diffusive processes. We assume that the decayrate is much slower than the collision frequency (this is reasonable since it iscollisions which give rise to diffusion in the first place).

In order that the plasma remain quasi-neutral, we require that the electronand ion fluxes in our one-dimensional system above are equal i.e. Γi = Γe = Γ.Since the electrons are more mobile, they will escape first. This establishes an

70

Figure 3.2: Schematic diagram showing plasma in a container of length 2L with

particle density vanishing at the wall. [4]

ambipolar electric field that enhances the rate at which the ions escape – it dragsthe ions out. Thus

Γ = µinE − Di∇n = µenE − De∇n

where quasineutrality ensures ne ≈ ni = n. Nevertheless, we can still solve for E(remember the plasma approximation) to obtain

E =Di − De

µi + µe

∇n

n. (3.19)

Now substitute back into our expression for the flux to obtain

Γ = µiDi − De

µi + µe

∇n − Di∇n

= −µeDi + µiDe

µi + µe∇n

≡ −Da∇n (3.20)

3.3 Plasma Decay by Diffusion 71

which is just Fick’s law but with ambipolar diffusion coefficient

Da =µeDi + µiDe

µi + µe. (3.21)

Noting that µe � µi (electrons much more mobile than ions) we can approx-imate

Da ≈ Di +µe

µiDe = Di +

Te

TiDi

where we have used Eq. (3.7). For Te ∼ Ti we obtain the simple result

Da ≈ 2Di. (3.22)

The ambipolar electric field enhances the diffusion rate by a factor of two. Therate is primarily controlled by the slower ions – the self consistent electric fieldretards the loss of the electron component.

3.3.1 Temporal behaviour

Combinig Fick’s law Eq. (3.20) with the equation of continuity gives a secondorder partial differential equation linking the temporal and spatial evolution ofthe density profile:

∂n

∂t= Da∇2n. (3.23)

This equation can be solved using separation of variables by setting n(r, t) =T (t)S(r) whereupon

SdT

dt= DaT∇2S

⇒ 1

T

dT

dt=

Da

S∇2S. (3.24)

Both sides of the equation have dimensions t−1 and are functions of differentvariables. We therefore equate left and right sides to the constant 1/τ . For theleft side we obtain

dT

dt= −T/τ

having solution

T = T0 exp (−t/τ) (3.25)

so that τ represents a diffusion time constant. For the right side we find (for 1-D)

∇2S =d2S

dx2= − S

Daτ(3.26)

72

whose solution isS = A cos

x

(Daτ)1/2+ B sin

x

(Daτ)1/2. (3.27)

Our boundary condition impies that B = 0 and for the “lowest order mode”

L

(Daτ)1/2=

π

2

so

τ =(

2L

π

)2 1

Da

. (3.28)

Combining equations (3.25), (3.27) and (3.28) finally gives

n = n0 exp (−t/τ) cos(

πx

2L

)(3.29)

which describes the lowest order diffusion mode decaying exponentially as a resultof collisions.

In general, Eq. (3.26) supports an infinte number of solutions (or Fouriermodes) that match the boundary conditions:

n = n0

∑l

al exp (−t/τl) cos(l + 1

2)πx

L+ n0

∑m

bm exp (−t/τm) sinmπx

L(3.30)

with

τj =

(L

jπ

)21

Da. (3.31)

Observe that the high spatial frequencies (higher j) decay much more rapidlythan the low frequency terms. This is consistent with Eq. (3.19) which showsthat the ambipolar electric field is proportional to the inverse of the density scalelength (∇n/n) and is shown schematically in Fig. 3.3

3.4 Plasma Decay by Recombination

When electrons and ions collide at low velocity (low temperature) there is a finiteprobability of recombination to a neutral atom with the emission of a photon.This is known as radiative recombination and is the inverse process to photo-ionization. Three body recombination involves a third particle for momentumconservation and with no emitted photon. The recombination rate is proportionalto nine = n2. The effect can be represented as a particle sink in the equation ofcontinuity which, ignoring diffusion, gives

∂n

∂t= −αn2 (3.32)

3.5 Plasma Breakdown 73

Figure 3.3: High spatial frequency features are quickly washed out by diffusion

as the plasma density relaxes towards its lowest order profile.[4]

where α is the recombination coefficient. The equation is nonlinear and hassolution

1

n(r, t)=

1

n0(r, t)+ αt (3.33)

so that n decays inversely with time.

3.5 Plasma Breakdown

To study the phenomenon of electric breakdown of a gas, consider the drift ofelectrons under the action of an external electric field (we do not consider ions inthis treatment due to their much smaller mobility). In steady state, and ignoringdiffusion, the electron drift velocity is given by Eq. (3.3)

ue =q

meνenE

=q

mennσen〈v〉E (3.34)

where 〈v〉 is the mean relative particle speed and σen is the cross-section forelastic electron-neutral collisions (we are assuming that the plasma is initially

74

very weakly ionized). The measured electron-neutral collision cross-section forvarious species is shown in Fig. 3.4 For scaling purposes, let us assume σen is

Figure 3.4: Elastic collision cross-section of electrons in Ne, A, Kr and Xe.[2]

velocity independent (though Fig. 3.4 would suggest otherwise!). We rearrangeEq. (3.34) to obtain

ue〈v〉 =q

meσ

E

n

=q

meEλmfp (3.35)

where λmfp = 1/nσ is the mean free path for e-n collisions [see Eq. (2.68)].The right side is proportional to the energy gained between electron collisionswith neutrals due to acceleration in the imposed electric field (KE = force .distance = qEλmfp). For this reason, the parameter E/p (or Eλmfp) where pis the gas pressure is a very important parameter for discussing phenomena ingaseous electronics.

If the drift speed is much larger than the thermal speed vth (i.e. the electrongas is cold) then 〈v〉 ∼ ue and u2

e ∝ E/p or ue ∼ (E/p)1/2. At low driftvelocities, vth > ue (i.e. 〈v〉 is independent of ue) then ue ∼ (E/p). As shownin Fig. 3.5, the measured dependence of the drift speed of electrons in hydrogenand deuterium as a function of E/p confirms these dependencies.

3.5 Plasma Breakdown 75

Figure 3.5: Drift velocity of electrons in hydrogen and deuterium.[2]

Once the drifting electron energy is greater than the ionization energy, ion-ization can occur (see Fig. 2.11). For an electron in hydrogen, for example, werequire meu

2e/2 = 13.6eV, or ue > 2.2 × 106 m/s. If an electron creates α new

electrons per unit length under the action of the electric field then

dne

dx= αne ⇒ n = n0 exp (αx) ⇒ I = I0 exp (αx) (3.36)

and provided α > 0, the electron current I grows exponentially with distance. αis known as the first Townsend coefficient and depends on the energy gained by anelectron in a mean free path and the ionization potential of the background gas.Since α is the number of ionization events per unit distance we take α = νi/ue

where

νi =1

ne

∫dv [nnσi(v)v] f(v) (3.37)

is the collision frequency (the term in square brackets) averaged over the dis-tribuion of electron velocities. Since the velocity moments of the distributionfunction are functions of (E/p) (and since p ∝ nn) it follows that α/p must alsobe a function of E/p:

α

p= f

(E

p

)= f(energy gained in λmfp)

=νi

ue

1

p

∼ A exp [−B(p/E)] (3.38)

76

where the A and B coefficients depend on the gas. The last result is derived indetail in the lab notes. The functional dependence of α/p on p/E is shown inFig. 3.6

Figure 3.6: The ionization coefficient α/p for hydrogen. Note the exponential

behaviour.[2]

With the exponential avalanche of electrons, the newly created ions drift tothe cathode where they release secondary ions with an efficiency γ – the secondTownsend coefficient (see Fig. 3.7). Typically this efficiency is between 1% and10%. The secondary population ns = γni or Is = γI allows the growth of thedischarge to continue. Thus the total current flowing, including the secondarycomponent is

I = (I0 + Is) exp (αx). (3.39)

Substituting for Is = γI and solving for I gives

I =I0 exp αx

1 − γ exp αx(3.40)

and breakdown occurs provided

1 − γ exp (αx) = 0. (3.41)

The current in the discharge is ultimately limited by an external resistor or thepower supply.

3.5 Plasma Breakdown 77

Figure 3.7: An electron avalanche as a function of time.[5]

3.5.1 Paschen’s law

Subsituting from Eq. (3.38) into Eq. (3.41) gives an expression for the breakdownvoltage in terms of gas pressure and electrode separation d:

VB =Bpd

ln [Apd/ ln(1/γ)](3.42)

where VB = EB/d, EB is the electric field for breakdown and, for a given gas,VB = VB(pd). In other words, breakdown voltage is constant when the productpd is constant. You will be investigating this behaviour (shown in Fig. 3.8) inlaboratory classes.

(a) (b)

Figure 3.8: (a) Paschen’s curve showing breakdown voltage as a function of the

product pd. (b) A plot of plasma current versus applied voltage. Note the

dramatic increase at the onset of breakdown. [5]

78

3.6 The Sheath

What happens to plasma in the vicinity of a wall? To answer this, we considera 1-D model with no magnetic field. Quasi-neutrality ensures that the electricpotential φ is zero in the plasma centre. However, since vthe � vthi, electrons arelost more quickly to the wall than ions and the wall acquires a negative potential(or the plasma is left with a net positive charge).

Because of Debye shielding, the potential variation occurs over a layer ofthickness a few Debye lengths called the sheath. The sheath forms a potentialbarrier that tends to confine the escaping electrons electrostatically. The heightof the barrier adjusts so that ambipolarity (Γi = Γe) is satisfied. The situation isdepicted schematically in Fig. 3.9

(a) (b)

Figure 3.9: (a) The plasma potential distribution for a plasma confined electro-

statically and (b) the corresponding density distribution of ions and electrons.

The ion density is higher than electrons near the wall due to the negative electric

field established there by the escaping electron flux.

We use the fluid equations to examine the electric potential variation in theregion of the sheath. We make some simplifying assumptions

(i) 1-D, no magnetic field

(ii) No collisions (λei ∼ 1m � λD at n ∼ ×1018 m−3, Te ∼ 5 eV)

(iii) Cold drifting ions ui � vthi (a beam like distribution with ion temperatureclose to that of the neutrals)

(iv) Boltzmann distributed electrons.

3.6 The Sheath 79

In equilibrium, and ignoring ionization and recombination in the body of theplasma (∂/∂t = 0), the ion particle flux is constant from the centre to the wall

niui = n0u0 (3.43)

Though we have ignored ionization, the ion flow u0 away from the sheath, thoughsmall, must be finite in order to observe particle balance between source (ion-ization) and sink (walls). The ions gain kinetic energy as they are acceleratedthrough the sheath to the negatively charged wall (use force equation for ions)

1

2miu

2i + φZe =

1

2miu

20. (3.44)

We have ignored the potential φ0 in the body of the plasma. This can be solvedfor the ion drift velocity in the region of non-zero potential

ui =

(u2

0 −2Zeφ

mi

)1/2

. (3.45)

Using continuity Eq. (3.43) we obtain

ni(x) = n0

(1 − 2Zeφ

miu20

)−1/2

. (3.46)

The electrons are Boltzmann distributed

ne(x) = n0 exp

(eφ

kBTe

). (3.47)

We can now solve for the potential φ using Poisson’s equation

∇2φ = − ρ

ε0

⇒ d2φ

dx2= e(ni − ne)

= en0

exp

(eφ

kBTe

)−(

1 − 2Zeφ

miu20

)−1/2 . (3.48)

This is a nonlinear differential equation for φ(x). To solve it approximately, welook at two limits – near the plasma edge of the sheath and near the wall (seeFig. 3.10).

Plasma edge of the sheath

We shall see later that the potential φw at the wall is a few times the thermalenergy so that eφw ∼ kBTe. Within the plasma edge of the sheath, we’ll take

80

Figure 3.10: The structure of the potential distribution near the plasma boundary.

the potential variation | eφ |� kBTe. Furthermore, since ui � vthi, we have|eφ |� miu

20 (to be later verified). Equation (3.48) can now be expanded to first

order in small quantities

exp

(eφ

kBTe

)≈ 1 +

eφ

kBTe(1 − 2Zeφ

miu20

)−1/2

≈ 1 +Zeφ

miu20

to give

d2φ

dx2= φ/X2 (3.49)

X2 = λ2D

(1 − kBTe

miu20

)(3.50)

where we have taken Z = 1 for simplicity. The solution to Eq. (3.49) withboundary condition φ = 0 at x = 0 (the body of the plasma) is

φ = Φ[1 − exp (−x/X)]. (3.51)

This shows that near the plasma edge of the sheath, the plasma potential de-creases exponentially (Φ < 0) since the plasma potential is negative at the wall(φw). Provided

kBTe < miu20 (3.52)

3.6 The Sheath 81

then X2 > 0 and the decay length is comparable with the Debye length. When thecondition Eq. (3.52) is violated, X would be imaginary and the electric potentialwould become an oscillating function of potential near the wall. This would trapparticles in the steady-state potential well. This doesn’t happen, as dissipativeprocesses destroy the ordered state. Thus Eq. (3.52), known as the Bohm sheathcriterion is satisfied in the wall edge of the sheath. It says that ions must enterthe sheath with a velocity greater than the acoustic velocity (thermal speed).

To obtain a directed velocity u0, there must be a small accelerating field in thebody of the plasma. The assumption that φ = 0 at x = 0 in obtaining Eq. (3.51)is only approximate and is made possible by d � λD. Ultimately, u0 is fixed bythe ion production rate (ionization).

The choice of the boundary at which ui = u0 is somewhat arbitrary. Inreaching this position in the plasma, the ions have fallen through some overallpotential drop which we denote φ00 (this was earlier assumed small and ignored).We hereafter take our starting position to be the plasma edge of the sheathwhich we define to commence when the electrostatic potential energy is equal tothe electron thermal energy, eφ0 = miu

20/2 = kBTe/2. In this case the initial ion

drift velocity is

u0 ≡ uBohm =

(kBTe

mi

)1/2

(3.53)

and uBohm is known as the Bohm speed. It is often the case that Te > Ti so thatui > u0 > vthi as was assumed early in the analysis. The region in the plasmaover which the potential drops slowly from the centre to the edge of the sheathis known as the pre-sheath.

The wall edge of the sheath

Near the wall, the electric potential is very negative and the electron density iscorrespondingly very low so that for the charge density we can write

ρ = −ene + eZni

≈ eZni

= n0u0Ze/ui (3.54)

where we have used continuity. The Poisson equation gives

d2φ

dx2= − n0u0Ze

ε0 [2Ze(φ0 − φ)/mi]1/2

(3.55)

where we have substituted eφ0 = miu20/2 in Eq. (3.45). Introducing the potential

drop V = φ0 − φ allows Eq. (3.55) to be written as

d2V

dx2=

g

V 1/2(3.56)

82

g =n0u0Ze

ε0(2Ze/mi)1/2≡ J

ε0(2Ze/mi)1/2(3.57)

We now develop the following equations towards the solution for V:

V ′′ = gV −1/2

×2V ′ ⇒ 2V ′V ′′ =gV ′

2V 1/2.4

integrate wrt x ⇒ (V ′)2 = 4gV 1/2

V ′ = 2√

gV 1/4

V ′/V 1/4 = 2√

g

integrate wrt x ⇒ 4

3V 3/4 = 2

√gx

g =4

9

V 3/2

x2∝ J (3.58)

where J is the ion current density in the sheath and Eq. (3.58) describes thevariation of the plasma potential in the region close to the wall. This variationis expressed explicitly in the Child-Langmuir Law:

V ∝ J2/3x4/3. (3.59)

Figure 3.11 compares the variation expressed by Eq. (3.59) with the variationthat would be expected for uniform and point source charge distributions.

Figure 3.11: Potential variation in the wall edge of the sheath compared with

that for uniform and point like charge distributions.

3.7 Plasma Potential and Wall Potential 83

3.7 Plasma Potential and Wall Potential

We have determined the thermal flux of particles across an interface to be givenby Eq. (2.38). For a given species α we have

Γsα = nα

(kBTα

2πmα

)1/2

. (3.60)

For ne = ni, Γse � ΓsI ⇒ φw is negative. The sheath potential is thus establishedby the mobility of the electrons due to their thermal motion. Since the electronsare Maxwellian (not drifting) we have

Γse = ne〈v〉/4

= (n0〈v〉/4) exp (eφ/kBTe). (3.61)

The ions on the other hand, are drifting

Γi ≈ n0vBohm. (3.62)

To determine the electric potential at the wall, we impose the equilibrium con-straint Γse = Γi to find

n0

(kBTe

mi

)1/2

≈ n0

(kBTe

2πme

)1/2

exp

(eφw

kBTe

)(3.63)

or

eφw

kBTe≈ 1

2ln(

2πme

mi

)

⇒ φw ≈ φ0 ln(

2πme

mi

)(3.64)

The right hand side factor takes values between 3 and 6 typically for argon plasmaand between 2 and 4 for hydrogen plasma (depending on choice of constants).The plasma wall potential is then several times the electron temperature. Thispotential is required ensure to ambipolarity. Note that both φ0 (the presheathdrop) and φw are negative. The total plasma potential at centre is thereforeφP = φ0 + φw ≈ φw. As an example, for a 1eV argon plasma φp ∼ 3 − 6 voltspositive with respect to the wall. If an external bias is applied the current flowsand Γse �= Γi. The situation is illustrated in Fig. 3.12

3.8 Langmuir Probes

We now study the behaviour of a conducting probe inserted into a plasma. Ini-tially, we assume the probe is biased negative such that all ions striking the

84

Figure 3.12: Schematic diagram showing the potential drops around the plasma

circuit.

probe are collected and electrons are repelled. The collected ion current densityis j = qΓ and the ion current is (assuming Z = 1)

I = n0u0eA (3.65)

where A is the collection area and u0 = uBohm. Now there is a potential dropφ00 − φ0 in the pre-sheath to accelerate the ions to the Bohm speed

−eφs = −e(φ0 − φ00) =1

2miu

20 =

1

2kBTe

where φs is the sheath edge potential. The electron density at the sheath edge is

n0 = n00 exp (eφs/kBTe) = n00 exp (−.5) = 0.61n00 ≈ 0.5n00

and the ion saturation current is

Isi = n0u0eA

=1

2n00eA

(kBTe

mi

)1/2

. (3.66)

Thus, given the electron temperature, we can use probes to measure the particledenisty in a plasma.

More generally, consider some probe potential V − Vp = V with respect tothe plasma potential Vp = φw. The current of electrons arriving at the probe is

I = AeΓse exp (eV /kBTe)

= Ise exp (eV /kBTe) (3.67)

3.8 Langmuir Probes 85

whereIse = Aen0〈v〉/4 (3.68)

is the electron saturation current. For a positive probe bias, essentially all theelectrons arriving at the probe are collected and I = Ise. Note that Ise/Isi ∼(mi/me)

1/2 � 1. As shown in Fig. 3.13 location of the knee of the V − I charac-teristic gives the plasma potential (V = Vp). A typical measurement arrangementis shown in Fig. 3.14.

Figure 3.13: The Langmuir probe I − V characteristic showing the electron and

ion contributions and the plasma.

Problems

Problem 3.1 Suppose that a so-called Q-machine has a uniform longitudinal mag-

netic field of 0.2 Tesla and a cylindrical plasma withTe = Ti = 0.2 eV. The density

profile is found experimentally to be of the form

n = n0 exp [exp(−r2/a2) − 1]

Assume the density obeys the Boltzmann relation n = n0 exp (eφ/kTe).

(a) Calculate the maximum vE if a = 1 cm.

86

Figure 3.14: The circuit shows the typical measurement arrangement using Lang-

muir probes. The probe bias is adjusted using the variable resistor.

(b) To what value can B be lowered before the ions of potassium (A=39, Z=1) have

a Larmor radius equal to a?