 # In this chapter we study the properties of a plasma in an electric ﬁeld Our treatment of magnetized plasmas will await consideration of individual charged particle orbits in spatially and time varying electric and magnetic ﬁelds presented in Chapter Thus in this chapter the Lorentz force is simple F qE We look at basic phenomena such as plasma breakdown equilibrium di usion and plasma wall interactions including sheath physics and Langmuir probes To commence let us look at plasma equilibrium in the presence of an E ﬁeld

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Niveau: Supérieur, Doctorat, Bac+8
Chapter 3 GASEOUS ELECTRONICS In this chapter we study the properties of a plasma in an electric ﬁeld. Our treatment of magnetized plasmas will await consideration of individual charged particle orbits in spatially and time varying electric and magnetic ﬁelds presented in Chapter 4. Thus in this chapter, the Lorentz force is simple F = qE. We look at basic phenomena such as plasma breakdown, equilibrium, di?usion and plasma-wall interactions, including sheath physics and Langmuir probes. To commence, 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 electric ﬁeld, ignoring the e?ects of collisions. Under these conditions, we retrieve the Boltzmann 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 plasma equation 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 di?usion processes). We simplify by further assuming that the system is in steady state (∂/∂t = 0) and that velocity gradients can be ignored, in which case the left side of the equation vanishes.

• ionized magnetized

• collision

• plasma

• di?usion coe?cient

• therefore very

• time between

• ions behind

• temperature plasma

• electron temperature

Sujets

##### Electron temperature

Informations

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Chapter 3
GASEOUS ELECTRONICS
In this chapter we study the properties of a plasma in an electric ﬁeld. Our
treatment of magnetized plasmas will await consideration of individual charged
particle orbits in spatially and time varying electric and magnetic ﬁelds presented
in Chapter 4. Thus in this chapter, the Lorentz force is simple F = qE.We
look at basic phenomena such as plasma breakdown, equilibrium, diﬀusion and
plasma-wall interactions, including sheath physics and Langmuir probes. To
commence, 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 electric
ﬁeld, ignoring the eﬀects of collisions. Under these conditions, we retrieve the
Boltzmann relation for a plasma immersed in a spatially varying electric potential
ˆ(electric ﬁeld). To show this, we take E = Ek.The z-component of the plasma
equation of motion Eq. (2.89) then reduces to

∂u ∂p
mn +(u.∇)u = qnE−
∂t ∂z
z
where we have ignored collisions (i.e. we ignore diﬀusion processes). We simplify
by further assuming that the system is in steady state (∂/∂t =0) and that
velocity gradients can be ignored, in which case the left side of the equation
vanishes. Substituting for∇p from Eq. (2.101) (the electrons have high thermal
conductivity) gives
∂n
qnE = k T .B
∂z66
Taking q =−e and E =−∂φ/∂z =0 gives
∂φ k T ∂nB e e
e =
∂z n ∂ze
whose solution is the previously stated Boltzmann relation

n = n exp (3.1)e 0
k TB e
where n is the density in the potential free region. This expresses the balance0
between electrostatic and pressure forces that must hold in a plasma (electrons
are mobile and respond to pressure forces). Thus, there is just enough charge
imbalance 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 forceF opposes the pressure gardient forceF .e p
Force balance – including collisions
In a real plasma, diﬀusion processes (collisions) will eventually ﬂatten or smooth
out density gradients unless they are supported by an external power source. To
show this we retain collisions with a suitably deﬁned collision frequency ν and
assume that a ﬂuid element does not move into a diﬀerent region of E or p in
less than a collision time so that the convective derivative can be ignored. Then
the force balance can be written
0=±enE− k T∇n− mnνuB3.1 Plasma in an Electric Field 67
where the± accounts for both ions and electrons. We can solve for the species
drift velocity

e k T ∇nB
u = ± E− (3.2)
mν mν n
∇n
≡±µE− D (3.3)
n
where
|q|
µ = (3.4)

is the particle mobility and
k TB
D = (3.5)

is the diﬀusion coeﬃcient.Notethat
2 2 2 2D∼ v /ν =(v /ν )ν = λ /τ (3.6)th th mfp
where λ is the distance between collisions and τ =1/ν is the collision time.mfp
The diﬀusion coeﬃcient therefore is proporitonal to the square of the step length
divided 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:
|q| Dj
µ = . (3.7)j
k TB j
−1/2 −1/2Using ν = nσv ∼ m we ﬁnd that µ ∼ m so that µ µ and theth e i
electrons are much more mobile than ions. This has signiﬁcant consequences for
plasma diﬀusion as shown below. The species particle ﬂux deﬁned by Eq. (2.14)
cannowbe writtenas
Γ = nu =±µ nE− D∇n. (3.8)j j j j
When either E = 0 or the particles are uncharged, we recover “Fick’s Law of
Diﬀusion”
Γ=−D∇n (3.9)
which shows that a net ﬂux of particles from a more dense to less dense region
occurs simply because there are more randomly moving particles in the dense
region.
In highly ionized magnetized plasma, Fick’s law needs to be reappraised.
Moreover, collective wave eﬀects and microturbulent convection can signiﬁcantly
enhance the rate of diﬀusion.68
3.2 Resistivity
To obtain an estimate for the plasma resistivty, we start with some simple deﬁ-
nitions
Ohmslaw E = V/L = IR/L (3.10)
ResistivityηR = 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 applied
voltage and and j is the current density.
Inside a plasma, electrons are being accelerated by the E ﬁeld and decelerated
by collisions. They acquire a net drift velocity given by Eq. (3.3) (and taking
∇n=0).
Γe = nue =±µenE (3.13)
i i i
Using the deﬁnition for j Eq. (2.103) we have
j = n qu + n eui i i e e
= e(Γ − Γ )i e
= ne(µ − µ )E. (3.14)i e
With µ µ , and in 1-D, we obtaine i
2ne
j = E (3.15)
m νe
which gives for the plasma resistivity
m νe
η = (3.16)
2ne
For a fully ionized plasma, ν = ν [Eq. (2.76)], so90ei
2 4m n Z e ln Λe i
η = .
22 2 3Zne 2πε m v0 e e
2Now in 3-D (three degrees of freedom) m v /2= 3k T /2and theCoulombe B ee
plasma resistivity can be expressed
2 1/2Ze m ln Λeη = √ . (3.17)
2 3/26 3πε (k T )B e0
ln Λ is only weakly dependent on plasma parameters and for the prupose of
studying the scaling of Eq. (3.17) can be regarded as constant. Thus
−3/2η∼ Te3.3 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. ν = nσ v and the two eﬀects cancel.ei ei
For a weakly-ionized plasma where ν is dominated by collisions with neutrals,
j =−n eu u =−µ E ⇒ j = n eµ Ee e e e e e
µ depends on the density of neutrals (not electrons) through the collision fre-e
quency ν so that now the current is proportional to the density n of chargeen e
carriers (electrons).
3.2.1 Ohmic dissipation
An easy way to heat a plasma is to pass a current through it. The power dissipated
2 2is I R (or power density = j η) and this appears as an increase in electron
temperature through frictional drag on the ion ﬂuid. This is known as Joule
−3/2or Ohmic heating. However, η ∼ T implies that the plasma is such a goode
conductor at thermonuclear temperatures (i.e. > 1 keV) that ohmic heating is
too slow - the plasma is eﬀectively collisionless.
Numerically
Z ln Λ−5η =5.2× 10 Ohm− m (3.18)
3/2
T (eV)e
The table below compares resistivity for a typical high-temperature plasma and
some well known metals.
η Ohm-m
−7H-1NF (100eV) 5× 10
−8Cu 2× 10
−7St. Steel 7× 10
−6Hg 1× 10
3.3 Plasma Decay by Diﬀusion
Consider the plasma container shown schematically in Fig. 3.2. As a boundary
condition we take n(±L) = 0 and use the ﬂuid equations to study the plasma
decay as a function of time due to diﬀusive processes. We assume that the decay
rate is much slower than the collision frequency (this is reasonable since it is
collisions which give rise to diﬀusion in the ﬁrst place).
In order that the plasma remain quasi-neutral, we require that the electron
and ion ﬂuxes in our one-dimensional system above are equal i.e. Γ =Γ =Γ.i e
Since the electrons are more mobile, they will escape ﬁrst. This establishes an70
Figure 3.2: Schematic diagram showing plasma in a container of length 2L with
particle density vanishing at the wall. 
ambipolar electric ﬁeld that enhances the rate at which the ions escape – it drags
the ions out. Thus
Γ= µ nE− D∇n = µ nE− D∇ni i e e
where quasineutrality ensures n ≈ n = n. Nevertheless, we can still solve for Ee i
(remember the plasma approximation) to obtain
D − D ∇ni e
E = . (3.19)
µ + µ ni e
Now substitute back into our expression for the ﬂux to obtain
D − Di e
Γ= µ ∇n− D∇ni i
µ + µi e
µ D + µ De i i e
= − ∇n
µ + µi e
≡−D ∇n (3.20)a3.3 Plasma Decay by Diﬀusion 71
which is just Fick’s law but with ambipolar diﬀusion coeﬃcient
µ D + µ De i i e
D = . (3.21)a
µ + µi e
Noting that µ µ (electrons much more mobile than ions) we can approx-e i
imate
µ Te e
D ≈ D + D = D + Da i e i i
µ Ti i
wherewehaveusedEq. (3.7). For T ∼ T we obtain the simple resulte i
D ≈ 2D . (3.22)a i
The ambipolar electric ﬁeld enhances the diﬀusion rate by a factor of two. The
rate is primarily controlled by the slower ions – the self consistent electric ﬁeld
retards 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 second
order partial diﬀerential equation linking the temporal and spatial evolution of
the density proﬁle:
∂n 2= D ∇ n. (3.23)a
∂t
This equation can be solved using separation of variables by setting n(r,t)=
T(t)S(r) whereupon
dT 2S = D T∇ Sa
dt
1 dT Da 2⇒ = ∇ S. (3.24)
T dt S
−1Both sides of the equation have dimensions t and are functions of diﬀerent
variables. We therefore equate left and right sides to the constant 1/τ.Forthe
left side we obtain
dT
=−T/τ
dt
having solution
T = T exp (−t/τ) (3.25)0
so that τ represents a diﬀusion time constant. For the right side we ﬁnd (for 1-D)
2d S S2∇ S = =− (3.26)
2dx D τa72
whose solution is
x x
S = A cos + B sin . (3.27)
1/2 1/2(D τ) (D τ)a a
Our boundary condition impies that B = 0 and for the “lowest order mode”
L π
=
1/2(D τ) 2a
so
22L 1
τ = . (3.28)
π Da
Combining equations (3.25), (3.27) and (3.28) ﬁnally gives

πx
n = n exp (−t/τ)cos (3.29)0
2L
which describes the lowest order diﬀusion mode decaying exponentially as a result
of collisions.
In general, Eq. (3.26) supports an inﬁnte number of solutions (or Fourier
modes) that match the boundary conditions:
1 (l + )πx mπx
2n = n a exp (−t/τ )cos + n b exp (−t/τ )sin (3.30)0 l l 0 m m
L Lml
with 2
L 1
τ = . (3.31)j
jπ Da
Observe that the high spatial frequencies (higher j) decay much more rapidly
than the low frequency terms. This is consistent with Eq. (3.19) which shows
that the ambipolar electric ﬁeld is proportional to the inverse of the density scale
length (∇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 ﬁnite
probability 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 momentum
conservation and with no emitted photon. The recombination rate is proportional
2to n n = n . The eﬀect can be represented as a particle sink in the equation ofi e
continuity which, ignoring diﬀusion, gives
∂n 2=−αn (3.32)
∂t3.5 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.
where α is the recombination coeﬃcient. The equation is nonlinear and has
solution
1 1
= + αt (3.33)
n(r,t) n (r,t)0
so that n decays inversely with time.
3.5 Plasma Breakdown
To study the phenomenon of electric breakdown of a gas, consider the drift of
electrons under the action of an external electric ﬁeld (we do not consider ions in
this treatment due to their much smaller mobility). In steady state, and ignoring
diﬀusion, the electron drift velocity is given by Eq. (3.3)
q
u = Ee
m νe en
q
= E (3.34)
m n σ ve n en
where v is the mean relative particle speed and σ is the cross-section foren
elastic electron-neutral collisions (we are assuming that the plasma is initially74
very weakly ionized). The measured electron-neutral collision cross-section for
various species is shown in Fig. 3.4 For scaling purposes, let us assume σ isen
Figure 3.4: Elastic collision cross-section of electrons in Ne, A, Kr and Xe.
velocity independent (though Fig. 3.4 would suggest otherwise!). We rearrange
Eq. (3.34) to obtain
q E
uv =e
m σ ne
q
= Eλ (3.35)mfp
me
=1/nσ is the mean free path for e-n collisions [see Eq. (2.68)].where λmfp
The right side is proportional to the energy gained between electron collisions
with neutrals due to acceleration in the imposed electric ﬁeld (KE = force .
distance = qEλ ). For this reason, the parameter E/p (or Eλ )where pmfp mfp
is the gas pressure is a very important parameter for discussing phenomena in
gaseous electronics.
If the drift speed is much larger than the thermal speed v (i.e. the electronth
2 1/2gas is cold) then v ∼ u and u ∝ E/p or u ∼ (E/p) .Atlowdrifte ee
velocities, v >u (i.e. v is independent of u )then u ∼ (E/p). As shownth e e e
in Fig. 3.5, the measured dependence of the drift speed of electrons in hydrogen
and deuterium as a function of E/p conﬁrms these dependencies.