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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

  • gradient can

  • collision

  • plasma

  • di?usion coe?cient

  • therefore very

  • time between

  • ions behind

  • temperature plasma

  • electron temperature


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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 .[4]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. [4]
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.[4]
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.[2]
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.