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Class notes for EE6318/Phys 6383 – Spring 2001
This document is for instructional use only and may not be copied or distributed outside of EE6318/Phys 6383
Lecture 7 Diffusion
Our fluid equations that we developed before are:
∂n 

f c =  ∇ r ∑(n v ) + 

∂t 
∂ v

+ v ∑∇ r v  = ∆M c −
mn
 ∂t
 123
momentum
lost via
collisions
mv fc
12
4 4
3
− ∇ r ∑P + qn(E + v ∧ B)
momentum change
via particle gain / loss
We have looked collisions and Lorentz force terms in the in the last section two sections. Now
we will look at the random and driven flows.
Often, we can approximate the collisional momentum loss as being proportional to the velocity.
Here we will assume that
− mnν m v = ∆M c − m v f c
123
12
4 4
3
momentum
lost via
collisions
momentum change
via particle gain / loss
where ν m is the effective momentum transfer collision frequency. (The negative sign is there
because we are assuming that we are losing energy to the other species.) Thus,
∂ v

mn
+ v ∑∇ r v  = − mnνv − ∇ r ∑P + qn(E + v ∧ B)
 ∂t

Assuming that all of the accounted for forces balance. Hence, the left hand side of the equation
is zero. This gives
0 = − mnνv − ∇ r ∑P + qn(E + v ∧ B)
= − mnνv − γkT∇ r n + qn(E + v ∧ B)
or
v=−
γkT
qn
∇r n +
(E + v ∧ B) - for now we will let B = 0
mnν m
mnν m
=−
γkT
qn
∇r n +
E
mnν m
mnν m
=−
D
∇ r n + µE
n
where
γkT
D=
mν m
q
- note that I have used a slightly different definition for µ
mν m
are the diffusion and the mobility respectively. This is a straight-forward equation to get to but it
has a lot of implications. First, we have assumed that the velocity was not a function of position
or time. This means that the fluid is flowing from one point to another put that there is also fluid
moving the other direction – in some sort of random walk type of fashion. We find – not
surprisingly – that this means that if we where to be able to ‘paint’ some of the particles in the
species in a certain area, we would see these painted particles drift – or diffuse – away from that
area. This diffusion is quite natural. The second thing that we see is that the velocity of charged
µ=
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Class notes for EE6318/Phys 6383 – Spring 2001
This document is for instructional use only and may not be copied or distributed outside of EE6318/Phys 6383
particles under the influence is limited. This again is not too surprising – think of terminal
velocity for a person who is sky diving (~186 mph). What else can we discover?
First, we can determine the flux of a given species.
Γs = ns v s = − Ds ∇ r ns + ns µ s E - for species ©s©
If there is no electric field we get what is known as ‘Fick’s Law’
Γs = ns v s = − Ds ∇ r ns
This is simply our random walk.
Ambipolar Diffusion
If one were to turn off a plasma, most of them will decay via a process known as ambipolar
diffusion. ‘Ambipolar’ comes from two words. ‘Ambhi’ which is Latin for ‘on both sides’ or
‘round’ and polar (Duh?). In essence it means that both positive and negative charged diffuse at
the same rate. Such a requirement is not particularly surprising as we have required and expect
that the density of the positive and negative species must be about the same.
Without an applied electric field we clearly find that
Γe = − De∇ r ne >> Γi = − Di ∇ r ni
This implies that we must have an electric field which pushes the ions out and serves to retain the
electron. Thus,
Γe = − De∇ r ne + ne µ e E = Γi = − Di ∇ r ni + ni µ i E
We can now solve for the E field.
= ni ≡ n 0
= ni
}
}
( Di − De )∇r ne = (µ i − µ e ) ne E
⇒E=
( Di − De ) ∇r n0
(µ i − µ e ) n0
Placing this into our common diffusion equation gives
( D − De ) ∇ n
Γa = − Di ∇ r n0 + µ i i
(µ i − µ e ) r 0
=
(µ e Di − µ i De ) ∇ n
(µ i − µ e ) r 0
= Da ∇ r n0
where
(µ D − µ i De )
Da = e i
(µ i − µ e )
is the ambipolar diffusion coefficient.
We can determine approximately what the ambipolar diffusion coefficient is by making some
simple estimates.
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Class notes for EE6318/Phys 6383 – Spring 2001
This document is for instructional use only and may not be copied or distributed outside of EE6318/Phys 6383
First µ i , e =
qi , e
. Now ν m ≈ nσv . Assuming that both the ions and the electrons have the
mi , eν m
ve
Mi
. (This is
≈
vi
me
actually an under estimate as often the electron energy is higher than the ion energy.) Further,
we will assume that the collision cross sections are on the same order of magnitude. This is not a
bad assumption as coulomb collisions will be very similar as well as collisions with neutrals.
Thus, we expect
µe
Mi
=−
µi
me
Then
µ
Da = Di − i De
µe
It just so happens that the ratio of
µ
q
=
D γkT
So that
µi
e γ e kTe
De =
Di
µe
γ i kTi −e
same energy, then the velocity of the electron will be on the order of
=−
γ e Te
Di - often γ e = 1
γ i Ti
Te
Di
γ i Ti
Plugging this into the above equation for Da gives
µ
Da = Di − i De
µe
=−

T 
= 1 + e  Di
 γ i Ti 
≈ (2 to 10) Di
Thus we see that the ambipolar diffusion is tied to the slower diffusion rate but which often
greatly exceeds that slower rate.
Finally, we have to this point ignored the continuity equation.
∂n 

f c =  ∇ r ∑(n v ) + 

∂t 
Ignoring the collisional term, and using the definition of the flux, it is easy to see that
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Class notes for EE6318/Phys 6383 – Spring 2001
This document is for instructional use only and may not be copied or distributed outside of EE6318/Phys 6383
∂ns
= -∇ r ∑(ns v s
∂t
)
but Γs = ns v s = − Ds ∇ r ns + ns µ s E
⇓
∂ns
= −∇ r ∑( − Ds ∇ r ns + ns µ s E)
∂t
= Ds ∇ 2r ns − ns µ s ∇ r ∑E - letting E = 0
= Ds ∇ 2r ns
Diffusion in a Slab
This last equation allows us to look closer at the decay of a plasma
First let’s let
ns (r.t ) = T (t )S(r)
Then using the standard separation of variable technique for solving PDEs we get
∂ns
= Ds ∇ 2r ns
∂t
∂TS
= Ds ∇ 2r TS
∂t
∂T
S
= TDs ∇ 2r S
∂t
1 ∂T 1
= Ds ∇ 2r S = Const because we have to vary t and r independently
T ∂t S
1
We will call that Const ≡
τ
Then
∂T
T
= - ⇒ T = T0 e − t / τ
∂t
τ
and
S
S
∇ r2 S = −
=− 2
Dsτ
Λ
⇓
S = S0 + e + i r / Λ + S0 − e − i r / Λ or
S = S0 A cos( r / Λ ) + S0 B sin( r / Λ )
Assuming that the plasma has a length of L, with a density of 0 at ± L/2, we find
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Class notes for EE6318/Phys 6383 – Spring 2001
This document is for instructional use only and may not be copied or distributed outside of EE6318/Phys 6383
S r = ± L / 2 = 0 = SA cos( r / Λ ) + SB sin( r / Λ )
⇓
SA cos( L / 2 Λ ) = SB sin( L / 2 Λ ) (from - L / 2)
= − SB sin( L / 2 Λ ) (from + L / 2)
⇓
SB = 0 and
π
L
= ± jπ
2Λ 2
Thus,
L
L2
−1
−2
Λ = (1 ± 2 j ) = Dsτ ⇒ τ j =
1 ± 2 j)
2 (
Ds π
π
and
 rπ

S = ∑ SAj cos
1 ± 2 j )
(
 L

j
Therefore
ns (r.t ) = T (t )S(r)
∞
 Ds π 2
2
 rπ
n
exp
(1 + 2 j )
 −t 2 (1 + 2 j )  cos
∑
0j


L
L
j = −∞
2
L
−2
1 ± 2 j ) it is readily apparent that the higher order modes will decay faster than
From τ j =
2 (
Ds π
lower order modes. Thus as a plasma shuts off, we would expect to see a ‘cos’ profile to the
density distribution.
=
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