Diffusion Plasma
-
Upload
priyanka-lochab -
Category
Documents
-
view
222 -
download
0
Transcript of Diffusion Plasma
-
7/31/2019 Diffusion Plasma
1/34
DIFFUSION IN WEAKLY
IONIZED PLASMA
Priyanka
Msc (final) PhysicsRoll no 19189
Miranda House
-
7/31/2019 Diffusion Plasma
2/34
What is diffusion ?
The migration of particles from highdensity region to a low densityregion, thereby tending to flatten outthe density gradient is calledDIFFUSION.
Any realistic plasma has a densitygradient , and so it tends to diffuse.
-
7/31/2019 Diffusion Plasma
3/34
weakly ionized plasma
It is a non-uniform distribution of ions and electronsin a dense background of neutrals.
The plasma spreads out due to pressure gradient
and electric field forces, the individual particlesundergo a random walk, colliding frequently with the
neutral atoms.
-
7/31/2019 Diffusion Plasma
4/34
Collision Parameters
When an electron collides with a neutral atom, it
may lose any fraction of its momentum, depending on theangle at which it rebounds.
probability of momentum loss = equivalent cross-section .
-
7/31/2019 Diffusion Plasma
5/34
A slab of area A andthickness dxcontainingnn neutral atoms per m
3
.
The electrons areincident on this slab.
Assumption : atoms areopaque spheres of
cross-sectional area .Fig : Illustration of definition of
cross section
-
7/31/2019 Diffusion Plasma
6/34
-
7/31/2019 Diffusion Plasma
7/34
In a distance m , the flux is decreased to 1/e of its
initial value.
The quantity mis the mean free pathfor
collisions:
m = 1/ (nn )
The mean timebetween the collisions is given by
= m/ v
-
7/31/2019 Diffusion Plasma
8/34
The fluid equation of motion including collisions
for any species is,
(1)
Where the + , - sign indicates the sign of the charge.
We take , v = constant.
we consider a steady state in which v/ t =0.
Diffusion Parameters
-
7/31/2019 Diffusion Plasma
9/34
Setting equation(1) equal to zero, we get:
The coefficients above are called the mobility and the diffusioncoefficient. These are different for different species.
-
7/31/2019 Diffusion Plasma
10/34
The transport coefficients and D are related by the
EINSTEIN RELATIONS.
= |q| D / KT
Using these definitions , flux J of the jth species can be
written as
(2)
-
7/31/2019 Diffusion Plasma
11/34
If E=0 or = 0 ,that is the particles are uncharged. ThenFrom eq(2) , we get
It shows that diffusion is a randomwalk process, in
which there is a net flux from dense regions to less dense
regions.
-
7/31/2019 Diffusion Plasma
12/34
From Ficks Law ,
flux is proportional to gradient of the density.
However , in plasmas Ficks Law may not be necessarily
obeyed. Because of the possibility of organized motions(plasma waves).
-
7/31/2019 Diffusion Plasma
13/34
Decay of a plasma by diffusion : Ambipolar Diffusion
Plasma created in a container decays by diffusion to the walls.
If the decay is slow, we need only the time derivative in the
continuity equation.
If collision frequency is large, time derivative in the equation
Of motion in eq(1) = negligible
So we get :
(3)
-
7/31/2019 Diffusion Plasma
14/34
If e and i are not equal , a serious charge imbalance would
soon arise. So ,the required electric field is found by setting
e = i =
So from equation (1), we can write ;
-
7/31/2019 Diffusion Plasma
15/34
This is Ficks law with a new diffusion coefficient.
called the ambipolar diffusion coefficient.
If Da is a constant , equation (3) becomes
the magnitude of Da can be estimated if we take e >> i
-
7/31/2019 Diffusion Plasma
16/34
so we can write Da as:
The effect of the ambipolar electric field is to enhance the diffusion ofions by a factor of two.
-
7/31/2019 Diffusion Plasma
17/34
Diffusion in a slab
The density distribution isa cosine.
This is called lowestdiffusion mode.
The peak density decaysexponentially.
-
7/31/2019 Diffusion Plasma
18/34
-
7/31/2019 Diffusion Plasma
19/34
Diffusion in a cylinder
The density decreases as 1/r. So it goes like a
damped cosine . This function is called BesselsFunction.
-
7/31/2019 Diffusion Plasma
20/34
-
7/31/2019 Diffusion Plasma
21/34
Steady state solutions
A plasma is maintained in a steady state by
continuous ionization or injection of plasma.
In this case, we must add a source term to the
equation of continuity :
In steady state we put n /t = 0 = Poisson
type equation for n (r).
-
7/31/2019 Diffusion Plasma
22/34
Constant Ionization Function
In weakly ionized gases, ionization is producedby energetic electrons in the maxwellian distribution.
Then, source term Q is proportional to the electron
density n.
Setting Q = Zn , where Z is the ionization function we
have
-
7/31/2019 Diffusion Plasma
23/34
Recombination
When an ion and an electron collide, they have a finiteprobability of recombining into a neutral atom.
For momentum conservation a third body must be
present.
If its a photon , we say radiative recombination
If its a particle, we say three-body recombination.
The loss by recombination = negative source term in theequation of continuity.
-
7/31/2019 Diffusion Plasma
24/34
In the absence of diffusion terms, the equation of continuity
becomes
n /t = - n2 (4)
The constant of proportionality is called recombination
coefficient and has units of m3/sec.
A non-linear equation for n. Its solution is
Where n (r, t) is the initial density distribution.
t
rtrn
n
)(
1
),(
1
-
7/31/2019 Diffusion Plasma
25/34
Diffusion across a magnetic field
The rate of plasma loss by
diffusion is decreased by
magnetic field.
Consider a weakly ionized plasma
in a magnetic field.
Charged particles move along B
by diffusion and mobility
parameters.
-
7/31/2019 Diffusion Plasma
26/34
Since B does not affect motion in the parallel direction. Thus
we have for each species.
If no collisions, particles would not diffuse at all in
perpendicular direction
they continue to gyrate about the
same line of force.
Like in a perfectly symmetric
cylinder
The drifts are harmless!
-
7/31/2019 Diffusion Plasma
27/34
When an ion collides with a
neutral atom, the ion
leaves the collision traveling in
a different direction.
The guiding centre shiftsposition in a collision andundergoes a randomwalk!
-
7/31/2019 Diffusion Plasma
28/34
Writing the perpendicular component of the fluid equation of
motion:
The x and y components being:
-
7/31/2019 Diffusion Plasma
29/34
These equations can be solved for Vx and VY.
We get values for the perpendicular mobility and diffusion coefficients as :
-
7/31/2019 Diffusion Plasma
30/34
we can write D as :
The square of the length over time , shows that diffusion is a random-
walk process with a step length of m
-
7/31/2019 Diffusion Plasma
31/34
Experimental verification : magnetic field reduces thetransverse diffusion
Lehnert and Hoh in sweden performed first confirmatory
experiment.
The Lehnert-Hoh experiment.
.
-
7/31/2019 Diffusion Plasma
32/34
The normalized longitudinal electric field measured as a function of B at two different pressures. [ fromF.C Hoh and B. Lehnert , phys. Fluids 3, 600(1960)]
-
7/31/2019 Diffusion Plasma
33/34
At low B fields, the experimental points closely follow the
predicted curve.
At a critical value of Bc = 0.2T, the experimental points departedfrom theory. They showed an increase in diffusion with B.
Critical field Bc increased with pressure, suggesting something was
wrong with the classicaltheory of diffusion.
Kadomtsev and Nedospasov, U.S.S.R : a plasma wave gets excitedby Ez field, and this causes enhanced radial losses.
-
7/31/2019 Diffusion Plasma
34/34
THANK YOU