Diffusion Plasma

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DIFFUSION IN WEAKLY

IONIZED PLASMA

Priyanka

Msc (final) PhysicsRoll no 19189

Miranda House


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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.


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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.


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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 .


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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


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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


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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


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Setting equation(1) equal to zero, we get:

The coefficients above are called the mobility and the diffusioncoefficient. These are different for different species.


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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)


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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.


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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).


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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)


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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 ;


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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


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so we can write Da as:

The effect of the ambipolar electric field is to enhance the diffusion ofions by a factor of two.


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Diffusion in a slab

The density distribution isa cosine.

This is called lowestdiffusion mode.

The peak density decaysexponentially.


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Diffusion in a cylinder

The density decreases as 1/r. So it goes like a

damped cosine . This function is called BesselsFunction.


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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).


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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


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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.


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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


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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.


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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!


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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!


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Writing the perpendicular component of the fluid equation of

motion:

The x and y components being:


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These equations can be solved for Vx and VY.

We get values for the perpendicular mobility and diffusion coefficients as :


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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


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Experimental verification : magnetic field reduces thetransverse diffusion

Lehnert and Hoh in sweden performed first confirmatory

experiment.

The Lehnert-Hoh experiment.

.


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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)]


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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.


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Diffusion Plasma

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