# A Ambipolar diffusion and sheaths in a bounded plasma

#### Decimal

Hello,

I am currently working through an introductory text book on plasma physics, and I have encountered two topics that I separately understand but seem to be at odds with one another. In a quasi neutral plasma in steady state, the following relation must hold, $$\Gamma_i = \Gamma_e.$$ In other words, the ion and electron flux must be equal. My textbook refers to this as the congruence assumption, which can be derived from the continuity equations for both ions and electrons. By using this assumption one can derive the equation for ambipolar diffusion, which ensures that this condition is always fulfilled.

In the next chapter however, it is explained that due to higher mobility of electrons they will tend to diffuse out of the plasma much more quickly than ions. Thus in this case $$\Gamma_i << \Gamma_e.$$This will cause any boundary of the plasma to become negatively charged, which leads to the formation of a positive sheath. This in isolation seems logical to me. However how does it not conflict with the statement made before about ambipolar diffusion?

If ambipolar diffusion in a plasma ensures that the two fluxes are always equal, then how can a sheath ever form. I realise that the quasi neutrality condition is violated inside the sheath, so here the congruence assumption no longer holds. This however still doesn't explain to me how the sheath could ever form in the first place.

It would be really helpful if someone could explain how these two concepts relate.

Thanks!

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#### tnich

Homework Helper
The first case is "a quasi neutral plasma in steady state".
The second case is a transient phenomenon. Note the words "due to higher mobility of electrons they will tend to diffuse out of the plasma much more quickly than ions". This is clearly not a system in steady state.

#### Decimal

The first case is "a quasi neutral plasma in steady state".
The second case is a transient phenomenon. Note the words "due to higher mobility of electrons they will tend to diffuse out of the plasma much more quickly than ions". This is clearly not a system in steady state.
Thanks for the response, that makes a lot of sense! Would it then be correct to say that in steady state an equilibrium builds up between the sheath region and the ambipolar diffusion region, where both phenomena ensure flux equality across the entire plasma? (Except for inside the sheath of course).

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