# Ambipolar diffusion and sheaths in a bounded plasma

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, $$Γ_i=Γ_e.$$ 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 $$Γ_i<<Γ_e.$$ 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!

• One condition relates to the bulk of the plasma, one to the boundary conditions. In the builk of the plasma carriers are diffusing in all directions - net fluxes can be equal. On the edge, the faster moving (lighter) charges are moving faster with no charges coming back the other way. – Jon Custer Sep 19 '19 at 12:37
• In that case, would it be correct to say the first condition effectively says that for a group of electrons diffusing in one direction, an equal group of ions will diffuse in the opposite direction, thus ensuring that the flux is equal? In that case I can see that if there are no ions to retreive from the boundary there will always be a much larger flux into the wall of electrons. – Jasper Sep 19 '19 at 13:02
• In the first case, the way I think about it, you have electrons moving in all directions and ions moving in all directions. But, in the bulk, you don't have all the electrons moving over here, and the ions moving over there. Kind of like how you can't sustain a field inside a conductor - you have electrostatically tied the electrons and ions, and while the whole thing can move they have to move in concert. Yet you know that individual electrons can move much faster than individual ions (and that happens to form the sheath until the field builds up). – Jon Custer Sep 19 '19 at 13:06
• Right this already makes much more sense to me now. Thanks a lot! – Jasper Sep 19 '19 at 13:09