icarus wrote:We could call it a "Carlson sheath" if you can pull it off.
Hey, I can do that! Ladies and gentlemen, hold on to your hats. For the first time in the free world, never before done, never before attempted, the death-defying mono-energetic electron sheath! Oh, heck, let's just call it the Carlson sheath for simplicity, OK?
The trick is I have found a mono-energetic distribution that is preserved as the electrons move through potential variations:
- f(v_x, v_y, v_z) = (1/4pi)*n_0*(1-e*phi/W_0)*delta(v)*|v_z|/v^3
where n_0 and W_0 are the electron density and energy upstream, where the reference voltage is also defined. v = sqrt(v_x^2+v_y^2+v_z^2), and the E field is in the z direction.
I'll leave the proof for another day. Better yet, I'll try this business model: Anyone who wants to see the proof send me 2 Euros and I'll mail it to them.
The neat thing, aside from the self-similarity, is the linear dependence on the potential. Actually, that's not so important, just neat. I think the derivation could be generalized to other distributions as well.
Anyway, at this point we can call Bohm back in. Basically, his condition says that, as the potential starts to drop, the electron density must drop faster than the ion density. Otherwise you would get an excess of electrons and the potential would curve in the wrong direction. When the electron distribution is Maxwellian, d n_e / d phi is n_e*e / T_e. Here it is n_e / W_0. For the ions, n_i = Gamma / v = Gamma / sqrt( v_z0^2 - 2*e*phi/m_i ), so d n_i / d phi = n_i*e/m_i. To require that d (n_e-n_i) / d phi > 0 is equivalent to v_z0 > sqrt(W_0/m_i).
Thus, even though the electron energy distribution is nowhere near Maxwellian, the Bohm condition still applies, with the electron energy playing the role of the electron temperature. This is the idea behind my claim that the ions, even if they are born with little energy, will be accelerated by the electric field in the plasma until they are zipping along when the leave through the cusps, and that this means a hardy energy drain.