Note on sheath formation

We have discussed a plasma-sheath transition as a function of the electric current delivered to the sheath, using an one-dimensional, collisionless model and a set of macroscopic equations. When the conditions 𝛾2=∑_e,i ms v^2 s0/∑_e,i kTs=1 and E0=0, (where the subscript s stands for e (electron) and i (ion), ms is the mass of a particle, vs0 the velocity at the plasma-sheath boundary, Ts the temperature and E0 the electric field at the boundary, (i) for miv^2 i0/kTi>1 (or mev^2 s0/kTe<1) the pure ion sheath is formed without any local excess of electron space charge, (ii) for miv^2 i0/kTi=1 (or mev2 s0/kTe=1) any sheath is not established and (iii) for miv^2 i0/kTi<1 (or mev^2 s0/kTe>1 ) the pure electron sheath is developed. When 𝛾2≠1 and E0=0 or ≠0 , then the formation of sheath is much complicated. Thus the conditions 𝛾=1 and E0=0 lead to the establishment of much simple sheathes among the various sheathes and are including the familiar Bohm criterion when ve0→0. Furthermore, one may find that the ideal Langmuir probe curve, in which the conditions 𝛾2=1 and E0=0 are imposed, deviates from the classical probe theory.