Field-driven ion migration against dead-stop collisional braking

The steady-state migration of ions, driven by a uniform electric field against full-stop collisions, is investigated in some detail. The required phase-space distribution is obtained very easily from Boltzmann's equation together with explicit recognition of energy conservation and population balance for the stagnant ion pool. We go on to decompose this aggregate solution into ion tiers classified by the number of background impacts previously endured. Such a decomposition permits us to detect the presence of Poisson statistics (as to collision number) lurking within the composite, thermalized Maxwellian, and likewise also a multiple-scattering hierarchy having the maiden, first-flight distribution for its natural kernel. Scattering-sequence accounting, in particular, allows a quantitative (even though unwieldy) distinction to be made between ions of varying residence times. A model of this sort is motivated by the technique of ion implantation through sample immersion within a plasma at higher electric potential. Numerical consequences of the solution obtained here reveal that both ion density and average kinetic energy relax to their terminal values within just a few mean free-path lengths. Such modest scaling of plasma-sheath extent evidently carries a beneficial implication for the technological ease with which surface properties (such as metal corrosion resistance and hardness) remain open to improvement via ion bombardment.