Ion Inertia Effects In Dynamic Plasma Sheaths

A new model for non-equilibrium bipolar sheath growth at the interface of a plasma and a conducting boundary is developed. Previous models
^1,2
are based on the Child-Langmuir bipolar relation, j
_e
/ = (M
_i
/ Zm
_e
)
^½
( or its relativistic version), and on an expression relating ion current density to the sheath motion of the form j
_i
= Zen
₀
u
₀
+ Zen
₀
dx
_s
dt. Here, u
₀
is the velocity of ions entering the sheath and x
_s
is the sheath edge position. These models ignore several significant effects when sheath growth is fast enough for ion inertia to be important. These effects are associated with assumptions made in deriving the equations above, namely that: a) ion energy is conserved, b) ion current density is uniform, and c) the total electron charge in the sheath is conserved. The conditions under which these assumptions are violated and the effects of the errors are described in this paper with emphasis on assumptions (a) and (b). In a companion paper
³
, the focus is on assumption (c). Taken together, these two papers develop generalized versions of the Child-Langmuir current density ratio and sheath opening equations shown above.