C.3.5 Models of current through the sheath

In a non perturbed plasma the plasma current impacting an infinitesimal surface A is related to the distribution functions of each species i, f_i(v), by

where is the velocity perpendicular to the surface A and q is charge.

For a Maxwellian Distribution

where

• n is density,

• k is the Boltzmann’s constant, and

• T is the temperature,

The current density related to each species can be expressed as:

When the temperature of the various species are close to each other the sum of the terms I_i is negative because the mass of the electron is much less than the one of the ions.

As shown in the clauses C.3.2 to C.3.4, the plasma around an object is, in general, perturbed and the amount of current that flows to the object is intricately related to the sheath structure. The sheath structure is determined in general numerical iterative algorithm to find a self-consistent solution of the Poisson and Vlasov system of equations. There are however limiting cases for which the current through the sheath can be easily calculated. Two of these limiting cases are described in C.3.6 and C.3.7.

It can also be noted for repelled Maxwellian species and when the geometry of the surface is symmetric enough the current collected is simply given by:

where V is the repelling voltage.