For the plasma sheath region formed in the vicinity of a plane wall immersed in a plasma,

Question:

For the plasma sheath region formed in the vicinity of a plane wall immersed in a plasma, assume that the ions at the plasma edge of the sheath can be described by a shifted Maxwellian distribution function

with drift velocity u₀ = u₀x̂. Prove that the ion flux Г_ix, at the edge of the sheath, is given by

where y = u₀[m_i/(2kT)]^1/2 and erf (y) is the error function, defined by

Calculate dГ_ix/dy. Note that the error function vanishes for y = 0, increases monotonically as y increases, and tends asymptotically to unity as y → ∞. Also note that