Power laws for the spatial dependence of electrical parameters in the high-voltage capacitive RF sheath

Following the analytical description of the collisionless high-voltage capa citive RF sheath in Lieberman (1988) and of the collisional high-voltage ca pacitive RF sheath in Lieberman (1989), electrical sheath parameters are ca lculated for a square-wave ie RF discharge current. The interest in the squ are-wave current lies in the analytical simplicity of the solutions, The sp atial dependences of the time-averaged electrostatic potential <(<Phi>)over bar> electric field (E) over bar, net-charge carrier density (n) over bar, and ion density ni are related to the distance a: from the plasma/sheath e dge by power laws. For the collisionless RF sheath, <(<Phi>)over bar> propo rtional to x(3/2), (E) over bar proportional to x(1/2), (n) over bar propor tional to x(-1/2), and n(i) proportional to x(-3/4) (the respective exponen ts for the collisionless de sheath are 4/3, 1/3, -2/3, -2/3), For the colli sional RF sheath, <(<Phi>)over bar> proportional to x(2), (E) over bar prop ortional to x(1), (n) over bar proportional to x(0), and n(i) proportional to x(-1/2) (the respective exponents for the collisional de sheath are 5/3, 2/3, - 1/3, - 1/3), The RF sheath solution is compared with Lieberman's so lution for the sinusoidal RF discharge current. The spatial dependences of <(<Phi>)over bar>, (E) over bar, (n) over bar, and n(i) are almost identica l for the two current waveforms, For fixed ionic sheath charge and ion curr ent, the sheath width and the time-averaged sheath voltage are similar for the two current waveforms.