Edge-To-center plasma density ratios in two-dimensional plasma discharges

Edge-To-center plasma density ratios-so-called h factors-are important parameters for global models of plasma discharges as they are used to calculate the plasma losses at the reactor walls. There are well-established theories for h factors in the one-dimensional (1D) case. The purpose of this paper is to establish h factors in two-dimensional (2D) systems, with guidance from a 2D particle-in-cell (PIC) simulation. We derive analytical solutions of a 2D fluid theory that includes the effect of ion inertia, but assumes a constant (independent of space) ion collision frequency (using an average ion velocity) across the discharge. Predicted h factors from this 2D fluid theory have the same order of magnitude and the same trends as the PIC simulations when the average ion velocity used in the collision frequency is set equal to the ion thermal velocity. The best agreement is obtained when the average ion velocity varies with pressure (but remains independent of space), going from half the Bohm velocity at low pressure, to the thermal velocity at high pressure. The analysis also shows that a simple correction of the widely-used 1D heuristic formula may be proposed to accurately incorporate 2D effects.