Semianalytical description of nonlocal secondary electrons in a radio frequency capacitively coupled plasma at intermediate pressures

A model for fast simulation of a radio-frequency capacitively coupled discharge in the pressure range of a few torr and higher has been developed. The semi-analytical description of the nonlocal ionization by secondary (gamma) electrons allows us to evaluate the discharge parameters in detail even when the local approach for gamma electrons is not valid. We focused on the effect of steep increase of the plasma density at the central part of the discharge, at the transition from alpha to gamma regime. The present study demonstrates that in the gamma regime the plasma density at the discharge center is proportional to the spatial scale of ionization decay by gamma electrons inside the plasma region. We have investigated the electron energy distribution function of gamma electrons both analytically and numerically. As a result, the analytical formulas for the spatial scales of ionization decay by gamma electrons were found in limiting cases.