Electron density and electron temperature measurement in a bi-Maxwellian electron distribution using a derivative method of Langmuir probes

In plasma diagnostics with a single Langmuir probe, the electron temperature T_e is usually obtained from the slope of the logarithm of the electron current or from the electron energy probability functions of current (I)-voltage (V) curve. Recently, Chen [F. F. Chen, Phys. Plasmas 8, 3029 (2001)] suggested a derivative analysis method to obtain T_e by the ratio between the probe current and the derivative of the probe current at a plasma potential where the ion current becomes zero. Based on this method, electron temperatures and electron densities were measured and compared with those from the electron energy distribution function (EEDF) measurement in Maxwellian and bi-Maxwellian electron distribution conditions. In a bi-Maxwellian electron distribution, we found the electron temperature T_e obtained from the method is always lower than the effective temperatures T_eff derived from EEDFs. The theoretical analysis for this is presented.