Theory of Langmuir probes in anisotropic plasmas

A theory has been developed for electron retardation by Langmuir probes of several geometries in a general anisotropic plasma with arbitrary probe orientation and valid for any sheath thickness. Electron densities and electron velocity distribution functions (EVDFs) are obtained from the second derivative of probe I-V curves, as in Druyvesteyn's original method, which was developed for isotropic plasmas. Fedorov had extended the latter method in the context of a thin sheath approximation, to axisymmetric plasmas, in which the EVDF is expanded in a series of Legendary polynomials. In the present work an expansion in a series of spherical harmonics is employed, and the coordinate transformations are handled using the irreducible representation of the three dimensional rotation group. It is shown that the Volterra integral equations that must be solved to obtain the expansion coefficients of the EVDF from the second derivative data are no more complicated in the general case that hose for the axisymmetric plasma. Furthermore in the latter case the results can be shown to be equivalent to Fedrov's thin sheath expression. For the case of planar probes a formulation based on first derivatives of the I-V curves has been obtained. If data is obtained at enough different probe orientation of a one sided planar disc probe, any number of spherical harmonic coefficient functions may be obtained by inverting a set of linear equations and the complete EVDF deduced. For a cylindrical probe or a two-sided planar disc probe the integration of the second derivative of the probe current gives the exact electron density with any arbitrary probe orientation and any degree of plasma anisotropy