Theory of electron retardation by Langmuir probes in anisotropic plasmas.

Abstract

The determination of electron densities and electron velocity distribution functions (EVDF's) from the current-voltage (I-V) characteristics in the electron repelling region is considered for cylindrical, spherical, one-sided planar, and two-sided planar Langmuir probes. Previous treatments of axisymmetric plasmas, in which the EVDF is expressed as a series in Legendre polynomials, are extended and generalized, including full consideration of orbital motion in the arbitrary sheath thickness case for cylindrical probes. An alternative formulation focusing on the first derivative of the I-V data, which is normally more noise immune than the usually used second derivative, is given for one-sided planar probes. A concept of an isotropic EVDF that would give the same probe current as the actual anisotropic one is defined for various probe geometries and used to clarify the physical meaning of parameters extracted from measurements with a single probe orientation. The theory is extended to a completely anisotropic plasma using an expansion of the EVDF in a series of spherical harmonic functions. The geometrical relationships between the various coordinate systems are expressed in terms of the group multiplication rule for the irreducible representations of the three-dimensional rotation group. A method for extracting the complete three-variable EVDF from probe I-V data at a sufficient number of probe orientations is given. The necessary Volterra integral equations are shown to be no more difficult than those arising in the axisymmetric case. Finally, it is shown that the original method of Langmuir or Druyvesteyn for finding electron densities by integrating the second derivative of the I-V characteristic is much more robust towards anistoropy of the plasma than previously realized. Specifically, the usual method, applied exactly as if the plasma were indeed isotropic, should with a single arbitrary orientation of a cylindrical or two-sided planar probe (or with a spherical probe) give the exact electron density, even in a completely anisotropic plasma, and this result is shown to be independent of the ratio of sheath radius to probe radius for cylindrical or spherical probes.