Rc. Woods et Id. Sudit, THEORY OF ELECTRON RETARDATION BY LANGMUIR PROBES IN ANISOTROPIC PLASMAS, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 50(3), 1994, pp. 2222-2238
The determination of electron densities and electron velocity distribu
tion functions (EVDF's) from the current-voltage (I-V) characteristics
in the electron repelling region is considered for cylindrical, spher
ical, 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, inclu
ding full consideration of orbital motion in the arbitrary sheath thic
kness 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-side
d planar probes. A concept of an isotropic EVDF that would give the sa
me probe current as the actual anisotropic one is defined for various
probe geometries and used to clarify the physical meaning of parameter
s extracted from measurements with a single probe orientation. The the
ory is extended to a completely anisotropic plasma using an expansion
of the EVDF in a series of spherical harmonic functions. The geometric
al relationships between the various coordinate systems are expressed
in terms of the group multiplication rule for the irreducible represen
tations of the three-dimensional rotation group. A method for extracti
ng the complete three-variable EVDF from probe I-V data at a sufficien
t number of probe orientations is given. The necessary Volterra integr
al equations are shown to be no more difficult than those arising in t
he 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 t
owards anisotropy of the plasma than previously realized. Specifically
, the usual method, applied exactly as if the plasma were indeed isotr
opic, should with a single arbitrary orientation of a cylindrical or t
wo-sided planar probe (or with a spherical probe) give the exact elect
ron density, even in a completely anisotropic plasma, and this result
is shown to be independent of the ratio of sheath radius to probe radi
us for cylindrical or spherical probes.