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88 Chapter 3 The sheath thickness at the wall depends on the plasma parameters and the potential difference between the plasma and the wall, and is found from the solution of Eq. (3.7-42). For the case of sheath potentials that are large compared to the electron temperature ( e >> kTe ), the current density through the sheath is described by the Child–Langmuir equation: 4eo 2e 1/2 3/2 J i = 9 M d 2 . ( 3 . 7 - 5 6 ) Finally, for the case of double sheaths where ion and electrons are counterstreaming across the boundary between two plasmas, the relationship between the two currents is J e = Mm J i . ( 3 . 7 - 5 7 ) If one boundary of the double layer is the sheath edge at a thermionic cathode, Eq. (3.7-51) can be used for the Bohm current to the opposite boundary to give the maximum emission current density as Je= nie kTe 1nee kTe. (3.7-58) 2m4m This is the maximum electron current density that can be accepted by a plasma due to space-charge effects at the cathode double sheath. For example, the maximum space-charge-limited cathode emission current into a xenon plasma with a density of 1018 m–3 and an electron temperature of 5 eV is about 3.8 A/cm2. These summary equations are commonly seen in the literature on the design and analysis of ion sources, plasma processing sources, and, of course, many electric propulsion thrusters. References [1] F.F.Chen,IntroductiontoPlasmaPhysicsandControlledFusion,vol.1, New York: Plenum Press, 1984. [2] M. Lieberman and A. Lichtenberg, Principles of Plasma Discharges and Materials Processing, New York: John Wiley and Sons, 1994.PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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