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Hall Thrusters 351 at high electron temperatures [30,31] because the incident electron flux can then equal or exceed the random electron flux along the magnetic field lines in the plasma. In reality, the sheath potential for a floating boundary can never go significantly more positive than the local plasma potential [37,38] for two reasons. First, the secondary electrons are ejected from the wall with very low energy (typically 1–2 eV). Any positive-going sheath (where the plasma is negative by one or two volts relative to the wall) will repel the secondary electrons and return them to the wall. This clamps the sheath potential to within a few volts positive with respect to the plasma. Second, the secondary electron emission is space charge– limited in the sheath. This effect was analyzed by Hobbs and Wesson [39], who showed that space charge limits the secondary electron current from the wall independently of the secondary electron yield. The local electron space charge in the sheath clamps the sheath voltage to a maximum value that is always negative relative to the plasma. The effects of space charge on the sheath potential at the wall can be analyzed [39] by solving Poisson’s equation for the potential in the sheath: 2 = 1 ( n e + n s n i ) , ( 7 . 3 - 3 1 ) x2 o where ns is the secondary electron density. Using a Maxwellian distribution for the electrons, the plasma density in the channel is ne =(no nso)ee /kT, (7.3-32) where no is the ion density at the sheath edge, nso is the secondary electron density at the sheath edge, and is the potential relative to the potential o at the wall. The ions are assumed to be cold and to have fallen through the pre- sheath to arrive at the sheath edge with an energy of E = 12 m v o2 , ( 7 . 3 - 3 3 ) where vo is the Bohm velocity modified for the presence of the secondary electrons. The ion density through the sheath is then E 1/2 ni =no E e . (7.3-34)PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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