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Hall Thrusters 353 Eo = 0.58 kTe . (7.3-39) e Equation (7.3-39) indicates that the Bohm sheath criterion will still approximately apply (within about 16%) in the presence of secondary electron emission. The value of the sheath potential for the space charge–limited case can be found by setting the electric field at the wall equal to zero in Eq. (7.3-37) and evaluating the potential using Eq. (7.3-38) and the current continuity equation: 1 m E 1/2 e 8kT 1/2 1 2E 1/2 1 exp o e = .(7.3-40) 4 1 Me kT m 1 M o e The space charge–limited sheath potential for xenon is found to be o = 1.02 kTe . (7.3-41) e The secondary electron yield at which the sheath becomes space-charge limited [39] is approximately m 1/2 o =1 8.3 M , (7.3-42) which for xenon is 0.983. This analysis shows that the sheath potential for a xenon plasma decreases from 5.97Te for walls where the secondary electron yield can be neglected to 1.02Te for the case of space charge–limited secondary electron emission that will occur at high plasma electron temperatures. The value of the sheath potential below the space-charge limit can be found exactly by evaluating the three equations, Eqs. (7.3-37), (7.3-38), and (7.3-40), for the three unknowns ( , , and E ). However, the value of the sheath potential relative to the plasma edge in the presence of the secondary electron emission can be estimated by evaluating Eq. (7.3-29) while accounting for each of three species [38]. Quasi-neutrality for the three species in the plasma edge dictates that ni = ne + ns , where ns is the secondary electron density, and the flux of secondary electrons is thePDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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