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354 Chapter 7 secondary electron yield times the flux of plasma electrons. Equating the ion flux to the net electron flux to the wall gives 1 4 8kT 1/2 ne(1 )e e m e Aexp s , (7.3-43) Iiw =nieviA=Iew(1 )= where the ion and electron densities are evaluated at the sheath edge. The sheath potential s relative to the plasma potential is then = kTe ln M ne vB (1 ) , (7.3-44) s e 2 mne+nsvi where vi is the modified ion velocity at the sheath edge due to the presence of kTe the secondary electrons and the ion density is the sum of the plasma and secondary electrons. This equation is useful up to the space charge–limited potential of o = 1.02TeV and provides good agreement with the results for xenon described above for nevB / nivi 0.5 . The sheath potential predicted by Eq. (7.3-44) is plotted in Fig. 7-13 for two wall materials. In the limit of no secondary electron emission ( = 0), the classic value for the sheath floating potential is obtained from Eq. (3.7-53). Once the electron temperature is sufficiently high to produce a yield approaching and even exceeding one, then the space charge–limited case of o = 1.02TeV is obtained. In between, the sheath potential depends on the electron temperature and material of the wall. Without the space charge–limited sheath regime predicted by Hobbs and Wesson, the potential would have continued along the thin dashed lines for the two cases and incorrectly resulted in very low sheath potentials and high power loadings at the wall. The total power to the wall of the Hall thruster is 1 8kTe 1/2 e /kT kTe P w = 4 m e n o A e s e 2 e + n o e v o A ( E s ) , ( 7 . 3 - 4 5 ) where the first term is due to electrons overcoming the repelling sheath potential and depositing 2Te on the wall, and the second term is due to ions that have fallen through the pre-sheath potential and then the full sheath potential. Note that no in this equation is the plasma density at the sheath edge and is roughly half the average plasma density in the center of the channel due to the radial pre-sheath. The cooling of the wall by the secondary electronPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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