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352 Chapter 7 The secondary electrons are assumed to be emitted with an energy that is small compared to the plasma electron temperature and are accelerated through the sheath. The equation of continuity for current at the sheath edge gives nsvs = novo, (7.3-35) 1 where vs is the secondary electron velocity. The secondary electron density through the sheath is then n=n m E . (7.3-36) s o 1 M o Equations (7.3-32), (7.3-34), and (7.3-36) are inserted into Poisson’s equation, Eq. (7.3-31), and evaluated by the usual method of multiplying through by (7.3-37) (7.3-38) d / dx and integrating to produce 1 d 2 2E e 1/2 2 nkT dx =kT 1 E 1 ooee 2 mEe 1/2 1/2 + o 1 1 1 MkTkT ee o mE 1/2 e + 1 1 M e e x p k T 1 . o e A monotonic sheath potential is found [39] for E=kTe+ m 1/2 E 3/2 kTe e . 2 1 M e o 2 o For the case of no secondary electron emission ( going to zero), the Bohm criteria solution of E kTe /2e is recovered. Due to the large electron-to-ion mass ratio for xenon, the right-hand term is always small and the ion velocity at the sheath edge for the case of finite secondary electron emission will be near the Bohm velocity. Hobbs and Wesson evaluated this minimum ion energy at the sheath edge for the case of space charge–limited emission of electrons at the wall, d o / dx = 0 in Eq. (7.3-37), and they foundPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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