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Hollow Cathodes 259 n Ji = nvr = Da = n(0)Da drift velocity at the wall is then vr = (2.4)2 e (TiV +TeV ). (6.4-7) 2R CEX no vscat M In the example above, the larger-diameter insert produces an electron temperature of about 1.4 eV at 1.8-torr internal xenon pressure. The effective ion velocity found near the wall outside the sheath is only 3.1 m/s due to the ion-neutral CEX collisions, which slows the ion velocity to significantly less than the 500-m/s ion thermal velocity and 1200-m/s xenon ion acoustic velocity. Since the pre-sheath potential that accelerates the ions to the Bohm velocity prior to entering the sheath extends only the order of the collision mean free path into the plasma, ions diffusing to the plasma edge are accelerated very close to the sheath to the Bohm velocity due to the high collisionality in the insert plasma. The density of the insert plasma can be estimated by a simple 0-dimensional (0-D) particle and energy balance model. These types of models assume a fairly uniform plasma in the insert region and so provide density estimates within factors of the order of two. In the insert plasma, heating of the plasma is balanced by the energy loss: It s+RIe2=IiU++52TeVIe+(2TeV+ s)Ire s TeV, (6.4-8) where It is the thermionic electron current, s is the cathode sheath voltage, R is the plasma resistance, Ie is the hollow cathode discharge current, Ii is the total ion current generated in the insert region, U+ is the ionization potential, TeV is the electron temperature (in volts), and Ir is the random electron flux at the sheath edge. In this case, excitation and radiation losses seen in the discharge chamber energy balance equations are ignored because the high density plasma inside the hollow cathode is optically “thick” and the radiated energy is reabsorbed by the plasma. The resistance, R, is the resistivity times the average conduction length, , divided by the cross-sectional area of the plasma: R= . (6.4-9) r2 r R 2R 01 J1( 01)= nDa Using the ambipolar diffusion coefficient from Eq. (6.4-4), the effective radial ( )2 01 . (6.4-6)PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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