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Ion Thruster Plasma Generators 113 Alternatively, the modified electric field given in Eq. (4.3-12) can be inserted into Eq. (4.3-10) to produce an expression for the transverse ion velocity: v2+ el 1+μ2B2– ei v–kTe=0. iμeM e e iM This quadratic equation can be easily solved to give 1 e 22 ei 24kTe vi=2 Mμe 1+μeB – e + M – e 1+μ2B2– ei . 2 M μ e e e (4.3-14) (4.3-15) The collision frequencies ( e = en + ei and = en / ei ) in these equations for xenon plasmas were given in Chapter 3, where the electron-neutral collision frequency is given in Eq. (3.6-12) and the electron-ion collision frequency is given in Eq. (3.6-14). It is possible to show that in the limit that B goes to zero and the flow is essentially collisionless, Eq. (4.3-15) reverts to the Bohm velocity. Defining an ion confinement factor fc vi , (4.3-16) vBohm and since the Bohm velocity is vBohm = kTe / M , it is a simple matter to calculate the reduction in the expected flux of ions going to the anode due to the reduction in the Bohm velocity at a given magnetic field strength B. The ion current transverse to the magnetic field between the cusps to the anode is then given by Iia=1nie kTe Aasfc, (4.3-17) 2M where Aas is the total surface area of the anode exposed to the plasma. There are two issues with using Eq. (4.3-17) to evaluate ion loss rate reduction between the cusps. First, the magnetic field in the ring-cusp geometry is not transverse to the wall everywhere. Near the cusp, the field transitions fromPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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