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Hollow Cathodes 285 where the diffusion is ambipolar with a diffusion coefficient given by Eq. (6.4-4). Using the electron momentum equation, Eq. (6.4-31), the electron current density on axis in the plume is given by J =1 E+ nTe , (6.7-2) e n where the resistivity for the case of electron-neutral and electron-ion collisions in the partially ionized gas is given by Eq. (6.4-35). The electron energy equation is given by Eq. (6.4-28) and includes convection, conduction, joule heating, pressure work, and ionization losses. Axial depletion of the neutral gas by ionization can be taken into account using a simple exponential attenuation model. The ionization rate of the neutral gas density is dno = none ive , (6.7-3) dt where no is the total neutral density in the plume. The neutral density is composed of the flowing component, n f , from the cathode and the background neutral density in the chamber, nc : no =nf (z)+nc. (6.7-4) The density of the flow gas decreases with distance away from the cathode orifice: dnf(z)=1dnf =1 dno dnc , (6.7-5) dz vodt vo dt dt where vo is the neutral gas velocity. Using Eq. (6.7-3), Eq. (6.7-5) can be written as 1 dnf(z)= 1 ne ive , (6.7-6) nf (z) dz vo which has a solution of the flow density decreasing exponentially away from the cathode. This is the same as the case analyzed in Section 3.6, in which the ionization mean free path is found to be one over the right-hand side of Eq. (6.7-6). The above equations [Eqs. (6.7-1) through (6.7-6)] are solved in a simple computational mesh shown in Fig. 6-30. The neutral gas is assumed to expand with a full angle of 45 deg. The code has been used to solve for the NEXISPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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