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94 Chapter 4 I i = 12 n i e v a A , ( 4 . 2 - 1 ) where ni is the ion density in the center of the volume; va is the ion acoustic velocity; A is the total ion loss area, which is assumed to be only the grid area; and the ions are assumed to be cold relative to the electrons. The ion beam current is then the total ion current to the grids multiplied by the effective grid transparency, Tg : I b = 12 n i e v a A T g , ( 4 . 2 - 2 ) where the current lost to the accel and decel grids has been neglected as small. Ions are assumed to be produced by ionization of neutral particles by the plasma electrons in the discharge chamber, with a rate given by Ip=nonee ive V, (4.2-3) where no is the neutral gas density, ne is the plasma electron density, i is the ionization cross section, ve is the electron velocity, and the term in the brackets is the reaction rate coefficient which is the ionization cross section averaged over the Maxwellian electron velocity distribution function. The formulation of the reaction rate coefficient was described in Chapter 3, and the values for xenon as a function of electron temperature are given in Appendix E. Power is conserved in the system, so the power put into the plasma is equal to the power that comes out in the form of charged particles and radiation. To first order, the power injected into the plasma goes into ionization and excitation of the neutral gas, heating of the electrons, and power that is carried to the walls and the grids by the ions and electrons. The power that is put into the system is then Pin = IpU+ + I*U* + Ii i + neV e , (4.2-4) where U+ is the ionization potential of the propellant gas, U* is the excitation potential of the gas, is the average electron confinement time, i is the ion energy carried to the walls, and e is the electron energy carried to the walls by the electrons leaving the plasma. The term I* is the excited neutral production rate, given byPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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