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Basic Plasma Physics 65 C2 + 2 = 01 , (3.6-50) R where 01 is the first zero of the zero-order Bessel function and R is the internal radius of the cylinder being considered. Setting = 0 , this eigenvalue results in an equation that gives a direct relationship between the electron temperature, the radius of the plasma cylinder, and the diffusion rate: R 2 na i(Te) 8kT e –D=0. (3.6-51) 01 m The physical meaning of Eq. (3.6-51) is that particle balance in bounded plasma discharges dominated by radial diffusion determines the plasma electron temperature. This occurs because the generation rate of ions, which is determined by the electron temperature from Eq. (3.6-43), must equal the loss rate, which is determined by the diffusion rate to the walls, in order to satisfy the boundary conditions. Therefore, the solution to the steady-state cylindrical diffusion equation specifies both the radial plasma profile and the maximum electron temperature once the dependence of the diffusion coefficient is specified. This result is very useful in modeling the plasma discharges in hollow cathodes and in various types of electric thrusters. 3.6.2.2 Ambipolar Diffusion Without a Magnetic Field. In many circumstances in thrusters, the flux of ions and electrons from a given region or the plasma as a whole are equal. For example, in the case of microwave ion thrusters, the ions and electrons are created in pairs during ionization by the plasma electrons heated by the microwaves, so simple charge conservation states that the net flux of both ions and electrons out of the plasma must be the same. The plasma will then establish the required electric fields in the system to slow the more mobile electrons such that the electron escape rate is the same as the slower ion loss rate. This finite electric field affects the diffusion rate for both species. Since the expression for the flux in Eq. (3.6-29) was derived for any species of particles, a diffusion coefficient for ions and electrons can be designated (because D contains the mass) and the fluxes equated to obtain μinE–Di n=–μenE–De n, (3.6-52) where quasi-neutrality ( ni ne ) in the plasma has been assumed. Solving for the electric field givesPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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