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Basic Plasma Physics 63 solution is symmetric ( B = 0 ) with the diffusion length equal to . The solution to Eq. (3.6-38) is then X = cos x . (3.6-40) d The lowest-order complete solution to the diffusion equation for the plasma density is then the product of Eq. (3.6-37) and Eq. (3.6-40): n=noe–t cos x . (3.6-41) d Of course, higher-order odd solutions are possible for given initial conditions, but the higher-order modes decay faster and the lowest-order mode typically dominates after a sufficient time. The plasma density decays with time from the initial value no , but the boundary condition (zero plasma density at the wall) maintains the plasma shape described by the cosine function in Eq. (3.6-41). While a slab geometry was chosen for this illustrative example due to its simplicity, situations in which slab geometries are useful in modeling ion and Hall thrusters are rare. However, solutions to the diffusion equation in other coordinates more typically found in these thrusters are obtained in a similar manner. For example, in cylindrical geometries found in many hollow cathodes and in ion thruster discharge chambers, the solution to the cylindrical differential equation follows Bessel functions radially and still decays exponentially in time if source terms are not considered. Solutions to the diffusion equation with source or sink terms on the right-hand side are more complicated to solve. This can be seen in writing the diffusion equation as n – D 2n = n , (3.6-42) t where the source term is described by an ionization rate equation given by n = na n ive na n i(Te)v , (3.6-43) and where v is the average particle speed found in Eq. (3.4-8) and i (Te ) is the impact ionization cross section averaged over a Maxwellian distribution of electrons at a temperature Te . Equations for the xenon ionization reaction rate coefficients averaged over a Maxwellian distribution are found in Appendix E.PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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