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Hollow Cathodes 257 operation and life. The collisional plasma will also tend to have a low electron temperature, which reduces the sheath voltages and further protects the low work function insert surface from damage or modification by the plasma. It is possible to describe [26] the insert plasma using simple particle and energy balance models and plasma diffusion models because the plasma transport inside the hollow cathode is dominated by collisions. In Chapter 3, the solution to the radial diffusion equation for ions in collisionally dominated plasmas in cylindrical geometry resulted in an eigenvalue equation with a unique dependence on the electron temperature: R 2 no i(Te) 8kT e –D=0, (6.4-1) 01 m where R is the internal radius of the insert, 01is the first zero of the zero-order Bessel function, no is the neutral density, i is the ionization cross section averaged over a Maxwellian electron temperature, and D is the diffusion coefficient. This means that the electron temperature is constrained to produce sufficient ions to offset the diffusion losses to the wall. The diffusion in the radial direction in the insert region is ambipolar, and the ion mobility is limited by resonant charge exchange (CEX) with the xenon neutral atoms. The average collisions frequency for the ions is then i = CEX no vscat , (6.4-2) where the effective velocity for scattering of the ions in the insert region is approximated by the ion thermal speed: vscat = kTi . (6.4-3) M Since the electron mobility is much higher than the ion mobility, the ambipolar diffusion coefficient Da from Eq. (3.6-58) for this case is then D=D 1+Te =e (TiV+TeV), (6.4-4) a i Ti M CEX no vscat where the ion and electron temperatures are shown in eV. As an example, take two hollow cathodes operating in xenon with different inside diameters of the insert. The neutral density inside the hollow cathode is described by Eq. (2.7-2) for a given pressure, determined by the gas flow and the orifice size. A simple analytical technique to estimate the neutral pressure in the insert region is givenPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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