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Ion Thruster Plasma Generators 161 Solving for the perturbed electron velocity, assuming plane waves, and inserting this into Eq. (4.6-15), the perturbed current is j1 = –nee oE1 . (4.6-17) i m Inserting Eq. (4.6-17) into Eq. (4.6-14) and solving for the frequency gives the dispersion relation for electromagnetic waves in a plasma: 2 = n e e 2 + c 2 k 2 = 2p + c 2 k 2 , ( 4 . 6 - 1 8 ) om where the definition of the electron plasma frequency 2p = nee2 / om has been used. This expression can be solved for the wavelength of the microwaves in the plasma = 2 c = c , (4.6-19) 2p – 2 f p2 – f 2 where fp is the real plasma frequency and f is the microwave frequency. If the microwave frequency exceeds the plasma electron frequency, the wavelength becomes infinitely long and the wave becomes evanescent (it will not propagate into the plasma) and is reflected. This condition, called cutoff, determines the maximum plasma density into which a microwave source can inject power to produce the plasma. Table 4-1 shows the cutoff frequency for a range of plasma densities and the ion current density from a xenon plasma at an electron temperature of 3 eV. As an example, if the ion thruster design requires an ion Table 4-1. Cutoff frequencies for several plasma densities, and the corresponding ion current density from a xenon plasma at Te = 3 eV. Plasma Density (cm–3) Cutoff Frequency (GHz) 109 0.285 1010 0.900 1011 2.846 1012 9.000 1013 28.460 J (mA/cm2) 0.0118 0.118 1.184 11.84 118.4PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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