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348 Chapter 7 the energies characteristic of Hall thrusters produces a significant number of secondary electrons, which reduces the sheath potential at the wall and increases the power loading. The requirement of local net current equal to zero and particle balance for the three species gives Iiw = Iew Iew = Iew (1 ), (7.3-28) where is the secondary electron yield from electron bombardment. Using Eq. (3.7-51) for the Bohm current of ions to the wall, Eq. (3.7-52) for the electron current to the wall, and neglecting the secondary electron velocity, Eq. (7.3-28) can be solved for the sheath potential s , including the effect of secondary electron emission: =kTeln (1 ) 2M . (7.3-29) se m This expression is slightly different than that found in the literature [30,31] because we have approximated e–1/2 = 0.61 0.5 for the coefficient in the expression for the Bohm current. Nevertheless, as the secondary electron yield increases, the sheath potential decreases from the classic floating potential described in Chapter 3 toward the plasma potential. Secondary electron yields reported in the literature [30,32,33] for several materials used for the walls of Hall thrusters are shown in Fig. 7-10. In this figure, the measurements were made using a monoenergetic electron gun. Generalizing these data for incident Maxwellian electron temperatures is accomplished by integrating the yield over the Maxwellian electron energy distribution function, which results in multiplying the secondary emission scaling by the gamma function [30]. An expression for the secondary electron yield from electron bombardment of materials is then = (2 + b)aT b , (7.3-30) eV where the electron temperature is in electron volts, (x) is the gamma function, and the coefficients a and b are found from fits to the data in Fig. 7-10. Values of the coefficients in Eq. (7.3-30) can be found in Table 7-1 for these materials, and the actual secondary electron yield for the Hall thruster walls is plotted versus plasma electron temperature in Fig. 7-11. It should be noted that due to reflection at the wall, the effective secondary electron yield does not go to zero for zero electron energy. This effect is accommodated by linear fits to the dataPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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