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Basic Plasma Physics 83 electron density through the pre-sheath and sheath, which results in the physically realistic monotonically decreasing potential behavior through the sheath. Figure 3-4 shows a plot of the sheath thickness d normalized to the Debye length versus the potential drop in the sheath normalized to the electron temperature. The criterion for a Debye sheath derived in Section 3.7.1 was that the potential drop be much less than the electron temperature ( e << kTe ), which is on the far left-hand side of the graph. The criterion for a Child– Langmuir sheath derived in Section 3.7.3 is that the sheath potential be large compared to the electron temperature ( e >> kTe ), which occurs on the right- hand side of the graph. This graph illustrates the rule-of-thumb that the sheath thickness is several Debye lengths until the full Child–Langmuir conditions are established. Beyond this point, the sheath thickness varies as the potential to the 3/2 power for a given plasma density. The reason for examining this general case is because sheaths with potential drops on the order of the electron temperature or higher are typically found at both the anode and insulating surfaces in ion and Hall thrusters. For example, it will be shown later that an insulating surface exposed to a xenon plasma will self-bias to a potential of about 6Te , which is called the floating potential. For a plasma with an electron temperature of 4 eV and a density of 1018m–3, the Debye length from Eq. (3.7-12) is 1.5 10–5m. Since the potential is actually 8 7 6 5 4 3 2 1 0 Child−Langmuir DeBye 0 2 4 6 8 10 12 14 16 18 20 χ = eφ/kTe Fig. 3-4. Normalized sheath thickness as a function of the normalized sheath potential showing the transition to a Child–Langmuir sheath as the potential becomes large compared to the electron temperature. =d/ DPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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