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76 Chapter 3 3.7.2 Pre-Sheaths In the previous section, the sheath characteristics for the case of the potential difference between the plasma and an electrode or boundary being small compared to the electron temperature ( e << kTe ) was analyzed and resulted in Debye shielding sheaths. What happens for the case of potential differences on the order of the electron temperature? Consider a plasma in contact with a boundary wall, as illustrated in Fig. 3-3. Assume that the plasma is at a reference potential at the center (which can be arbitrarily set), and that cold ions fall through an arbitrary potential of o as they move toward the boundary. Conservation of energy states that the ions arrived at the sheath edge with an energy given by 12 M v o2 = e o . ( 3 . 7 - 1 4 ) This potential drop between the center of the plasma and the sheath edge, o , is called the pre-sheath potential. Once past the sheath edge, the ions then gain an additional energy given by 12 Mv2 = 12 Mvo2 – e (x) , (3.7-15) where v is the ion velocity in the sheath and is the potential in the sheath (becoming more negative relative to the center of the plasma). Using Eq. (3.7-14) in Eq. (3.7-15) and solving for the ion velocity in the sheath gives Φ no Φo Edge Sheath Plasma Fig. 3-3. Plasma in contact with a boundary. SheathPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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