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Basic Plasma Physics 77 v= 2e[ o– ]1/2. (3.7-16) M However, from Eq. (3.7-14), vo = 2e o M , so Eq. (3.7-16) can be rearranged to give Using Eq. (3.7-17) in Eq. (3.7-18), the ion density in the sheath is vo = o v o – , (3.7-17) which represents an acceleration of the ions toward the wall. The ion flux during this acceleration is conserved: (3.7-18) (3.7-19) niv = novo n = n vo . iov n =n o i o o– . Examining the potential structure close to the sheath edge such that is small compared to the pre-sheath potential o , Eq. (3.7-19) can be expanded in a Taylor series to give n =n 1– 1 +... , (3.7-20) i o 2 o where the higher-order terms in the series will be neglected. The electron density through the sheath is given by the Boltzmann relationship in Eq. (3.5-9). If it is also assumed that the change in potential right at the sheath edge is small compared to the electron temperature, then the exponent in Eq. (3.5-9) can be expanded in a Taylor series to give n = n exp e = n 1 e + ... . (3.7-21) e o k T e o k T e Using Eqs. (3.7-20) and (3.7-21) in Poisson’s equation, Eq. (3.7-8), for singly charged ions in one dimension givesPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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