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Basic Plasma Physics 79 fallen to a value of kTe / 2e relative to the plasma potential where the density is no (far from the edge of the plasma). The electron density at the sheath edge is then n =n exp e o =n exp e –kTe e o kTe o kTe 2e (3.7-27) = 0.606 no. Therefore, the plasma density at the sheath edge is about 60% of the plasma density in the center of the plasma. The current density of ions entering the sheath at the edge of the plasma can be found from the density at the sheath edge in Eq. (3.7-27) and the ion velocity at the sheath edge in Eq. (3.7-25): Ji = 0.6 noevi 1 ne kTe , (3.7-28) 2M where n is the plasma density at the start of the pre-sheath, which is normally considered to be the center of a collisionless plasma or one collision-mean-free path from the sheath edge for collisional plasmas. It is common to write Eq. (3.7-28) as Ii=1ne kTeA, (3.7-29) 2M where A is the ion collection area at the sheath boundary. This current is called the Bohm current. For example, consider a xenon ion thruster with a 1018m–3 plasma density and an electron temperature of 3 eV. The current density of ions to the boundary of the ion acceleration structure is found to be 118 A/m2, and the Bohm current to an area of 10–2 m2 is 1.18 A. 3.7.3 Child–Langmuir Sheaths The simplest case of a sheath in a plasma is obtained when the potential across the sheath is sufficiently large that the electrons are repelled over the majority of the sheath thickness. This will occur if the potential is very large compared to the electron temperature ( >> kTe / e ). This means that the electron density goes to essentially zero relatively close to the sheath edge, and the electron space charge does not significantly affect the sheath thickness. The ion velocity through the sheath is given by Eq. (3.7-16). The ion current density is thenPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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