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86 Chapter 3 where is a constant that varies from 0.8 to 0.2 for Te /Ti changing from 2 to about 20. For typical thruster plasmas where Te / Ti 10 , k is about 0.5. While the presence of free-standing double layers in the plasma volume in thrusters is often debated, the sheath at a thermionic cathode surface certainly satisfies the criteria of counter-streaming ion and electron currents and can be viewed as a double layer. In this case, Eq. (3.7-49) describes the space-charge- limited current density that a plasma can accept from an electron-emitting cathode surface. This is useful in that the maximum current density that can be drawn from a cathode can be evaluated if the plasma parameters at the sheath edge in contact with the cathode are known (such that Ji can be evaluated from the Bohm current), without requiring that the actual sheath thickness be known. Finally, there are several conditions for the formation of the classic double layer described here. In order to achieve a potential difference between the plasmas that is large compared to the local electron temperature, charge separation must occur in the layer. This, of course, violates quasi-neutrality locally. The current flow across the layer is space-charge limited, which means that the electric field is essentially zero at both boundaries. Finally, the flow through the layer discussed here is collisionless. Collisions cause resistive voltage drops where current is flowing, which can easily be confused with the potential difference across a double layer. 3.7.6 Summary of Sheath Effects It is worthwhile to summarize here some of the important equations in this section related to sheaths because these will be very useful later in describing thruster performance. These equations were derived in the sections above, and alternative derivations can be found in [1–3]. The current density of ions entering the sheath at the edge of the plasma is given by Ji =0.6 nevi 1 ne kTe , (3.7-50) 2M where n is the plasma density at the start of the pre-sheath far from the boundary, which was considered to be the center of the plasma by Langmuir for his collisionless plasmas. The convention of approximating the coefficient 0.6 as 1/2 was made by Bohm in defining what is now called the “Bohm current.”PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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