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78 Chapter 3 d 2 = – e (n – n ) = – eno 1 – 1 – 1 + e dx2 o i e o 2 o kTe =eno 1 – e . o 2 o kTe (3.7-22) In order to avoid a positive-going inflection in the potential at the sheath edge, which would then slow or even reflect the ions going into the sheath, the right- hand side of Eq. (3.7-22) must always be positive, which implies 1 > e . 2 o kTe This expression can be rewritten as through a potential in the plasma of at least Te / 2 before entering the sheath to produce a monotonically decreasing sheath potential. Since vo = 2e o M , Eq. (3.7-24) can be expressed in familiar form as vo kTe . (3.7-25) M This is usually called the Bohm velocity for ions entering a sheath. Equation (3.2-25) states that the ions must enter the sheath with a velocity of at least kTe / M (known as the acoustic velocity for cold ions) in order to have a stable (monotonic) sheath potential behavior. The plasma produces a potential drop of at least Te / 2 prior to the sheath (in the pre-sheath region) in order to produce this ion velocity. While not derived here, if the ions have a temperature Ti , it is easy to show that the Bohm velocity will still take the form of the ion acoustic velocity given by vo= ikTi+kTe. (3.7-26) M It is important to realize that the plasma density decreases in the pre-sheath due to ion acceleration toward the wall. This is easily observed from the Boltzmann behavior of the plasma density. In this case, the potential at the sheath edge has (3.7-23) (3.7-24) which is the Bohm sheath criterion [10] that states that the ions must fall o > kTe , 2ePDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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