A Moving Boundary Problem in Plasma Physics
To a large degree, the electromagnetic properties of capacitive rf discharges are determined by the space-charge sheaths along the electrode boundaries of the rf discharge plasma. Understanding the behavior of the electrode sheaths and predicting their properties and parameters are crucial in the analysis of capacitive rf discharges. Different models have been developed to describe the characteristics of the rf sheaths. A review of these models can be found in Ref.1, A self-consistent hydrodynamic model for the sheaths of a symmetric rf discharge was formulated in Ref.2. In Ref.1, this model has been generalized and fully worked out. In the model, under the action of the rf field, the position of the plasma-sheath interface at any given time θ can be described by a periodic function λ(θ) that achieves its maximum λ1 at some time θ = θ 1 and its minimum λ2 at θ = θ 2. In other words, the plasma-sheath interface oscillates between its two extreme positions λl, the maximal sheath width, and λ2, the minimal sheath width. To the right of the moving interface (in the plasma), the electron density is equal to the ion density. To the left of the moving interface (in the sheath), the electron density is equal to zero. In Ref.1, the generalized model from Ref.2 was worked out under the assumption that the function λ is symmetric with respect to its minimum λ2. A comparison of the analytical and numerical solutions of the mathematical model with the available experimental results for mercury vapor have demonstrated a good agreement in corresponding values. Ref.1, however, did not include the mathematical analysis, which underlies the numerical results that were obtained. The purpose of the present paper is to provide more of the details.
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