Diffusion Theory of the Electrodeless Ring Discharge

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Abstract
The Schottky theory of the positive column is applied to the inductive type of a high frequency discharge for cases where the electron collision frequency νc is large compared to the field frequency ω. Simultaneous solutions of the equations for the induced field and for the electron balance are obtained under the following assumptions: (1) Mobilities of electrons and ions in the discharge volume are constant; (2) the distributions of ionization rate and diffusion coefficient across the discharge tube are determined by the absolute value of the induced electric field at the inner tube radius |ER| and by its gradient at the same position; (3) the distribution of E can be obtained on the basis of an equivalent uniform conductivity σ̄ which yields the same impedance for the discharge as the actual σ distribution. The characteristic equation relating |ER| to tube radius R contains (compared to the positive column) an extra term which takes into account the nonuniformity of |E|. This term is expressed as a function of the parameter ρR≡2R/δ, where δ is the skin depth. Solving for R with assumed values of |ER| and ρR thus yields δ and by the known value of ω also σ̄. From the impedance relationship one then determines the value of the magnetic field HR that must be applied to produce particular values of σ̄ and |ER| with given ω and R. The distribution of electron density n is obtained in relation to the undetermined value at the tube axis n0. On the basis of constant electron mobility a relationship between σ̄ and n0 is obtained from the impedance equivalence which allows one to calculate absolute distributions of σ and n. These distributions are compatible with the |E| distribution for cases where ρR≤1.4. The method is illustrated by examples for hydrogen, where empirical data for νc, electron energy, ionization, and diffusion rates are used. The results are generalized in the way common for glow discharges by introduction of gas pressure p as similarity parameter. They indicate that the maximum obtainable electron density depends primarily upon the product ωHR, only moderately upon p and not upon R.