Diffusion controlled breakdown of gases in a rectangular microwave cavity

The boundary-value problem for diffusion-controlled breakdown of gases in microwave cavities is converted to a variational principle for the square of the reciprocal of the ‘effective’ diffusion length. For a rectangular cavity excited in the TE_αγ0 mode, the resultant variational principle was minimized by an iterativevariational procedure. This was done using the high-frequency ionization coefficient of Herlin & Brown. However, any other functional dependence on the r.m.s. value of the electric field could have been used for this purpose. The resultant hierarchy of approximations to the variational principle obtained by this iterative-variational procedure for a cavity of square transverse crosssection is applied to the TE₁₁₀ and TE₂₂₀ modes of excitation. Adequate convergence is obtained with the first six approximations. Resultant plots of the ratio of the ‘effective’ to the characteristic diffusion length as a function of the Herlin & Brown parameter β, for various ratios of longitudinal to effective transverse cavity dimensions, imply that diffusion loss is greater for the TE₂₂₀ mode of excitation. Also, from this theoretical investigation it is felt that the first ten approximations to the variational principle should provide adequate convergence for most situations encountered in practice with regard to β, as well as cavity dimensions.