Upper and lower bounds on gas breakdown in inhomogeneous fields

The prediction of the electric field strength at which gas breakdown will take place is a difficult and crucial problem for many applications. In the case of microwave antennas, breakdown in the gas or plasma surrounding the antenna first occurs in the inhomogeneous near-field region close to the antenna surface. Recently, variational techniques have been used to estimate breakdown field strengths in such cases. These techniques have one drawback in that only upper bounds are provided, i.e., a field strength above which breakdown definitely takes place. Depending upon the accuracy of a particular calculation, breakdown may occur for an appreciable range below the estimated field strength. What is needed is a lower bound upon the breakdown field strength. In this paper a formulation of variational theory is used which allows lower bounds to be calculated. In addition, a variational-iterative technique is presented by which it is possible to systematically improve upon a calculated lower bound should this be necessary. These techniques are then applied to breakdown problems already considered by other authors to demonstrate the upper-bound calculations.