Approximate formulas for crossed-field potential-minimum parameters

A parabolic potential distribution is assumed in order to study the static properties of the crossed-field potential minimum. The half-Maxwellian normal velocity distribution function and the full-Maxwellian transverse velocity distribution function are integrated over the initial velocity plane with the proper boundary conditions to find an expression for the ratio of beam current to emission current. For conditions satisfying most practical cases, this current-sorting expression is reduced to a very simple form. Using the current-sorting equation together with the expression which relates the potential minimum position to the minimum potential for zero magnetic field, an approximate method of determining the potential-minimum parameters is devised. The values obtained in this manner agree closely with the results obtained by other more exact (and more complicated) numerical methods.