Application of a nonlinear similarity method for calculation of space charge factor parameters for relativist electron beams with different forms of cross-section

For calculation of a space charge factor for the shield beams such a solid beam, a hollow beam and a tape beam the approximate analytical expressions were obtained. Utilization these expressions allow manages without application of the transcendental equation and could be useful for determination optimal electrical and geometrical parameters of a relativistic and non-relativistic klystronpsilas beams.