Third-order treatment of combined effects of space charge and external fields on cylindrical ion and electron beams

Analytical treatments of the space-charge spread of cylindrical beams have been based upon a number of restrictive assumptions. Most of them can be removed and a higher accuracy attained with the help of a suitable relation connecting motion and Maxwell's equations. As previous treatments appear as paraxial approximations, third-order formulas applying to arbitrary field forms have been carried out.