On the Space-Charge Divergence of an Axially Symmetric Beam†

An examination is made of the assumptions customarily used in estimating the effect of space-charge in an axially symmetric beam. It is shown that, subject to certain restrictions, the assumptions arc consistent with a coaxial, cylindrical conducting boundary. For small amounts of divergence in an initially parallel beam, a hyperbola solution is proposed as an alternative to the usual solution which involves a tabulated function. Lt is then argued that in some oases there is reason for believing that the hyperbola solution may be as good, or better, an approximation than the usual solution, oven when the divergence is not small