The spreading of an axisymmetric ion beam by space charge and diffusion

Abstract

An analysis is presented of the evolution of a narrow ion beam, produced for instance by an electrospray source, in ambient pressure air. To minimize beam broadening, an intense axial electric field E is externally imposed. Still, radial beam broadening due to the ionic diffusivity D and space charge must be accounted for. Ignoring axial contributions of diffusion and space charge, a simple second order partial differential equation is obtained for the space charge induced field. In the planar problem, this field is governed by Burgers's equation, which is exactly solvable via the Cole-Hopf transformation. The axisymmetric problem is not similarly tractable by any familiar transformation, but may be reduced to an ordinary differential equation in terms of a similarity variable, yielding the complete beam structure under all regimes of interest. The beam broadening effect of space charge is governed by the ratio between the beam current I and the group 2πε0DE, where ε0 is the permittivity of the gas.