The first-order theory of quadrupole lenses including the effects of space charge

The space charge forces tending to expand an ion beam of elliptical cross section are found using Green's function. It is assumed that the charge density is constant over any beam cross section. These space charge forces are then included in the equations governing the trajectory of the beam envelope in the two halves of a strong-focusing quadrupole lens and also in the drift space separating them. Next it is shown that with proper injection the beam will maintain its elliptical cross section so the trajectories of the major and minor axes of the beam constitute a complete description of the envelope. From these equations, one finds that the force of space charge remains finite even when the minor axis of the beam becomes zero. Thus, beam cross-over is possible for beams of elliptical cross section, but not for those of circular cross section. Sample calculations for the axis trajectories are included for varioua ratios of focusing to defocusing forces.