Theoretical studies of ion trajectories in quadrupole systems

A brief introduction to the mathematical theory of quadrupole systems is given and the configuration of hyperbolic and circular cross-section rods is considered. Application of the theory of electrostatics to the latter case indicates that multipole contributions to the field occur which influence the ion trajectories. We have considered motion in the x-z plane of a static quadrupole (DC-only) for the simplest case of a 1:1 combination of quadrupole and dodecapole contributions. The quadrupole contribution to ion motion is simple harmonic and may be described as a sine function, whereas the dodecapole contribution may only be obtained by solving the appropriate elliptic integral. This solution gives rise to a sine (amplitude) or sn function. In the case of collision cells (RF-only) we have attempted to solve the Mathieu equation both analytically and numerically. Both stable and unstable trajectories may be calculated from these solutions and are presented in various formats. Some consideration is also given to the problems of collisional interaction and the associated difficulties likely to be encountered when the study is expanded to include this phenomenon.