CERTAIN THEORETICAL ASPECTS OF ION TRAJECTORIES IN MULTIPOLE SYSTEMS

The motion of single ions through a perfect quadrupole is governed by linear differential equations in both static (d.c. only) and radio-fre quency cases. However, motion in distorted quadrupole fields and highe r multipoles (e.g. hexapoles, octapoles etc.) is determined by non-lin ear differential equations with coupled (x, y) terms in Cartesian coor dinates. In imperfect quadrupoles with electrodes of circular cross-se ction this non-linearity is ''weak''. Motion in multipoles, in contras t, is entirely non-linear and it is to these multipoles that we turn o ur attention. We are aware that this work may also have significance f or quadrupoles. Trajectory plotting can be successfully achieved by nu merical methods, e.g. Runge-Kutta and the objective of any such studie s is to discover the functional properties of the device apparent in t he behaviour of charged particles during transit. In multipole and dis torted systems the application of numerical methods is complicated by the substantially increased number of parameters involved. This mitiga tes against achieving the objective stated above. We have found that a simplification may be introduced which allows a semi-analytical treat ment of the static cases and furthermore reduces the number of paramet ers required for the numerical analysis of the radio-frequency cases. Despite the simplification, the r.f. special case trajectories retain many of the features that occur in general cases. The simplification i nvolves removing the (x, y) product terms from the differential equati ons. This is not unreasonable as the equations then describe motion co nfined to the symmetry planes of the multipole. Results are shown that demonstrate, semi-analytically, the types of function encountered in the approximate solutions for the r.f. case over small increments of t ime. Trajectory plots are shown for the r.f. case, for which the simpl ification described above allows the production of phase-space diagram s.