ION SEPARATION IN IMPERFECT FIELDS OF THE QUADRUPOLE MASS ANALYZER .2. ION-BEAM DYNAMICS IN THE PHASE-SPACE OF THE FRINGING FIELD

The theoretical aspects of ion separation taking account of fringing fields in the quadrupole mass analyzer are analyzed by applying beam dynamics in phase-space. The mathematical simulation of fringing fields is developed by the principles of statistical mechanics. Numerical solutions of differential equations (the Hill equation) for ion motion in the fringing field are considered using matrix conversions with Galerkin-Slezkin's integral averaging in the iteration procedure. Analysis of beam dynamics in the phase-space where the law of phase volume conservation (Liouville's theorem) is valid indicates that, for stable ion motion, beam boundaries are described by a family of ellipses. The parameters of the ellipses are determined by the elements of the state transition matrix for an arbitrary set of initial conditions. Theoretical analysis indicates that beam divergence has a major influence on transmission. Transmission is an exponential function of relative transit time. The optimum time of transit through the fringing field has exponential dependences on the initial values of radial and angular envelopes of a beam with an asymptotic value of 1.77 pi.