Ion trajectories in exactly determined quadrupole fields

The use of multi-electrode structures as mass analysers is now very widespread. These devices are employed as partial pressure analysers, mass filters and ion traps. The first of these devices to utilise an electric quadrupole field for separating ions according to their mass-to-charge ratio was introduced in the 1950s. The operation of these devices is based on the nature of particle trajectories between the electrode rods. The lack of an exact theory up to now has precluded full exploitation of the device. In the quadrupole mass filter the relevant stability theory depends directly on the precise structure of the multipole potential expansion. The exact coefficients in this expansion have been determined. A number of trajectory simulations are performed using these coefficients. In addition a simple relationship is found for which certain key coefficients vanish. A set of scaling rules has also been discovered which enables immediate calculation of each higher order coefficient from a fixed set of values for the corresponding scaled coefficients.