New ion-optical devices utilizing oscillatory electric fields. I. Principle of operation and analytical theory of multipole devices with two-dimensional electric fields

Abstract

This paper describes the principle of operation and the analytical theory of multipole devices obtained by generalizing the geometry of the well-known quadrupole mass analyzer. The resulting field potential for a multipole consisting of 2 n electrodes ( n = 2, 3, 4, ...) is calculated by solving analytically the Laplace equation. If the axis of symmetry for one of the positively biased electrodes coincides with the positive X axis, the equation of motion is ▪ Here, the displacement Z = X + iY = R e iθ , its complex Z , the time T, and the initial phase T 0 are dimensionless, as given in a per-unit system. ▪ and ▪ are parameters. The remaining quantities are: electronic charge ( e), d.c. voltage ( U), zero-to-peak rf voltage ( V) applied between opposite sets of electrodes, charged particle mass ( m), angular frequency (ω) of the rf voltage, and field radius ( r 0). The equations of motion for the quadrupole analyzer may be obtained by setting n = 2. A few results obtained by computer simulation of the equations of motion are also discussed. Since the restoring force acting upon a charged particle is proportional to the ( n - 1)th power of the displacement, Z , a very strong focusing effect can be obtained.