A simulation study of coupled secular oscillations in nonlinear Paul trap mass spectrometers

Abstract

This paper presents the results of a simulation study of coupled axial and radial secular motion in nonlinear Paul trap mass spectrometers. The nonlinearities included in the simulation are external excitation and hexapole and octopole field aberrations. The equations of motion have the form of an inhomogeneous-coupled Duffing oscillator with quadratic, cubic nonlinearities and periodic forcing function in the axial direction. The study used a numerical technique to probe the role of field inhomogeniety and external excitation in the appearance of coupled frequencies in axial and radial directions. The multiple scales perturbation technique was used to develop an analytical expression for frequency perturbation. It is seen that the strength of coupling can be characterized by the magnitude of the coefficient of the cross terms in the equations of motion and the appearance of coupled frequencies in each direction is a function of this coupling strength. Hexapole superposition affects in the coupling strength of only the axial direction and its influence in coupled oscillations is minimal. Octopole superposition plays the predominant role in the manifestation of coupled frequencies in both directions. In the presence of external excitation, a larger number of frequencies appear in the frequency spectrum because of the mixing of fundamental and harmonic frequencies in the axial and radial directions. In the presence of external excitation, the axial secular motion is amplitude and phase modulated, and results in a beat frequency corresponding to the difference between the perturbed axial frequency and the applied excitation frequency.