Quadrupole ion trap with dipolar DC excitation: motivation, nonlinear dynamics, and simple formulas

We study the dynamics of a trapped ion in a mass spectrometer under the action of both the usual quadrupolar RF and dipolar DC excitation. The relevant governing equation, derived from electrostatic field calculations for realistic geometries, is a classical Mathieu equation perturbed with a constant inhomogeneous term and a small quadratic nonlinearity. An early paper by Plass examined the case without the quadratic term using variation of parameters. Here, we note a significantly simpler particular solution than used by Plass, include the quadratic term, and develop a second-order averaging-based approximation. The averaging results show that a particular underlying simple periodic solution is stable. We then show that a two-frequency approximation matches that solution well for practical purposes. Finally, we present and validate an easy iterative calculation for obtaining that two-frequency solution and quantify the effect of the quadratic nonlinearity.

Automated summary

This paper studies the dynamics of a trapped ion in a mass spectrometer under the influence of quadrupolar RF and dipolar DC excitation. By calculating the electrostatic field for realistic geometries, we derive a classical Mathieu equation with a perturbation term and quadratic nonlinearity. We simplify the solution proposed by Plass, include the quadratic term, and develop a second-order averaging-based approximation. The results show that a stable periodic solution can be obtained using a two-frequency approximation, and we quantify the effect of the quadratic nonlinearity.