Escape velocity and resonant ion dynamics in Paul trap mass spectrometers

Abstract

This paper examines the role of field inhomogeneity in altering trapping strength and ion dynamics within the nominally stable region of Paul trap mass spectrometers. The concept of escape velocity, the minimum velocity required for escape by an ion starting at the center of the trap, has been used to numerically investigate and understand trapping strength variations reported in the literature. The governing equations of motion, in our study, have the form of a pair of weakly coupled and nonlinear Mathieu equations with higher order terms corresponding to hexapole and octopole superpositions. An analytical study of a single (decoupled but representative) nonlinear Mathieu equation has also been carried out to shed light on qualitative aspects of ion dynamics near two important resonances. The numerical study shows sharp drops in escape velocity near specific β values ( β is related to the two Mathieu parameters a and q), with the two largest drops occurring near β=2/3 and 1/2. It also shows that the hexapole nonlinearity contributes to the resonance at β=2/3, while the octopole nonlinearity does so at β=1/2. The analytical study indicates that at the β=2/3 resonance the ion is inherently unstable and decreasing nonlinearity has no effect on the net reduction in trapping strength, though it does narrow the region of reduced trapping strength. In the case of the β=1/2 resonance, however, the phase portrait indicates only bounded solutions with “escape” occurring when the ion encounters the geometric restriction of the trap. In particular, the reduction in trapping strength near this resonance has been interpreted in terms of the location of a separatrix in the averaged phase space, and its relation to trap size.