Nonlinear Dynamics Of Resonances In, And Ejection From Paul Traps

Abstract

This thesis presents results of investigations that have been carried out to understand dynamics in nonlinear Paul trap mass spectrometers. Of the three problems that have been taken up for study in this thesis, the first concerns understanding early/delayed ejection of ions in mass selective boundary ejection experiments. The second looks at the differential resolution observed in forward and reverse scan resonance ejection experiments. The third study explores a coupled nonlinear resonance within the nominally stable region of trap operation.

The method of multiple scales has been to elucidate dynamics associated with early and delayed ejection of ions in mass selective ejection experiments in Paul traps. We develop a slow flow equation to approximate the solution of a weakly nonlinear Mathieu equation to describe ion dynamics in the neighborhood of the stability boundary of ideal traps (where the Mathieu parameter qz = qz* = 0.908046). For positive even multipoles in the ion trapping field, in the stable region of trap operation, the phase portrait obtained from the slow flow consists of three fixed points, two of which are saddles and the third is a center. As the qz value of an ion approaches qz*, the saddles approach each other, and a point is reached where all nonzero solutions are unbounded, leading to an observation of early ejection. The phase portraits for negative even multipoles and odd multipoles of either sign are qualitatively similar to each other and display bounded solutions even for qz > qz*, resulting in the observation of delayed ejection associated with a more gentle increase in ion motion amplitudes, a mechanism different from the case of the positive even multipoles.

The second study investigates constraints on pre-ejection dynamical states which cause differential resolution in resonance ejection experiments using Paul traps with stretched geometry. Both analytical and numerical computations are carried out to elucidate the role of damping and scan rate in influencing coherence in ion motion associated with the forward and reverse scan. It has been shown that in the forward scan experiments, for a given damping, low scan rates result in coherent motion of ions oof a given mass at the jump point. At this point, the amplitude and phase of ions of a given mass, starting at different initial conditions, become effectively identical. As the scan rate is increased, coherence is destroyed. For a given scan rate, increasing damping introduces coherence in ion motion, while decreasing damping destroys this coherence. In reverse scan experiments, for a given damping, very low scan rates will cause coherent ion motion. Increasing the scan rate destroys this coherence. The effect of damping in reverse scan experiments is qualitatively similar to that in the forward scan experiments, but settling times in the forward scan are shorter, leading to improved coherence and resolution. For mass spectrometrically relevant scan rates and damping values, significantly greater coherence is obtained in the forward scan. In the third study we investigate the weakly coupled and nonlinear Mathieu equations governing ion motion in axial and radial directions in a Paul trap in the neighborhood of a nonlinear resonance point at az* = -0.2313850427 and qz* = 0.9193009931$. Using harmonic balance based approximate averaging up to second order; we obtain a slow flow that, we numerically demonstrate, approximates the actual ion dynamics. We find that the slow flow is Hamiltonian. We study the slow flow numerically with the objective of exploring and displaying some of the possible types of interesting ion motions. In particular, we choose specific but arbitrary parameter values; study the stability of the individual radial and axial motion invariant manifolds; examine the rather large times associated with escape of ions; notice regions in the averaged phase space wherein trajectories do not, in fact, escape; observe apparently chaotic dynamics preceding escape for ions that do escape; and note that trajectories that do not escape appear to be confined to 4-tori. We conclude with some comments on the implications for practical operation of the Paul trap near this resonant point.