Multiple scales analysis of early and delayed boundary ejection in Paul traps

We use the method of multiple scales 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). The method of multiple scales enables us to incorporate higher order multipoles, extend computations to higher orders, and generate phase portraits through which we view early and delayed ejection.Our use of the method of multiple scales is atypical in two ways. First, because we look at boundary ejection, the solution to the unperturbed equation involves linearly growing terms, requiring some care in identification and elimination of secular terms. Second, due to analytical difficulties, we make additional harmonic balance approximations within the formal implementation of the method.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 qzqz 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.