Ion Trajectories in Quadrupole Mass Filters fora andq Parameters Near the Stability Region Tip

Abstract

The ion motion inside quadrupole mass filters is described for a and q values close to those corresponding to the stability region tip. The conventional solutions involving the constant-amplitude series of sine functions, which vanish for the stability region tip, were replaced by Mathieu functions of the second kind. Trajectories are completely defined as functions of time, or of the longitudinal coordinate, by the initial conditions, including the radiofrequency phase when the ion enters the filter, by the operating line slope and the value of q. For the stability region tip, ions with trajectories in a 12.5 cm long filter with 5.6 mm interelectrode distance, starting with x = 2.2 mm (y = 1.41 mm) or making an angle α x = 0.35° (α y = 0.48°), are predicted to be successful in traversing the quadrupole and to thus be detected. © 1997 by John Wiley & Sons, Ltd. Received 23 June 1997; Accepted 8 July 1997 Rapid. Commun. Mass Spectrom. 11, 1383-1386 (1997 ). This fact may be thought to be of minor importance because the operating line passes under the stability tip. However, as will be shown below, the contribution of trajectories with parameters corresponding to the tip is not always negligible. Also, it is not obvious how to connect the mathematical discontinuity of the stable solutions to a meaningful ion trajectory behaviour. It is somewhat uncertain whether or not the stable solutions can correctly describe ion trajectories near the stability region tip. In fact this question is relevant for most operating quadrupole filters because unit mass resolution at mass 100 u assumes differences for a and q from the tip values of only δa = ± 0.0012 and δq = ± 0.0035, respectively.The purpose of the following derivations is to establish a bridge over this region of possible controversies. The relations developed are a first order approximation in δa and δq, and their validity is limited by this to a narrow range. This range is less restrictive for shorter analysers in which ions are subjected to fewer radiofrequency cycles. Finally, both x and y ion coordinates are connected by analytical expressions to the time, to q, to the initial phase and to the operating line slope (as a parameter). ION MOTION ALONG THE x COORDINATE AXISThe motion of an ion in a quadrupole mass filter is described by differential equations of the Mathieu type.12 Along the x axis (Fig. 1)

Ion energy in quadrupole mass spectrometry

“…The ϭ 0 case is considered in this work to compare the derived ion energy with other results [4,6]. In this instance for ions injected on the quadrupole axis (x i ϭ 0, K 2 ϭ 0 or parallel to it (ẋ i ϭ 0, K 1 ϭ 0), a trajectory is accepted provided the following conditions are met:…”

“…The analytical method of the solution of the Mathieu equation in conjunction with algebraic presentation of the Mathieu functions allows introduction of simplifications and generalizations, delivering an alternative method to both the numerical solution of the Mathieu equation and to the matrix method. Previous attempts to use the Mathieu functions were too general due to the absence of simple algorithms or too local due to the necessity to use expansion series around small quantities (see [4] and references therein). The analytical method, in contrast to the matrix method, utilizes a single solution for a complete ion trajectory.…”