Abstract
In this paper we have used a modified homotopy perturbation method used previously by A. Belendez and his coworkers, for calculation of axial secular frequencies of a nonlinear ion trap with hexapole, octopole and decapole superpositions. We transform the motion of the ion in a rapidly oscillating field to the motion in an effective potential and obtain a nonlinear differential equation in the form of a Duffing-like equation. With only octopole superposition the resulted nonlinear equations are symmetric; however, in the presence of hexapole and decapole superpositions, they are asymmetric. For asymmetric oscillators, it has been pointed out that the angular frequency for positive amplitudes is different from the angular frequency for negative amplitudes. Considering this problem, the modified homotopy perturbation method is used for solving the resulted nonlinear equations. As a result, the ion axial secular frequencies as a function of nonlinear field parameters are obtained. The calculated secular frequencies are compared with the results of modified Lindstedt-Poincare approximation and the exact results. There is an excellent agreement between the results of this paper and the exact results.
Highlights
In an ideal ion trap the potential is pure quadrupole and the main properties of the movement of an ion are obtained by the solution of Mathieu equation [1]
In a recent study [11], this technique was used in the framework of the parameter expanding or modified Lindstedt-Poincare method [12,13,14] for calculation of secular axial frequencies in a nonlinear ion trap with hexapole, octopole and decapole superpositions
As is seen in the table 1-4, the results of this paper are in excellent agreement with the exact results and are closer to the exact results compared with the results of modified Lindstedt-Poincare approximation obtained in [11]
Summary
In an ideal ion trap the potential is pure quadrupole and the main properties of the movement of an ion are obtained by the solution of Mathieu equation [1]. In a recent study [11], this technique was used in the framework of the parameter expanding or modified Lindstedt-Poincare method [12,13,14] for calculation of secular axial frequencies in a nonlinear ion trap with hexapole, octopole and decapole superpositions. The modified homotopy perturbation method [26,27,28] is used for solving the two auxiliary nonlinear differential equations and the ion secular frequencies are calculated.
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