Calculation of secular axial frequencies in a nonlinear ion trap with hexapole, octopole and decapole superpositions by a modified Lindstedt–Poincare method

Abstract

In this paper the modified Lindstedt–Poincare method is used for calculation of axial secular frequencies of a nonlinear ion trap with hexapole, octopole and decapole superpositions. The motion of the ion in a rapidly oscillating field is transformed to the motion in an effective potential. The equations of ion motion in the effective potential are 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 Lindstedt–Poincare method is used for solving the resulted nonlinear equations. As a result, the ion secular frequencies as a function of nonlinear field parameters are obtained. The calculated secular frequencies are compared with the results of some other methods and the exact results. There is an excellent agreement between the results of this paper and the exact results.