Weak Multipole Field Effects on Axial Secular Frequency in a Nonlinear Dual-Frequency Paul Trap

The axial secular frequency in a dual-frequency Paul trap has been evaluated using the modified Lindstedt-Poincare method. The ion motion in the presence of two rapidly oscillating frequencies is transformed to a motion under the influence of an effective potential resulting in a Duffing-like equation with contributions from multipole fields, namely, hexapole, octopole, and decapole fields. The resulting equation is asymmetric, and the secular frequency is different for positive and negative amplitudes of the ion motion. The voltage and frequency ratios have a profound influence on the axial secular frequency. The axial secular frequency increases with an increase in the voltage ratio and decreases with an increase in the frequency ratio. The octopole field has the strongest influence on the axial secular frequency for a given voltage and frequency ratio, whereas hexapole field, the least.