Effective potential and the ion axial beat motion near the boundary of the first stable region in a nonlinear ion trap

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Small higher-order field imperfections of the main trapping quadrupole field are well known to have a strong influence on the performance of modern quadrupole mass spectrometers. Mass selectivity is usually achieved by means of the stability region boundaries. The stability diagram for ion motion is the area on the plane of voltage parameters for which the quadrupole field is able to trap ions of a given mass. Hence, the trapping efficiency of the quadrupole field is equal to zero at a boundary of the stability region. In this case, the trapping properties of the RF field depend on higher field imperfections regardless of how small they are compared to the quadrupole field. The ion motion with parameters close to the boundary β z = 1 is investigated in this article. The influence of nonlinear field imperfections is taken into account. A treatment similar to trajectory averaging in a pseudopotential is possible in this case. The ion motion has the characteristics of a beat with a fast oscillation at half the frequency of the RF and a slowly varying envelope. The ion motion is described by a dynamic equation for the envelope. This equation has the form of a Newton equation for the motion of a particle in a potential field. The effective potential function of the envelope is derived and investigated. The effective potential well is rather different for the cases of negative and positive even-order higher fields. The results are applied to the mass-selective axial instability scan of an ion trap. The influence of negative higher field harmonics explains the ejection delay and poor mass resolution of the Paul trap with truncated electrodes. Positive even-field imperfections are shown to be beneficial to the mass selective axial instability scan. This explains why stretched or hyperbolic angle modified traps give improved performance. Stable ion motion outside of the first stable region is predicted. This motion has the character of a limit cycle, and all ions move coherently in the radio frequency field.