A Theoretical Method for Characterizing Nonlinear Effects in Paul Traps with Added Octopole Field

Abstract

Abstract. In comparison with numerical methods, theoretical characterizations of ion motion in the nonlinear Paul traps always suffer from low accuracy and little applicability. To overcome the difficulties, the theoretical harmonic balance (HB) method was developed, and was validated by the numerical fourth-order Runge-Kutta (4th RK) method. Using the HB method, analytical ion trajectory and ion motion frequency in the superimposed octopole field, ε, were obtained by solving the nonlinear Mathieu equation (NME). The obtained accuracy of the HB method was comparable with that of the 4th RK method at the Mathieu parameter, q=0.6, and the applicable q values could be extended to the entire first stability region with satisfactory accuracy. Two sorts of nonlinear effects of ion motion were studied, including ion frequency shift, Δβ, and ion amplitude variation, Δ(C 2n /C 0 ) (n≠0). New phenomena regarding Δβ were observed, although extensive studies have been performed based on the pseudo-potential well (PW) model. For instance, the |Δβ| at ε=0.1 and ε=-0.1 were found to be different, but they were the same in the PW model. This is the first time the nonlinear effects regarding Δ(C 2n /C 0 ) (n≠0) are studied, and the associated study has been a challenge for both theoretical and numerical methods. The nonlinear effects of Δ(C 2n /C 0 ) (n≠0) and Δβ were found to share some similarities at q<0.6: both of them were proportional to ε, and the square of the initial ion displacement, z(0) 2 .

Miniature and Fieldable Mass Spectrometers: Recent Advances

“…198 Since simulations of multiple ions can be very computationally intensive, Xu et al have implemented GPU acceleration, wherein a computer’s graphics card is used as a parallel processing unit, in order to decrease computational time of space charge calculations by at least two orders of magnitude compared to a single core CPU. 199 Similar GPU-assisted algorithms were used to study the rates of ion/ion reactions.…”

Linear ion trap with added octopole field component: the property and method

“…[24,27,28] The greater the amount of the octopole field is, the greater the frequency shift is. [27,44] At a low q value, the sign of the octopole field does not affect the magnitude of the frequency shift, [45] so CID efficiency using sCRIT-3x is comparable to that using sCRIT-3y. This is consistent with the previous experiments, [27] in which the results with excitation between smaller rods are similar to those with excitation between larger rods in the linear quadrupole with a 4% added octopole field.…”

Nonlinear Ion Harmonics in the Paul Trap with Added Octopole Field: Theoretical Characterization and New Insight into Nonlinear Resonance Effect

“…In order to keep the advantages simultaneously, it is highly desired that there is a theoretical method that is as accurate as the numerical methods. The HB method is such a high-accuracy theoretical method [11]. In the HB method, a trial solution, which is the addition of ion motion harmonics, is employed and substituted into the NME.…”

“…The weight can increase to 10% or higher in some Paul trap geometrical variations [5], such as cylindrical [6] and rectilinear ion traps [7]. The nonlinear fields inevitably introduce a series of nonlinear effects into the ion motion, such as frequency shift [8][9][10], amplitude variation [11], nonlinear resonance [12][13][14][15][16], and thus have great impacts on the trap performance, in terms of mass resolution [8], mass accuracy [9], sensitivity [12], and tandem MS (MS/MS) efficiency [10,15], etc.…”

“…However, the PW model still suffers from some limitations because of the omission of the small-amplitude harmonics. The small-amplitude harmonics are closely related to the Bfine structure^of the nonlinear effects [11]. For instance, in the PW method, the magnitudes of ion secular frequency shift, |Δβ|, in the superimposed octopole fields, e.g., ε = ±0.1, are undistinguishable.…”