Characteristics of stability boundary and frequency in nonlinear ion trap mass spectrometer

Abstract

In this article, the Poincare-Lighthill-Kuo (PLK) method is used to derive an analytical expression on the stability boundary and the ion trajectory. A multipole superposition model mainly including octopole component is adopted to represent the inhomogeneities of the field. In this method, both the motional displacement and secular frequency of ions have been expanded to asymptotic series by the scale of nonlinear term , which represents a weak octopole field. By solving the zero and first-order approximate equations, it is found that a frequency shift exists between the ideal and nonlinear conditions. The motional frequency of ions in nonlinear ion trap depends on not only Mathieu parameters, a and q, but also the percentage of the nonlinear field and the initial amplitude of ions. In the same trap, ions have the same mass-to-charge ratio (m/z) but they have different initial amplitudes or velocities. Consequently, they will be ejected at different time through after a mass-selective instability scan. The influences on the mass resolution in quadrupole ion trap, which is coupled with positive or negative octopole fields, have been discussed respectively. where u is the related motional frequency in the axial and radial direction, ⍀ is the frequency of the driven radio-frequency (rf) potential, ␤ u is a parameter dependent on Mathieu parameters a u and q u , and the subscript u refers to axial and radial coordinates. Boundary ejection methods take the advantage of ejecting ions with a mass-selective instability scan, with which the rf amplitude is scanned linearly to cause the secular frequency of ions to increase until they become unstable [2-4], when the ions reach the boundary at a ϭ 0, q eject ϭ 0.908, ␤ z is equal to one, the ion has reached its stability limit. When a buffer gas was first used to improve the resolution of the mass spectrum by Stafford et al. [4], the stability boundary would move towards the positive direction on q-axis as bath gas pressure increase [2,5,6]. For the particle under the background at about 40 mTorr, the delay ejection also occurred at a large q value in the audio-frequency ion trap mass spectrometer [7][8][9][10][11].In practical traps, however, the electric field distribution is nonlinear, and the main causes contributing to field distortions are the misalignment of the trap [12], the truncation of electrodes [12], and the space charge [13,14]. As the existence of high order terms in electric field, the stability boundary deviates from the ideal value and the equation of motion for the ions will be nonlinear Mathieu equation.Most papers [15][16][17][18][19] deal with the time-dependent nonlinear terms with the method of pseudopotential well approximation [20].Using this approximation, high order terms become time-independent, and the nonlinear Mathieu equation is a normal nonlinear equation known as Duffing equation. The solution of such kind of equations has been well-studied [21]. When q u Ͻ 0.4 [2,22], this approximation accords well with the physical truth, some useful...

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

“…Ion motion in the quadrupole field with superimposed nonlinear field is described by the nonlinear Mathieu equation (NME) [45], which can be solved by using both theoretical methods and numerical methods. In comparison with the numerical methods, the theoretical methods can provide better understanding of the associated nonlinear effects of ion motion.…”

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

“…It is meaningful to predict the rf working voltage (represented by the q value), at which the nonlinear resonance occurs. However, because of the frequency shift, the q value at the nonlinear resonance condition (i.e., β = 0.5), q nr , is also dependent on the superimposed octopole, ε, and the initial 2 [17,22,26]. Figure 6a shows the β as a function of q under positive superimposed octopole fields: ε = 0.1 (black curve), ε = 0.05 (red curve), and ε = 0.01 (blue curve).…”

“…Equation 3 is the NME [17], in which a z and q z are dimensionless Mathieu parameters, ξ is dimensionless time and they are defined as:…”

“…The nonlinear fields make the ion motion even more complex. They not only change the frequencies and amplitudes of the natural harmonics but also produce nonlinear harmonics (marked in red) superimposed into the natural harmonics of ion motion [17,18]. Generally, among the harmonics, the ion fundamental secular oscillation has the largest amplitude and is therefore the most important [2,3].…”

“…First, they are the fingerprints of the nonlinear fields. For example, the octopole field mainly generates the characteristic 3β harmonic series ( Figure 1) [17,18]. From these characteristic harmonics, the detailed composition of the nonlinear fields in the trap can be readily determined.…”