Nonlinear Effects in Paul Traps Operated in the Second Stability Region

Nonlinear Effects in Paul Traps Operated in the Second Stability Region:

Analytical Analysis and Numerical Verification

Caiqiao Xiong1,2, Xiaoyu Zhou1,2, Ning Zhang1,2, Lingpeng Zhan1,2, Suming Chen1,2, and Zongxiu Nie1,2,3*

1. Key Laboratory of Analytical Chemistry for Living Biosystems, Institute of Chemistry Chinese Academy of Sciences, Beijing 100190, China;

2. Beijing National Laboratory for Molecular Sciences, Beijing 100190, China;

3. Beijing Center for Mass Spectrometry, Beijing 100190, China.

Error evaluation for fourth-order Runge–Kutta (4th R-K)

4th R-Kis famous for its high accuracy and fast computational speed[1]. To evaluate the numerical error of 4th R-K, its numerical solution is compared with the exact solution of ion trajectory in ideal quadrupole field[2-4]. As shown in Fig. S1a, it is difficult to see the difference between the numerical and exact ion trajectories.Therefore, the secular frequency of the ion motion is further calculated using Fourier transform of ion trajectory(Fig. S1b). By comparing the numerical value (βz=1.665) with its exact value (βz=1.6677), the error of 4th R-Kis determined as 0.16%. Herein, the error of Fourier transform is counted in that of 4th R-K.

Fig. S1

Fig. S1: Numerical (black curve) and exact (red curve) ion trajectories in ideal quadrupole field at az=2.7 andqz=3.85. (b) Power spectrum calculated from Fourier transform of the numerical ion trajectory.The initial displacement and velocity of the ion are 0.1 r0 and 0, respectively.This initial condition is used throughout this supporting material, unless otherwise specified.

Calculation of ion amplitude, umax

For weak octopole field, the ion amplitude therein is very close to that in ideal quadrupole field (Fig. 2). Using this approximation, umax can be derived from the ion trajectory in the ideal quadrupole field, which is described by the solution of Mathieu equation (Eq. 14)[2-4]. From the umax shown in Fig. S2, the effective au' and qu' in octopole field can be readily obtained from Eqs. (15-18).

Fig. S2

Fig. S2The ion amplitudes, umax, in the (a) z direction and (b) r direction as a function of az and qz in ideal quadrupole field.

Nonlinear frequency shift of ion motion in positive and negative octopole fields

The nonlinear octopole field gives rise to the shift of ion motional frequency (Fig. 3, S3). Positive octopole field can positively shift the ion secular frequency, β, and vice versa[5]. As indicated in Eqs. (15-18), the magnitude of the nonlinear disturbance depends on ion amplitude. It can be foundthe shifts for both positive and negative octopole fields are greater in the z direction than that in the r direction because the larger ion amplitude in the z direction (Fig. 3).

Fig. S3

Fig. S3 Power spectra in the (c) z direction and (d) r direction with different octopole fields: ε=0 (black curve)ε=0.025 (red curve) and ε=-0.025 (blue curve).

Ion amplitude, umax,in the boundary of the stability region

The ion trajectories become instable in the stability boundary of the stability regions. As a result, an effectiveumaxin these instable az and qz points should be defined to study the nonlinear effects. With the effective umax, the effective az' and qz' in the octopole field can be obtained. The effective umaxcan be found by the best fitting of the analytical and the numerical results. The effective umaxvalues in the four boundary curves, βr=0, βr=1,βz=1 and βz=2, of the second stability region are given in Tab. S1.

Tab. S1 The umax in the boundary of the second stability region.

Note: these umax values are valid for this conditiononly:the initial displacement and velocity of the ion are 0.1 r0 and 0, respectively.

Nonlinear shift of the first stability region

The nonlinear shift of the first stability region (Fig. S4) has been well studied [5], and is used herein for comparison with that in the second stability region (Fig. 5).

Fig. S4

Fig. S4Diagram of the second stability regions with different octopole fields: ε=0 (black curve)ε=0.025 (red curve) and ε=-0.025 (blue curve) calculated using 4th R-K.

References

1.Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. 1992. Numerical Recipes in Fortran 77, The Art of Scientific Computing. Vol. 1 of Fortran Numerical Recipes. New York: Press Syndicate of the University of Cambridge.

2.Dawson, P. H. 1976. Quadrupole mass spectrometry and its applications. Amsterdam: Elsevier.

3.Hoffmann, E. D.; Stroobant, V. 2007. Mass spectrometry: principles and applications. Chichester: John Wiley & Sons.

4.March, R. E.; Todd, J. F. J. 1995. Practical Aspects of Ion Trap Mass Spectrometry. Vol. I Chap. 3. Fundamentals of Ion Trap Mass Spectrometry. New York: CRC Press.

5.Zhou, X.; Zhu, Z.; Xiong, C.; Chen, R.; Xu, W.; Qiao, H.; Peng, W.-P.; Nie, Z.; Chen, Y. Characteristics of stability boundary and frequency in nonlinear ion trap mass spectrometer. J. Am. Soc. Mass Spectrom.2010,21, 1588-1595.

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