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Quadrupole ion trap with dipolar DC excitation: motivation, nonlinear dynamics, and simple formulas

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Abstract

We study the dynamics of a trapped ion in a mass spectrometer under the action of both the usual quadrupolar RF and dipolar DC excitation. The relevant governing equation, derived from electrostatic field calculations for realistic geometries, is a classical Mathieu equation perturbed with a constant inhomogeneous term and a small quadratic nonlinearity. An early paper by Plass examined the case without the quadratic term using variation of parameters. Here, we note a significantly simpler particular solution than used by Plass, include the quadratic term, and develop a second-order averaging-based approximation. The averaging results show that a particular underlying simple periodic solution is stable. We then show that a two-frequency approximation matches that solution well for practical purposes. Finally, we present and validate an easy iterative calculation for obtaining that two-frequency solution and quantify the effect of the quadratic nonlinearity.

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Notes

  1. Such methods, which do not require the expansion parameter to be small, can sometimes give very good results although they are not asymptotic methods.

  2. Physically, e.g., in a laboratory, we set q and \(\beta \) is determined as a result. For analytical work, it is simpler to choose a \(\beta \) of interest and then compute the required q. In terms of results obtained, there is no difference.

  3. With parametric excitation of the quadratic nonlinear term, which we do not have here, the resonance is felt at first order.

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Acknowledgements

DDT thanks Bidhayak Goswami for help with Maple.

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We have not received any financial support for conducting this research.

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Authors and Affiliations

  1. Mechanical Engineering, Indian Institute of Technology, Kanpur, 208016, India

    D. D. Tandel & Anindya Chatterjee

  2. Instrumentation and Applied Physics, Indian Institute of Science, Bangalore, 560012, India

    Atanu K. Mohanty

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  1. D. D. Tandel
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  2. Anindya Chatterjee
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Contributions

D. D. Tandel contributed to detailed execution of analysis and writing. Anindya Chatterjee was involved in conception, aspects of analysis, guidance, and writing. Atanu K. Mohanty contributed to conception, aspects of analysis, execution of analysis, and writing.

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Correspondence to D. D. Tandel.

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Tandel, D.D., Chatterjee, A. & Mohanty, A.K. Quadrupole ion trap with dipolar DC excitation: motivation, nonlinear dynamics, and simple formulas. Nonlinear Dyn 111, 15837–15852 (2023). https://doi.org/10.1007/s11071-023-08706-1

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