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Matrix methods for the calculation of stability diagrams in quadrupole mass spectrometry

Focus: Quadrupole Ion Traps Account and Perspective
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Abstract

The theory of the computer calculation of the stability of ion motion in periodic quadrupole fields is considered. A matrix approach for the numerical solution of the Hill equation and examples of calculations of stability diagrams are described. The advantage of this method is that it can be used for any periodic waveform. The stability diagrams with periodic rectangular waveform voltages are calculated with this approach. Calculations of the conventional stability diagram of the 3-D ion trap and the first six regions of stability of a mass filter with this method are presented. The stability of the ion motion for the case of a trapping voltage with two or more frequencies is also discussed. It is shown that quadrupole excitation with the rational angular frequency ω = NΩ/P (where N, P are integers and Ω is the angular frequency of the trapping field) leads to splitting of the stability diagram along iso-β lines. Each stable region of the unperturbed diagram splits into P stable bands. The widths of the unstable resonance lines depend on the amplitude of the auxiliary voltage and the frequency. With a low auxiliary frequency splitting of the stability diagram is greater near the boundaries of the unperturbed diagram. It is also shown that amplitude modulation of the trapping RF voltage by an auxiliary signal is equivalent to quadrupole excitation with three frequencies. The effect of modulation by a rational frequency is similar to the case of quadrupole excitation, although splitting of the stability diagram differs to some extent. The methods and results of these calculations will be useful for studies of higher stability regions, resonant excitation, and non-sinusoidal trapping voltages.

Keywords

Parametric Resonance Excitation Parameter Stability Diagram Mass Filter Paul Trap 
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References

  1. 1.
    Ince, E. Researches into the Characteristic Numbers of the Mathieu Equation. Proceedings of the Royal Society of Edinburgh; Neiland Co. Ltd.: Edinburgh, 1925–1926; XLVI, 20–29.Google Scholar
  2. 2.
    McLachlan, N. W. Theory and Applications of Mathieu Functions; Clarendon: Oxford, 1947.Google Scholar
  3. 3.
    Quadrupole Mass Spectrometry and its Applications; Dawson, P. H., Ed.; Elsevier: Amsterdam, 1976; reissued by AIP Press: Woodbury, NY, 1995, pp 65–77.Google Scholar
  4. 4.
    March, R. E., Hughes, R. J., Todd, J. F. J. Quadrupole Storage Mass Spectrometry, Wiley Interscience, 1989, pp 31–73.Google Scholar
  5. 5. (a)
    Dawson, P. H.; Yu, B. The Second Stability Region of the Quadrupole Mass Filter. II. Experimental Results. Int. J. Mass Spectrom. Ion Proc. 1984, 56, 41–50.CrossRefGoogle Scholar
  6. 5. (b)
    Konenkov, N. V.; Kratenko, V. I. Characteristics of a Quadrupole Mass Filter in the Separation Mode of a Few Stability Regions. Int. J. Mass Spectrom. Ion Proc. 1991, 108, 115–136.CrossRefGoogle Scholar
  7. 5. (c)
    Du, Z.; Douglas, D. J.; Konenkov, N. V. Elemental Analysis with Quadrupole Mass Filters Operated in Higher Stability Regions. J. Anal. Atom. Spectrom. 1999, 14, 1111–1119.CrossRefGoogle Scholar
  8. 6. (a)
    Paul, W.; Reinhard, H. P.; Von Zahn, U. Das Elektrische Massenfilter als Massenspectrometer und Isotopentrenner. Z. Phys. 1958, 152, 143–182.CrossRefGoogle Scholar
  9. 6. (b)
    Vedel, F.; Vedel, M.; March, R. E. New Schemes for Resonant Ejection in RF Quadrupolar Ion Traps. Int. J. Mass Spectrom. Ion Proc. 1990, 99, 125–138.CrossRefGoogle Scholar
  10. 6. (c)
    Alfred, R. L.; Londry, F. A.; March, R. E. Resonance Excitation of Ions Stored in a Quadrupole Ion Trap. Part IV. Theory of Quadrupolar Excitation. Int. J. Mass Spectrom. Ion Proc. 1993, 124, 171–185.CrossRefGoogle Scholar
  11. 6. (d)
    Makarov, A. Resonance Ejection from the Paul Trap: A Theoretical Treatment Incorporating a Weak Octopole Field. Anal. Chem. 1996, 68, 4257–4263.CrossRefGoogle Scholar
  12. 7. (a)
    Richards, J. A.; Huey, R. M.; Hiller, J. A New Operating Mode for the Quadrupole Mass Filter. Int. J. Mass Spectrom. Ion Proc. 1973, 12, 317–339.CrossRefGoogle Scholar
  13. 7. (b)
    Sheretov, E. P.; Terentyev, V. A. The Fundamentals of Theory of Quadrupole Mass Spectrometry with Pulsed Voltage Waveform. Russ. J. Tech. Phys. 1972, 42, 953–962.Google Scholar
  14. 8.
    Floquet, G. Sur les Equations Differentielles Lineares a Coefficients Periodiques. Ann. Ecole Norm. Sup. Paris 1883, 12, 47–89.Google Scholar
  15. 9. (a)
    Hale, J. K. Ordinary Differential Equations; Wiley Inter-science: New York, 1969, pp 117–131.Google Scholar
  16. 9. (b)
    Hartman, P. Ordinary Differential Equations; 2nd ed., Burkhauser, Boston, 1982, pp 60–62.Google Scholar
  17. 10.
    Pipes, L. A. Matrix Solution of Equations of the Mathieu-Hill Type. J. Appl. Phys. 1953, 24, 902–910.CrossRefGoogle Scholar
  18. 11.
    Waldren, R. M.; Todd, J. F. J. The Use of Matrix Methods and Phase-Space Dynamics for the Modelling of Quadrupole-Type Device Performance. In Dynamic Mass Spectrometry; Price, D.; Todd, J. F. J., Eds.; Heyden: London, 1978; Chap V; 14–40.Google Scholar
  19. 12. (a)
    Dawson, P. H. Numerical Calculation. In Quadrupole Mass Spectrometry and its Applications. Elsevier: Amsterdam, 1976; 84.Google Scholar
  20. 12. (b)
    Humphries, S., Jr.; Principles of Charged Particle Acceleration; John Wiley and Sons: New York, 1986.Google Scholar
  21. 12. (c)
    Arnol’d, V. I. Ordinary Differential Equations, 3rd ed.; Springer-Verlag: Berlin, 1991; pp 256–264.Google Scholar
  22. 13.
    Symon, K. R. Mechanics. Addison-Wesley Publishing Co. Inc.: Reading, 1964; p 399.Google Scholar
  23. 14.
    Ding, L.; Kumashiro, S. Ion Motion in the Rectangular Wave Quadrupole Field and Digital Operation Mode of a Quadrupole Mass Spectrometer. Chinese Vac. Sci. Tech. 2001, 3, 176–181.Google Scholar
  24. 15.
    Sudakov, M. Stability Diagrams of Ions in Radio Frequency Mass Spectrometry. Russ. J. Tech. Phys. 1994, 65, 170–178.Google Scholar
  25. 16. (a)
    Kaiser, R. E., Jr.; Cooks, R. G.; Stafford, G. C.; Syka, J. E. P.; Hemberger, P. H. Operation of a Quadrupole Ion Trap Mass Spectrometer to Achieve High Mass/Charge Ratios. Int. J. Mass Spectrom. Ion Proc. 1991, 106, 79–115.CrossRefGoogle Scholar
  26. 16. (b)
    Schwartz, J. C.; Syka, J. E. P.; Jardine, I. High Resolution in a Quadrupole Ion Trap Mass Spectrometer. J. Am. Soc. Mass Spectrom. 1991, 2, 198–204.CrossRefGoogle Scholar
  27. 16. (c)
    Londry, F. A.; Wells, G. J.; March, R. E. Enhanced Mass Resolution in a Quadrupole Ion Trap. Rapid Commun. Mass Spectrom. 1993, 7, 43–45.CrossRefGoogle Scholar
  28. 17.
    Sevugarajan, S.; Menon, A. G. Frequency Perturbation in Nonlinear Paul Traps: A Simulation Study of the Effect of Geometric Aberration, Space Charge, Dipolar Excitation, and Damping on Ion Axial Secular Frequency. Int. J. Mass Spectrom. 2000, 197, 263–278.CrossRefGoogle Scholar
  29. 18.
    Alheit, R.; Kleineidam, S.; Vedel, F.; Vedel, M.; Werth, G. Higher Order Non-Linear Resonances in a Paul Trap. Int. J. Mass Spectrom. Ion Proc. 1996, 154, 155–169.CrossRefGoogle Scholar
  30. 19.
    Chen, W.; Collings, B. A.; Douglas, D. J. High-Resolution Mass Spectrometry with a Quadrupole Operated in the Fourth Stability Region. Anal. Chem. 2000, 72, 540–545.CrossRefGoogle Scholar
  31. 20. (a)
    Landau, L. D.; Lifshitz, E. M. Mechanics, 3rd ed. Pergamon Press: Oxford, 1960, pp 80–83.Google Scholar
  32. 20. (b)
    Bogolubov, N. N.; Mitropol’skii, Y. A. Problems of the Asymptotic Theory of Non-stationary Vibrations; Israel Program for Scientific Translations: Jerusalem, 1965.Google Scholar
  33. 21.
    Sudakov, M.; Konenkov, N.; Douglas, D. J.; Glebova, T. Excitation Frequencies of Ions Confined in a Quadrupole Field With Quadrupole Excitation. J. Am. Soc. Mass Spectrom. 2000, 11, 11–18.CrossRefGoogle Scholar
  34. 22.
    Kozo, M. Quadrupole Mass Spectrometer, US patent 5, 227, 629, July 13, 1993.Google Scholar
  35. 23. (a)
    Konenkov, N. V.; Cousins, L. M.; Baranov, V. I.; Sudakov, M. Y. Quadrupole Mass Filter Operation with Auxiliary Quadrupolar Excitation: Theory and Experiment. Int. J. Mass Spectrom 2001, 208, 17–27.CrossRefGoogle Scholar
  36. 23. (b)
    Baranov, V. I.; Konenkov, N. V.; Tanner, S. D. QMF Operation with Quadrupole Excitation. In Plasma Source Mass Spectrometry, the New Millenium; Holland, G.; Tanner, S. D., Eds.; Royal Society of Chemistry: Cambridge, 2001, pp 63–72.CrossRefGoogle Scholar
  37. 24.
    Du, Z.; Olney, T.; Douglas, D. J. Inductively Coupled Plasma Mass Spectrometry with a Quadrupole Mass Filter Operated in the Third Stability Region. J. Am. Soc. Mass Spectrom. 1997, 8, 1230–1236.CrossRefGoogle Scholar
  38. 25. (a)
    Collings, B. A.; Douglas, D. J. Observation of Higher Order Quadrupole Excitation Frequencies in a Linear Ion Trap. J. Am. Soc. Mass Spectrom. 2000, 11, 1016–1022.CrossRefGoogle Scholar
  39. 25. (b)
    Collings, B. A.; Sudakov, M.; Londry, F. Resonance Shifts in the Excitation of the n = 0, K = 1 to 6 Quadrupolar Resonances for Ions Confined in a Linear Ion Trap. J. Am. Soc. Mass Spectrom., in press.Google Scholar
  40. 26.
    Sudakov, M. Stability Diagram of Secular Motion of Ions Trapped in RF Quadrupole Field with Auxiliary Harmonic Excitation (in Russian). Pisma JTP 2000, 26, 46–51.Google Scholar
  41. 27.
    Sheretov, E. P.; Gurov, V. S.; Kolotilin, B. I. Modulation Parametric Resonances and Their Influence on Stability Diagram Structure. Int. J. Mass Spectrom. 1999, 184, 207–216.CrossRefGoogle Scholar
  42. 28.
    Arnol’d, V. I. Comments on Perturbation Theory for Mathieu-Like Problems. Russ. Uspehi Math. Sci. 1983, 38, 189–203.Google Scholar
  43. 29.
    Cha, B.; Sudakov, M.; Douglas, D. J. unpublished.Google Scholar
  44. 30. (a)
    Razvi, M. A.; Chu, X. Z.; Alheit, R.; Werth, G.; Blumel, R. Fractional Frequency Collective Parametric Resonance of an Ion Cloud in a Paul Trap. Phys. Rev. A 1998, 58, R34-R37.CrossRefGoogle Scholar
  45. 30. (b)
    Chu, X. Z.; Holzki, M.; Alheit, R.; Werth, G. Observation of High-Order Motional Resonances of an Ion Cloud in a Paul Trap. Int. J. Mass Spectrom. Ion Proc. 1998, 173, 107–112.CrossRefGoogle Scholar
  46. 31.
    Rettinghaus, G. Ph.D. Thesis, Bonn University, 1965, unpublished.Google Scholar
  47. 32.
    Alheit, R.; Chu, X. Z.; Hofer, M.; Holzki, M.; Werth, G.; Blumel, R. Nonlinear Collective Oscillations of an Ion Cloud in a Paul Trap. Phys. Rev. A 1997, 56, 4023–4031.CrossRefGoogle Scholar

Copyright information

© American Society for Mass Spectrometry 2002

Authors and Affiliations

  1. 1.Department of General PhysicsRyazan State Pedagogical UniversityRyazanRussia
  2. 2.Department of ChemistryUniversity of British ColumbiaVancouverCanada

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