Subharmonic excitation of the eigenmodes of charged particles in a Penning trap

  • G. Tommaseo
  • P. Paasche
  • C. Angelescu
  • G. WerthEmail author


When parametrically excited, a harmonic system reveals a nonlinear dynamical behaviour which is common to non-deterministic phenomena. The ion motion in a Penning trap -- which can be regarded as a system of harmonic oscillators -- offers the possibility to study anharmonic characteristics when perturbed by an external periodical driving force. In our experiment we excited an electron cloud stored in a Penning trap by applying an additional quadrupole r.f. field to the endcaps. We observed phenomena such as individual and center-of-mass oscillations of an electron cloud and fractional frequencies, so-called subharmonics, to the axial oscillation. The latter show a characteristic threshold behaviour. This phenomenon can be explained with the existence of a damping mechanism affecting the electron cloud; a minimum value of the excitation amplitude is required to overcome the damping. We could theoretically explain the observed phenomenon by numerically calculating the solutions of the damped differential Mathieu equation. This numerical analysis accounts for the fact that for a weak damping of the harmonic system we observed an even-odd-staggering of the the different orders of the subharmonics in the axial excitation spectrum.


Harmonic Oscillator Electron Cloud Excitation Amplitude Fractional Frequency Threshold Behaviour 
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  1. 1.
    R.S. van Dyck Jr, Phys. Rev. Lett. 59, 26 (1987)CrossRefGoogle Scholar
  2. 2.
    G. Werth, Phys. Scripta T59, 206 (1995)Google Scholar
  3. 3.
    H. Häffner, Phys. Rev. Lett. 85, 5308 (2000)CrossRefGoogle Scholar
  4. 4.
    R.S. van Dyck Jr, in: Atomic Physics at Accelerators, edited by D. Lunney, G. Audi, H.-J. Kluge (Kluwer Acad. Press, 2001), p. 163Google Scholar
  5. 5.
    S. Rainville, in: Atomic Physics at Accelerators, edited by D. Lunney, G. Audi, H.-J. Kluge (Kluwer Acad. Press, 2001), p. 215Google Scholar
  6. 6.
    T. Fritjoff, in: Atomic Physics at Accelerators, edited by D. Lunney, G. Audi, H.-J. Kluge (Kluwer Acad. Press, 2001), p. 231Google Scholar
  7. 7.
    J.J. Bollinger, Phys. Rev. A 48, 525 (1993)CrossRefGoogle Scholar
  8. 8.
    References in: Proc. Non-Neutral Plasma Workshop, edited by F. Anderegg, L. Schweikhardt, F. Driscoll, AIP Conf. Proc. 606 (2002)Google Scholar
  9. 9.
    J. Tan, G. Gabrielse, Phys. Rev. A 48, 3105 (1993)CrossRefGoogle Scholar
  10. 10.
    C.H. Tseng, G. Gabrielse, Appl. Phys. B 60, 95 (1995)Google Scholar
  11. 11.
    G. Tommaseo, Eur. Phys. J. D 25, 113 (2003)Google Scholar
  12. 12.
    G. Brown, G. Gabrielse, Rev. Mod. Phys. 58, 233 (1986)CrossRefGoogle Scholar
  13. 13.
    P. Paasche, Eur. Phys. J. D 18, 295 (2002)CrossRefGoogle Scholar
  14. 14.
    D.J. Bate, J. Mod. Opt. 39, 305 (1991)Google Scholar
  15. 15.
    M. Kretzschmar, Phys. Scripta 46, 544 (1992)Google Scholar
  16. 16.
    R. Alheit, Phys. Rev. A 56, 4023 (1997)CrossRefGoogle Scholar
  17. 17.
    M. Vedel, Appl. Phys. B 66, 191 (1998)CrossRefGoogle Scholar
  18. 18.
    B. Schäfer, Diploma thesis, Inst. f. Phys., Mainz (1999)Google Scholar
  19. 19.
    L.D. Landau, E.M. Lifschitz, Lehrbuch der theoretischen Physik - Bd. I Mechanik, Hrsg. von G. Heber (Akademie-Verlag, Berlin, 1969)Google Scholar
  20. 20.
    M. Minkorsky, Nonlinear Oscillations (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1962)Google Scholar
  21. 21.
    E. Butikov, Comput. Sci. Engineer. 1, 76 (1999)CrossRefGoogle Scholar
  22. 22.
    L. Mandelstam, N. Papalexi, Z. Phys. 73, 223 (1932)zbMATHGoogle Scholar
  23. 23.
    L. Mandelstam, J. Tech. Phys. (U.S.S.R.) 2, 81 (1934)zbMATHGoogle Scholar
  24. 24.
    H. Poincaré, Méthodes nouvelles de la mécanique céleste (Gauthier-Villars, Paris, 1892), tome 1, p. 79Google Scholar
  25. 25.
    A.M. Ljapunow, Das allgemeine Problem der Stabilität einer Bewegung (Onti, 1935)Google Scholar
  26. 26.
    N.W. McLachlan, Theory and Applications of Mathieu Functions (Oxford University Press, New York, 1947)Google Scholar
  27. 27.
    J. Meixner, F.W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen (Springer-Verlag, Berlin, 1954)Google Scholar
  28. 28.
    M.J.O. Strutt, Math. Ann. 99, 625 (1928)zbMATHGoogle Scholar
  29. 29.
    E.T. Whittaker, G.N. Watson, A Course of Modern Analysis (Cambridge at the University Press, 1958), chapter XIXGoogle Scholar
  30. 30.
    G. Kotowski, Z. Angew. Math. Mech. 23, 213 (1943)MathSciNetzbMATHGoogle Scholar
  31. 31.
    R.D. Knight, Int. J. Mass Spectrom. Ion Phys. 51, 127 (1983)CrossRefGoogle Scholar
  32. 32.
    N.N. Bogoljubow, J.A. Mitropolski, Asymptotische Methoden in der Theorie der nichtlinearen Schwingungen (Akademie-Verlag, Berlin, 1965)Google Scholar
  33. 33.
    G. Rettinghaus, Ph.D. thesis, Bonn University (1965), unpublishedGoogle Scholar
  34. 34.
    D. Wineland, H.G. Dehmelt, Int. J. Mass Spectrom. Ion Phys. 16, 338 (1975)CrossRefGoogle Scholar
  35. 35.
    M.A.N. Razvi, Phys. Rev. A 58, R34 (1998)Google Scholar
  36. 36.
    B.A. Collings, D.J. Douglas, J. Am. Soc. Mass Spectrom. 11, 1016 (2000)CrossRefGoogle Scholar
  37. 37.
    R.S. van Dyck Jr, Phys. Rev. A 40, 6308 (1989)CrossRefGoogle Scholar
  38. 38.
    J.V. Porto, Phys. Rev. A 64, 023403 (2001)CrossRefGoogle Scholar
  39. 39.
    D.J. Wineland, H.G. Dehmelt, J. Appl. Phys. 46, 919 (1975)CrossRefGoogle Scholar
  40. 40.
    P. Hagedorn, Z. Angew. Math. Mech. 48, T256 (1968)Google Scholar
  41. 41.
    C.H. Tseng, Phys. Rev. A 59, 2094 (1999)CrossRefGoogle Scholar
  42. 42.
    J.P. Schermann, F.G. Major, Appl. Phys. 16, 225 (1978)Google Scholar
  43. 43.
    C.H. Kwak, E.-H. Lee, Electr. Telecomm. Res. Inst. J. 16, 13 (1995)zbMATHGoogle Scholar
  44. 44.
    N.V. Kukhtarev, Ferroelectrics 22, 949 (1979)Google Scholar
  45. 45.
    G. Gabrielse, Phys. Rev. A 29, 462 (1984)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  • G. Tommaseo
    • 1
  • P. Paasche
    • 1
  • C. Angelescu
    • 1
  • G. Werth
    • 1
    Email author
  1. 1.Institut für PhysikJohannes Gutenberg UniversitätMainzGermany

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