Theory of the elliptical Penning trap

An ideal “Elliptical Penning Trap” is an ideal cylindrically symmetric Penning trap with an additional electrostatic quadrupolar potential ∝κ(x2−y2)∝κ(x2−y2). This configuration is here investigated for arbitrary strength κκ of the additional potential. Aside from the decoupled axial motion the system is characterized by a generalized cyclotron and a generalized magnetron frequency. While the former depends only weakly on κκ, the magnetron frequency decreases rapidly with increasing κκ, vanishing at a maximum value κmax⁡κmax⁡ which represents the stability limit for the magnetron motion. Magnetron orbits are elliptical, with their numerical excentricity tending toward unity as κκ approaches its maximum value. A complete and rigorous description of the dynamics of the ideal elliptical trap is given, its approximate physical realization by use of a segmented ring electrode is discussed, and the frequency shifts expected for a real elliptical trap on account of anharmonic potential terms and of image charges are estimated by means of classical canonical perturbation theory. The accompanying paper by Breitenfeldt et al. [M. Breitenfeldt, S. Baruah, K. Blaum, A. Herlert, M. Kretzschmar, F. Martinez, G. Marx, L. Schweikhard, N. Walsh, Int. J. Mass Spectrom. (2008) this issue] reports corresponding experimental investigations.