# Charged Particle Traps: Physics and Techniques of Charged by Fouad G. Major, Viorica N. Gheorghe, Günther Werth

By Fouad G. Major, Viorica N. Gheorghe, Günther Werth

This publication offers an creation and advisor to fashionable advances in charged particle (and antiparticle) confinement by way of electromagnetic fields. Confinement in several capture geometries, the impression of catch imperfections, classical and quantum mechanical description of the trapped particle movement, various equipment of ion cooling to low temperatures, and non-neutral plasma homes (including Coulomb crystals) are the most topics. They shape the root of such functions of charged particle traps as high-resolution optical and microwave spectroscopy, mass spectrometry, atomic clocks, and, possibly, quantum computing.

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18) into the radial variable r = x2 + y 2 , we obtain 2 2 + R− + 2R+ R− cos(ω1 t + ϕ+ − ϕ− )]1/2 . 19) is bounded by the cylinder |R+ − R− | ≤ r ≤ R+ + R− , |z| < Rz . 18) is a ﬁgure which can be described as an epitrochoid (see also Appendix B). The x − y projection of the motion is periodic if ω+ /ω− is rational; otherwise if ω+ /ω− is irrational, the radial motion is quasiperiodic. Some trajectories with the initial conditions x(0) = x0 = 0, y(0) = 0, and vx (0) = 0 are illustrated in Figs. 3.

54 3 The Penning Trap Fig. 2. 5R+ . Periodic orbits for (a) ω+ /ω √− = 2; (b) ω+ /ω− = 8; (c) ω+ /ω− = 9/2; (d) quasiperiodic orbit for ω+ /ω− = 2 17 Fig. 3. 2R+ . (a) Periodic orbit for ω+ /ωz = 6; (b) quasiperiodic orbit for ω+ /ωz = 35 The energy content of the diﬀerent motions can be derived using the Hamiltonian formalism. We choose the symmetric gauge with the vector potential A = B0 (−y/2, x/2, 0). 1 Theory of the Ideal Penning Trap 55 where p = M v is the momentum vector and the axial component of the canonical angular momentum is Lz = xpy − ypx .

32) ωs = 2QV0 /M Ω(r02 + 2z02 ) , and Φsc (r) is the space charge potential that is itself determined by the density through the Poisson equation. 33) where R is the value of r at which the cloud density is reduced to 1/e of its center value, and can be roughly considered as the cloud radius. The ion oscillation frequency, shifted by space charge, now depends on the distance from the trap center. The numerical results agree with those previously 32 2 The Paul Trap Fig. 15. 2 [69]. Copyright (2003) by the American Physical Society obtained by a statistical model using similar density distributions [69] and graphically represented in Fig.