Mathematics

Charged Particle Traps: Physics and Techniques of Charged by F. G. Major;Viorica N. Gheorghe;G????nther Werth

By F. G. Major;Viorica N. Gheorghe;G????nther Werth

This e-book presents an creation and consultant to trendy advances in charged particle (and antiparticle) confinement through electromagnetic fields. Confinement in several seize geometries, the impression of catch imperfections, classical and quantum mechanical description of the trapped particle movement, various tools of ion cooling to low temperatures, and non-neutral plasma homes (including Coulomb crystals) are the most topics. They shape the foundation of such functions of charged particle traps as high-resolution optical and microwave spectroscopy, mass spectrometry, atomic clocks, and, very likely, quantum computing.

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Additional resources for Charged Particle Traps: Physics and Techniques of Charged Particle Field Confinement (Springer Series on Atomic, Optical, and Plasma Physics)

Example text

For convenience we introduce the dimensionless variables x1 = MΩ x, 2 x2 = MΩ y, 2 x3 = MΩ z, 2 τ= 1 Ωt . 75) j = 1, 2, 3 . 76) where the functions gj are π-periodic. 77) = 2iδj , dτ dτ where δj > 0. 78) where ρj = |wj |, γj = arg wj , the characteristic exponents βj are positive and the functions vj are π-periodic. 79) where n1 , n2 , and n3 are nonnegative integers. 80) γj x2j , where Hn are Hermite polynomials. The quasienergy of ψn1 n2 n3 is 3 E n1 n2 n3 = 1 1 Ω βj nj + 2 2 j=1 . 79) is complete and orthonormal.

Most important for practical purposes is the stable region near the origin (Fig. 5) which has been exclusively used for ion confinement. 4 qz Fig. 5. The lowest stability domain of the Paul trap including lines of constant values for βr and βz where Aj and Bj are constants depending on the initial conditions. For the stability parameter β one obtains a continued fraction expression: βj2 = aj + fj (βj ) + fj (−βj ) , fj (βj ) = qj2 (2 + βj )2 − aj − qj2 (4+βj )2 −aj −··· . 8) For the coefficients c2n which are the amplitudes of the Fourier components of the particle motion, we have the following recursion formula: c2n, j qj =− .

The effect of heavy collision partners on the storage time has been investigated in some detail by Moriwaki and Shimizu [80]. In order to derive a more quantitative description of the ion motion in a Paul trap in the presence of a background of very light particles, the effect of the collisions can be modeled as a viscous damping force proportional to the ion velocity, thus F = −Dv. If we define a damping constant as b = D/M Ω, then the equation of motion of an ion in a Paul trap reads d2 u du + 2b + (au − 2qu cos 2τ )u = 0 .

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