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Open Access Publications from the University of California

Analytic Solution of the Envelope Equations for an Undepressed Matched Beam in a FODO Quadrupole Channel


In 1958, Courant and Snyder analyzed the problem of alternating-gradient beam transport and treated a model without focusing gaps or space charge. Recently we revisited their work and found the exact solution for matchedbeam envelopes in a linear quadrupole lattice [O.A. Anderson and L.L. LoDestro, Phys. Rev. ST Accel. Beams, 2009]. We extend that work here to include the effect of asymmetric drift spaces. We derive the solution and show exact envelopes for the first two solution bands and the peak envelope excursions as a function of the phase advance sigma up to 360o. In the second stable band, decreased occupancy requires higher focus field-strength and accentuates the remarkable compression effect predicted for the FD (gapless) model.

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