Abstract
Longitudinal focusing for a bunched beam is provided by both the change in path length with particle energy and by the time-dependent accelerating voltage. Usually one employs a smooth approximation, i.e., one ignores the discrete locations of the rf cavities, in describing the particle motion. The longitudinal motion can then be modelled by second order differential equations. For small oscillation amplitudes these equations simplify to those of harmonic oscillators.
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Minty, M.G., Zimmermann, F. (2003). Longitudinal Optics Measurement and Correction. In: Measurement and Control of Charged Particle Beams. Particle Acceleration and Detection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-08581-3_7
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