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Neutrino Factory/Muon Collider Document 132-v1

Simple criteria for orbit stability in periodic focusing channels

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Document type:
MuCool Note
Submitted by:
John Urish
Updated by:
John Urish
Document Created:
07 Jun 2000, 12:00
Contents Revised:
07 Jun 2000, 12:00
Metadata Revised:
07 Jun 2000, 12:00
Actually Revised:
14 Sep 2006, 13:35
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Simple criteria for orbit stability in periodic focusing channels

Author: Chun-xi Wang and Kwang-Je Kim


Periodic focusing channels are widely used to provide focusing
for beam transport, in which orbit stability generally occurs as a
basic issue. Hill's equation $x''+\kappa(s) x=0$ is the
fundamental equation that governs the linear transverse beam
dynamics. For a general periodic focusing strength $\kappa(s)$,
we provide an efficient and exact method to determine the
stability boundaries and the betatron tune of stable orbits
directly from the Fourier coefficients of $\kappa$. Approximate
formulae are also given for quick estimate of stability
boundaries. The analytic expression explicitly reveals the
dependency of stability on particle energy as well as other
design parameters, which is important for understanding and
optimizing the performance of a periodic focusing channel.
Applications for muon cooling channels are illustrated.

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Chun-xi Wang and Kwang-Je Kim
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