JOHN J. BARNARD, BOJAN LOSIC (LLNL & LBNL)
For a particle beam propagating in an alternating gradient focusing system, envelope equations are often employed to describe the evolution of the beam radii in the two directions transverse to the direction of propagation, and aligned with the principle axes of the alternating gradient system. When the beams have zero net angular momentum and when the alternating gradient focusing is approximated by a continuous focusing system, there are two normal modes to the envelope equations: the 'breathing' mode and a 'quadrupole' mode in which the two radii oscillate in phase, and 180 degrees out of phase, respectively. In this paper, we extend the analysis to beams which have a finite angular momentum. We perturb the moment equations of ref. , wherein it assumed that space charge is a distributed in a uniform density ellipse. Two additional modes are obtained. The breathing mode remains, but the quadrupole mode is split into two modes, and a new low frequency mode appears. We calculate the frequencies and eigenmodes of these four modes as a function of tune depression and a dimensionless net angular momentum. These modes can be excited by rotational errors of the quadrupoles in an alternating gradient focusing channel.
 John J. Barnard, Proc. 1995 Particle Accelerator Conference, (Dallas, TX), vol. 5, 3241, (1996).
*Work performed under the auspices of the US Department of Energy under contract W-7405-ENG-48.
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