|WEPPP061||A Method to Obtain the Frequency of the Longitudinal Dipole Oscillation for Modeling and Control in Synchrotrons with Single or Double Harmonic RF Systems||2846|
Funding: This work was partly funded by GSI Helmholtzzentrum für Schwerionenforschung GmbH
In a heavy-ion synchrotron the bunched beam can perform longitudinal oscillations around the synchronous particle (single bunch dipole oscillation, SBDO). If disturbances/instabilities exciting the SBDO exceed the rate of Landau damping, the beam can become unstable. Furthermore, Landau damping is accompanied by an increase of the beam emittance which may be undesired. Thus, control efforts are taken to stabilize the beam and to keep the emittance small. It is known that for a single harmonic cavity and a small bunch the SBDO oscillates with the synchrotron frequency* if the oscillation amplitudes are small. For a larger bunch or a double harmonic RF systems that introduces nonlinearities**, this is no longer valid. This work shows how the frequency of the SBDO can be determined in general. As a result, the SBDO can again be modeled as a harmonic oscillator with an additional damping term to account for Landau damping. This model can be used for feedback designs which is shown by means of a simple example. As the frequency of the SBDO and the damping rate depend on the size of the bunch in phase space, it is shown how this information can be obtained from the measured beam current.
* F. Pedersen and F. Sacherer, IEEE Transactions on Nuclear Science, 24:1296–1398, 1977
** A. Hofmann and S. Myers, Proc. of the 11th International Conference on High Energy Acceleration, 1980