J A Ellison et al 2007 New J. Phys. 9 32 doi:10.1088/1367-2630/9/2/032
A new model for the collective beam–beam interaction
Focus on Accelerator and Beam PhysicsJ A Ellison1, A V Sobol2 and M Vogt3,4
Show affiliationsPart of Focus on Accelerator and Beam Physics
The collective beam–beam interaction is studied in the framework of maps with a 'kick-lattice' model in the four-dimensional (4D) phase space of the transverse motion. A novel approach to the classical method of averaging is used to derive an approximate map which is equivalent to a flow within the averaging approximation. The flow equation is a continuous-time Vlasov equation which we call the averaged Vlasov equation (AVE), the new model of this paper. The power of this approach is evidenced by the fact that the AVE has exact equilibria and the associated linearized equations have uncoupled azimuthal Fourier modes. The equation for the Fourier modes leads to a Fredholm integral equation of the third kind and the setting is ready-made for the development of a weakly nonlinear theory to study the coupling of the π and σ modes. The π and σ eigenmodes are calculated from the third kind integral equation. These results are compared with the kick-lattice model using our weighted macroparticle tracking code and a newly developed, density tracking, parallel, Perron–Frobenius (PF) code.
- PACS
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29.27.Bd Beam dynamics; collective effects and instabilities
- Subjects
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Accelerators, beams and electromagnetism
- Dates
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Issue 2 (February 2007)
Received 29 June 2006
Published 19 February 2007
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A new model for the collective beam–beam interaction
J A Ellison et al 2007 New J. Phys. 9 32
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