Abstract
The study of collective effects in circular accelerators can be tackled by solving numerically the Vlasov equation or by using tracking codes. The two methods are obtained with different approaches: Vlasov solvers consider a continuous distribution function and describe the beam with coherent oscillation modes in frequency domain (ending up usually with an eigenvalue system to solve), while tracking codes use macroparticles and wakefields in time domain. In this paper we present two Vlasov solvers for the study of collective effects (from impedances/wakefields only) which evaluate the frequency shift of coherent oscillation modes and possible mode coupling instability in the single-bunch case for both longitudinal and transverse planes. In the longitudinal plane the Vlasov solver also takes into account the potential well distortion due to the wakefields under some conditions. In parallel to this theoretical approach, tracking codes, which include collective effects, have been used as benchmark. In particular, starting from their results, we also propose a new method to study the frequency shift of coherent modes and compare it with the output of the Vlasov solvers.
Highlights
Collective effects in circular accelerators can be studied by taking into account, in a self-consistent way and in addition to external guiding fields, the effects of selfinduced wakefields [1,2,3]
For the study of beam dynamics, it is convenient to distinguish between the longitudinal and transverse planes. This is true for many accelerators [4,5,6] except when synchrobetatron resonances become important. Another distinction which helps to simplify the study of collective effects is related to single-bunch or multibunch beam dynamics generated by short or long-range wakefields, respectively
In this paper, focusing on protons, we present two Vlasov solvers which allow to obtain the frequencies of coherent oscillation modes describing the beam motion in the single-bunch case for the longitudinal and transverse planes, and compare their results with a novel method which analyzes directly the macroparticle motion obtained as output from tracking simulation codes, and allows determining the analogous coherent frequencies without recurring to Vlasov solvers
Summary
University of Rome La Sapienza and INFN, Sez. Roma1, 00185 Rome, Italy (Received 1 May 2020; accepted 23 June 2020; published 6 July 2020). The two methods are obtained with different approaches: Vlasov solvers consider a continuous distribution function and describe the beam with coherent oscillation modes in frequency domain (ending up usually with an eigenvalue system to solve), while tracking codes use macroparticles and wakefields in time domain. In this paper we present two Vlasov solvers for the study of collective effects (from impedances/wakefields only) which evaluate the frequency shift of coherent oscillation modes and possible mode coupling instability in the single-bunch case for both longitudinal and transverse planes. In the longitudinal plane the Vlasov solver takes into account the potential well distortion due to the wakefields under some conditions. In parallel to this theoretical approach, tracking codes, which include collective effects, have been used as benchmark. Starting from their results, we propose a new method to study the frequency shift of coherent modes and compare it with the output of the Vlasov solvers
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