Van Kampen modes for bunch longitudinal motion
Conditions for existence, uniqueness and stability of bunch steady states are considered. For the existence uniqueness problem, simple algebraic equations are derived, showing the result both for the action and Hamiltonian domain distributions. For the stability problem, van Kampen theory is used. Emerging of discrete van Kampen modes show either loss of Landau damping, or instability. This method can be applied for an arbitrary impedance, RF shape and beam distribution function Available areas on intensity-emittance plane are shown for resistive wall wake and single harmonic, bunch shortening and bunch lengthening RF configurations. Language of van Kampen modes is a powerful tool for studying beam stability. Its unique efficiency reveals itself in those complicated cases, when the dielectric function cannot be obtained, as it is for the longitudinal bunch motion. Emergence of a discrete mode means either loss of Landau damping or instability. By definition, the discrete modes lie outside the continuous incoherent spectrum, but they still may stay within the bucket. In the last case, the discrete mode would disappear after a tiny portion of resonant particles would be added. However, if the discrete mode lie outside the bucket, the Landau damping cannot be restored by tiny perturbation of the particle distribution; LLD is called radical in that case. For a given bunch emittance and RF voltage, the intensity is limited either by reduction of the bucket acceptance or by (radical) LLD. In this paper, results are presented for longitudinal bunch stability in weak head-tail approximation and resistive wall impedance; three RF configurations are studied: single harmonic, bunch shortening and bunch lengthening. It is shown that every RF configuration may be preferable, depending on the bunch emittance and intensity.
- Research Organization:
- Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- AC02-07CH11359
- OSTI ID:
- 991772
- Report Number(s):
- FERMILAB-CONF-10-376-AD; TRN: US1007562
- Resource Relation:
- Conference: Presented at 46th ICFA Advanced Beam Dynamics Workshop HB2010, Morschach, Switzerland, Sep 27 - Oct 1 2010
- Country of Publication:
- United States
- Language:
- English
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