Dancing bunches as Van Kampen modes
Van Kampen modes are eigen-modes of Jeans-Vlasov equation [1-3]. Their spectrum consists of continuous and, possibly, discrete parts. Onset of a discrete van Kampen mode means emergence of a coherent mode without any Landau damping; thus, even a tiny couple-bunch wake is sufficient to drive instability. Longitudinal instabilities observed at Tevatron [4], RHIC [5] and SPS [6] can be explained as loss of Landau damping (LLD), which is shown here to happen at fairly low impedances. For repulsive wakes and single-harmonic RF, LLD is found to be extremely sensitive to steepness of the bunch distribution function at small amplitudes. Based on that, a method of beam stabilization is suggested. Emergence of a discrete van Kampen mode means either loss of Landau damping or instability. Longitudinal bunch stability is analysed in weak head-tail approximation for inductive impedance and single-harmonic RF. The LLD threshold intensities are found to be rather low: for cases under study all of them do not exceed a few percent of the zero-amplitude incoherent synchrotron frequency shift, strongly decreasing for shorter bunches. Because of that, LLD can explain longitudinal instabilities happened at fairly low impedances at Tevatron [4], and possibly for RHIC [5] and SPS [6], being in that sense an alternative to the soliton explanation [5, 20]. Although LLD itself results in many cases in emergence of a mode with zero growth rate, any couple-bunch (and sometimes multi-turn) wake would drive instability for that mode, however small this wake is. LLD is similar to a loss of immune system of a living cell, when any microbe becomes fatal for it. The emerging discrete mode is normally very different from the rigid-bunch motion; thus the rigid-mode model significantly overestimates the LLD threshold. The power low of LLD predicted in Ref. [17] agrees with results of this paper. However, the numerical factor in that scaling low strongly depends on the bunch distribution function. Particularly, for inductive impedance above transition and three examined distributions, the highest LLD threshold intensity exceeds the lowest one by a factor of {approx}100. Based on that observation, proper RF phase shaking as a method of beam stabilization is suggested.
- Research Organization:
- Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC)
- DOE Contract Number:
- AC02-07CH11359
- OSTI ID:
- 1013737
- Report Number(s):
- FERMILAB-CONF-11-121-AD; TRN: US1102530
- Resource Relation:
- Conference: Presented at 2011 Particle Accelerator Conference (PAC'11), New York, NY, 28 Mar - 1 Apr 2011
- Country of Publication:
- United States
- Language:
- English
Similar Records
Loss of Landau Damping for Bunch Oscillations
Introduction to collective instabilities----longitudinal and transverse