Study of microwave instability
The microwave instability is usually described by linearizing Vlasov equation in the angle-action variables I, {phi} and assuming that the interaction of azimuthal harmonics {rho}{sub n}(I) of the distribution function {rho} is weak. The argument implied here is that the Hamiltonian flow smears out particles over invariant tori characterized by the action variables, and the remaining azimuthal dependence of the distribution function is small. Indeed, such an approach successfully describes bunch spectrum and the threshold of the microwave instability. However, recently there have been interesting observations of bunch centroid and bunch shape oscillations above instability threshold at LEP and the damping ring at SLAC. There are also indicates that the oscillations sometimes occur in localized region in the longitudinal coordinate instead of affecting the entire longitudinal distribution as one expects by an action-angle analysis. In this paper the authors describe an alternative approach to the problem of bunch stability using decomposition of the Fokker-Plank equation in the system of nonlinear equations for the moments of the distribution function. In particular, this approach allows them to avoid the conventional action-angle decomposition. The physical quantities they are interested in, the moments, are expressed in the Cartesian z {minus} {delta} phase space. To close the infinite hierarchy of moments equations, the authors assume that higher order correlations are small. Although both the action-angle and the Cartesian languages must be equivalent before truncation, they may have different speed of convergence depending on the problem being studied. It is hoped that Cartesian expansion approach would converge faster for the cases corresponding to those observed recently above threshold. The recent experimental observations made them interested in it again. This note is a progress report of their work.
- Research Organization:
- Stanford Linear Accelerator Center, Menlo Park, CA (United States)
- Sponsoring Organization:
- USDOE Office of Energy Research, Washington, DC (United States)
- DOE Contract Number:
- AC03-76SF00515
- OSTI ID:
- 491604
- Report Number(s):
- SLAC-PUB--7390; CONF-9609198--7; ON: DE97006459
- Country of Publication:
- United States
- Language:
- English
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