Loss of Landau Damping for Bunch Oscillations

Abstract: Conditions for the existence, uniqueness and stability of self-consistent bunch steady states are considered. For the existence and uniqueness problems, simple algebraic criteria are derived for both the action and Hamiltonian domain distributions. For the stability problem, van Kampen theory is used. The onset of a discrete van Kampen mode means the emergence of a coherent mode without any Landau damping; thus, even a tiny couple-bunch or multi-turn wake is sufficient to drive the instability. The method presented here assumes an arbitrary impedance, RF shape, and beam distribution function. Available areas on the intensity-emittance plane are shown for resistive wall wake and single harmonic, bunch shortening and bunch lengthening RF configurations. Thresholds calculated for the Tevatron parameters and impedance model are in agreement with the observations. These thresholds are found to be extremely sensitive to the small-argument behaviour of the bunch distribution function. Accordingly, a method to increase the LLD threshold is suggested. This article summarizes and extends recent author's publications.