Longitudinal Modes of Bunched Beams with Weak Space Charge

TL;DR

This paper analytically describes the longitudinal collective modes of bunched beams with weak space charge, revealing the loss of Landau damping and the conditions for beam instability due to coupled-bunch interactions.

Contribution

It provides a parameter-free integral equation framework for analyzing longitudinal modes and derives analytical growth rates for instabilities across all multipolarities.

Findings

Discrete spectrum consists of infinite modes with real tunes

Landau damping is lost despite RF nonlinearity

Even minimal coupled-bunch interaction causes instability

Abstract

Longitudinal collective modes of a bunched beam with a repulsive inductive impedance (the space charge below transition or the chamber inductance above it) are analytically described by means of reduction of the linearized Vlasov equation to a parameter-less integral equation. For any multipolarity, the discrete part of the spectrum is found to consist of infinite number of modes with real tunes, which limit point is the incoherent zero-amplitude frequency. In other words, notwithstanding the RF bucket nonlinearity and potential well distortion, the Landau damping is lost. Hence, even a tiny coupled-bunch interaction makes the beam unstable; such growth rates for all the modes are analytically obtained for arbitrary multipolarity. In practice, however, the finite threshold of this loss of Landau damping is set either by the high-frequency impedance roll-off or intrabeam scattering. Above the threshold, growth of the leading collective mode should result in persistent nonlinear oscillations.