Dancing bunches as Van Kampen modes

TL;DR

This paper explores Van Kampen modes in beam physics, showing how discrete modes can cause instabilities in particle accelerators and proposing a stabilization method based on the sensitivity of Landau damping loss.

Contribution

It provides a new analysis of Van Kampen modes' role in beam instabilities and introduces a method for beam stabilization considering the distribution function's steepness.

Findings

Discrete Van Kampen modes can induce instabilities without Landau damping.

Longitudinal instabilities at major accelerators are linked to loss of Landau damping at low impedances.

Beam stabilization can be achieved by controlling the bunch distribution function's steepness.

Abstract

Van Kampen modes are eigen-modes of Jeans-Vlasov equation. 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, RHIC and SPS 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.