Modes of Space Charge Interaction

Abstract

The incoherent space charge effects discussed in the previous chapter are connected with matched beams, e.g. beams in – or nearly in – “equilibrium” with the focusing lattice. Beams deviating from such a matched state – for various reasons – are a source of coherent space charge effects. This chapter is dedicated to the diverse causes of coherent motion – from mismatch to instability – and a discussion of their characteristic properties. The linearised analytical Vlasov approach in constant focusing approximation is used to derive eigenmodes of coherent motion in different orders and including anisotropy, which are helpful to gain theoretical insight into the modes of space charge interaction. The concept of “negative energy modes” – analogous to the well-known “negative energy waves” of plasma streams – is introduced to characterize unstable behaviour, and the relevance of Landau damping of these modes examined by using simulation examples.