Instabilities and beam intensity limitations in circular accelerators

Abstract

The main aim of these lectures will be to give an insight into the physics of the mechanisms of instabilities and beam intensity limitations in circular accelerators. Three different techniques will be used to evaluate the various instabilities. The first will be using ‘few-particle models’, the second will use matrix techniques involving eigenvalues, and the third technique will use Sacherer’s modal analysis technique. For each instability the threshold or the growth rate will be evaluated with particular attention to the parameter dependence. 1. CALCULATION TECHNIQUES 1.1 Using matrix techniques (Eigenvalues) The position of a particle (U) in two-dimensional phase space can be defined by its instantaneous position u and angle uc (du/dt). Consequently n particles may be represented by a column matrix with N (2n) rows. A transition matrix may be derived which describes the transition of the n particles from one situation to another, i.e.