Gaussian approximation of the bunch lengthening in electron storage rings with a typical wake function

S Petracca,Th Demma,K Hirata

doi:10.1103/physrevstab.8.074401

Abstract

The validity of the Gaussian approximation of the distribution function in the study of the equilibrium bunch length in electron storage rings is considered with specific reference to the inductive wake. The Gaussian approximation can be used to describe not only localized wakes, but also uniformly distributed wakes, for which the longitudinal profiles at equilibrium are ruled by the potential-well-distortion (Haissinski) equation. Comparison of Gaussian approximation based results with those obtained from Haissinski equation shows good qualitative agreement in their common range of validity, while shedding light on the nature of the unstable regimes.

Highlights

The wake force describes space-charge and image interactions which affect the particles distribution in a bunched beam [1]
The main simplifying assumption is that if the initial distribution function in synchrotron phase space is Gaussian, it can still be approximated by a Gaussian after the action of the wake force, represented by a ‘‘kick.’’ The distribution function can be parametrized by its first and second order moments only, describing the beam envelope in synchrotron phase space
The above stochastic mapping is equivalent to an infinite hierarchy of deterministic maps in the moments of the distribution function x~, i.e., x i hxi i; ij h xi x j xj x j i;, we can split the maps for the second order moments into three parts, representing the effect of radiation, wake force, and synchrotron oscillation, respectively, as follows: 012 12 ; 022 2 22 1 2

Summary

INTRODUCTION

The wake force describes space-charge and image interactions which affect the particles distribution in a bunched beam [1]. The main simplifying assumption is that if the initial distribution function in synchrotron phase space is Gaussian, it can still be approximated by a Gaussian after the action of the wake force, represented by a ‘‘kick.’’ The distribution function can be parametrized by its first and second order moments only, describing the beam envelope in synchrotron phase space The evolution of these moments from turn to turn is obtained by applying a (nonlinear) mapping at each turn. For a wake function (resistive beam coupling impedance) a solution of the Haissinski equation always exists [7], and the Gaussian approximation (map) approach discloses multistable states in some regions of parameter space [8]. In this paper the Gaussian approximation (map) and Haissinski (equation) approach will be compared and contrasted for the specific case of a 0 wake function (inductive beam coupling impedance).

THE GAUSSIAN APPROXIMATION

THE HAISSINSKI EQUATION

COMPARISON

REGULARIZED WAKE

CONCLUSIONS