Bunch Lengthening Calculations for the Slc Damping Rings’

The problem of bunch lengthening in electron storage rings has been treated by many people, and there have been many experiments. In the typical experiment, the theory is used to determine the impedance of the ring. What has been lacking thus far, however, is a calculation of bunch lengthening that uses a carefully calculated ring impedance (or wakefield). In this paper we begin by finding the potential well distortion due to some very simple impedance models, in order to illustrate different types of bunch lengthening behavior. We then give a prescription for extendin the turbulent regime once the t % potential well calculations into reshold is known. Then finally, using the wakefield calculated in Ref. 1 for the SLC damping rings, combined with the measured value of the threshold given in Ref. 2, we calculate bunch lengthening for the damping rings, and compare the results with the measurements. 1. POTENTIAL WELL DISTORTION The self-consistent beam current distribution in an electron machine, below the turbulent threshold, is given by3 with as the natural bunch length, pr, the slope of the rf voltage at the position of the bunch and I& the transient induced voltage. In our notation a smaller value of t signifies an earlier point in time, with t = 0 the synchronous point for a low current beam. The induced voltage I& is given by 00 i’&(t) = I W(t’)l(t t’) dt’ , (2)