Analytical Treatment of Single Bunch Transverse Dynamics in Linacs with Wakefields

In this paper we present an analytical treatment of the equation of motion of single bunch particles traveling in the linac of a linear collider, in the presence of wakefields. Using a somewhat simplified model, the equation is solved for a Gaussian distribution of charge, a linear variation of the wakefield along the bunch, a smooth focusing and in the absence of acceleration. The specificity of the method consists of preventing the appearance of artificial secular terms and keeping at any stage the intrinsic tune-shift that characterizes the problem and stabilizes the motion. Hence, a first order perturbation becomes sufficient provided that a non-standard perturbation expansion, specially developed for this analysis, be however applied. Solutions for the particle off-sets within the bunch are obtained and their contributions to the effective emittance calculated. The results explain the observed features of beam breakup and BNS damping, and reproduce the bunch dipole-oscillations visible on the animated graphics after simulations with the code MUSTAFA. This treatment provides in addition a closed expression for the tune shift along the bunch and confirms the existence of an optimum BNS damping setting which differs from autophasing in a single bunch mode.