Theory of single bunch stability and dynamics in linacs with strong wakefields and misalignments

The basic method we propose in order to solve analytically the equation of motion of a relativistic single-bunch travelling in a linac, in the presence of wakefields, has been summarised in a preceding report [1]. The extended treatment presented here includes the quadrupole transverse displacements, the chromatic variation of the magnetic focusing, the energy spread along the bunch and possible microwave quadrupoles. It deals with a Gaussian distribution of charge, linear variation of the wakefields within the bunch and smooth focusing. The energy is assumed to be constant in linac sectors, but increases from one sector to the next to simulate acceleration. The longitudinal and transverse equations of motion are solved,the second by using the perturbation method with partial expansions developed for this theory. The localised nature of the misalignment kicks and their superposition property are preserved by using thin lenses. The causality of the downstream oscillations due to these kicks is introduced via Heaviside functions. These ideas make it possible to build an analytical model for quadrupole misalignments and correlated displacements due to trajectory corrections. The resulting theory provides algebraic expressions for BNS damping, tune shifts, transverse off-sets and emittance dilution. It represents a significant break-through complementing the simulations and reproducing the oscillations observed numerically.