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 presenc e of wakefields, has been summarized in a preceding report (Guignard and Ha gel, CERN-SL-98-015 (AP) and CLIC Note 362, 1998). The extended treatment p resented here includes the quadrupole transverse displacements, the chromat ic variation of the magnetic focusing, the energy spread along the bunch an d possible microwave quadrupoles. It deals with a Gaussian distribution of charge, linear variation of the wakefields within the bunch and smooth focu sing. 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 perturba tion method with partial expansions developed for this theory. The localize d nature of the misalignment kicks and their superposition property are pre served by using thin lenses. The causality of the downstream oscillations d ue to these kicks is introduced via Heaviside functions. These ideas make i t possible to build an analytical model for quadrupole misalignments and co rrelated 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 com plementing the simulations and reproducing the oscillations observed numeri cally. (C) 1999 Elsevier Science B.V. All rights reserved.