Analytical treatment of single bunch stability in a linac

Single bunch stability is analysed by solving the equation of motion of the particles travelling in a linac, for a Gaussian distribution of charge, a linear variation of the transverse wakefield along the bunch, a smooth focusing and negligible acceleration. The treatment is based on a non-standard perturbation expansion that has been specifically developed for this study and preserves at each order the intrinsic detuning likely to stabilise the resonant beam break-up. It provides a closed expression for the tune shift along the bunch resulting from BNS damping and autophasing, methods proposed in the past to control the emittance, and a first­order solution for the transverse off­sets within the bunch. The analytic result obtained makes it possible to study the behaviour of the solution and compute the emittance dilution in specific cases. The present theory is a useful complement to the numerical simulations done with the MUSTAFA code in the Compact Linear Collider scheme (CLIC). It also gives an interesting as well as comprehensive view of the physics involved in the single-bunch motion and the damping of the instability.