John Christopher Hagel

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

Abstract 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 summarized in a preceding report (Guignard and Hagel, CERN-SL-98-015 (AP) and CLIC Note 362, 1998). 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 localized 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.

Beam characteristics versus cavity models in CLIC

The luminosity requested for linear colliders with center-of-mass energies exceeding 500 GeV, implies a multibunch operation be considered in order to limit the RF power consumption. In the Compact Linear Collider (CLIC), though the repetition rate is high, a train of at least 20 bunches is necessary to obtain the performance needed for the experiments. Since at high RF frequency the wakefields are large, beam break-up is critical and stability simulations have been carried out for different pre-estimated models of detuned and/or damped cavities. The results obtained give indications about the wakefield level that should not be exceeded along the train, to avoid significant emittance growth. They also show the sensitivity to some specific parameters and the dependence on the scaling of focusing with energy. Eventually, they are used as guide lines for accelerating structure development and as a basis for a possible set of CLIC parameters.

A new approach to potential well bunch deformation

A stationary solution to the Vlasov equation fulfils a nonlinear integral equation of the Volterra type known as Haissinskis equation. It describes the shape of a bunch of particles as function of the current and the impedance of the surrounding structure. An analytic technique is described for solving this integral equation which is based on transforming the Volterra integral equation into one of Fredholm type with fixed integration limits, using a Fourier expansion for the discontinuous integrand. The results are applied to LEP (Large Electron-Positron colliding beam accelerator) at injection energy, and are found to be in excellent agreement with measurements of the bunch length in the potential well regime.<<ETX>>

New theory of single bunch stability in a linac with quadrupole displacements

The analytical treatment previously described by Guingard and Hagel (1998) has been extended to include the important effect of magnetic quadrupole transverse displacements, the chromatic variation of the magnetic focusing, the energy spread along the bunch and possible microwave quadrupoles, the last two in relation to BNS damping. Both, the longitudinal and transverse equations of motion are solved, the second by using the perturbation method with partial expansions developed for this theory. The localized nature of the quadrupole displacements is preserved by using thin lenses and the superposition principle for the kick effects. The causality principle applied to the downstream beam oscillations due to the kicks is introduced via Heaviside functions. The treatment presented provides formulae for the tuneshift in the bunch and first-order solutions for the transverse beam off-sets within the bunch. It presents a break-through in the recent efforts to solve the problem of the bunch stability theoretically, with realistic beam and linac models.