Time evolution of coupled-bunch modes from beta function variation in storage rings

Abstract

We present an analytical and numerical study of the equations of motion for bunches coupled by transverse wakefields. We base our study on a recent lattice design for the damping rings in the baseline configuration of the International Linear Collider. Using the macroparticle model, and assuming resistive wall wakefield coupling, we present numerical results on the time evolution of the multibunch modes. Decay modes display growth after initial decay, and mode amplitudes exhibit high-frequency oscillations. These phenomena are not expected if the beta function is assumed to have a constant, averaged value. We show analytically that they can come from coupling between modes caused by variation of the beta function in a real lattice. The effect is shown to be comparable to the effect of a nonuniform fill pattern and significantly larger than that of the higher-order mode wakefield localized in the rf cavities. Turning to the case of constant beta function, we develop a more complete treatment of the equations of motion. We derive general formulas for the bunch trajectories, and show that such formulas can only be valid in the limit of small wakefield coupling.

Transient jitter from injection in storage rings

“…Changes in focusing strength around the ring may be accounted to some extent by weighting the impedance at each location around the ring by the local beta function. However, variations in the beta function can lead to additional dynamical effects not taken into account in the usual macroparticle model, and these effects can result in some deviation in the growth rates from the predictions of the simple formulas [13].…”

“…This is how the history has to be stored, in order to apply the Fourier convolution. It is a different format from the straightforward integration of the equations of motion that is used in [13], in which bunch displacements at the same point in time are stored at each integration step. This new format is in fact not a disadvantage.…”

“…Numerical simulations have the potential to provide greater accuracy, and are also more appropriate for the transient effects with which we are presently concerned. The simulation technique we shall apply has been described previously [13]. Briefly, we evolve a system of pointlike bunches distributed around the ring in time steps corresponding to the bunch spacing.…”

“…In Ref. [13], the Fourier modes were computed from the normalized action of every bunch at the same instant in time. With the new format, the same Fourier modes can be computed by multiplying the normalized action of each bunch by a corresponding phase factor in time.…”

“…Characterizing this effect is the main theme and goal of the work in this paper. Work has previously been published on the resistive-wall wakefield in a vacuum chamber with finite wall thickness [12], and on the effects of variations of focusing strength with position around the ring [13]; and also on techniques for efficient computation of wakefield effects [14]. However, since these issues are highly relevant for our conclusions, we include some discussion as appropriate.…”