Abstract
We calculate the impedance and wake functions for laminated structures with parallel-plane and circular geometries in the ultrarelativistic limit. We critically examine the approximations used in the literature for the coupling impedance in laminated chambers and find that most of them are not justified because the wall surface impedance is large. A comparison between flat and circular geometry impedances is presented. We apply our calculation in a state-of-the-art beam dynamics simulation of the Fermilab Booster which includes nonlinear optics, laminated wakefields, and space charge impedance. The latter can have a significant effect away from the ultrarelativistic limit. Even though the simulations and the comparison with the experiment are done at the Booster injection energy, where the relativistic factor $\ensuremath{\gamma}=1.42$, we find good agreement between our calculation of the coherent tune shift and recent experimental measurements.
Highlights
Wakefields are an important component of beam dynamics in high-intensity accelerators, being a potential cause for losses and instabilities
The surface impedance is calculated by solving the Maxwell equations inside the lamination and the crack regions
The impedance is evaluated on a large frequency range, the knowledge of both the low the high frequency behavior being necessary for the calculation of the wake functions
Summary
Wakefields are an important component of beam dynamics in high-intensity accelerators, being a potential cause for losses and instabilities. In this paper we calculate the impedance and the wake functions for laminated structures with parallel-plane and circular geometries for ultrarelativistic beams. Since the effect on the transverse tunes is specific to geometries without circular symmetries [3], these measurements stress the importance of choosing the right chamber geometry for any calculation which addresses the wakes in the Booster magnets. The resistive-wall specific relation they use to connect the wall surface impedance to the coupling impedance is not valid for structures with large surface impedance and flat chambers such as the laminated magnets in the Booster. For the sake of simplicity, he assumes a beam which extends to infinity in the horizontal direction While this approximation works for metallic pipes, it produces wakes which are too large for laminated chambers.
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