Coupling impedance and wake functions for laminated structures with an application to the Fermilab Booster

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 ¼ 1:42, we find good agreement between our calculation of the coherent tune shift and recent experimental measurements.

Impedances of Laminated Vacuum Chambers

“…The transverse impedance and the flat geometry case were addressed in more recent paper of K. Y. Ng [4]. The latest calculations of A. Macridin et al [5] revealed some disagreement with Ref. [4]; this fact stimulated us to get our own results on that matter.…”

Nonperturbative algorithm for the resistive wall impedance of general cross-section beam pipes

“…as can be deduced from the analytical expression of the impedance for the circular and parallel-plane geometries [13,16]. For typical metallic pipes the small-frequency regime is relevant for distances larger than z * 10 6 -10 7 m, while the large frequency regime is relevant at short range, z & 10-100 m. One might argue that these length scales make only the perturbative region of interest for beam dynamics.…”

“…See, for example, [1][2][3] and the references therein. With a few exceptions, the vast majority of impedance studies address cylindrical [3][4][5][6][7][8][9][10][11][12] and parallel-plane [13][14][15][16] beam pipes. Since these systems are highly symmetric, characteristic modes can be decoupled and analytical expressions for the impedance can be derived.…”

“…Another important difference from the existing literature is that our method does not impose an ultrarelativistic approximation. The ability to calculate the impedance for nonultrarelativistic beams and for systems with large wall surface impedance is extremely important for machines like the Fermilab Booster synchrotron, which has laminated magnets characterized by very large surface impedance [13,16] and an injection energy of 400 MeV ( ¼ 1:42).…”

The Legacy of the Tevatron in the Area of Accelerator Science

“…Several methods to suppress the instabilities were implemented, including various damping systems.The vacuum chamber of the Booster synchrotron is formed by the laminated poles of the combined function magnets. The magnet laminations contribute a total transverse impedance of about Z  40-55 M/m and affect beam stability(62). In operation, coherent synchro-betatron instabilities were suppressed by large chromaticities Q'~(10)(11)(12)(13)(14)(15)(16) (63), which lead to deterioration of the dynamic aperture and beam lifetime reduction.…”