Coupling impedances of small discontinuities: Dependence on beam velocity

Sergey S Kurennoy

doi:10.1103/physrevstab.9.054201

Abstract

The beam coupling impedances of small discontinuities of an accelerator vacuum chamber have been calculated [e.g., S.S. Kurennoy, R.L. Gluckstern, and G.V. Stupakov, Phys. Rev. E 52, 4354 (1995)] for ultrarelativistic beams using the Bethe diffraction theory. Here we extend the results to an arbitrary beam velocity. The vacuum chamber is assumed to have an arbitrary, but uniform along the beam path, cross section. The longitudinal and transverse coupling impedances are derived in terms of series over cross-section eigenfunctions, while the discontinuity shape enters via its polarizabilities. Simple explicit formulas for two important particular cases - circular and rectangular chamber cross sections - are presented. The impedance dependence on the beam velocity exhibits some unusual features: for example, the reactive impedance, which dominates in the ultrarelativistic limit, can vanish at a certain beam velocity, or its magnitude can exceed the ultrarelativistic value many times. In addition, we demonstrate that the same technique, the field expansion into a series of cross-section eigenfunctions, is convenient for calculating the space-charge impedance of uniform beam pipes with arbitrary cross section.

Highlights

A common tendency in design of modern accelerators is to minimize beam-chamber coupling impedances to avoid beam instabilities and reduce heating
The analytical approach presented above provides a general picture of the coupling impedances for small discontinuities of the vacuum chamber with an arbitrary cross section in a wide frequency range, up to frequencies well above the cutoff
The upper limit on the frequency is imposed by the applicability of the Bethe theory: the wavelength must be large compared to the typical size h of the discontinuity

Summary

INTRODUCTION

A common tendency in design of modern accelerators is to minimize beam-chamber coupling impedances to avoid beam instabilities and reduce heating. A general analytical approach for calculating the beam coupling impedances of small discontinuities on the walls of an accelerator vacuum chamber has been developed in [1,2] for ultrarelativistic beams. A more refined theory that takes into account the reaction of radiated waves back on the hole was developed in [1] In the latter approach, the beam coupling impedances of a discontinuity come out as perturbative series in polarizabilities, more exactly, in small parameters =b3 and =b3 : the reactive impedance is the first order effect, while the real part of the impedance has the second order. The beam coupling impedances of a small discontinuity on the walls of a vacuum chamber with any connected cross section are derived for an arbitrary beam velocity v c.

FIELDS

BEAM COUPLING IMPEDANCE OF A SMALL

Transverse impedance

SPACE-CHARGE IMPEDANCE

DISCUSSION