Abstract
We study the impedance of a tapered transition at small frequencies for an arbitrary shape of the transition cross section. Our approach does not require a symmetry axis in the system (unlike round geometry). We show that the calculation of the impedance reduces to finding a few auxiliary potential functions that satisfy two-dimensional Poisson equations with Dirichlet boundary conditions. In simple cases such solutions can be obtained analytically; for more complicated geometries they can easily be found numerically. We apply our method to axisymmetric geometry and reproduce results known from the literature. We then calculate the impedance of a taper with rectangular cross section in which the vertical dimension of the cross section is a slowly changing function of the longitudinal coordinate. Finally, we find a transverse kick experienced by a beam passing near a conducting wall with a variable distance from the beam to the wall.
Highlights
The calculation of the impedance for the elements of a vacuum chamber system and the associated calculation of beam dynamics effects, such as beam instabilities or wakefield induced emittance growth, are important elements in the design of a modern accelerator
Various discontinuities between segments of the vacuum chamber are common in practice
Computer simulations of tapers are not always easy to carry out, especially in cases when the taper cross section is strongly elongated in one direction
Summary
Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309, USA (Received 30 July 2007; published 20 September 2007). We study the impedance of a tapered transition at small frequencies for an arbitrary shape of the transition cross section. Our approach does not require a symmetry axis in the system (unlike round geometry). We show that the calculation of the impedance reduces to finding a few auxiliary potential functions that satisfy two-dimensional Poisson equations with Dirichlet boundary conditions. In simple cases such solutions can be obtained analytically; for more complicated geometries they can be found numerically. We calculate the impedance of a taper with rectangular cross section in which the vertical dimension of the cross section is a slowly changing function of the longitudinal coordinate. We find a transverse kick experienced by a beam passing near a conducting wall with a variable distance from the beam to the wall
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