Abstract
The coupling impedance of beam tubes is a long-standing important topic for particle accelerators that many authors have addressed. The present study was initiated in view of a specific problem, but its novel approach is broadly applicable to the longitudinal and transverse coupling impedances of coated beam tubes or multilayer tubes. The matrix method presented here derives the wall impedance by treating the radial wave propagation of the beam-excited electromagnetic fields in full analogy to longitudinal transmission lines. Starting from the Maxwell equations, the radially transverse magnetic field components are described for monopole and dipole modes by a $2\ifmmode\times\else\texttimes\fi{}2$ matrix. Assuming isotropic material properties within one layer, the transverse field components at the inner boundary of a layer uniquely are determined by matrix transfer of the field components at its outer boundary. By imposing power-flow constraints on the matrix, wave impedance mapping and field matching between layers is enforced and replaced by matrix multiplication. The longitudinal and transverse coupling impedances are derived from the wall impedance at the innermost boundary, and the different procedures for its determination are discussed. The matrix method is demonstrated via selected yet representative examples of the well-documented cases of a stainless-steel tube, and of a graphite collimator.
Highlights
Many authors have addressed the coupling impedance of beam tubes, a long-standing topic of importance for particle accelerators
While studying the coated ceramic beam tube in the Relativistic Heavy Ion Collider (RHIC) injection kicker, Hahn noticed the analogy of the radial with axial wave transmission, and developed the concept wherein the matching of the sequential wave impedance is replaced with multiplication of the appropriate matrices relating the electric and magnetic field components in each layer [5]
This paper presents the alternative matrix method wherein in full analogy to the treatment of longitudinal transmission lines the field matching is replaced by the multiplication of radial transfer matrices
Summary
Many authors have addressed the coupling impedance of beam tubes, a long-standing topic of importance for particle accelerators. The solutions for multilayer structures typically are based on an algorithm involving field matching at the boundary layers, and sequential matching of radial wave impedances [4]. In principle, this method allows many layers, the numerical implementation becomes increasingly complex and can be simplified by the codified use of the matrix method presented here. The goal for this paper was to develop a formally simpler method to find the longitudinal and transverse impedance of infinitely long multilayer circular beam tubes based on a rigorous model valid at all frequencies and beam energies, and to transfer the actual numerical work to the Wolfram MATHEMATICA program. The paper concludes with a short summary of the pertinent properties of the matrix method
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