Matrix solution for the wall impedance of infinitely long multilayer circular beam tubes

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 Â 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 welldocumented cases of a stainless-steel tube, and of a graphite collimator.

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

“…jðx; y; z; tÞ ¼jðx; y; z À ctÞ ¼ ðx À x 0 Þðy À y 0 Þcẑe ið!tÀkzÞ : (10) We are looking for synchronous solutions Eðx; y; z; tÞ ¼Ẽðx; yÞe ið!tÀkzÞ ;…”

“…In Fourier space ðx; y; k; !Þ, for the charge and the current given by Eqs. (9) and (10), the potential equations, Eqs. (3) and (4), read…”

“…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.…”

“…Since these systems are highly symmetric, characteristic modes can be decoupled and analytical expressions for the impedance can be derived. Of particular interest is the calculation of impedance in multilayer beam pipes; the problem has been addressed in the literature for both cylindrical [8][9][10][11][12] and parallel-plane [14,15] geometries. Beam pipes of general cross section have also been addressed in the literature [17][18][19], but only in the ultrarelativistic approximation and for single-layered metallic pipes in the frequency region where perturbation theory with respect to the penetration skin depth is valid.…”

Beam Instabilities in Hadron Synchrotrons

“…Infinite beam pipe boundary conditions were implemented by van Rienen [86], Balk [81] and Doliwa [87], Floquet (quasi-periodic) boundary conditions by Niedermayer [88]. Approximations to neglect the charge completely result in quasi-stationary models and are discussed in [89,90]. Solutions can be obtained by the FIT, where Doliwa [87,91] implemented a special low frequency stabilization technique based on Neumann series expansion of the divergence corrected (Helmholtz decomposition) system matrix.…”