Impedance of a Rectangular Beam Tube with Small Corrugations

We consider the impedance of a structure with rectangular, periodic corrugations on two opposing sides of a rectangular beam tube. Using the method of field matching, we find the modes in such a structure. We then limit ourselves to the the case of small corrugations, but where the depth of corrugation is not small compared to the period. For such a structure we generate analytical approximate solutions for the wave number $k$, group velocity $v_g$, and loss factor $\kappa$ for the lowest (the dominant) mode which, when compared with the results of the complete numerical solution, agreed well. We find: if $w\sim a$, where $w$ is the beam pipe width and $a$ is the beam pipe half-height, then one mode dominates the impedance, with $k\sim1/\sqrt{w\delta}$ ($\delta$ is the depth of corrugation), $(1-v_g/c)\sim\delta$, and $\kappa\sim1/(aw)$, which (when replacing $w$ by $a$) is the same scaling as was found for small corrugations in a {\it round} beam pipe. Our results disagree in an important way with a recent paper of Mostacci {\it et al.} [A. Mostacci {\it et al.}, Phys. Rev. ST-AB, {\bf 5}, 044401 (2002)], where, for the rectangular structure, the authors obtained a synchronous mode with the same frequency $k$, but with $\kappa\sim\delta$. Finally, we find that if $w$ is large compared to $a$ then many nearby modes contribute to the impedance, resulting in a wakefield that Landau damps.Comment: 18 pages, 6 figures, 1 bibliography fil

Calculation of wakefields in 2D rectangular structures

We consider the calculation of electromagnetic fields generated by an electron bunch passing through a vacuum chamber structure that, in general, consists of an entry pipe, followed by some kind of transition or cavity, and ending in an exit pipe. We limit our study to structures having rectangular cross-section, where the height can vary as function of longitudinal coordinate but the width and side walls remain fixed. For such structures, we derive a Fourier representation of the wake potentials through one-dimensional functions. A new numerical approach for calculating the wakes in such structures is proposed and implemented in the computer code ECHO(2D). The computation resource requirements for this approach are moderate and comparable to those for finding the wakes in 2D rotationally symmetric structures. Numerical examples obtained with the new numerical code are presented.Comment: 31 pages, 10 figure

Intrabeam Scattering Analysis of ATF Beam Measurements

At the Accelerator Test Facility (ATF) at KEK intrabeam scattering (IBS) is a strong effect for an electron machine. It is an effect that couples all dimensions of the beam, and in April 2000, over a short period of time, all dimensions were measured as functions of current. In this report we derive a simple relation for the growth rates of emittances due to IBS. We apply the theories of Bjorken-Mtingwa, Piwinski, and a formula due to Raubenheimer to the ATF parameters, and find that the results all agree (if in Piwinski's formalism we replace the dispersion squared over beta by the dispersion invariant). Finally, we compare theory, including the effect of potential well bunch lengthening, with the April 2000 measurements, and find reasonably good agreement in the energy spread and horizontal emittance dependence on current. The vertical emittance measurement, however, implies that either: there is error in the measurement (equivalent to an introduction of 0.6% x-y coupling error), or the effect of intrabeam scattering is stronger than predicted (35% stronger in growth rates).Comment: 4 pages, 3 figures, Presented at IEEE Particle Accelerator Conferenc