Wakefield and the Diffraction Model Due to a Flat Beam Moving past a Conducting Wedge∗

A collimator is often used to clean a beam of its excessive tail particles. If the beam intensity is high enough or if the beam is brought too close to the collimator, however, the wakefields generated by the beam-collimator interaction can cause additional beam tails to grow, thus defeating, or even worsening, the beam-tail cleaning process. The wakefield generated by a sheet beam moving past a conducting wedge has been obtained in closed form by Henke using the method of conformal mapping [1]. This result is applied in the present work to obtain the wake force and the transverse kick received by a test charge moving with the beam. For the beam to be approximated as sheet beams, it is assumed to be flat and the collimator is assumed to have an infinite extent in the flat dimention. We derive an exact expression for the transverse wake force delivered to particles in the beam bunch. Implication of emittance growth as a beam passes closely by a collimator is discussed. We consider two idealized wedge geometries: In Section 2, when the wedge has the geometry as a disrupted beam pipe, and in Section 3, when it is like a semi-infinite screen. Unfortunately, we do not have solutions for more realistic collimator geometrie, such as when it is tapered to minimize the wakefield effects. However, our results should still serve as pessimistic limiting cases. An interesting opportunity is offered by our exact calculation of the wakefields: it can be used to confront the diffraction model [2,3,4] used to estimate the highfrequency impedance of a cavity structure. It is shown that the field pattern, as well as the impedance, agrees with those obtained by the diffraction model in appropriate limits. We would like to thank K. Bane, R. Warnock, and P. Wilson for their help.