In this study, we present calculations of the longitudinal wakefields at cryogenic temperatures for extremely short bunches, characteristic for modern x-ray free electron lasers. The calculations are based on the equations for the surface impedance in the regime of the anomalous skin effect in metals. This paper extends and complements an earlier analysis of B. Podobedov, Phys. Rev. ST Accel. Beams 12, 044401 (2009). into the region of very high frequencies associated with bunch lengths in the micron range. We study in detail the case of a rectangular bunch distribution for parameters of interest of LCLS-II with a superconducting undulator.
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