Csr Wake for a Short Magnet in Ultrarelativistic Limit∗

When an electron moves in a bending magnet it emits synchrotron radiation and generates a wakefield that effects other electrons in the beam. This wake is usually called a Coherent Synchrotron Radiation (CSR) wake and for a steady state case it has been computed in Refs. [2, 3]. In practice, bending magnets are often not long enough to assume that the wake is in the steady state. In the case when the formation length for the radiation is comparable or larger than the magnet length the transient effects associated with the entrance to and exit from the magnet should be taken into account. The analysis of such wake for large, but finite, values of the relativistic factor γ has been carried out in Ref. [1]. The results of this paper can be used in computation for moderate values of γ ∼ 10 − 10, however, for very large γ, calculations involve extremely small spatial scale ∼ R/γ, where R is the bending radius. In many cases, when γ ∼ 10, such scale cannot be resolved numerically and the algorithm fails. For such large values of γ a good approximation for the wake would be the limit γ → ∞. One such formula is given in Ref. [1], but only for the wake at the entrance to the magnet.