- - . - on the Impedance Due to Synchrotron Radiations

'Abstrac.t A qualitative consideration of the impedance caused by the synchrotron radiation is given. The rigorous results such tithe value of the threshold frequency and the maximum value of the impedance are obtained in a simple way. I. INTRODUCTION-The problem of the synchrotron radiation of a charge in a conductive vacuum chamber has been considered many times, see references in recent publications [1,2,3]. The rigorous consideration is based on the exact solution of the wave equation in a particular geometry (a charge moving between two conductive planes or in a toroidal chamber j and involves rather cumbersome calculations. Providing very useful reference models, these solutions call for a-more simple heuristic picture of the physics involved which would clarify the situation, especially in cases where the exact sdl'ution is unknown. Consider, for example, the results [2] for a charge moving along a circle with the radius R, in a pillbox cavity with the radius b and the height h = 2s. The real part of the impedanc_e is given as a sum of b-functions due to the excitation-of eigen modes of the cavity. The threshold frequency Wth is much higher than the cutoff frequency wcut = nclh, 312 >>l,-and the maximum value of the impedance [ 1 Re z(n) n maz N 300 i Ohm is independent of b, see Appendix. This indicates that c%nsideration based on the modal analysis is superfluous while the threshold frequency is a result of the intrinsic properties of the synchrotron radiation in a waveguide. The threshold frequency may be obtained in the following way. As it is well known, the harmonic n of the synchrotron radiation can be radiated only within the small angle 0 with the plane of motion: e-e n-II3. (1) That follows from the intensity of the n-th harmonic of the synchrotron radiation [4] f 0 an ultrarelativistic particle y>>l: dW(n,B) = g $ where e2 = l/y2+8 2. The int,ensity rolls off exponentially for harmonics * Work supported by Departrnent of Energy contract DE-AC03-76SF00515. (3) in agreement with Eq. (1). The result Eq. (1) is the direct result of the Lorentz transformation of the dipole radiation in the moving frame of a particle and is an intrinsic feature of the synchrotron radiation. Consider now a particle moving in (2, y) plane between two conductive planes separated by the distance 11 = 2y. The radiated wave propagates between the planes as in …