Longitudinal geometric loss factor and impedance of a step-out discontinuity at arbitrary beam energy in a round cylindrical beam-pipe

Using exact field matching, the longitudinal loss factor and impedance of a step-out in a round cylindrical beam-pipe at arbitrary beam energy have been obtained analytically. The beam-pipe sections have perfectly conducting walls which are connected coaxially at z=0 such that b 2 < b 1 , where b2 and b1 are the corresponding radii. Comparison between the step-out and step-in impedances at ultrarelativistic energies shows that both impedances peak at the same eigenmode frequencies. The imaginary parts of the step-out and step-in impedances are negative, slightly different in magnitude up to few Ohms, and have a similar frequency evolution. The real part of the step-out starts at positive values below the first cut-off, remains positive and tends toward a positive value at higher frequencies. At low frequencies below the first cut-off frequency, the real parts of the step-out and step-in impedances are equal in magnitude, shifted up to a positive value for the step-out and down below zero to the same value for the step-in. Numerical examples of the variation of step-out impedance with beam energy have also been given. The loss factor of the step-out is positive indicating energy loss, is higher in magnitude than the negative loss factor of the step-in, and therefore, the net effect will be a slowing down of the beam.