Analytical calculation of the longitudinal space charge and resistive wall impedances in a smooth cylindrical pipe

The longitudinal space charge and resistive wall impedances have been investigated in a smooth cylindrical beam pipe. At any point from the beam axis, we obtained an expression for the total impedance, which at the beam surface r=a for infinite pipe wall conductivity gives the expression for the total impedance that was derived by Zotter and Kheifets in studying the impedance of uniform beams in concentric cylindrical wall chambers, when a single cylindrical chamber is considered [B. W. Zotter and S. A. Kheifets, Impedances and Wakes in High-Energy Particle Accelerators (World Scientific, Singapore, 1998), Chap. 6]. A fitting formula for the space-charge impedance at the beam surface (r=a), which is valid for arbitrary wavelengths, is given. Rather than calculating the impedance with the field on the axis [Joseph J. Bisognano, Fifth European Particle Accelerator Conference (EPAC96), edited by S. Myers, A. Pacheco, R. Pascual, Ch. Petit-Jean-Genaz, and J. Poole (Institute of Physics, Bristol, 1996), Vol. 1, p. 328], the fitting formula is obtained by averaging over the transverse beam distribution. We also give another approach for the calculation of the resistive wall impedance using the flux of the Poynting vector at the pipe wall and then compare it with the expression obtained from the volume integral over the beam distribution.