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

Abstract

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 rϭa 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, 1998͒, Chap. 6͔. A fitting formula for the space-charge impedance at the beam surface (rϭa), 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 ͑EPAC96͒, edited by S. Myers, A. Pacheco, R. Pascual, Ch. Petit-Jean-Genaz, and J. Poole ͑Institute of Physics, Bristol, 1996͒, Vol. 1, p. 328͔, 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.

Resistive-wall wake and impedance for nonultrarelativistic beams

“…[7] Al-khateeb et al calculated the longitudinal resistive-wall impedance by a different approach and defined it via the energy loss of a uniform beam with transverse size a. We can identify the parameter of [7] with our variable k r . Also setting a r, Al-khateeb's complete expression for the resistive impedance [Eqs.…”

“…An expression for the longitudinal impedance due to space charge and the resistive wall for a beam of finite transverse size can be found in Ref. [7]. A general formalism for computing impedances of nonrelativistic beams, including resistive-wall boundaries, was given by Gluckstern [8].…”

A study of fast bunch rotation in the negative mass region

“…It appears that the agreement between Al-Khateeb's formula for the space charge impedance and the numerically evaluated one is excellent, which confirms that the corrected expression for the space charge impedance coming from Ref. [7] can be used for the analytical estimates in this study.…”

“…While |Z ||sc (n)| increases linearly with n for low harmonic numbers, a more refined model is needed for large n. Bisognano [6] obtained a Lorentzian shape for |Z ||sc (n)|/n, which defines a cut-off at high mode numbers, but its application is limited to ultra-relativistic beams and to ratios R p /R b of about 2. A general expression for the space charge impedance at all harmonic numbers and arbitrary γ and R p /R b has recently been obtained by AlKhateeb et al [7]. In Fig.1 the curves that are compared with a numerical evaluation from PATRIC (dots) are the Bisognano formula (red dashed curve) and the newly corrected formula [7] (blue curve).…”

“…A general expression for the space charge impedance at all harmonic numbers and arbitrary γ and R p /R b has recently been obtained by AlKhateeb et al [7]. In Fig.1 the curves that are compared with a numerical evaluation from PATRIC (dots) are the Bisognano formula (red dashed curve) and the newly corrected formula [7] (blue curve). It appears that the agreement between Al-Khateeb's formula for the space charge impedance and the numerically evaluated one is excellent, which confirms that the corrected expression for the space charge impedance coming from Ref.…”