A rigorous study on the longitudinal beam coupling impedance of a cylindrical lossy pipe in normal and anomalous regimes

TL;DR

This work develops a rigorous, energy- and regime-agnostic theory of longitudinal beam coupling impedance for a cylindrical lossy pipe, bridging normal and anomalous conductivity regimes. It builds a fundamental description from a Boltzmann transport equation to obtain a nonlocal conductivity $\sigma(k_r,\omega)$ and, from there, a general cylindrical surface impedance $Z_s(\omega)$ that includes two anomalous cases $p=0$ and $p=1$, governing carrier reflection. Using $Z_s(\omega)$, the authors derive a comprehensive longitudinal impedance $Z'(\omega)$ and associated wake potentials for on-axis charges and finite-energy beams, covering both DC/AC normal conductivity and the fully nonlocal anomalous regime; they also validate against established results and numerical tools like IW2D. The framework is applied to bunched beams and cryogenic structures, with implications for FELs and cryogenic accelerators, where ultrashort bunches and anomalous conductivity strongly influence energy loss and beam quality.

Abstract

One of the primary issues in designing particle accelerators is the effect of energy losses and collective instabilities caused by the conductivity of the vacuum chamber. Due to its relevance, many authors have long focused on studying the coupling impedance of lossy pipes with various geometries and metals. Most studies were developed for relativistic beams and room-temperature conductivity. In recent years, there has been increasing interest in the high-frequency impedance behavior of ultra-short bunches, as in the case of FELs, and in the anomalous conductivity at cryogenic temperatures of the vacuum chambers. This work introduces a new unifying theoretical framework for analyzing electromagnetic interactions in cylindrical pipes excited by charged particles traveling on the axis. In this context, a key achievement is the derivation of a rigorous expression for the electric conductivity, based on the Boltzmann theory of a Fermi gas, which forms the foundation for subsequent developments. New formulas for the surface impedance of cylindrical pipes are presented, from which a more general impedance expression is derived. Our approach captures the system's behavior across various energy and frequency regimes. Eventually, the anomalous regime is studied rigorously, providing analytical solutions for the electromagnetic field at cryogenic temperatures. Indeed, new and exact expressions for the anomalous surface impedance for carriers' reflection coefficients p = 0 and p = 1 are obtained, offering deeper insight into the electromagnetic properties of such systems under extreme conditions.

Figures (11)

• Figure 1: Real (a) and imaginary (b) part of coupling impedance $Z^\sigma$ vs frequency for different materials at normal condition, assuming DC electric conductivity ($b=2.5\ mm$, $\gamma=10$).
• Figure 2: Real (a) and imaginary (b) part of coupling impedance $Z'^\sigma$ vs frequency for different materials at normal condition, assuming AC electric conductivity ($b=2.5\ mm$, $\gamma=1000$). For copper, it is also provided a direct comparison with the coupling impedance calculated through the IW2D code.
• Figure 3: Relative loss rate $\chi$ (between finite and infinite $\gamma$ theories, $\chi=1$ for infinite $\gamma$) versus $\gamma$, for the case of a cylindrical beam pipe with radius $b=2.5\ mm$.
• Figure 4: Examples of wake potential calculated for a beam pipe made of copper ($b=2.5\ mm$) versus the $\gamma$ parameter. The chosen normalization shows consistency with the beam loading theorem.
• Figure 5: Longitudinal beam impedance for the case of a copper beam pipe of radius $b=2.5\ mm$, Lorentz factor $\gamma=1000$ and normal temperature. The continuous line considers the anomalous effect through Eq. \ref{['Zs0']}, while the dashed lines correspond to the same curves while neglecting the dependence of $\sigma$ upon $k_r$ in Eq. \ref{['Zs0']}.