Reactive impedance of a smooth toroidal chamber below the resonance region.

Abstract

We evaluate the longitudinal coupling impedance of a smooth toroidal vacuum chamber in the domain of frequencies below the first synchronous resonant mode. The chamber has rectangular cross section. With infinite wall conductivity, as assumed here, the nonresonant impedance is purely reactive. It consists of a space-charge term proportional to $\frac{1}{{\ensuremath{\gamma}}^{2}}$ and a curvature term which survives even when $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$. In the entire subresonant domain, the curvature term is well represented as a quadratic function of frequency: namely, $\frac{Z}{n}=i{Z}_{0}{(\frac{h}{\ensuremath{\pi}R})}^{2}[A\ensuremath{-}3B{(\frac{\ensuremath{\nu}}{\ensuremath{\pi}})}^{2}]$, where $h$ is the height of the chamber and $R$ is the trajectory radius and $\ensuremath{\nu}=\frac{\ensuremath{\omega}h}{c}$. The constants $A$ and $B$ are of order 1, being nearly equal to 1 if the width of the chamber is somewhat greater than its height. Thus, $\frac{\mathrm{Im}Z}{n}$ from curvature is typically a very small fraction of an ohm below the resonance domain, which begins when $\ensuremath{\nu}>{(\frac{R}{h})}^{\frac{1}{2}}$. Consequences for beam stability, if any, arise from high-frequency resonances which can produce values of several ohms for $\frac{Z}{n}$.