Bloch's formula

Impedance in Accelerator Physics is a quantity that characterizes the self interaction of a charged particle beam, mediated by the beam environment, such as the vacuum chamber, RF cavities, and other elements encountered along the accelerator or storage ring.

Definition in terms of wakefunction

The impedance is defined as the Fourier transform of the Wakefunction.

$Z_0^{\left|\right|}(\omega )=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{dz}{c}{e}^{-i\omega z/c}{W}_0^{\text{'}}(z)$

Important sources of impedance

The impedance is defined at all positions along the beam trajectory. The beam travels through a vacuum chamber. Substantial impedance is generated in transitions, where the shape of the beam pipe changes. The RF cavities are another important source.

Impedance models

In the absence of detailed geometric modeling, one can use various models to represent different aspects of the accelerator beam pipe structure.

One such model is the

Broadband resonator

For the longitudinal case, one has

$Z_{\left|\right|}(\omega )=R_s\frac{1-iQ(\frac{{\omega }_r}{\omega }-\frac{\omega }{{\omega }_r})}{1+Q^2{(\frac{{\omega }_r}{\omega }-\frac{\omega }{{\omega }_r})}^2}$

with $R_s$ the shunt impedance, $Q$, the quality factor, and ${\omega }_r$ the resonant frequency.

Resistive Wall

Given a circular beam piper of radius $b$, and conductivity $\sigma$, the impedance is given by

$Z(\omega )=\frac{1-i}{cb}\sqrt{\frac{\omega }{2\pi \sigma }}$

The corresponding longitudinal wakefield is approximately given by

$W(s)=\frac{q}{2\pi b}\sqrt{\frac{c}{\sigma }}\frac{1}{s^{3/2}}$

See Also

https://impedance.web.cern.ch/impedance/

References

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