Impedance (accelerator physics)

The impedance is defined as the Fourier transform of the Wakefunction.^[1]

${\displaystyle Z_{0}^{||}(\omega )=\int _{-\infty }^{\infty }{\frac {dz}{c}}e^{-i\omega z/c}W_{0}^{'}(z)}$

From this expression and the fact that the wake function is real, one can derive the property:

${\displaystyle Z^{*||}(\omega )=Z^{||}(-\omega )}$

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.

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

One such model is the

For the longitudinal case, one has

${\displaystyle Z_{||}(\omega )=R_{s}{\frac {1-iQ({\frac {\omega _{r}}{\omega }}-{\frac {\omega }{\omega _{r}}})}{1+Q^{2}\left({\frac {\omega _{r}}{\omega }}-{\frac {\omega }{\omega _{r}}}\right)^{2}}}}$

with ${\displaystyle R_{s}}$ the shunt impedance, ${\displaystyle Q}$ , the quality factor, and ${\displaystyle \omega _{r}}$ the resonant frequency.

Given a circular beam piper of radius ${\displaystyle b}$ , and conductivity ${\displaystyle \sigma }$ , the impedance is given by^[2]

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

The corresponding longitudinal wakefield is approximately given by ^[3]

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

The transverse wake-function from the resistive wall is given by

${\displaystyle W(s)\approx {\frac {1}{s^{1/2}}}}$

The impedance acts back on the beam and can cause a variety of effects, often considered deleterious for accelerator functioning. In general, impedance effects are classified under the category of "collective effects" due to the fact that the whole beam must be considered together, and not just a single particle. The whole beam may, however, cause particular changes in the dynamics of individual particles such as tune shifts and coupling. Whole beam changes include emittance growth and instabilities that can lead to beam loss.

https://impedance.web.cern.ch/impedance/ Archived 2018-07-04 at the Wayback Machine