In accelerator physics, impedance 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 | | ( ω ) = ∫ − ∞ ∞ d z c e − i ω z / c W 0 ′ ( z ) {\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:
Z ∗ | | ( ω ) = Z | | ( − ω ) {\displaystyle Z^{*||}(\omega )=Z^{||}(-\omega )}
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 | | ( ω ) = R s 1 − i Q ( ω r ω − ω ω r ) 1 + Q 2 ( ω r ω − ω ω r ) 2 {\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 R s {\displaystyle R_{s}} the shunt impedance, Q {\displaystyle Q} , the quality factor, and ω r {\displaystyle \omega _{r}} the resonant frequency.
Resistive Wall Given a circular beam piper of radius b {\displaystyle b} , and conductivity σ {\displaystyle \sigma } , the impedance is given by
Z ( ω ) = 1 − i c b ω 2 π σ {\displaystyle Z(\omega )={\frac {1-i}{cb}}{\sqrt {\frac {\omega }{2\pi \sigma }}}}
The corresponding longitudinal wakefield is approximately given by
W ( s ) = q 2 π b c σ 1 s 3 / 2 {\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
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