Closed Form Formulas of the Indirect Space Charge Wake Function for Axisymmetric Structures
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N. Mounet, E. Dadiani, E. Métral, C. Zannini
CERN, Geneva, Switzerland
A. Rahemtulla
EPFL, Lausanne, Switzerland
Indirect space charge contributes significantly to the impedance of non ultrarelativistic machines such as the LEIR, PSB and PS, at CERN. While general expressions exist in frequency domain for the beam coupling impedance, the time domain wake function is typically obtained numerically, thanks to an inverse Fourier transform. An analytical expression for the indirect space charge wake function, including the time dependence as a function of particle velocity, is nevertheless highly desirable to improve the accuracy of time domain beam dynamics simulations of coherent instabilities. In this work, a general formula for the indirect space charge wake function is derived from the residue theorem. Moreover, simple approximated expressions reproducing the time and velocity dependence are also provided, which can even be corrected to recover an exact formula, thanks to a numerical factor computed once for all. The expressions obtained are successfully benchmarked with a purely numerical approach based on the Fourier transform.
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N. Biancacci, X. Buffat, N. Mounet, E. Métral, D. Valuch
CERN, Meyrin, Switzerland
A. Oeftiger
GSI, Darmstadt, Germany
Landau damping is an essential mechanism for ensuring collective beam stability in particle accelerators. Precise knowledge of the strength of Landau damping is key to making accurate predictions on beam stability for state-of-the-art high-energy colliders. We demonstrate an experimental procedure that would allow quantifying the strength of Landau damping and the limits of beam stability using an active transverse feedback as a controllable source of beam coupling impedance. In a proof-of-principle test performed at the Large Hadron Collider, stability diagrams for a range of Landau octupole strengths have been measured. In the future, the procedure could become an accurate way of measuring stability diagrams throughout the machine cycle.
