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WEPSB021 |
McMillan Map and Its Application for Accelerator Physics | |
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McMillan map is an important discrete time model of 1D transverse nonlinear accelerator lattice. We will discuss a full analytical theory based on parametrization of individual canonical biquadratic curves*. Using the normal forms provided in* we were able to generalize this result to entire phase-plane of finite trajectories and calculate mechanical action-angle coordinates. In addition we will present an alternative way of analytical extraction of phase-advance variable out of map's invariant of motion (Danilov Theorem). The connection of these results with possible 2D generalizations, axially symetric and 2D-magnetostatic McMillan lenses, is presented.
Iatrou, A., & Roberts, J. A. (2002). Integrable mappings of the plane preserving biquadratic invariant curves II. Nonlinearity, 15(2), 459. |
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WEPSB022 |
Vanishing TMCI | |
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| The space charge (SC) is known to be one of the major limitations for the collective transverse beam stability. When SC is strong (i.e. space charge tune shift is much greater than synchrotron tune) the problem allows an exact analytical solution. When machine chromaticity is not equal to zero the beam is unstable for any wake amplitude. On the other hand, when chromaticity is equal to zero the instability has a threshold and respectively called TMCI (transverse mode coupling instability). The question of convergence of TMCI threshold is still open. In this presentation we will consider various longitudinal distribution functions of a bunch (bi-Gaussian and Hoffman-Pedersen distributions, Boxcar model and bunch in a square potential well) under the action of driving and detuning wake forces. We will discuss the appearance of instability for different families of wake functions. | ||
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