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WEA1CO06 |
Analytical theory for McMillan map | |
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McMillan map is an important discrete time model of 1D transverse nonlinear accelerator lattice. We provide 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. The bifurcation map for canonical McMillan map including stability of fixed points is provided. In addition, we discuss the connection of these results with possible 2D generalizations - axially symetric and 2D-magnetostatic McMillan lenses.
Iatrou, A., & Roberts, J. A. (2002). Integrable mappings of the plane preserving biquadratic invariant curves II. Nonlinearity, 15(2), 459. |
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Slides WEA1CO06 [5.151 MB] | |
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| THPOA30 | SCHARGEV 1.0 - Strong Space Charge Vlasov Solver | 1164 |
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| The space charge (SC) is known to be one of the major limitations for the collective transverse beam stability. When space charge is strong, i.e. SC tune shift much greater than synchrotron tune, the problem allows an exact analytical solution. For that practically important case we present a fast and effective Vlasov solver SCHARGEV (Space CHARGE Vlasov) which calculates a complete eigensystem (spatial shapes of modes and frequency spectra) and therefore provides the growth rates and the thresholds of instabilities. SCHARGEV 1.0 includes driving and detuning wake forces, and, any feedback system (damper). In the next version we will include coupled bunch interaction and Landau damping. Numerical examples for FermiLab Recycler and CERN SPS are presented. | ||
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Poster THPOA30 [1.493 MB] | |
| DOI • | reference for this paper ※ https://doi.org/10.18429/JACoW-NAPAC2016-THPOA30 | |
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| THPOA31 | Sector Magnets or Transverse Electromagnetic Fields in Cylindrical Coordinates | 1167 |
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| Laplace's equation in normalized cylindrical coordinates is considered for scalar and vector potentials describing electric or magnetic fields with invariance along the azimuthal coordinate (arXiv:1603.03451). A series of special functions are found which when expanded to lowest order in power series in radial and vertical coordinates (rho=1 and y=0) replicate harmonic homogeneous polynomials in two variables. These functions are based on radial harmonics found by Edwin M. McMillan forty years ago. In addition to McMillan's harmonics, a second family of radial harmonics is introduced to provide a symmetric description between electric and magnetic fields and to describe fields and potentials in terms of the same functions. Formulas are provided which relate any transverse fields specified by the coefficients in the power series expansion in radial or vertical planes in cylindrical coordinates with the set of new functions. This result is important for potential theory and for theoretical study, design and proper modeling of sector dipoles, combined function dipoles and any general sector element for accelerator physics and spectrometry. | ||
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Poster THPOA31 [2.274 MB] | |
| DOI • | reference for this paper ※ https://doi.org/10.18429/JACoW-NAPAC2016-THPOA31 | |
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