Abstract
AC dipoles in accelerators are used to excite coherent betatron oscillations at a drive frequency close to the tune. These beam oscillations may last arbitrarily long and, in principle, there is no significant emittance growth if the AC dipole is adiabatically turned on and off. Therefore the AC dipole seems to be an adequate tool for non–linear diagnostics provided the particle motion is well described in presence of the AC dipole and non–linearities. Normal Forms and Lie algebra are powerful tools to study the non–linear content of an accelerator lattice. In this article a way to obtain the Normal Form of the Hamiltonian of an accelerator with an AC dipole is described. The particle motion to first order in the non– linearities is derived using Lie algebra techniques. The dependence of the Hamiltonian terms on the longitudinal coordinate is studied showing that they vary differently depending on the AC dipole parameters. The relation is given between the lines of the Fourier spectrum of the turn–by–turn motion and the Hamiltonian terms.
Highlights
Ac dipoles in accelerators are used to excite coherent betatron oscillations at a drive frequency close to the tune
The ac dipole seems to be an adequate tool for nonlinear diagnostics provided the particle motion is well described in the presence of the ac dipole and nonlinearities
Normal forms and Lie algebra are powerful tools to study the nonlinear content of an accelerator lattice
Summary
Where x and px are the transverse canonical coordinates, s is the longitudinal coordinate, Kxsis the focusing strength, and ds, tis the time-dependent kick of the ac dipole placed at the location sD given by the expression ds, t . Where BL is the integrated field amplitude, ͑B0ris the rigidity, QD and c0 are the tune and initial phase of the ac dipole, and dDiracs 2 sDis the Dirac delta function. Using the Courant-Snyder variablesx , px͒ this solution is written as a function of the turn number T as p x ͑T 2 ipx͑T 2J ei2pQxT1fx0͒. Where J and fx0 are the linear invariant and the initial phase given by the initial conditions and d2 and d1 are defined as d6. The equivalent expression of Eq (3) at the longitudinal location s is given by p x ͑T 2 ipx͑T 2J ei2pQxT1fx0͒. Notice that fD has a discontinuity at the ac dipole since right before the ac dipole fD is zero and right after it fD is equal to 2pQx
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