An Algebraic Approach to the Evolution of Emittances upon Crossing the Linear Coupling Difference Resonance
One of the hallmarks of linear coupling is the resonant exchange of oscillation amplitude between the horizontal and vertical planes when the difference between the unperturbed tunes is close to an integer. The standard derivation of this phenomenon (known as the difference resonance) can be found, for example, in the classic papers of Guignard [1, 2]. One starts with an uncoupled lattice and adds a linear perturbation that couples the two planes. The equations of motion are expressed in hamiltonian form. As the difference between the unperturbed tunes approaches an integer, one finds that the perturbing terms in the hamiltonian can be divided into terms that oscillate slowly and ones that oscillate rapidly. The rapidly oscillating terms are discarded or transformed to higher order with an appropriate canonical transformation. The resulting approximate hamiltonian gives equations of motion that clearly exhibit the exchange of oscillation amplitude between the two planes. If, instead of the hamiltonian, one is given the four-by-four matrix for one turn around a synchrotron, then one has the complete solution for the turn-by-turn (TBT) motion. However, the conditions for the phenomenon of amplitude exchange are not obvious from a casual inspection of the matrix. These conditions and those that give rise to the related sum resonance are identified in this report. The identification is made using the well known formalism of Edwards and Teng [3, 4, 5] and, in particular, the normalized coupling matrix of Sagan and Rubin [6]. The formulae obtained are general in that no particular hamiltonian or coupling elements are assumed. The only assumptions are that the one-turn matrix is symplectic and that it has distinct eigenvalues on the unit circle in the complex plane. Having identified the conditions of the one-turn matrix that give rise to the resonances, we focus on the difference resonance and apply the formulae to the evolution of the horizontal and vertical emittances of a beam distribution upon passing through the resonance. Exact and approximate expressions for the TBT evolution of the emittances are derived and applied to a number of examples.
- Research Organization:
- Brookhaven National Lab. (BNL), Upton, NY (United States)
- Sponsoring Organization:
- Doe - Office Of Science
- DOE Contract Number:
- DE-AC02-98CH10886
- OSTI ID:
- 939991
- Report Number(s):
- BNL-81604-2008-IR; R&D Project: 18031; KB0202011; TRN: US0807064
- Country of Publication:
- United States
- Language:
- English
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