Theory of the tune shift due to linear coupling
The presence of skew quadrupole fields will linearly couple the x and y motions. The x and y motion can then be written as the sum of two normal modes which have the tunes {nu}{sub 1},{nu}{sub 2}, in the absence of the skew quadrupole fields. New beta functions, {Beta}{sub 1} and {Beta}{sub 2}, can be defined which are the beta functions of the normal modes and which are different from {Beta}{sub x} and {Beta}{sub y}, the beta functions of the unperturbed accelerator. This paper presents analytical perturbation theory results for {nu}{sub 1},{nu}{sub 2}. The results for {nu}{sub 1},{nu}{sub 2} are first found correct to lowest order in the skew quadrupole fields. The results for {nu}{sub 1},{nu}{sub 2} are then carried one step further to include the next higher order terms in the skew quadrupole fields. Results for {Beta}{sub 1},{Beta}{sub 2} will be given in a future paper. These analytical results show that for the higher order shift in tune the important harmonics of the skew quadrupole field are the harmonics near {nu}{sub x} + {nu}{sub y}. However the harmonics closest to {nu}{sub x} + {nu}{sub y} do not contribute to the higher order tune splitting, {vert bar}{nu}{sub 1}{minus}{nu}{sub 2}{vert bar}, as they shift {nu}{sub 1} and {nu}{sub 2} about equally. This results in lack of a dominant harmonic for the higher order contribution of {vert bar}{nu}{sub 1}{minus}{nu}{sub 2}{vert bar}, which complicates the understanding and correction of the higher order contribution to {vert bar}{nu}{sub 1}{minus}{nu}{sub 1}{vert bar}. Analytical results are found for the residual tune splitting which is the {vert bar}{nu}{sub 1}{minus}{nu}{sub 2}{vert bar} that remains after the driving term of the nearby difference resonance has been corrected. 4 refs.
- Research Organization:
- Brookhaven National Lab. (BNL), Upton, NY (United States)
- Sponsoring Organization:
- USDOE; USDOE, Washington, DC (United States)
- DOE Contract Number:
- AC02-76CH00016
- OSTI ID:
- 5611334
- Report Number(s):
- BNL-46415; AD/RHIC-100; ON: DE91015845; TRN: 91-021613
- Country of Publication:
- United States
- Language:
- English
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