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RF considerations for the dEDM ring

RF considerations for the dEDM ring. Require coherent longitudinal oscillations with synchrotron tune around 0.2 Superconducting RF needed to produce such a tune. Cannot modulate RF phase at 0.2*frev so need another cavity with frequency offset from a revolution line by 0.2*frev

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RF considerations for the dEDM ring

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  1. RF considerations for the dEDM ring Require coherent longitudinal oscillations with synchrotron tune around 0.2 Superconducting RF needed to produce such a tune. Cannot modulate RF phase at 0.2*frev so need another cavity with frequency offset from a revolution line by 0.2*frev With superconducting RF all amplitudes and phases vary slowly compared to synchrotron oscillations and an adiabatic excitation is required

  2. Longitudinal dynamics Thin lens equations of motion for energy deviation on turn n just before entering the cavity at time Initially, consider only two voltage terms The approximation is only for analytic work

  3. Analytic considerations Set so that Where Next set the momentum variable to Dynamics are described by the hamiltonian Where

  4. Analytic considerations 2 Now go over to action (J) angle () variables that rotate at The new hamiltonian is where small or fast terms have been dropped. Going back to Cartesian Final hamiltonian

  5. Analytic considerations 3 Final hamiltonian Fixed points where , and stable when Plot of as is raised with fixed drive . The stable fixed point migrates to larger x. This is fully adiabatic.

  6. Analytic considerations 4 Intuitive picture of raising tune

  7. Simulations Applied voltage. 360 MHz for main harmonic drive at 176.3 MHz 2nd harmonic 720 MHz Final bunches Expect slight changes with extended field region.

  8. Scaling laws for other RF frequencies Synchrotron tune is fixed by Maximum velocity excursion Inexpensive RF power in FM radio band and above 350 MHz Cavities get larger as frequencies drop

  9. Conclusions • Principle appears to be sound • Need to decide on frequencies and drive phases (90 .vs. 180) • Power and RF control considerations remain • Full system cost of 3 to 5 million dollars (fully loaded with, design, over head, etc. and assuming no surprises)

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