1. 1 Monday Week 1 Lecture Jeff Eldred Longitudinal Dynamics, RF manipulations

2. Jeffrey Eldred (TA) Graduated with PhD December 2015 Indiana University working with Shyh-Yuan Lee Worked at Fermilab on PIP-II upgrade Slip-stacking and Electron Cloud. Postdoc at Fermilab on IOTA experiment Nonlinear transverse dynamics, Landau damping, space-charge compensation, modern ring-design 2 Kilean Hwang (grader) Graduating with PhD this semester Indiana University working with Shyh-Yuan Lee Dipole fringe fields Electrostatic storage rings using EDM.

3. 3 Overview 1.RF Cavities 2.Longitudinal Dynamics 3.RF bucket phase-space 4.RF acceleration 5.Other RF dynamics 3

4. 4 Longitudinal Motion of Particle Beams 4

5. 5 Credit: FNAL Rookie Book

6. 6 Numerically Calc. Eigenmodes Credit: University of Rostock

7. 7 Credit: Q. Wu, S. Belomestnykh W. Xu

8. 8 Energy in one pass through cavity

9. Fractional Momentum: RF Acc. Per Pass: Change Momentum per unit time: 9 Change in Momentum Sinesoidal potential:

10. 10 Phase-Slip Factor η The arrival time of the particle depends on the momentum: Higher momentum particles may arrive earlier or later than lower momentum particles: We can write the change in phase per unit time using the phase-slip factor:

11. 11 Longitudinal Focusing

12. 12 Phase-space Motion

13. 13 Hamiltonian & Separatrix Separatrix: Hamiltonian: Stable Phase-space Area:

14. 14 Perturbation of Synchrotron Motion 14

15. 15 Stable Beams

16. 16 Perturbation of Stable Beams

17. 17 Slipping Beams

18. 18 Perturbation of Slipping Beams

19. 19 Accelerating Buckets 19

20. 20 A fixed frequency beam longitudinally focuses the beam into a several beam “bunches” in individual RF “buckets”. Particles in the bucket can be accelerated by adiabatically changing the RF frequency, the other particles are lost. RF Acceleration

21. 21 Credit: X. Kang SY. Lee

22. 22 0 45 30 15

23. 23 Stable Phase-space Area as a function of φ s

24. 24 Other Examples of Longitudinal RF Dynamics 24

25. 25 Transition Crossing 25

26. 26 Phase-Focusing & Acceleration

27. 27 Phase-space at Transition

28. 28 RF + Harmonic RF Cavities 28

29. 29 Harmonic RF 2 nd Harmonic RF: In general you could imagine:

30. 30 Harmonic RF for H- injection Credit: JPARC

31. 31 Harmonic RF for Ph-Sp Dilution Credit: A. Pham

32. 32 Bunch Rotation by Quadrupole Resonance 32

33. 33

34. 34 Credit: X. Yang