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

2. Jeffrey Eldred (TA) Kilean Hwang (grader)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
Kilean Hwang (grader)
Graduating with PhD this semester
Indiana University working with Shyh-Yuan Lee
Dipole fringe fields
Electrostatic storage rings using EDM. 2 2 2 2

3. Overview RF Cavities Longitudinal Dynamics RF bucket phase-space3
Overview
RF Cavities
Longitudinal Dynamics
RF bucket phase-space
RF acceleration
Other RF dynamics 3 3 3 3 3

4. Longitudinal Motion of4
Longitudinal Motion of
Particle Beams 4 4 4 4 4

5. Credit: FNAL Rookie Book5 5 5

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

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

8. Energy in one pass through cavity8 8 8

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

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: 10 10 10 10

11. Longitudinal Focusing11 11 11 11

12. Phase-space Motion 12 12 12 12

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

14. 14
Perturbation of
Synchrotron Motion 14 14 14 14 14

15. Stable Beams 15 15 15 15

16. Perturbation of Stable Beams16 16 16 16

17. Slipping Beams 17 17 17 17

18. Perturbation of Slipping Beams18 18 18 18

19. 19
Accelerating Buckets 19 19 19 19 19

20. RF Acceleration
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. 20 20 20 20

21. Credit:
X. Kang
SY. Lee 21 21 21 21

22. 15 30 45 22 22 22 22

23. Stable Phase-space Area as a function of φs23 23 23 23

24. Longitudinal RF Dynamics24
Other Examples of
Longitudinal RF Dynamics 24 24 24 24 24

25. 25
Transition Crossing 25 25 25 25 25

26. Phase-Focusing & Acceleration26 26 26 26

27. Phase-space at Transition27 27 27 27

28. RF + Harmonic RF Cavities28
RF + Harmonic RF Cavities 28 28 28 28 28

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

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

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

32. 32
Bunch Rotation by
Quadrupole Resonance 32 32 32 32 32

33. 33 33 33 33

34. 34
Credit: X. Yang 34 34 34