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

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

4 Longitudinal Motion of Particle Beams 4

5 Credit: FNAL Rookie Book

6 Numerically Calc. Eigenmodes Credit: University of Rostock

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

8 Energy in one pass through cavity

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

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 Longitudinal Focusing

12 Phase-space Motion

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

14 Perturbation of Synchrotron Motion 14

15 Stable Beams

16 Perturbation of Stable Beams

17 Slipping Beams

18 Perturbation of Slipping Beams

19 Accelerating Buckets 19

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 Credit: X. Kang SY. Lee

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

24 Other Examples of Longitudinal RF Dynamics 24

25 Transition Crossing 25

26 Phase-Focusing & Acceleration

27 Phase-space at Transition

28 RF + Harmonic RF Cavities 28

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

30 Harmonic RF for H- injection Credit: JPARC

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

32 Bunch Rotation by Quadrupole Resonance 32

33

34 Credit: X. Yang