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

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

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

Longitudinal Motion of 4 Longitudinal Motion of Particle Beams 4 4 4 4 4

Credit: FNAL Rookie Book 5 5 5

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

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

Energy in one pass through cavity 8 8 8

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

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

Longitudinal Focusing 11 11 11 11

Phase-space Motion 12 12 12 12

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

14 Perturbation of Synchrotron Motion 14 14 14 14 14

Stable Beams 15 15 15 15

Perturbation of Stable Beams 16 16 16 16

Slipping Beams 17 17 17 17

Perturbation of Slipping Beams 18 18 18 18

19 Accelerating Buckets 19 19 19 19 19

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

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

15 30 45 22 22 22 22

Stable Phase-space Area as a function of φs 23 23 23 23

Longitudinal RF Dynamics 24 Other Examples of Longitudinal RF Dynamics 24 24 24 24 24

25 Transition Crossing 25 25 25 25 25

Phase-Focusing & Acceleration 26 26 26 26

Phase-space at Transition 27 27 27 27

RF + Harmonic RF Cavities 28 RF + Harmonic RF Cavities 28 28 28 28 28

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

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

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

32 Bunch Rotation by Quadrupole Resonance 32 32 32 32 32

33 33 33 33

34 Credit: X. Yang 34 34 34