1 Longitudinal Dynamics RF Capture Jeffrey Eldred Classical

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1 Longitudinal Dynamics & RF Capture Jeffrey Eldred Classical Mechanics and Electromagnetism June 2018

1 Longitudinal Dynamics & RF Capture Jeffrey Eldred Classical Mechanics and Electromagnetism June 2018 USPAS at MSU

Overview 1. RF Cavities 2. Longitudinal Dynamics 3. RF Bucket Phase-space 4. RF Acceleration

Overview 1. RF Cavities 2. Longitudinal Dynamics 3. RF Bucket Phase-space 4. RF Acceleration 5. Other RF Dynamics 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

3 Longitudinal Motion of Particle Beams 3 Classical Mechanics and Electromagnetism | June 2018

3 Longitudinal Motion of Particle Beams 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

4 2/22/2021 Book Credit: FNAL Rookie Classical Mechanics and Electromagnetism | June 2018 USPAS

4 2/22/2021 Book Credit: FNAL Rookie Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

Numerically Calculated Eigenmodes 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

Numerically Calculated Eigenmodes 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

6 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

6 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Energy in one pass through cavity 7 Classical Mechanics and Electromagnetism | June 2018

Energy in one pass through cavity 7 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

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

Change in Momentum Fractional Momentum: RF Acc. Per Pass: Change Momentum per unit time: Sinesoidal potential: 8 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Phase-Slip Factor η The arrival time of the particle depends on the momentum: Higher

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: Momentum compaction factor: We can write the change in phase per unit time using the phaseslip factor: 9 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Longitudinal Focusing Synchrotron Tune 10 10 Classical Mechanics and Electromagnetism | June 2018 USPAS

Longitudinal Focusing Synchrotron Tune 10 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Phase-space Motion 11 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

Phase-space Motion 11 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Hamilton & Separatrix Hamiltonian: Separatrix: Stable Phase-space Area: 12 12 Classical Mechanics and Electromagnetism

Hamilton & Separatrix Hamiltonian: Separatrix: Stable Phase-space Area: 12 12 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

13 Accelerating Buckets 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

13 Accelerating Buckets 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

RF Acceleration A fixed frequency beam longitudinally focuses the beam into a several beam

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. 14 14 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

RF Capture 15 15 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

RF Capture 15 15 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Bucket Portraits by Synchronous Phase 16 16 Classical Mechanics and Electromagnetism | June 2018

Bucket Portraits by Synchronous Phase 16 16 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Stable Phase-space Area as a function of Φs 17 17 Classical Mechanics and Electromagnetism

Stable Phase-space Area as a function of Φs 17 17 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

18 Other Examples of Longitudinal RF Dynamics 18 Classical Mechanics and Electromagnetism | June

18 Other Examples of Longitudinal RF Dynamics 18 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

19 Transition Crossing 19 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

19 Transition Crossing 19 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Phase-Focusing & Acceleration 20 20 Classical Mechanics and Electromagnetism | June 2018 USPAS at

Phase-Focusing & Acceleration 20 20 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Phase-space at Transition 21 21 Classical Mechanics and Electromagnetism | June 2018 USPAS at

Phase-space at Transition 21 21 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

22 RF + Harmonic RF Cavities 22 Classical Mechanics and Electromagnetism | June 2018

22 RF + Harmonic RF Cavities 22 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Harmonic RF 2 nd Harmonic RF: In general you could imagine: 23 23 Classical

Harmonic RF 2 nd Harmonic RF: In general you could imagine: 23 23 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Harmonic RF for H- Injection 24 24 Classical Mechanics and Electromagnetism | June 2018

Harmonic RF for H- Injection 24 24 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Harmonic RF for Phase-space Dilution 25 25 Classical Mechanics and Electromagnetism | June 2018

Harmonic RF for Phase-space Dilution 25 25 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

26 Bunch Rotation by Quadrupole Resonance 26 Classical Mechanics and Electromagnetism | June 2018

26 Bunch Rotation by Quadrupole Resonance 26 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Quadrupole Resonance Drives a resonance: 27 27 Classical Mechanics and Electromagnetism | June 2018

Quadrupole Resonance Drives a resonance: 27 27 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

Quadrupole Resonance 28 28 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

Quadrupole Resonance 28 28 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 2/22/2021

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