1. Longitudinal Dynamics & RF Capture1 1
Longitudinal Dynamics & RF Capture
Jeffrey Eldred
Classical Mechanics and Electromagnetism
June 2018 USPAS at MSU 1 1 1 1 1 1

2. Overview RF Cavities Longitudinal Dynamics RF Bucket Phase-space
RF Acceleration
Other RF Dynamics 2
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3. Longitudinal Motion of3 3
Longitudinal Motion of
Particle Beams 3
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4. Credit: FNAL Rookie Book4 4
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5. Numerically Calculated Eigenmodes5
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6. 6
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7. Energy in one pass through cavity7
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8. Change in Momentum Fractional Momentum: RF Acc. Per Pass:
Change Momentum per unit time:
Sinesoidal potential: 8
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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:
Momentum compaction factor: 9
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10. Longitudinal Focusing
Synchrotron Tune 10
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11. Phase-space Motion 11
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12. Stable Phase-space Area:
Hamilton & Separatrix
Hamiltonian:
Separatrix:
Stable Phase-space Area: 12
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13. 13 13
Accelerating Buckets 13
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14. 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
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15. RF Capture 15
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16. Bucket Portraits by Synchronous Phase16
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17. Stable Phase-space Area as a function of Φs17
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18. Longitudinal RF Dynamics18 18
Other Examples of
Longitudinal RF Dynamics 18
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19. 19 19
Transition Crossing 19
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20. Phase-Focusing & Acceleration20
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21. Phase-space at Transition21
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22. RF + Harmonic RF Cavities22 22
RF + Harmonic RF Cavities 22
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23. Harmonic RF 2nd Harmonic RF: In general you could imagine: 23 23 23 23
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24. Harmonic RF for H- Injection24
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25. Harmonic RF for Phase-space Dilution25
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26. Bunch Rotation by Quadrupole Resonance 26 26 26 26 26 26 26 26 26
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27. Quadrupole Resonance Drives a resonance: 27 27 27 27 27
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28. Quadrupole Resonance 28
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