1. Review of Accelerator Physics Concepts1 1
Review of Accelerator Physics Concepts
Jeffrey Eldred
Classical Mechanics and Electromagnetism
June 2018 USPAS at MSU 1 1 1 1 1 1

2. 2 2
Betatron Motion 2
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3. Linear Focusing We can solve the linear Hill’s equation: 3 3 3 3 3
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4. Transfer Matrices
The final position and slope is a linear combination of the initial position and initial slope. We can use matrices: 4
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5. Example: FODO Cell
The transfer matrix for a sequence of elements can be obtained by multiplying the matrices for the components. 5
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6. Courant-Snyder (TWISS) Parameters
The transfer matrix for a stable ring can be parametrized:
The transfer matrix for a general transfer matrix:
For example, we can calculate TWISS for a FODO ring: 6
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7. TWISS Plots 7
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8. TWISS & Betatron Oscillation
The general transfer matrix can be decomposed:
An inverse transformation, a rotation, and transformation.
This allows us to understand what the TWISS parameters mean for the particle motion: 8
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9. Stability over many revolutons
The ring transfer matrix at location s1:
This is the “one-turn map”, the effect on the beam as starts at s1, travels around the ring, and returns to s1.
The effect of N turns:
The eigenvalues of M are:
they must be unimodular reciprocals. 9
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10. Betatron Phase Advance
Betatron Tune: 10
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11. Betatron Oscillation Wille Emittance: Spot size:
Relativistic Adiabatic
Damping:
Wille 11
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12. Transverse Phase-space12
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13. Phase-space from Courant-Snyder13
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14. Betatron Motion Bartolini 14 14 14 14 14
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15. Two Beam Position Monitor Measurement
The correlation between X1 and X2, can be used to see the betatron phase advance between those points.
S.Y. Lee 15
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16. 16 16
Nonlinearities 16
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17. Betatron Tune Resonance
Perturbation will accumulate if tune is a fraction corresponding to the symmetry of the applied fields. 17
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18. Tune Diagrams Barletta
The tune is carefully picked to avoid resonances.
The tune for the beam occupies a finite space:
- Laslett space-charge tune spread.
- Amplitude-dependent tune shift.
- Tune-change during acceleration.
- Chromaticity
- Beam-beam effects
Barletta 18
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19. Phase-space Distortions19
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20. Nonlinear Decoherence
Injection errors, instabilities, and sudden lattice changes may cause a phase-space mismatch.
Tune spread causes the beam to fill-out along the phase-space contours.
Kain 20
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21. Off-Momentum Particles21 21
Off-Momentum Particles 21
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22. Dispersion Barletta Dispersion: Spot Size: 22 22 22 22 22
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23. TWISS Plots 23
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24. Chromaticity Barletta Change in tune with momentum: Chromaticity: 24
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25. Sextupoles & Chromaticity Correction
Dispersion is position offset dependence on momentum.
Chromaticity is tune dependence on momentum.
Sextupoles provide tune-shift depending on position offset.
FNAL Rookie Book 25
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