Review of Accelerator Physics Concepts 1 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 Betatron Motion 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 2 2 2 2 2 2

Linear Focusing We can solve the linear Hill’s equation: 3 3 3 3 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 3 3 3 3

Transfer Matrices The final position and slope is a linear combination of the initial position and initial slope. We can use matrices: 4 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 4 4 4 4

Example: FODO Cell The transfer matrix for a sequence of elements can be obtained by multiplying the matrices for the components. 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 5 5 5 5

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 6 6 6 6

TWISS Plots 7 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 7 7 7 7

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 8 8 8 8

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 9 9 9 9

Betatron Phase Advance Betatron Tune: 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 10 10 10 10

Betatron Oscillation Wille Emittance: Spot size: Relativistic Adiabatic Damping: Wille 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 11 11 11 11

Transverse Phase-space 12 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 12 12 12 12

Phase-space from Courant-Snyder 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 13 13 13 13

Betatron Motion Bartolini 14 14 14 14 14 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 14 14 14 14

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 15 15 15 15

16 16 Nonlinearities 16 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 16 16 16 16 16 16

Betatron Tune Resonance Perturbation will accumulate if tune is a fraction corresponding to the symmetry of the applied fields. 17 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 17 17 17 17

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 18 18 18 18

Phase-space Distortions 19 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 19 19 19 19

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 20 20 20 20

Off-Momentum Particles 21 21 Off-Momentum Particles 21 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 21 21 21 21 21 21

Dispersion Barletta Dispersion: Spot Size: 22 22 22 22 22 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 22 22 22 22

TWISS Plots 23 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 23 23 23 23

Chromaticity Barletta Change in tune with momentum: Chromaticity: 24 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 24 24 24 24

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/5/2018 25 25 25 25