1 Review of Accelerator Physics Concepts Jeffrey Eldred

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

2 Betatron Motion 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

Linear Focusing We can solve the linear Hill’s equation: 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

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 11/24/2020

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 11/24/2020

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 11/24/2020

TWISS Plots 7 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

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 11/24/2020

Stability over many revolutons The ring transfer matrix at location s 1: This is the “one-turn map”, the effect on the beam as starts at s 1, travels around the ring, and returns to s 1. 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 11/24/2020

Betatron Phase Advance Betatron Tune: 10 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

Betatron Oscillation Emittance: Spot size: Relativistic Adiabatic Damping: Wille 11 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

Transverse Phase-space 12 12 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

Phase-space from Courant-Snyder 13 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

Betatron Motion Bartolini 14 14 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

Two Beam Position Monitor Measurement The correlation between X 1 and X 2, can be used to see the betatron phase advance between those points. S. Y. Lee 15 15 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

16 Nonlinearities 16 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

Betatron Tune Resonance Perturbation will accumulate if tune is a fraction corresponding to the symmetry of the applied fields. 17 17 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

Tune Diagrams The tune is carefully picked to avoid resonances. Barletta 18 18 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 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 11/24/2020

Phase-space Distortions 19 19 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

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 20 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

21 Off-Momentum Particles 21 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

Dispersion: Spot Size: Barletta 22 22 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

TWISS Plots 23 23 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

Chromaticity Change in tune with momentum: Chromaticity: Barletta 24 24 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020

Sextupoles & Chromaticity Correction Dispersion is position offset dependence on momentum. Chromaticity is tune dependence on momentum. Sextupoles provide tuneshift depending on position offset. FNAL Rookie Book 25 25 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/24/2020