1 Nonlinear Accelerator Resonances Jeffrey Eldred Classical Mechanics

Linear Transverse Focusing See Lecture 8. 2 Classical Mechanics and Electromagnetism | June 2018

3 Accelerator Action-Angle 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

Action-Angle Coordinates for Linear Accelerator Equations of Motion & Hamiltonian: Courant-Snyder / Floquet /

Local Hamiltonian to Global Hamiltonian Local Hamiltonian: Integrated Hamiltonian: 5 Classical Mechanics and Electromagnetism

6 Sextupole Resonances 6 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

Sextupole Resonances Regular Sextupole Term: 2 D Hamiltonian with Sextupole: 7 Classical Mechanics and

Sextupole Resonances (cont. ) Local Hamiltonian with Action-Angle: 8 Classical Mechanics and Electromagnetism |

Sextupole Resonances (cont. ) Local Hamiltonian with Action-Angle: Averaged over many turns, the cosine

Sextupole Resonances (cont. ) Integration over s: G 30 is evaluated only at sextupoles,

Nonlinear Equations of Motion (cont. ) Hamiltonian: Equations of Motion: Fixed Points: Normalization by

Nonlinear Equations of Motion (cont. ) 12 12 Classical Mechanics and Electromagnetism | June

Nonlinear Equations of Motion (cont. ) 13 13 Classical Mechanics and Electromagnetism | June

14 3 rd-Order Resonance Slow-Extraction 14 Classical Mechanics and Electromagnetism | June 2018 USPAS

3 rd-Order Slow-Extraction All diagrams taken from Marco Pullia thesis Chapter 3. Adiabatically increasing

Extraction Septa The Electric Septa is thin and delivers a kick to provide a

Steinbach Diagrams Steinbach diagrams are useful for determining the uniformity of the spill and

Method 1: Move the Tune 18 18 Classical Mechanics and Electromagnetism | June 2018

Method 2: Move the Beam 19 19 Classical Mechanics and Electromagnetism | June 2018

Method 3: Excite the Beam Operation experience shows this is the method that provides

21 Further Reading on Resonances Boris Chirikov “A Universal Instability of Many. Dimensional Oscillator

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1 Nonlinear Accelerator Resonances Jeffrey Eldred Classical Mechanics and Electromagnetism June 2018 USPAS at MSU

Linear Transverse Focusing See Lecture 8. 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

3 Accelerator Action-Angle 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Action-Angle Coordinates for Linear Accelerator Equations of Motion & Hamiltonian: Courant-Snyder / Floquet / TWISS / Action-Angle Coordinate Transformation: New Hamiltonian: 4 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Local Hamiltonian to Global Hamiltonian Local Hamiltonian: Integrated Hamiltonian: 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

6 Sextupole Resonances 6 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Sextupole Resonances Regular Sextupole Term: 2 D Hamiltonian with Sextupole: 7 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Sextupole Resonances (cont. ) Local Hamiltonian with Action-Angle: 8 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Sextupole Resonances (cont. ) Local Hamiltonian with Action-Angle: Averaged over many turns, the cosine terms go to zero, unless… So consider the case where: 9 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Sextupole Resonances (cont. ) Integration over s: G 30 is evaluated only at sextupoles, which may add up constructively or destructively. Super-periodicity of accelerator rings cause the resonances to cancel out naturally. Families of sextupoles are also used to change chromaticity without driving resonances. 10 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Nonlinear Equations of Motion (cont. ) Hamiltonian: Equations of Motion: Fixed Points: Normalization by Fixed Points: 11 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Nonlinear Equations of Motion (cont. ) 12 12 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Nonlinear Equations of Motion (cont. ) 13 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

14 3 rd-Order Resonance Slow-Extraction 14 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

3 rd-Order Slow-Extraction All diagrams taken from Marco Pullia thesis Chapter 3. Adiabatically increasing the sextupole strength deforms the linear circular trajectory into a triangular trajectory surrounded by the separatrix. 15 15 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Extraction Septa The Electric Septa is thin and delivers a kick to provide a large enough gap that the Magnetic Septa can fully extract. The phase advance between the two septa determines the rotation of the separatrix. 16 16 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Steinbach Diagrams Steinbach diagrams are useful for determining the uniformity of the spill and the momentum spread of the spill. Chromaticity relates the particle momenta to the tune. 17 17 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Method 1: Move the Tune 18 18 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Method 2: Move the Beam 19 19 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

Method 3: Excite the Beam Operation experience shows this is the method that provides the most fine-control of the spill uniformity. Method 4: Change the Sextupoles? This method is not recommended, because the spill is very nonuniform and not all particles may be removed. 20 20 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020

21 Further Reading on Resonances Boris Chirikov “A Universal Instability of Many. Dimensional Oscillator Systems” 21 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 12/3/2020