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

Linear Transverse Focusing See Lecture 8. 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 2 2 2 2

Accelerator Action-Angle 3 3 Accelerator Action-Angle 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 3 3 3 3 3 3

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 1/12/2019 4 4 4 4

Local Hamiltonian to Global Hamiltonian Integrated Hamiltonian: 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 5 5 5 5

6 6 Sextupole Resonances 6 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 6 6 6 6 6 6

Sextupole Resonances Regular Sextupole Term: 2D Hamiltonian with Sextupole: 7 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 7 7 7 7

Sextupole Resonances (cont.) Local Hamiltonian with Action-Angle: 8 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 8 8 8 8

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 1/12/2019 9 9 9 9

Sextupole Resonances (cont.) Integration over s: G30 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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 10 10 10 10

Nonlinear Equations of Motion (cont.) Hamiltonian: Equations of Motion: Fixed Points: Normalization by Fixed Points: 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 11 11 11 11

Nonlinear Equations of Motion (cont.) 12 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 12 12 12 12

Nonlinear Equations of Motion (cont.) 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 13 13 13 13

3rd-Order Resonance Slow-Extraction 14 14 14 14 14 14 14 14 14 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 14 14 14 14 14 14

3rd-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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 15 15 15 15

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 16 16 16 16

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 17 17 17 17

Method 1: Move the Tune 18 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 18 18 18 18

Method 2: Move the Beam 19 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 19 19 19 19

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 non-uniform and not all particles may be removed. 20 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 1/12/2019 20 20 20 20

Further Reading on Resonances 21 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 1/12/2019 21 21 21 21 21 21