Thursday Week 1 Lecture Jeff Eldred Nonlinear Sextupole Resonance 1 1

Overview Nonlinear Sextupole Resonance: 2 Overview Nonlinear Sextupole Resonance: Linear Betatron motion in Action-Angle CT to orbital angle frame Addition of sextupole perturbation Fourier analysis of sextupole perturbation CT to resonance-rotating frame Motion of sextupole-driven nonlinear beam Application to slow extraction 2 2 2 2 2

3 Action-Angle to Orbital Angle Frame 3 3 3 3 3

Linear Action-Angle Coordinates Shown in Gregg’s “Lecture 8” yesterday: 4 4 4 4

CT to Orbital-Angle Coordinate 5 5 5 5

What is the Orbital-Angle Coord.? 6 6 6 6

Nonlinear Sextupole Resoance 7 Nonlinear Sextupole Resoance 7 7 7 7 7

Sextupole Perturbation We add a term corresponding to a sextupole: We transform to the Action-Angle Orbital-Angle Frame: 8 8 8 8

3rd-Order Nonlinear Resonance Near a 3nu = l resonance we can focus on: Integrating over s we obtain the Fourier coefficient: 9 9 9 9

Fourier coefficient 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. Credit: USPAS Wolski & Newton 10 10 10 10

11 Nonlinear Motion 11 11 11 11 11

Resonance-Rotating Frame With a given Fourier coefficient: 12 12 12 12

Nonlinear Equations of Motion 13 13 13 13

Nonlinear Motion 14 14 14 14

Nonlinear Motion 15 15 15 15

Nonlinear Motion X, P x, x’ 16 16 16 16

17 3rd-Order Resonance Slow-Extraction 17 17 17 17 17

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. 18 18 18 18

Two Septa to Extraction Line The ES is thin and delivers a kick to provide a large enough gap that the MS can fully extract. The phase advance between the two septa determines the rotation of the separatrix. 19 19 19 19

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. 20 20 20 20

Method 1: Move the Tune 21 21 21 21

Method 2: Move the Beam 22 22 22 22

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. 23 23 23 23

Hardt Condition The line of unstable particle trajectories should coincide for particles of different momenta. Hardt Condition relates dispersion, phase-advance, chromaticity, and sextupole strength. 24 24 24 24

Septa Locations The Hardt condition, the rotation of the separatrix, and the aperture constraints determine the septa locations. 25 25 25 25