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Chromaticity Correction & Dynamic Aperture in MEIC Ion Ring

Chromaticity Correction & Dynamic Aperture in MEIC Ion Ring. Fanglei Lin MEIC Detector and Interaction Region Designing Mini-Workshop, Oct. 31 , 2011. Outline. Fundamental Concepts MEIC Ion Collider Ring Lattice Function Chromaticity Correction Studies

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Chromaticity Correction & Dynamic Aperture in MEIC Ion Ring

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  1. Chromaticity Correction & Dynamic Aperture in MEIC Ion Ring Fanglei Lin MEIC Detector and Interaction Region Designing Mini-Workshop, Oct. 31 , 2011

  2. Outline • Fundamental Concepts • MEIC Ion Collider Ring • Lattice Function • Chromaticity Correction Studies • Dynamic Aperture and Frequency Map • Summary

  3. Fundamental Concepts • Chromaticity Aberration • The dependence of the focusing strength on the momentum of a circulating particle. A higher (lower) energy particle has a weaker (stronger) effective focusing strength. Furthermore, the gradient error arising from the chromatic abberation is propotional to the designed focusing function and is a “systematic” error causing major perturbation in the designed betatron amplitude functions and reduce the dynamical aperture for off-mementum particle. • Chromaticity • Defined as the derivative of the betatron tunes vs fractional momentum deviation: • “Natural chromaticity” arises solely from quadrupoles and depends on the lattice design, given as • Lead to the tune spread in the beam with the momentum spread, resulting in tunes overlapping a nonlinear resonance and cause particles loss. • Lead to an energy dependent increase in the spot size with Δσx,y/σx,y~ Cx,yσδ. • Chromatic Correction • Sextupole magnets provide focusing function increasing linearly with momentum to compensate the loss of cocusing in quadrupoles. • First order chromaticity can be obtained from the contribuation of quadrupoles and sextupoles:

  4. Fundamental Concepts • Nonlinear Effects of Chromatic Sextupoles and Correction • Second order chromaticity driven by the first order chromatic beta wave ∂βx,y/∂δ and dispersion wave ∂Dx/∂δ (dependent on the first orde sextupole strength). • First order geometric resonances νx, ν3x,νx±2νy(dependent on the first order sextupole strength). • Tune shift with amplitude ∂νx,y/∂Jx,y (dependent on the second order sextupole strength). • Nonlinear effects can be minimized/optimized by properly arranged sextupole families around the ring. • Dynamic Aperture • Characterized by the area in horizontal and vetical space into which particles may be injected and survive as stored beam. • Determined by tracking particles with increasing initial horizontal and vertical amplitues until the boundary between survial and loss is found. • Momentum Aperture • Characterized by the maximum momentum displacement that a particle can undergo and still survive. • Determined by tracking particles with increasing positive or negative momentum kicks at an selected interesting element until the boundary between survial and loss is found. • Presented by the diffusion rates in the frequency map, which is a numerical method based on Fourier techniques providing insight into the global dynamcis of multi-dimension systmes.

  5. MEIC at JLAB Pre-booster (up to 3 GeV) Large booster to collider ring transfer beamline Ion source Large booster (warm) (up to 20 GeV/c) • Energy: e- 3 to 11 GeV, p 20 to 100 GeV, ion 12 to 40 GeV/u • Luminosity: 1035 cm-2 s-1 (e-nucleon) per interaction point • Detectors: One full-acceptance detector (primary) + One high luminosity detector (secondary) with 7 m and 4.5 m between IP & 1st final focusing quad, respectively • Polarization: Figure-8 shape is adoped for preservation of polarization >70% desirable SRF linac Three Figure-8 rings stacked vertically Ion collider ring (cold) (up to 100 GeV/c) Electron ring (3 to 11 GeV) Medium energy IP with horizontal crab crossing Injector 12 GeV CEBAF

  6. MEIC Ion Collider Ring (1) (3) (2) (4) (1) (2) (3) Interaction Region: (1) Final Focusing Block (FFB) (2) Chromaticity Compensation Block (CCB) (3) Beam Extension Section (BES) (4) IP Dispersion Suppressor Arc end with dispersion suppression Short straight ARC FODO CELL

  7. Chromaticity Budget In Ion Ring • Reducing the whole chromaticity (by reducing in BES) helps lowering the required stregth of sextupoles for chromaticity correction. This may also help reduce the nonlinear effect introduced by sextupoles, such as second order chromaticity, geometric abberation and tune shift with amplitude.

  8. Beam Extension Section • Large x,y,nat at BES • Small x,y,nat at BES

  9. Chromaticity Correction • First order chromaticity can be easily corrected by using two sextupoles (families). • Beta wave ∂βx,y/∂δ can be exactly out of phase if two sources are π/2 apart in phase. • Dispersion wave ∂Dx/∂δcan be cancelled if two sources are π apart in phase. • First order geometric resonance terms can be removed with π phase advance between the memembers in a family. • Tune shift with amplitude depends on both phase advance and betatron tunes. π ~ π π/2 ~ π IP Separated Sextupoles Close Sextupoles

  10. Beta Wave and Dispersion Wave • Wx,y is chromatic amplitude function, given as • Dx’ is chromatic derivative of dispersion Dx, given as • before sext. correction • after sext. correction

  11. Summary of Correction Studies

  12. Tune vs. Momentum (σδ) (νx,νy)=(25.28,21.31) LS LC LC: Large chromaticity at BES and Close sextupole in the middle of CCB LS: Large chromaticity at BES and Separatedsextupole in the middle of CCB 0.02 0.02 0.01 0.01 SS SC • (νx,νy)=(23.273/21.285) SC: Small chromaticity at BES and Close sextupole in the middle of CCB SS: Smallchromaticity at BES and Separatedsextupole in the middle of CCB

  13. Frequency Map (νx,νy)=(25.28,21.31) LS LC SC SS • (νx,νy)=(23.273/21.285)

  14. Dynamic Aperture (νx,νy)=(25.28,21.31) LS LC • (νx,νy)=(23.273/21.285) SC SS

  15. Tune Shift With Amplitude LC LS SC SS

  16. Search for Optimum Woring Point • Genetic Algorithm- Using principles of natural selection: mutation, recombination, evolution- Survival and reproduction of the fittest- Ideal for solving non-linear optimization problems in many dimensions- Particularly well-suited for this problem, because resonance-induced loss of DA makes the problem intractable using standard methods (CG, steepest descent, etc…) • Automated GA-based search found an optimal working point

  17. Summary and Prospect • Comprehensive studies for chromatic correction based on the current MEIC ion ring lattice. • Chromaticity can be compensated up to the second order by an arrangement of symmetric sextupoles in the chromaticity compensation block. • This symmetry concept also has an additional advantage of removing the geometric abbration. • Tune shift with amplitude correction needs considering higher order compensation. This can be done either by arranging sextupoles in a certain phase advance and chosing a proper working point or by adding octupoles • Searching for optimum working work will help us to understand the nonlinear effect introduced by sextupoles, as well as for future nonlinear optimiation. • Optimize (Minimize) the nonlinear driving terms for the current solution: geometric terms (5) , second order chromaticities (2) and tune shift with amplitude(3). • Search another possible tunes using genetic algorithm.Using genetic algorithm for optimizing nonlinear properties. This has been implemented in LBNL by using the diffusion rate as objective.

  18. Back Up

  19. Beam Extension Section • Large x,y,nat at BES • Small x,y,nat at BES

  20. Chromaticities Correction • To minimize their strength, chromatic sextupoles are located near quads, where βxDx and βxDy are maximum. • To control chromaticity independently, a large ration of βx/βy and βy/βy for focusing and defocusing sextupole respectively. • To minimizenonlinear resonance stregth, families of sextupoles are properly arranged. Separated Sextupoles (SS) Close Sextupoles (CS)

  21. Tune vs. Momentum (δ) LC LS (νx,νy)=(25.28,21.31) SC SS • (νx,νy)=(23.273/21.285)

  22. MEIC Ion Collider Ring Dispersion Suppressor ARC FODO CELL Arc end with dispersion suppression Short straight A. Bogacz & V. Morozov

  23. Interaction Region: Ions Beam Extension Section (BES) Final Focusing Block (FFB) Chromaticity Compensation Block (CCB) • Distance from the IP to the first FF quad = 7 m • Maximum quad strength at 100 GeV/c • 64.5 T/m at Final Focusing Block • 88.3 T/m at Chromaticity Compensation Block • 153.8 T/m at Beam Extension Section • Symmetric CCB design (both orbital motion & dispersion) required for efficient chromatic correction 7 m βymax ~ 2700 m βx* = 10 cm βy* = 2 cm Whole Interaction Region: 158 m

  24. Dynamic Aperture and Frequency Map

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