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Chromaticity Correction & Dynamic Aperture in MEIC Ion
Ring
Fanglei Lin
MEIC Detector and Interaction Region Designing Mini-Workshop, Oct. 31 , 2011
• Fundamental Concepts• MEIC Ion Collider Ring
– Lattice Function– Chromaticity Correction Studies– Dynamic Aperture and Frequency Map
• Summary
Outline
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:
dssDsSsKC yxyxyx )]()()([4
1,,
1,
ddC yxyx /)( ,,
dsKC yxyxnatyx ,,,, 4
1
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.
Pre-booster(up to 3 GeV) Ion source
Three Figure-8rings stacked
vertically
Large booster to collider ring transfer
beamline
Medium energy IP withhorizontal crab crossing
Electron ring (3 to 11 GeV)
Injector
12 GeV CEBAF
SRF linac
Large booster (warm)(up to 20 GeV/c)
Ion collider ring (cold)(up to 100 GeV/c)
MEIC at JLAB
• 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
ARC FODO CELL Dispersion Suppressor
Short straight Arc end with dispersion suppression
MEIC Ion Collider Ring
Interaction Region: (1) Final Focusing Block (FFB)(2) Chromaticity Compensation Block (CCB)(3) Beam Extension Section (BES)(4) IP
(1) (2) (3)(1)(2)(3) (4)
Beam parameters BES with large x,y,nat BES with small x,y,nat
x,nat y,nat x,nat y,nat
FFB (4) -18.41838018 -41.16790492 -18.41838018 -41.16790492
CCB (4) -27.13163424 -8.290146313 -27.13163424 -8.290146313
BES (4) -28.44876855 -44.29066406 -17.99838050 -12.05564551
LST (2) -3.124883582 -2.960790993 -3.124883582 -2.960790993
ARC (2) -8.720764078 -8.034955307 -8.720764078 -8.034955307
Natral Chromaticity x,y,nat -319.6864272 -396.9863538 -277.884875 -268.046279
Tune Spread (δrms=3e-4) 0.096 0.119 0.083 0.081
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.
Beam Extension Section
Large x,y,nat at BES Small x,y,nat at BES
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/∂δ can 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.
Close Sextupoles Separated Sextupoles
IPπ/2
π~ π ~ π
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
2,
2,, yxyxyx baW
yxyxyxa ,,2, /)/(
)/(*)//( ,,,,2, yxyxyxyxyxb
/2/)/( 22'xx DxD
after sext. correction
Beam parameters Large x,y,nat at BES Small x,y,nat at BES
LC LS SC SS
Tunes (H/V) 25.28/21.31 25.28/21.31 23.273/21.285 23.273/21.285
x,y,max (m) 1864 / 2450 1864 / 2450 2225 / 2450 2225 / 2450
Natural chromaticities x,y,nat -320 / -397 -320 / -397 -278/-268 -278/-268
1st order chromaticity 1x,y 0/0 0/0 0/0 0/0
2nd order chromaticity 2x,y 2603/7761 1122/6658 1342/3303 331/2853
dνx/dJx (*10e6) -2.3 -2.7 -1.7 -1.9
dνx/dJy (*10e6) -0.025 -0.5 0.12 -0.12
dνy/dJx (*10e6) 2.0 1.1 1.3 0.77
h21000 40.8 44.9 34.8 37.8
h30000 24.9 27.6 24.3 26.4
h10110 46.2 51.2 30.8 31.1
h10020 17.8 3.8 18.1 10.4
h10200 24.7 21.4 30.2 25.4
1st Sextupole Strength (1/m3) 1.37 1.83 1.19 1.53
2nd Sextupole Strength (1/m3) 1.96 2.19 1.44 1.61
Summary of Correction Studies
Tune vs. Momentum (σδ)
LC(νx,νy)=(25.28,21.31)
LS
SC SS(νx,νy)=(23.273/21.285)
LC: Large chromaticity at BES and Close sextupole in the middle of CCBLS: Large chromaticity at BES and Separated sextupole in the middle of CCB
SC: Small chromaticity at BES and Close sextupole in the middle of CCBSS: Small chromaticity at BES and Separated sextupole in the middle of CCB
0.02 0.02
0.01 0.01
Frequency Map
LC LS
SC SS
(νx,νy)=(25.28,21.31)
(νx,νy)=(23.273/21.285)
Dynamic Aperture
LC LS
SCSS
(νx,νy)=(25.28,21.31)
(νx,νy)=(23.273/21.285)
Tune Shift With Amplitude
LC LS
SC SS
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
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.
Back Up
Beam Extension Section
Large x,y,nat at BES Small x,y,nat at BES
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.
Close Sextupoles (CS) Separated Sextupoles (SS)
Tune vs. Momentum (δ)
LC LS
SC SS
(νx,νy)=(25.28,21.31)
(νx,νy)=(23.273/21.285)
ARC FODO CELLDispersion
Suppressor
Short straight
Arc end with dispersion suppression
Circumference m 1340.92
Total bend angle/arc deg 240
Figure-8 crossing angle deg 60
Averaged arc radius m 93.34
Arc length m 391
Long and short straight m 279.5 / 20
Lattice base cell FODO
Cells in arc / straight 52 / 20
Arc/Straight cell length m 9 / 9.3
Phase advance per cell m 60 / 60
Betatron tunes (x, y) 25.501 /25.527
Momentum compaction 10-3 5.12
Transition gamma 13.97
Dispersion suppression Adjusting quad strength
Dipole 144Length m 3
Bending radius M 53.1Bending Angle deg 3.236
Field @ 60 GeV T 3.768Quad 298
Length M 0.5Strength @ 60 GeV T/m 92 / 89
A. Bogacz & V. Morozov
MEIC Ion Collider Ring
βx* = 10 cm
βy* = 2 cm
βymax ~ 2700 m
Final Focusing Block (FFB)
Chromaticity Compensation Block (CCB)
Beam Extension Section (BES)
Whole Interaction Region: 158 m• 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
Interaction Region: Ions
Dynamic Aperture and Frequency Map