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 ,,,, 41

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 -41.16790492CCB (4) -27.13163424 -8.290146313 -27.13163424 -8.290146313BES (4) -28.44876855 -44.29066406 -17.99838050 -12.05564551LST (2) -3.124883582 -2.960790993 -3.124883582 -2.960790993ARC (2) -8.720764078 -8.034955307 -8.720764078 -8.034955307Natral 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 SSTunes (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 1.532nd 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 AmplitudeLC 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