MEIC Detector and Interaction Region Designing Mini-Workshop, Oct. 31 , 2011

• 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 x , y
 d (
  x , y
) / d

– “Natural chromaticity” arises solely from quadrupoles and depends on the lattice design, given as
C x , y , nat


1
4

  x , y
K x , y ds
– 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 Δσ x,y
/σ x,y
~ C x,y
σ
δ.
• 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:
C
1 x , y


4

1   x , y
[ K x , y
( s )  S ( s ) D ( s )] ds

• Nonlinear Effects of Chromatic Sextupoles and Correction
– Second order chromaticity driven by the first order chromatic beta wave ∂β x,y
∂D x
/∂δ (dependent on the first orde sextupole strength).
/∂δ and dispersion wave
– First order geometric resonances ν x
, ν
3x
, ν x
±2ν y
(dependent on the first order sextupole strength).
– Tune shift with amplitude ∂ν x,y
/ ∂ J x,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.

Large booster (warm)
(up to 20 GeV/c)
Three Figure-8 rings stacked vertically
Large booster to collider ring transfer beamline
Pre-booster
(up to 3 GeV)
Injector
Medium energy IP with horizontal crab crossing
Ion source
SRF linac
Electron ring
(3 to 11 GeV)
Ion collider ring (cold)
(up to 100 GeV/c)
12 GeV CEBAF
• Energy: e 3 to 11 GeV, p 20 to 100 GeV, ion 12 to 40 GeV/u
• Luminosity: 10 35 cm -2 s -1 (e-nucleon) per interaction point
• Detectors: One full-acceptanc e detector (primary) + One high luminosity detector (secondary) with 7 m and 4.5 m between IP & 1 st final focusing quad, respectively
• Polarization: Figure-8 shape is adoped for preservation of polarization >70% desirable

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

Beam parameters
FFB (4)
CCB (4)
BES (4)
LST (2)
ARC (2)
Natral Chromaticity  x,y,nat
Tune Spread (δ rms
=3e-4)
BES with large  x,y,nat
 x,nat
 y,nat
BES with small 
 x,nat x,y,nat
 y,nat
-18.41838018
-41.16790492
-18.41838018
-41.16790492
-27.13163424
-8.290146313
-27.13163424
-8.290146313
-28.44876855
-44.29066406
-17.99838050
-12.05564551
-3.124883582
-2.960790993
-3.124883582
-2.960790993
-8.720764078
-8.034955307
-8.720764078
-8.034955307
-319.6864272
-396.9863538
0.096
0.119
-277.884875
-268.046279
0.083
0.081
• 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.

Large  x,y,nat at BES Small  x,y,nat at BES

• First order chromaticity can be easily corrected by using two sextupoles (families).
• Beta wave ∂β x,y
/∂δ can be exactly out of phase if two sources are π/2 apart in phase.
• Dispersion wave ∂D x
/∂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.
π
~ π
π/2
IP
~ π
Close Sextupoles Separated Sextupoles

• W x,y is chromatic amplitude function, given as
W x , y
 2 a x , y
 2 b x , y a
2 x , y

(
   x , y
/

) /
 x , y
2 b x , y

(
  x , y
/
     x , y
/
  x , y
) * (
  x , y
/
 
)
• D x
’ is chromatic derivative of dispersion D x
, given as
D x
' 
(
 2 x /
 
2
) / 2
 
D x
/
  before sext. correction after sext. correction

Beam parameters
Tunes (H/V)
 x,y,max
(m)
Natural chromaticities  x,y,nat
1 st order chromaticity  1 x,y
2 nd order chromaticity  2 x,y dν x
/dJ x
(*10e6) dν x
/dJ y
(*10e6) dν y
/dJ h
21000 x
(*10e6) h
30000 h
10110 h
10020 h
10200
1 st Sextupole Strength (1/m 3 )
2 nd Sextupole Strength (1/m 3 )
Large  x,y,nat at BES
LC LS
25.28/21.31
Small 
SC x,y,nat at BES
SS
25.28/21.31
23.273/21.285
23.273/21.285
1864 / 2450
-320 / -397
0/0
2603/7761
-2.3
-0.025
2.0
40.8
24.9
46.2
17.8
24.7
1.37
1.96
1864 / 2450
-320 / -397
0/0
1122/6658
-2.7
-0.5
1.1
44.9
27.6
51.2
3.8
21.4
1.83
2.19
2225 / 2450
-278/-268
0/0
1342/3303
-1.7
0.12
1.3
34.8
24.3
30.8
18.1
30.2
1.19
1.44
2225 / 2450
-278/-268
0/0
331/2853
-1.9
-0.12
0.77
37.8
26.4
31.1
10.4
25.4
1.53
1.61

LC
LS
0.02
(ν x
,ν y
)=(25.28,21.31)
LC : Large chromaticity at BES and Close sextupole in the middle of CCB
LS : Large chromaticity at BES and Separated sextupole in the middle of CCB
SS SC
0.01
(ν x
,ν y
)=(23.273/21.285)
SC : Small chromaticity at BES and Close sextupole in the middle of CCB
SS : Small chromaticity at BES and Separated sextupole in the middle of CCB
0.02
0.01

LC
(ν x
,ν y
)=(25.28,21.31)
LS
SC
(ν x
,ν y
)=(23.273/21.285)
SS

LC
(ν x
,ν y
)=(25.28,21.31)
LS
(ν x
,ν y
)=(23.273/21.285)
SS
SC

LC LS
SC SS

• 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

• 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.

Large  x,y,nat at BES Small  x,y,nat at BES

• To minimize their strength, chromatic sextupoles are located near quads, where β x are maximum.
D x and β x
D y
• To control chromaticity independently, a large ration of β x defocusing sextupole respectively.
/β y and β y
/β y for focusing and
• To minimizenonlinear resonance stregth, families of sextupoles are properly arranged.
Close Sextupoles (CS)
Separated Sextupoles (SS)

LC
LS
(ν x
,ν y
)=(25.28,21.31)
SC
(ν x
,ν y
)=(23.273/21.285)
SS

Dipole
Length
Bending radius
Bending Angle
Field @ 60 GeV
Quad m
T
144
3
M 53.1
deg 3.236
3.768
298
Length M 0.5
Strength @ 60 GeV T/m 92 / 89
Circumference
Total bend angle/arc
Figure-8 crossing angle
Averaged arc radius
Arc length
Long and short straight
Lattice base cell
Cells in arc / straight
Arc/Straight cell length
Phase advance per cell
Betatron tunes (  x,
 y
)
Momentum compaction
Transition gamma
Dispersion suppression m deg deg m m m m m
10 -3
1340.92
240
60
93.34
391
279.5 / 20
FODO
52 / 20
9 / 9.3
60 / 60
25.501 /25.527
5.12
13.97
Adjusting quad strength
A. Bogacz & V. Morozov
ARC FODO CELL
Short straight
Dispersion
Suppressor
Arc end with dispersion suppression

7 m
Final Focusing Block
(FFB)
Chromaticity Compensation Block
(CCB)
Beam Extension Section
(BES)
β y max ~ 2700 m
β
β x
* y
*
= 10 cm
= 2 cm
• 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
Whole Interaction Region: 158 m