1. Design Study of CEPC Booster and Mainring Lattice
Tianjian Bian1, Jie Gao1, Yunhai Cai2, Michael Koratzinos3, Chuang Zhang1, Xiaohao Cui1, Yiwei Wang1, Sha bai1, Dou Wang1, Feng Su1, Ming Xiao1
1Institute of High Energy Physics, Beijing, China,
2SLAC National Accelerator Laboratory, CA, USA,
3University of Geneva, Geneva, Switzerland
Mail to:

2. Outline Design Goal of CEPC Booster. Possible Options:
Normal Bend Scheme.
Wiggling Bend Scheme.
CEPC Mainring Design
Optimization Code Preparation
Moola：Modular&parallel Optics Optimization for LAttice
Summary

3. Design Goal Consideration of injection Injected Beam p/2
Inject in X direction.
Closed orbit kickers
Injected Beam
Circulating Beam
p/2

4. Design Goal Consideration of injection
Emit in mainring is 2E-9 m*rad, asuming beta_x=590 meter in the injection point.
Asuming DA_x=20 in the mainring.
Asuming beta_x=590 meter in the injection point.
The total space for injection:
8 sigma is retained for revolution beam to get enough quantum life time:
6 sigma is retained for injection beam to loss less particles:
3.2mm is lefted for septum and emit of is 3.5E-9 m*rad.

5. Design Goal Design Goal Linac parameters
For injection, emit of should be about 3.5E-9 m*rad.
1 percent energy acceptance for enough quantum lifetime.
DA_x and DA_y should bigger than 5~6 sigma for injection.
Linac parameters
From : Li Xiaoping, Pei Guoxi, etc, "Conceptual Design of CEPC Linac and Source".
Parameter
Symbol
Unit
Value
E- beam energy
Ee-
GeV 6
E+ beam energy
Ee+
Repetition rate
frep Hz 50
E- bunch population
Ne-
2×1010
E+ bunch population
Ne+
Energy spread (E+/E-) σE
<1×10-3
Emitance (E-)
0.3 mm mrad
Emitance (E+)

6. Normal Bend Scheme Introduction of Normal Bend Scheme
The inject energy is 6GeV.
The earth field is about 0.5Gs, so in the normal bend scheme start at may be difficult.
Shielding and correcting are needed.
With earth field, booster is a broken ring. So the first turn orbit correction is important. After the first turn orbit correction, the circular beam is existed and the closed orbit correction can be done.

7. Linear Optics
90 degree FODO
FODO length:70 meter

8. Chromaticity Optimization
Sextupole scheme
Non-interleaved sextupoles are used.
Another pair of sextupole with the same strength apart by 90°phase advance for cancel the second order chromaticity automatically.
8 sextupole families, optimize using both symbolic differential algebra and numeric way.
Only optimizing the 8 sextupole families is not enough.
Appropriate phase advance between arcs is necessary.
SF SF1 SF SF2 SF SF3 SF SF4
SD SD1 SD SD2 SD SD3 SD SD4

9. Chromaticity Optimization
Optimization result
before optimization
after optimization

10. Earth field Orbit Correction
First turn orbit correction
As we have said, the first turn orbit correction is important. It is similar to the closed orbit correction. This code is finished using Matlab.

11. Earth field Orbit Correction
Closed orbit correction
After the first turn orbit correction, the closed orbit is existed.
Closed orbit after first turn orbit correction
Closed orbit after first turn orbit correction and closed orbit correction

12. CEPC Booster Error Estimate
From Bai Sha

13. DA result Tune:190.61/190.88 and cavity on
With error and orbit correction, dp=0.0
With error and orbit correction, dp=0.01

14. Booster Parameters Parameter List for Normal Bend Scheme. Parameter
Unit
Value
Beam energy [E]
GeV 6
Circumference [C] km
63.84
Revolutionfrequency[f0]
kHz
4.69
SR power / beam [P] MW
2.16
Beam off-set in bend cm
Momentum compaction factor[α]
1.91E-05
Strength of dipole Gs
25.8
nB/beam 50
Lorentz factor [g]
Magnetic rigidity [Br]
T·m 20
Beam current / beam [I] mA
0.92
Bunchpopulation[Ne]
2.44E10
Bunch charge [Qb] nC
emittance-horizontal[ex] inequilibrium
m·rad
0.91E-11
injected from linac
3E-7
emittance-vertical[ey] inequilibrium
0.046E-11
Parameter
Unit
Value
RF voltage [Vrf] GV
0.2138
RF frequency [frf]
GHz
1.3
Harmonic number [h]
276831
Synchrotronoscillationtune[ns]
0.21
Energy acceptance RF %
4.995
SR loss / turn [U0]
GeV
1.47E-5
Energyspread[sd] inequilibrium
7.47E-05
injected from linac
0.1
Bunch length[sd] inequilibrium mm
5.85E-05
~1.5
Transversedampingtime[tx] s
174
turns
Longitudinaldampingtime[te]

15. Booster Parameters Parameter List for Normal Bend Scheme. Parameter
Unit
Value
Beam energy [E]
GeV
120
Circumference [C] km
63.84
Revolutionfrequency[f0]
kHz
4.69
SR power / beam [P] MW
2.16
Beam off-set in bend cm
Momentum compaction factor[α]
2.54E-5
Strength of dipole Gs
516.71
nB/beam 50
Lorentz factor [g]
Magnetic rigidity [Br]
T·m
400
Beam current / beam [I] mA
0.92
Bunchpopulation[Ne]
2.44E10
Bunch charge [Qb] nC
emittance-horizontal[ex] inequilibrium
m·rad
3.61E-9
injected from linac
3E-7
emittance-vertical[ey] inequilibrium
0.1083E-9
Parameter
Unit
Value
RF voltage [Vrf] GV 6
RF frequency [frf]
GHz
1.3
Harmonic number [h]
276831
Synchrotronoscillationtune[ns]
0.21
Energy acceptance RF %
4.57
SR loss / turn [U0]
GeV
2.34
Energyspread[sd] inequilibrium
0.12
injected from linac
0.1
Bunch length[sd] inequilibrium mm
1.36
~1.5
Transversedampingtime[tx] ms
21.76
Longitudinaldampingtime[te]

16. Conclusion
With error, orbit correction, cavities on and tune 0.61/0.88,
With error, orbit correction, cavities on and tune 0.61/0.88,
Contrast with the design goal we have proposed in previous section, this design is reasonable and meet requirements.

17. Wiggling Bend Scheme Introduction of Wiggling Bend Scheme
The inject energy is 6GeV.
If all the dipoles have the same sign, may cause problem.
In wiggling bend scheme, adjoining dipoles have different sign to avoid the low field problem.
The picture below shows the FODO structure.
The wiggler scheme using the same sextupole scheme and magnet error and the wiggler scheme has little or no effect on dynamics.

18. Linear Optics
90 degree FODO
FODO length:70 meter

19. DA result Tune:190.61/190.88 and cavity on with error, dp=0

20. Booster Parameters
Parameter List for Alternating Magnetic Field Scheme.
Parameter
Unit
Value
Beam energy [E]
GeV 6
Circumference [C] km
63.84
Revolutionfrequency[f0]
kHz
4.69
SR power / beam [P] MW
2.16
Beam off-set in bend cm
1.2
Momentum compaction factor[α]
2.33E-5
Strength of dipole Gs /
nB/beam 50
Lorentz factor [g]
Magnetic rigidity [Br]
T·m 20
Beam current / beam [I] mA
0.92
Bunchpopulation[Ne]
2.44E10
Bunch charge [Qb] nC
emittance-horizontal[ex] inequilibrium
m·rad
6.38E-11
injected from linac
3E-7
emittance-vertical[ey] inequilibrium
0.191E-11
Parameter
Unit
Value
RF voltage [Vrf] GV
0.2138
RF frequency [frf]
GHz
1.3
Harmonic number [h]
276831
Synchrotronoscillationtune[ns]
0.21
Energy acceptance RF %
5.93
SR loss / turn [U0]
GeV
5.42E-4
Energyspread[sd] inequilibrium
0.0147
injected from linac
0.1
Bunch length[sd] inequilibrium mm
0.18
~1.5
Transversedampingtime[tx] ms
4.71
turns
Longitudinaldampingtime[te]

21. Booster Parameters
Parameter List for Alternating Magnetic Field Scheme.
Parameter
Unit
Value
Beam energy [E]
GeV
120
Circumference [C] km
63.84
Revolutionfrequency[f0]
kHz
4.69
SR power / beam [P] MW
2.16
Beam off-set in bend cm
Momentum compaction factor[α]
2.54E-5
Strength of dipole Gs
516.71
nB/beam 50
Lorentz factor [g]
Magnetic rigidity [Br]
T·m
400
Beam current / beam [I] mA
0.92
Bunchpopulation[Ne]
2.44E10
Bunch charge [Qb] nC
emittance-horizontal[ex] inequilibrium
m·rad
3.61E-9
injected from linac
3E-7
emittance-vertical[ey] inequilibrium
0.1083E-9
Parameter
Unit
Value
RF voltage [Vrf] GV 6
RF frequency [frf]
GHz
1.3
Harmonic number [h]
276831
Synchrotronoscillationtune[ns]
0.21
Energy acceptance RF %
4.57
SR loss / turn [U0]
GeV
2.34
Energyspread[sd] inequilibrium
0.12
injected from linac
0.1
Bunch length[sd] inequilibrium mm
1.36
~1.5
Transversedampingtime[tx] ms
21.76
Longitudinaldampingtime[te]

22. Conclusion
In the wiggling bend scheme, strength of dipole increase from 30Gs to / Gs.
Shorter damping times are obtained, which is 4.7 seconds.
A ramping method of is alternating magnetic field booster proposed.
Serveal tunes has been test. Some tunes are sensitive to magnet errors, some are not. 0.61/0.88 seems good.
With error, cavities on and tune 0.61/0.88,
With error, cavities on and tune 0.61/0.88,

23. CEPC Mainring Design Pre-CDR H-high lumi. H-low power W Z From:Dou
Pre-CDR
H-high lumi.
H-low power W Z
Number of IPs 2
Energy (GeV)
120 80
45.5
Circumference (km) 54
SR loss/turn (GeV)
3.1
2.96
0.59
0.062
Half crossing angle (mrad) 15
Piwinski angle
2.5
2.6 5
7.6
Ne/bunch (1011)
3.79
2.85
2.67
0.74
0.46
Bunch number 50 67 44
400
1100
Beam current (mA)
16.6
16.9
10.5
26.2
45.4
SR power /beam (MW)
51.7
31.2
15.6
2.8
Bending radius (km)
6.1
6.2
Momentum compaction (10-5)
3.4
2.2
2.4
3.5
IP x/y (m)
0.8/0.0012
0.25/
0.268 /
0.1/0.001
Emittance x/y (nm)
6.12/0.018
2.45/0.0074
2.06 /0.0062
1.02/0.003
0.62/0.0028
Transverse IP (um)
69.97/0.15
24.8/0.1
23.5/0.088
10.1/0.056
7.9/0.053
x/IP
0.118
0.03
0.032
0.008
0.006
y/IP
0.083
0.11
0.074
0.073
VRF (GV)
6.87
3.62
3.53
0.81
0.12
f RF (MHz)
650
Nature z (mm)
2.14
3.0
3.25
3.9
Total z (mm)
2.65
4.1
4.0
3.35
HOM power/cavity (kw)
3.6
1.3
0.99
Energy spread (%)
0.13
0.09
0.05
Energy acceptance (%)
Energy acceptance by RF (%) 6
2.1
1.7
1.1 n
0.23
0.47
0.3
0.24
Life time due to beamstrahlung_cal (minute) 47 36 32
F (hour glass)
0.68
0.82
0.92
0.95
Lmax/IP (1034cm-2s-1)
2.04
2.01
3.09
From:Dou

24. CEPC Mainring Design CEPC Mainring Design
The chromaticity formulae are derived for X m, bend angle alpha=2*pi/1788 and thin lens condition

25. CEPC Mainring Design
CEPC Mainring Design
The inject energy is 6GeV.

26. CEPC Mainring Design
CEPC Mainring Design
The inject energy is 6GeV.

27. CEPC Mainring Design 【 For CEPC Mainring, large energy
acceptance is needed. Even more than
2%. So chromaticity optimization is
important.
Finally, we find a sextupole module with
20 sextupole families that can zero out the
chromaticity order by
order. 【

28. CEPC Mainring Design
In the booster, dynamic aperture drop about 28.3% for 1% off-momentum particles.
But dynamic aperture for mainring drop about 66.7% for 1% off-momentum particles.
Why?
X=72.8sigma
Y=1237.2sigma
@0%
X=48.6sigma
Y=824.8sigma
@0.5%
X=24.3sigma
Y=421.4sigma
@1%

29. CEPC Mainring Design
Booster
V.S.
Mainring

30. CEPC Mainring Design CEPC Mainring Design From dynamic apture：
From nonlinear parameter：

31. CEPC Mainring Design CEPC Mainring Design
In this case, we zero out the 2nd and 3rd order chromaticity, but the 4th order is so big that limit the energy acceptance because we didn't control the 4th chromaticity in the optimization process.
So, we need to optimize the 2nd, 3rd and 4th order together to get better result.
For further study, more and more nonlinear parameters will be included.
Maybe the multi-objective optimization algorithm is a good choice.
That is also one of the motivations to develope new code.

32. Moola：Modular¶llel Optics Optimization for LAttice
Introduction of Moola
In the lattice design process, especially in the challenging project like CEPC, we want to control every thing(such as the twiss, high order nonlinear parameter and so on), and using them for optimization.
Lattice design code like Mad and Sad can complete some thing very well, but they don't let us do what ever we want, beacuse we can't handle the code.
Based on this requirement, we need own code.
Moola is a c++ library, which is modular&parallel.
Moola is linked to Zlib.

33. Moola：Modular¶llel Optics Optimization for LAttice
Element Simulator
High Order Fields Error Simulator
Fringing Fields Simulator*
Alignment
Simulator*
Beamline
Modify
Beamline
LinkList
Radiation Parameters*
Orbit correction*
Twiss*
Tracker
Dynamic
Aperture
Frequency Map Analysis
Arbitrary Order One Turn Map（OTM）
Optimization Algorithm
Zlib
Arbitrary Order Nonlinear Parameters
Optimization*
*in the developing process

34. Moola：Modular¶llel Optics Optimization for LAttice
Main physics base
Hamilton and canonical coordinates
Fourth-order symplectic integrator

35. Moola：Modular¶llel Optics Optimization for LAttice
CEPC mainring results from Moola

36. Moola：Modular¶llel Optics Optimization for LAttice
CEPC mainring results from Moola
2500 particles, 256 turns in 109 seconds

37. Moola：Modular¶llel Optics Optimization for LAttice
CEPC mainring results from Moola
2500 particles, 512 turns in 219 seconds

38. Moola：Modular¶llel Optics Optimization for LAttice
CEPC mainring results from Moola
Arbitrary order chromaticity
Detuning terms
Arbitrary order one-turn-map
Linear optics

39. Moola：Modular¶llel Optics Optimization for LAttice
Introduction of Moola
Moola is still preliminary, many functions is waiting to be added.
With all the nonlinear parameters in my computer memory, we can call them in any optimization algorithm(like all kinds of evolution algorithm).
In the booster further design and the mainring design, Moola will play a more important role.

40. Summary
In this report, we proposed a design of both normal scheme and wiggler scheme.
Normal scheme:
With error, orbit correction, cavities on and tune 0.61/0.88,
With error, orbit correction, cavities on and tune 0.61/0.88,
Wiggler scheme:
With error, cavities on and tune 0.61/0.88,
With error, cavities on and tune 0.61/0.88,
Contrast with the design goal we have proposed in previous section, both of the two design are reasonable and meet requirements.
For the mainring design, we have a good idea for chromaticity correction, further study is needed.
Moola, an optimization code is in developing process for further study.

41. Thanks！

42. Summary
Optimization of lattice: reference lattice design, Lcell=47.2m vs. Lcell=70.8 m. (ZC, CXH)
Linear optics
Chromaticity correction and dynamic aperture
Machine errors and correction
Sawtooth effect and their correction;
Low field at injection and its tolerance: test, simulation and mitigation; (BTJ)
Consideration of a pre-booster
Instability: further simulation study.(BTJ)
Injection: physical design. (ZC)
Ejection: work together with collider injection. (CXH)
Design of transfer line from linac to booster. (ZC)
Design of transfer line from booster to collider. (CXH)
Provide a full parameter list of reference design including field, aperture, RF, kicker pulse length, vacuum, diagnostics, and tolerances.

43. CEPC Mainring Design Achromatic module
Sextupole pair for the 3rd order, but it can't be used in the mainring. So 2nd and 3rd order chromaticity can only cancelled by appropriate sextupole families.
Octupole pair for the 4rd order

44. Plan There are 6 months left, our plan is: Design goal of CDR
Optimization code month
Lattice redesign, with injection and ejection month
The process of ramping month
Design goal of CDR
For injection, emit of should be about 3.5E-9 m*rad.
1 percent energy acceptance for enough quantum lifetime.
DA_x and DA_y should bigger than 7~8 sigma for injection for both on-momentum and off-momentum.