1. Design study of CEPC Alternating Magnetic Field Booster
Tianjian Bian Jie Gao Michael Koratzinos (CERN) Chuang Zhang Xiaohao Cui Sha bai Dou Wang Yiwei Wang Feng Su

2. 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.
Shorten the Damping times greatly.
The picture below shows the FODO structure.

3. Linear Optics

4. Linac Parameters 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+)

5. Booster Parameters
Contrast With the Alternating Magnetic Field Scheme.
Main difference in parameters caused by wiggling bend scheme.
Parameter
U0 [MeV/turn]
0.019
0.448
Damping times(x/y) [s]
 115.61
4.86
Emittances(x) [pi nm]
0.015
0.098
Strength of dipole [Gs] 33 /
Beam offset in dipole[cm]
0.56
Length of dipole [m]
19.6*1
1.75*8
Length of FODO [m]
47.2

6. Booster Parameters
Parameter List for Alternating Magnetic Field Scheme.
Parameter
Unit
Value
Beam energy [E]
GeV 6
Circumference [C] km
Revolutionfrequency[f0]
kHz
5.5135
SR power / beam [P] MW
6.41E-04
Beam off-set in bend cm
0.56
Momentum compaction factor[α]
3.28E-5
Strength of dipole [Gs] 33 /
nB/beam 50
Lorentz factor [g]
Magnetic rigidity [Br]
T·m
20.01
Beam current / beam [I] mA
0.92
Bunchpopulation[Ne]
2.08E+10
Bunch charge [Qb] nC
3.34
emittance-horizontal[ex] inequilibrium
m·rad
0.98E-10
injected from linac
3.00E-07
emittance-vertical[ey] inequilibrium
0.98E-12
Parameter
Unit
Value
RF voltage [Vrf] GV
RF frequency [frf]
GHz
1.3
Harmonic number [h]
235800
Synchrotronoscillationtune[ns]
0.210
Energy acceptance RF [h] %
5.4
SR loss / turn [U0]
GeV
4.48E-04
Energyspread[sd] inequilibrium
0.0159
injected from linac
0.1
Bunch length[sd] inequilibrium mm
0.057
~1.5
Transversedampingtime[tx] ms
turns
26784
Longitudinaldampingtime[te]
13388

7. Booster Parameters Ramping scheme Angle of dipole v.s. time
Magnetic field of dipole v.s. time

8. Booster Parameters Ramping scheme U0 v.s. time
Phase slipping factor v.s. time

9. Booster Parameters
Ramping scheme
Vrf v.s. time
Phase v.s. time

10. Booster Parameters Ramping scheme
Change rate of phase due to ramping v.s. time
Change rate of phase due to wiggler scheme v.s. time

11. Nonlinear Optimization and Sextupole Scheme
Challenges we face
We choose 60° FODO with interleaved sextupoles.
It is a big ring. Nonlinear optimization for big ring is much harder than small ring.
SSRF booster(only 180 meters) is also made up with interleaved FODO structure. SSRF booster's dynamic is 11sigma in horizontal and 42sigma in vertical without any sextupole optimization[1].
Without sextupole optimization, What we have is: 3.6sigma in horizontal and 1.7sigma in vertical.
There is so many sextupoles that the tune shift with amplitude is serious. Even the second order tune shift effect is remarkable.
In FODO structure, we can not place harmonic sextupoles easily as DBA structure do.
No released code for the sextupole optimization.

12. Nonlinear Optimization and Sextupole Scheme
Optimization algorithm
There are so many sextupoles in the booster. So, the tune shift effect is serious,
In the paper[2], tune shift with amplitude is derived. We can see that it is related to the sextupole strength, beta function, working point,etc.
We choose the three value as our goal function, and genetic algorithm is used in the optimization process.

13. Nonlinear Optimization and Sextupole Scheme
Step1:Divide sextupoles into diferent families.
See three FODOs as a cell, and there are six sextupoles in a cell as the picture below.
We have 320 cells in the whole ring. Every cell use the same sextupoles. So there are six sextupole famliies in total.
The most important task for sextupole is to correct the linear chromaticity and this is the constraint condition.
In this step,minimize the goal function is not important, the most important task is choosing the direction of tune shift.

14. Nonlinear Optimization and Sextupole Scheme
Step 2:Fake harmonic sextupole
It is difficult to add harmonic sextupole in a cell, because there is no zero dispersion point.
We arrange SH1, SH2, SH3, SH4, SH5, SH6 as fake harmonic sextupole, use zero chromaticity contribution as their constraint condition. And this is why we call them fake harmonic sextupoles.
Smaller goal function values do not means bigger DA, plot DA in the process of optimization and choose the best result.

15. Nonlinear Optimization and Sextupole Scheme
Tune shift optimization results
Direction of tune shift is more important.
With 1920 sextupoles, second order tune shift play a more important role.

16. Nonlinear Optimization and Sextupole Scheme
Optimization of working point
Working point effects the DA greatly.
vx= , vy=

17. Nonlinear Optimization and Sextupole Scheme
First order tune shift optimization is suitable for CEPC booster?
Plot tune shift as a function of Jx.
CEPC booster FODO
HEPS DBA

18. Nonlinear Optimization and Sextupole Scheme
My understanding
Why second order tune shift play an important role?
First order tune shift in CEPC:
Second order tune shift in CEPC:
Second order tune shift in HEPS:

19. Nonlinear Optimization and Sextupole Scheme
60° FODO with interleaved sextupoles(DP=0)

20. Nonlinear Optimization and Sextupole Scheme
60° FODO with interleaved sextupoles(DP=0.001)

21. Nonlinear Optimization and Sextupole Scheme
60° FODO with interleaved sextupoles(DP=0.003)

22. Nonlinear Optimization and Sextupole Scheme
60° FODO with interleaved sextupoles(DP=0.004)

23. Nonlinear Optimization and Sextupole Scheme
60° FODO with non-interleaved sextupoles
The sextupole arrange: SF SF SD SD -I

24. Nonlinear Optimization and Sextupole Scheme
60° FODO with non-interleaved sextupoles(DP=0)

25. Nonlinear Optimization and Sextupole Scheme
60° FODO with non-interleaved sextupoles(DP=0.001)

26. Nonlinear Optimization and Sextupole Scheme
60° FODO with non-interleaved sextupoles(DP=0.002)

27. Nonlinear Optimization and Sextupole Scheme
60° FODO with non-interleaved sextupoles(DP=0.003)

28. Nonlinear Optimization and Sextupole Scheme
60° FODO with non-interleaved sextupoles(DP=0.004)

29. Nonlinear Optimization and Sextupole Scheme
60° FODO with non-interleaved sextupoles
The first order chromaticity is both 0.5 in X and Y plane.
The second order chromaticity will play a big role.
Plot tune shift as a function of energy dispersion.

30. Nonlinear Optimization and Sextupole Scheme
60° FODO with non-interleaved sextupoles
The optimization of high order chromaticity is important in non-interleaved case.
In order to optimize quickly,We need formulae for second order chromaticity.
LieMath help me a lot, one turn map can be derived analytically.
If one turn map is obtained, we will get all the driving terms, as detuning terms(C22000,C11110,C00220, ...), chromaticities terms(C11001,C00111,C11002,C00112, ...)

31. Nonlinear Optimization and Sextupole Scheme
60° FODO with non-interleaved sextupoles
Then we do an analytical caculation for a cell in booster. One cell contain 9 FODO.
We can get the tune as a function of energy dispersion and action quantity:
Contrast with MADX:

32. Nonlinear Optimization and Sextupole Scheme
60° FODO with non-interleaved sextupoles
What we want is to express the second order chromaticities as a function of sextupoles strength.
For exampole:
C00112=C00112(Ks1, Ks2, Ks3....)
C11002=C11002(Ks1, Ks2, Ks3....)
Then we can optimize them using Ga algorithm quickly.
This work is in process.

33. Nonlinear Optimization and Sextupole Scheme
60° FODO with non-interleaved sextupoles
Driving terms from LieMath

34. Nonlinear Optimization and Sextupole Scheme
71m 60° FODO with non-interleaved sextupoles
Linear optics

35. Nonlinear Optimization and Sextupole Scheme
71m 60° FODO with non-interleaved sextupoles
FMA results
DP=0.0‰ DP=0.5‰ DP=1.0‰

36. Conclusion
The low field problem is solved by the wiggling bend scheme. Strength of dipole increase from 33Gs to / Gs.
Shorter damping times are obtained, which is 4.8 seconds.
A ramping method of is alternating magnetic field booster proposed.
DA of interleaved sextupole scheme is still a problem. The "second order tune shift" idea is proposed and waiting to try and carefully thought.
DA of non-interleaved sextupole scheme is good when Dp=0. But for off-momentum particles, optimization is needed.
Booster using 71m FODO also need optimization for off-momentum particles.

37. Reference
[1] 张满洲. 上海光源增强器束流稳定性研究[D]. 中国科学院上海应用物理研究所, [2] Bengtsson J. The Sextupole Scheme for the Swiss Light Source SLS! An Analytic Approach[J] [3] Nadolski L S. Methods and Tools to Simulate and Analyze Non-linear Dynamics in Electron Storage Rings[J] [4] 彭月梅. 基于纵向变磁场二极铁的 BAPS 储存环 Lattice 设计[D].中国科学院高能物理研究所, [5] 焦毅. FMA 在环形加速器动力学分析中的应用[D]. 中国科学院高能物理研究所，2008. [6] Bryant P J. Planning Sextupole Families in a Circular Collider[J]

38. Thanks for your attention!