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
0.70
Damping times(x/y) [s]
 115.61
3.12
Emittances(x) [pi nm]
0.015
0.11
Strength of dipole [Gs] 33
-164.3/+229.9
Beam offset in dipole[cm]
2.3
Length of dipole [m]
19.6*1
4.9*4
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
2.30E+00
Momentum compaction factor[α]
2.70E-05
Bending radius [r] m
nB/beam 50
Lorentz factor [g]
Magnetic rigidity [Br]
T·m
20.01
Beam current / beam [I] mA
0.9197
Bunchpopulation[Ne]
2.08E+10
Bunch charge [Qb] nC
3.34
emittance-horizontal[ex] inequilibrium
m·rad
1.11E-10
injected from linac
3.00E-07
emittance-vertical[ey] inequilibrium
1.11E-12
Parameter
Unit
Value
RF voltage [Vrf] GV
RF frequency [frf]
GHz
1.3
Harmonic number [h]
235800
Synchrotronoscillationtune[ns]
Energy acceptance RF [h] %
SR loss / turn [U0]
GeV
6.97E-04
Energyspread[sd] inequilibrium
injected from linac
0.1
Bunch length[sd] inequilibrium mm
0.05
~1.5
Transversedampingtime[tx] ms
3124.6
turns
17228
Longitudinaldampingtime[te]
1.6 9

7. Booster Parameters
Parameter List for Alternating Magnetic Field Scheme.
Angle of dipole v.s. time
Field of dipole v.s. time

8. Booster Parameters
Parameter List for Alternating Magnetic Field Scheme.
U0 v.s. time
η v.s. time

9. Booster Parameters
Parameter List for Alternating Magnetic Field Scheme.
Vrf v.s. time
Phase v.s. time

10. Nonlinear Optimization and Sextupole Scheme
Challenges we face
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 FODO structure. SSRF booster's dynamic is 11sigma in horizontal and 42sigma in vertical without any sextupole optimization[].
Without sextupole optimization, What we have is: 3.6sigma in horizontal and 1.7sigma in vertical.
We have long straight section that we can correct the chromaticity locally.
In FODO structure, we can not place harmonic sextupoles easily as DBA structure do.
No released code for the sextupole optimization.

11. 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[], 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.

12. 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.

13. 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.

14. Nonlinear Optimization and Sextupole Scheme
Tune shift optimization results
The working opint is not good.
Direction of tune shift

15. Nonlinear Optimization and Sextupole Scheme
Change working point
Working point effects the DA greatly.

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

17. 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:

18. Conclusion
The low field problem is solved by the wiggling bend scheme. Strength of dipole increase from 33Gs to / Gs.
A ramping method of is alternating magnetic field booster proposed.
Shorter damping times are obtained, which is 3.12 seconds.
DA is still a problem. The "second order tune shift" idea is proposed and waiting to try and carefully thought.

19. Thanks for your attention!