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  The inject energy is 6GeV.  If all the dipoles have the same sign, 33Gs@6GeV 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. Introduction of Wiggling Bend Scheme

3. Linear Optics

4. Linac Parameters Linac parameters ParameterSymbolUnitValue E - beam energyE e- GeV6 E + beam energyE e+ GeV6 Repetition ratef rep Hz50 E - bunch populationN e- 2×10 10 E + bunch populationN e+ 2×10 10 Energy spread (E + /E - )σEσE <1×10 -3 Emitance (E - ) 0.3 mm  mrad Emitance (E + ) 0.3 mm  mrad  From : Li Xiaoping, Pei Guoxi, etc, "Conceptual Design of CEPC Linac and Source".

5. Booster Parameters Contrast With the Alternating Magnetic Field Scheme.  Main difference in parameters caused by wiggling bend scheme. Old@6GeVNew@6GeV Parameter U0 [MeV/turn]0.0190.448 Damping times(x/y) [s] 115.614.86 Emittances(x) [pi nm]0.0150.098 Strength of dipole [Gs]33-138.04/+230.08 Beam offset in dipole[cm]00.56 Length of dipole [m]19.6*11.75*8 Length of FODO [m]47.2

6. Booster Parameters Parameter List for Alternating Magnetic Field Scheme. ParameterUnitValue Beam energy [E]GeV6 Circumference [C]km54.3744 Revolutionfrequency[f 0 ]kHz5.5135 SR power / beam [P]MW6.41E-04 Beam off-set in bendcm0.56 Momentum compaction factor[α]3.28E-5 Strength of dipole [Gs]33-138.04/+230.08 n B /beam50 Lorentz factor [g]11742.9 Magnetic rigidity [Br]T·m20.01 Beam current / beam [I]mA0.92 Bunchpopulation[N e ]2.08E+10 Bunch charge [Q b ]nC3.34 emittance-horizontal[e x ] inequilibrium m·rad0.98E-10 injected from linacm·rad3.00E-07 emittance-vertical[e y ] inequilibriumm·rad0.98E-12 injected from linacm·rad3.00E-07 ParameterUnitValue RF voltage [Vrf]GV0.213867 RF frequency [frf]GHz1.3 Harmonic number [h]235800 Synchrotronoscillationtune[n s ]0.210 Energy acceptance RF [h]%5.4 SR loss / turn [U0]GeV4.48E-04 Energyspread[s d ] inequilibrium%0.0159 injected from linac%0.1 Bunch length[s d ] inequilibriummm0.057 injected from linacmm~1.5 Transversedampingtime[t x ]ms4857.90 turns26784 Longitudinaldampingtime[t e ]ms2428.27 turns13388

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

8. Booster Parameters Ramping scheme U0 v.s. timePhase 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 wiggler scheme v.s. time Change rate of phase due to ramping v.s. time

11. Nonlinear Optimization and Sextupole Scheme Outline of booster sextupole scheme  So far, our best result comes from 70 meter FODO with 60° phase advance and interleaved sextupole scheme.  Advantage: interleaved scheme can cancel chromaticities locally and it is good for off- momentum particles.  Disadvantage: Geometry terms is produced and it makes the performance of on- momentum particles worse.  Because the energy spread of linac is 1 ‰, so non-interleaved sextupole scheme is considered.  Advantage: Geometry terms is cancelled very well and it have good performance for on-momentum particles.  Disadvantage: Chromaticities can't be cancelled locally and higher order Chromaticities is produced, so it is dangerous for off-momentum particles.

12. Nonlinear Optimization and Sextupole Scheme Outline of booster sextupole scheme  From reference[6], we can't tell whether 1 ‰ can be consider very small or not. So non-interleaved sextupole lattice is designed.

13. Nonlinear Optimization and Sextupole Scheme Outline of booster sextupole scheme  From the plot below, we can see the DA of on- momentum is very good. But for the 0.8 ‰ off- momentum, even tune can't be found under periodic boundary condition.  Also we spand a lot of time to optimize sextupole strength to obtain good performance for off- momentum, but no better solution.

14. Nonlinear Optimization and Sextupole Scheme Outline of booster sextupole scheme  We choose 70 meters FODO with 60° phase advance and interleaved sextupoles.  Only two families sextupoles and we obtain the DA_x: 6.6sigma and DA_y: 6.3sigma. DP=0 DP=5 ‰ DP=-5 ‰

15. Nonlinear Optimization and Sextupole Scheme Driving terms expressed in analytical formulae for booster  The optimization of driving terms is an effective way for DA optimization.  In order to optimize quickly, We need formulae for driving terms.  LieMath help us 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(C 22000,C 11110,C 00220,...), chromaticities terms(C 11001,C 00111,C 11002,C 00112,...)

16. Nonlinear Optimization and Sextupole Scheme Driving terms expressed in analytical formulae for booster  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:

17. Nonlinear Optimization and Sextupole Scheme Driving terms expressed in analytical formulae for booster  What we want is to express the driving terms as a function of sextupoles strength.  For exampole: C 00112 =C 00112 (Ks1, Ks2, Ks3....) C 11002 =C 11002 (Ks1, Ks2, Ks3....)  Then we can optimize them using Ga algorithm quickly.  Lie factor :F: decide the DA aperture directly. So if we make F as a function of sextupoles strength, then our optimization will be efficient and directly.  But this process need a lot of computer resource. I make series of approximate in my caculation.

18. Nonlinear Optimization and Sextupole Scheme Driving terms expressed in analytical formulae for booster

19. 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 apture@injection is 11sigma in horizontal and 42sigma in vertical without any sextupole optimization[1].  Without sextupole optimization, What we have is: 6.6sigma in horizontal and 6.3sigma 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.

20. 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 C 4 0022,0, C 4 2200,0, C 4 1111,0 as our goal function, and genetic algorithm is used in the optimization process.

21. Nonlinear Optimization and Sextupole Scheme 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 104 cells in the whole ring. Every cell use the same sextupoles. So there are six sextupole famlies in total.  The most important task for sextupole is to correct the linear chromaticity and this is the constraint condition.

22. Nonlinear Optimization and Sextupole Scheme Optimization results  GA optimization toolbox is used. Three goal functions constitute a matrix M. It's eigenvalue is the fitness function.  For on-Momentum particles, DA is 1.5 times bigger.  But for off-Momentum particles, even for 2 ‰, the lattice is unstable.  Maybe we can't hope a good result only optimize geometry terms, achromaticity terms must be considered.

23. 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

24. Working History Booster 47 meter FODO Wotking point optimization interleaved non-interleaved "Fake" harmonic sextupoles Lattice for harmonic sextupoles Whole ring optimization partial ring optimization Higher order chrom optimization geometry terms optimization GA Toolbox ga gamultiobj fminimax fmincon kinds of sextupole families 70 meter FODO one turn map 3th order geometry terms 2th order chromaticity terms 2th order tune shift DBA lattice for test kinds of goal function

25. Conclusion The low field problem is solved by the wiggling bend scheme. Strength of dipole increase from 33Gs to -138.04/+230.08 Gs. Shorter damping times are obtained, which is 4.8 seconds. A ramping method of is alternating magnetic field booster proposed. At present, interleaved sextupole scheme is most suitable for booster. DA of interleaved sextupole scheme is still a problem. Optimization of one turn map seems effective, but more geometry and achromaticity terms are waiting to be added. The "second order tune shift" idea is proposed and waiting to try and carefully thought.

26. 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, 2009. [2] Bengtsson J. The Sextupole Scheme for the Swiss Light Source SLS! An Analytic Approach[J]. 1997. [3] Nadolski L S. Methods and Tools to Simulate and Analyze Non-linear Dynamics in Electron Storage Rings[J].2011. [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, 2011. [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]. 1995.

27. Thanks for your attention!