1. Single particle dynamics in the ThomX ring Jianfeng zhang 12/02/2015 LAL

2. Outline  Improve the Dynamic Aperture (DA) and Momentum Acceptance (MA) by adding octupoles in the ThomX ring  Definition of DA & MA  Why we need large DA & MA  DA & MA of the ThomX ring with and without multipole field errors  Hamiltonian of the single particle dynamics in the accelerators  Resonance driving terms of sextupoles  Compensations of second order sextupole resonance driving terms by octupoles  Touschek beam lifetime in the ThomX ring  Beam lifetime in an electron ring  Touschek lifetime and MA  Vacuum chamber size and MA  Touschek lifetime and the install location of the septum in the ThomX ring  Conclusions and outlook  Acknowledgement 2

3. Basic parameters of single particle dynamics in accelerator rings  On momentum dynamics:  DA (Dynamic Aperture) : Region in the transverse plane (x,z) where the particle motion is stable for a given number of turns  Physical aperture : Aperture size of the vacuum chamber (ThomX: 40 mm in x & 28 mm in z )  Off momentum dynamics:  MA (Momentum Acceptance): Region in the longitudinal plane where the particle motion is stable for a given number of turns  RF MA, determined by RF cavity  Lattice MA, lattice optics dependent, longitudinal position dependent  ThomX ring: Dominated by DA & lattice MA 3 40 mm 28 mm

4. Why we need large DA & MA (1) ?  DA & MA : determine the beam injection efficiency and beam lifetime  ThomX ring injection:  Single-turn, on-axis injection  injection rate 50 Hz ===> high injection efficiency (100 %)  Good news: one turn on-axis injection, less stress to the DA  Bad news:  septum/kicker angle errors (field errors, time jitters), beam can't injected on the closed orbit ==> betatron oscillations ===> DA  Intra-beam scattering, other collective effects ==> increase energy spread quickly ===> MA  Small DA & MA ===> beam loss ==> radiation damage (mirrors in Fabry- Perot cavity, etc.) & increase background noise of the X-ray 4

5. Why we need large DA & MA (2) ?  Operation modes of the ThomX machine :  User mode (laser on): Provide user required X-ray 20 ms storage time (330 000 turns) Round beam (  x =  z = 50 nm.rad) Full current (16.7 mA)  Machine study mode (laser off): Commissionning & improve accelerator performances Flat beam (quantum excitation + damping + flat ring + couping) Full current or half current or even less current operation? ( Less beam current higher beam lifetime & less collective effects ) Beam lifetime ===> several minutes (response time of the high level physics applications of MML) ==> large DA & MA 5

6. Dynamic aperture (DA) without multipole field errors  6D particle tracking ( TRACY3, 4 th order symplectic integrator for long term tracking, including correct dipole fringe field model for small low energy ring* )  Vacuum chamber size (half): 20 mm (x), 14 mm (z)  Beam size @ injection (middle of septum): σ x (0.48 mm), σ z (0.35 mm)  Very good DA in the horizontal and vertical directions  Most particles will be safe!!! Ideal case! DA size: 30 σ x & 57 σ z 6 * J. Zhang, A. Loulergue, “dipole fringe field effects in the ThomX ring”, IPAC 2013.

7. Momentum acceptance (MA) without multipole field errors  6D particle tracking ( TRACY3 )  MA big enough for the user operation mode: energy spread 0.3% (initial) energy spread 0.6% (final) Most particles will be safe!!! Ideal case! MA size: -3% ~ 3% 7

8. The ThomX ring  Compact size (C = 18 m), low energy (50 MeV) ===> non linear dynamics dominated machine  If no non linear dynamics in the ring, DA & MA are infinite large!  Need sextupoles to correct chromaticities!  Sextupole is the #1 source to reduce DA & MA !  Sensitive to various perturbations  Systematic (magnet design) and random (magnet construction) multipole field errors  Systematic and random misalignment errors  Power supply ripples  Ground vibrations  Thermal expansions (Momentum compaction factor of the ThomX ring is large, ~10 -2 )  Etc. ===> impact on DA & MA 8

9. DA & MA with systematic multipole errors  Magnets in the ThomX ring : 12 dipoles, 24 quadrupoles, 12 sextupoles, 12 H/V correctors  Systematic multipole field errors Errors in dipoles, quadrupoles, sextupoles; but no errors in correctors Same errors in the same type magnets Real magnets : different error in each magnet  Random multipole field errors (NO)  Misalignment errors (NO)  Power supply ripples (NO)  Other perturbation sources: ground motion, thermal expansion…. (NO) 9

10. DA with multipole errors in the ThomX ring  Systematic multipole field errors (design)  Most dangerous multipoles: sextupole errors in dipoles. (1st order chrom. correction - MML). 1X multipole errors 10 DA size: 28 σx & 29 σz ↓ 2*σx ↓ 28*σz ↓ 4*σx ↓ 29*σz ↓ 5*σx ↓ 30*σ 2X multipole errors 3X multipole errors

11. MA with multipole errors in the ThomX ring  Systematic multipole field errors (design )  Most dangerous multipoles: sextupole errors in dipoles. (1st order chrom. correction - MML). 11 2X multipole errors MA size: -2.1% ~ 2.3% MA size: -2.0% ~ 2.2% MA size: -1.9% ~ 2.0% 3X multipole errors 1X multipole errors

12. How to improve DA & MA ?  Optics optimization ( rematch sextupoles )  Add high-order multipole correctors ( Octupoles, decapoles … ) Reduce number of resonance driving terms Reduce strengths resonance driving terms 12 Qx Qz Working point (3.175, 1.64) Tune map of the ThomX ring mQ x + nQ z = l m, n, l are arbritrary integers. Resonance condition:

13. Hamiltonian and resonance driving terms  H = (p x 2 +p z 2 )/2 + V(x,z,s) ( Hamiltonian ) o Quadrupole: V(x,z,s) = K(s)/2*(x 2 +z 2 ) o Sextupole : V(x,z,s) = S(s)/3*(x 3 -3xz 2 ) o Octupoles, decapoles, other high order multipoles …  Main non linear dynamics driving sources: sextupoles! o Greatly reduce DA & MA  Normal form + Lie Algebra + TPSA  Hamiltonian with sextupole reosnance driving terms h jklmp (first order) & h jklmp *h’ jklmp (second order) 13

14. Non linear driving terms of the sextupoles Driving termResonance h 21000 = h* 12000 Qx h 30000 = h* 03000 3Qx h 10110 = h* 01110 Qx h 10200 = h* 01020 Qx+2Qz h 10020 = h* 01200 Qx-2Qz h 20001 = h* 02001 2Qx h 00201 = h* 00021 2Qz Driving termResonance h 40000 =h* 04000 4Qx h 00400 =h* 00040 4Qz h 20200 =h* 02020 2Qx+2Qz h 20020 =h* 02200 2Qx-2Qz h 31000 =h* 13000 2Qx h 20110 =h* 02110 2Qx h 00310 =h* 00130 2Qz h 01110 =h* 10110 2Qz 14  First order chromaticities: h 11001 (H), h 00111 (V) First order effects of sextupoles Second order effects of sextupoles & first order effects of octupoles  Second order chromaticities  3 types of the tune shift with amplitude Second order sextupole resonance driving terms can be compensated by octupoles!

15. ThomX ring layout with 4 octupoles 15 RF cavity Def. octupole Focus octupole  Optimize 1 st order sextupole resonances (must also correct chromaticities!)  Compensate 2 nd order sextupole resonances using octupoles  ThomX ring lattice: DBA, 4 folder  Keep the symmetry of the ring (reduce the resonances)

16. Resonance driving terms optimization 16 Find the minimum total resonance strength! Resonance terms Reosonance strengths sextupole octupole decapole

17. DA & MA of the ThomX ring with 4 octupoles 17 DA size: 36 σx & 34 σz ↑ 8*σx ↑ 5*σz MA size: no obvious change

18. Longitudinal dynamics Touschek lifetime, MA, and install location of the septum 18

19. Beam lifetime in the ThomX ring  3 types of beam lifetimes in an electron ring:  Quantum lifetime  Vacuum lifetime  Touschek lifetime (low energy)  Touschek lifetime determined by 3 effects: IBS (intra beam scattering) Quantum excitation Radiation damping The 3 effects determine the equilibrium state of the beam 19

20. Touschek lifetime (theory)  q: bunch charge  γ: relativistic beam energy  σ x,z, : horizontal and vertical beam sizes  σ s : longitudinal bunch length  σ x‘: beam divergence in x  δ acc : MA (Momentum Acceptance) 20

21. Install location of the septum in the ThomX ring  Beam stay clear region (inside and outside septum)  Installation location of the septum : ??? mm to the center of the vacuum chamber 21 septum Center of the vacuum chamber x

22. Vacuum chamber of the ThomX ring 22 All particles will finally lost on the vacuum chambers! First half of septum Second half of septum

23. Lattice MA (obtained by 6D tracking...) 23  Tracking taking into account of the vacuum chamber size!  The installation location of the septum has effects on the lattice MA  Most particles are lost on the septum  RF MA: ~10% for the ThomX ring! Result of low beam energy (low radiation lost)! U 0 = 1.5 eV/turn; V RF = 300 kV, phase syn = 180 degrees  ThomX ring: dominated by lattice MA!

24. Touschek lifetime V.S. install location of the septum to the vacuum chamber (MA) 24 Simulation parameters: ε x = 50 nm.rad ε z = 50 nm.rad q = 1 nC  s = 4 mm δ = 0.006

25. Septum in the ThomX ring  Beam stay clear region (inside and outside septum)  Installation location of the septum : 8 ~ 10 mm to the center of the vacuum chamber 25 septum Center of the vacuum chamber x

26. Conclusions and outlook  adding octupole correctors DA of ThomX ring  MA is not significant improved (locations of octupoles are dispersion free), but high order momentum compaction factor  Will re-calculate DA & MA of the ThomX ring with the new multipole errors  Further studies need to be done, such as, change the locations of the octupoles (put extra coils in quadrupoles)?  Non linear resonance driving terms can be measured when the machine is ready (BPM + FFT) reduce the resonances improve DA & MA  Measure DA (Injection off axis and measure captured number of electrons)  From the Touschek lifetime, the install locations of the septums can be determined to keep a high injection rate and reasonable storage time  IBS and Touschek lifetime need to be further studied for the ThomX ring (will have the help from J. Bengtsson (BNL, USA) and collaborations from the international AT group) 26

27. Acknowledgement  Special thanks to P. Bambade (LAL) and A. Loulergue (SOLEIL) for their initiations and helpful discussions and advices of this talk  C. Bruni (LAL) for her provisions of multipole field errors.  A. Faus-Golfe (LAL) for her helpful advices.  J. Bengtsson (BNL, USA), B. Nash (ESRF, France), S. Leemann (MaxIV, Sweden) for their helpful discussions. 27