1. Bocheng Jiang SSRF AP group 2014.8.14 @Lanzhou
12th Symposium on Accelerator Physics 
Simulation of longitudinal beam dynamics with the third harmonic cavity for SSRF phase II project
Bocheng Jiang
SSRF AP group
@Lanzhou

2. Outline Recent progresses of SSRF SSRF phase II project.
3rd harmonic cavity beam dynamics.
Conclusions

3. SSRF accelerator complex
Linac :158 MeV
Booster :180 m, 158~3.5GeV, FODO, 2HZ
Storage ring：432 m, 3.5GeV, DBA
Bean line Phase I：7 (5 IDs+2 Bend)
Open to user since May 2009

4. Beam Parameters (Operation mode)
Parameter / unit
Design value
Measured value
Beam energy / GeV
3.50
3.50±0.02
Beam current / mA
200~300
210 (operation current)
300 (achievable)
Tune (H, V)
22.22, 11.29
22.220, (±0.002)
Natural emittance / nm.rad
3.89
3.8±0.2
Coupling 1%
0.3%
Natural chromaticity (H, V)
-55.7, -17.9
-55.8, (LOCO model)
-50, -15 (direct measurement)
Corrected chromaticity (H, V)
1.5, 0.5
RMS energy spread
9.845×10-4
0.001
Energy loss per turn / MeV
1.435
~1.45 (without ID, from RF power)
Momentum compaction factor
4.27×10-4
(4.2±0.2)×10-4
RF voltage / MV
4.0
1.51, 1.55, 1.54 (Three cavities)
RF frequency / MHz
(depend on machine conditions)
Synchrotron frequency
(VRF=4.0MV)
0.0075±0.0002
Natural bunch length / ps 13
14±2
Injection efficiency
>95% (from BS DCCT to SR DCCT)
Beam lifetime / hrs
>10
~17 (0.3% coupling, 210 mA)

5. Topup injection and increasing I
2012年运行Topup注入200mA
2013年230mA
2014年240mA

6. 轨道快反馈 轨道快反馈系统2013年投入用户运行，40BPM*60Corrector，@10KHz 对轨道抑制带宽 100Hz，
和轨道慢反馈系统兼容，长期轨道稳定性（40 ID BPM）水平、垂直方向分别达到0.26微米和0.25微米（RMS值）

7. Very Low Effective Emittance Optics
A question: how mach the available lowest effective emittance is, on which SSRF can be operated well with the current hardware.
MOGA technique developed in ALS can help us.
Optimal front between the horizontal tune and the effective emittance.
A solution: the effective emittance is 4.15 nm.rad, the natural emittance is 2.88 nm.rad, reduced by about 20%. The brightness will be increased by 20%~40%.

8. The measured beam parameters agree with the design ones, beside the natural chromaticity.
This optics has been operated for users’ experiments for four days. Brightness are recorded to increase by 10%~30%, in the beamlines.
The beam energy of 3.0 GeV is also implemented with this optics, whose natural emittance is 2.12 nm.rad.
Parameter / unit
Designed value
Measured value
Beam energy / GeV
3.50
3.50±0.02
3.00
2.99±0.02
Tune (H, V)
23.310,
23.309,
23.313,
Natural emittance / nm.rad
2.88
2.9±0.2
2.12
2.0±0.3
Natural chromaticity (H, V)
-74.5, -26.7
-67, -23
-67, -24
Corrected chromaticity (H, V)
------
2.0, 3.0
3.0, 4.0
Momentum compaction factor
4.13×10-4
(4.2±0.2) ×10-4
(3.9±0.3) ×10-4
Synchrotron frequency
(4MV)
0.0074±0.0002
(VRF>4.3 MV)
(4MV)
0.0078±0.0003
Coupling
0.5%
0.4%
Beam current / mA
210
Beam lifetime / hrs
15.0
8.5
Injection efficiency
~50%
RMS beta beatings (H, V)
0.7%, 0.8%
0.6%, 0.8%

9. Recent progress 6 users beam lines have been built since 2009
5 beamlines are dedicated to protein sciences within the program of National Facility for Protein Sciences
1 beamline is founded by Institute of Physics. CAS

10. SSRF Phase-II project

11. SSRF Phase-II Project
+16 beam lines

12. SSRF Phase-II Project Storage ring upgrade
Superbends + double mini beta long straight
Superconducting 3rd harmonic cavity
Superconducting Wiggler
Cryogenic system (4K, 650W)

13. 3rd harmonic cavity beam dynamics

14. 3rd harmonic cavity beam dynamics
Passive 3rd harmonic cavity
Get better beam lifetime for single bunch operation
Decrease loss factor, less chamber heating.
Landau damping
Working f
MHz
Working T
2.2K
Working V
1.5~1.8MV
Bunch lengthening
~3.7 Q0
>

15. 3rd harmonic cavity beam dynamics
the RF voltage is excited by the wakefields of the beam passing through the cavity.
For lengthening , the detuning angle is positive, Robinson instability is a concern.
Avoid Robinson Instability:
Z--Z+ <7.6*104 Ω -3 -2 -1 1 2 3
x 10 4 6 8 10 12 14
fs0=5KHz
+3rd cavity at optimum voltage, fs=0~2KHz
V ≈ Ib (Rs/Q) (fr / Δ f) Rs/Q =180
300mA, Δf=65.9KHz ±2KHz
200mA, Δf=43.9KHz ±2KHz
To get 1.5MV
Z--Z+ <<7.6*104 Ω

16. 3rd harmonic cavity beam dynamics
Passive cavity only one tuning parameters:
Ψ or fres
𝑡𝑔 Ψ= 2𝑄 Δ𝑓 𝑓 𝑟𝑒𝑠 , Q>>1, Ψ≈𝜋/2
LLRF to control Ψ is impractical。
Control V3rd through fres

17. 3rd harmonic cavity beam dynamics
𝑉 ∆𝑡 =−2𝑘𝑞 𝑒 (𝜔 𝜎 𝑙 ) 2 /2 𝑒 [𝑗𝜔−(𝜔/2𝑄)]∆𝑡
∆𝑡>0
𝑉 0 =−𝑘𝑞 𝑒 𝜔 𝜎 𝑙
𝜔,𝑄 are the resonance angular frequency and quality factor of the 3rd harmonic cavity respectively. q is the bunch charge excites the wakefield. 𝑘=𝜔𝑅/(2𝑄), R is the shunt impedance of the cavity
𝑉 𝑡𝑜𝑡𝑎𝑙 = ∆𝑡=0 ∞ 𝑉(∆𝑡
Tracking Code is transplant from a code to evaluate longitudinal coupled bunch instability
φ 𝑛+1 = φ 𝑠 + 𝜔 𝑟𝑓 ∆ 𝜏 𝑛 ,
∆U=2 ε 𝑛 𝑇 0 / 𝜏 𝑠 ,
𝜀 𝑛+1 = 𝜀 𝑛 − 𝑈 0 + 𝑉 𝑟𝑓 𝑐𝑜𝑠 𝜑 𝑛+1 −∆𝑈+ 𝑉 𝑡𝑜𝑡𝑎𝑙 𝑛+1 ,
∆ 𝜏 𝑛+1 =∆ 𝜏 𝑛 +𝛼∗ ε 𝑛+1 ∗ 𝑇 0 / 𝐸 0 ,

18. 3rd harmonic cavity beam dynamics
Q> 108 for SC cavity,
Impossible to record the wakefield within 108 /H turns (where H is the harmonic number of the ring)
Impossible tracking 108 /H turns for hundreds of bunches.
A reduction of Q will not change the transient loading effects [J.M. Byrd et al., Phys. Rev. ST Accel. Beams, 5, (2002)]]
reduce the Q to 104~105 which will greatly reduce the turns of wakefield that need take into account for tracking and turns for convergence of the longitudinal oscillations.

19. 3rd harmonic cavity beam dynamics
f=65KHz
Longitudinal motion is damped

20. 3rd harmonic cavity beam dynamics
For SSRF a practical filling pattern is a 500-bunch train filled in 720 buckets
0.5
∆φ≈ 𝐼 2 𝑉 𝑐 𝑅 𝑠 𝑄 𝜔 𝑟𝑓 ∆𝑡 ≈0.08
The average lengthening factor is 2.3 which is smaller than optimized lengthening factor 3.7 (Bunch length is characterized in full width at half maximum).

21. 3rd harmonic cavity beam dynamics
∆φ≈ 𝐼 2 𝑉 𝑐 𝑅 𝑠 𝑄 𝜔 𝑟𝑓 ∆𝑡 ≈0.018
0.036
the average lengthening factor is 4.3, much bigger than a long bunch train. In this case, bunches are a little bit over stretched to bi-Gaussian shape which makes the lengthening factor even grate than optimized value.

22. 3rd harmonic cavity beam dynamics
The single bunch at the center of the empty-bucket-train gets the optimal harmonic phase and gets the biggest lengthening effect.

23. 3rd harmonic cavity beam dynamics
TMCI
Yong Ho Chin, EPAC94

24. 3rd harmonic cavity beam dynamics

25. 3rd harmonic cavity beam dynamics
Ith≈2*Ith0

26. Conclusions
Passive 3rd harmonic cavity can operate at optimized condition without Robinson instability.
Beam transient loading will great reduce the lengthening factor, The phase shift of tracking result is bigger than formula prediction. A symmetry filling pattern is preferred.
Bi-Gaussian beam shows some interesting properties, yet needs more works.

27. Thanks!