1. Longitudinal accumulation in triple RF systems
Gang Xu
IHEP, Beijing , China
Workshop on Injection and Injection System, Berlin, Germany, Aug. 28~30, 2017

2. Topics Longitudinal dynamics Potential(Plot) Phase-space(Plot)
Bunch lengthening
RF Errors effects
Case with less momentum acceptance
Summary

3. Longitudinal dynamics
h=720，E0=6000，αp=3.667*10-5，U0=1.995，Φ=π h αp E0
δacc=0.03，Vacc=δacc2 Φ，Taking ϕs=π will not lost generality

4. 1st and 3rd RF combination
Form two separate RF stable buckets for the injection bunch and storage bunch
To merge two bunches RF parameters must be changed(ramp)
There is a potential barrier between the two Separate RF stable buckets
Making bunch lengthening to increase beam lifetime

5. Potential A local minimum is a bunch center
Potential barrier make the two bunches separate

6. 1st, 2nd and 3rd RF combination
In order to remove the potential barrier, the 2nd harmonic can be introduced
There will increase two parameters(voltage and phase) , the solution will be not sole as for only 1st and 3rd combination
Using simple program to find the solutions

8. Local maximum hints the momentum acceptance and bucket width
There is not any point make V’[φ]=0 between local minimum and local maximum, so only one bunch in one period (for the fundamental RF frequency)

9. Bunch length 4.2cm
2.1ns
2.5ns

10. Bunch lengthening
Transverse beam size very small, bunch lengthening will increase the beam lifetime
V’[φ]=0 in the mean time V’’[φ]=0
There are many solution can satisfy these conditions

11. Potential Locally zoom Green line is just the curve in the page 8
Longer bunch will make the distance decrease between injection point and the bunch center

12. Red one a little bit better than “original” one(longer bunch and farther distance between injection bunch and bunch center )

13. Errors effects nearer farther
two types effects: (Voltage 0.3%, Phase 0.3 ͦ Random seeds )
bunch shorten and the distance between inject point and the center change
nearer
farther

14. Distance between inject point and bunch center
Red line is the original，left(nearer) 40%，right(farther) 50.63%。
This means the bunch center change continuously.

15. Bunch length change( original 4.2cm):1.2cm~4.2cm

16. Errors effects: 2nd type, one stable bucket splits into two buckets
V’=0 at two points

17. Errors effects on different bunch length

18. Errors effects for short bunch case(V’’=0, 1.2cm)
Small bunch occurs: 5.6%.
94.4% is the bunch center shaking

19. Find the new solutions with V’=0, but V’’≠0, bunch length 1.05cm
Do errors effects again

20. Bunch center shaking, small bunch occurs less than 0.01%。
Bunch center shaking less than 0.05rad(<48ps)。

21. δacc=2.5% solution with less momentum acceptance, the distance between inject point and bunch center
is almost same as δacc=3%

22. Summary
With 1st, 2nd, and 3rd harmonic RF systems combination(fundamental frequency 166.6MHz), one can get a stable RF buckets with far enough distance between injecting and circulating bunches
The distance(about 3.06rad/2.9ns) is matched with our kicker system(6ns bottom width)
RF system does not need ramping as the 1st and 3rd harmonic combination to avoid RF aging and bunch length changing
The solution still need optimization.
More simulation including to errors of KICKERs system, beam collective effects, etc.

23. Thanks for your attention!