Longitudinal accumulation in triple RF systems Gang Xu

Longitudinal accumulation in triple RF systems Gang Xu IHEP, Beijing 100049, China Workshop on Injection and Injection System, Berlin, Germany, Aug. 28~30, 2017

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

Longitudinal dynamics h=720，E 0=6000，αp=3. 667*10 -5，U 0=1. 995，Φ=π h αp E 0 δacc=0. 03，Vacc=δacc 2 Φ，Taking ϕs=π will not lost generality

1 st and 3 rd 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

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

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

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)

Bunch length 4. 2 cm 2. 1 ns 2. 5 ns

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

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

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

Errors effects two types effects: (Voltage 0. 3%, Phase 0. 3 Random seeds 100000) bunch shorten and the distance between inject point and the center change nearer farther

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.

Bunch length change( original 4. 2 cm): 1. 2 cm~4. 2 cm

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

Errors effects on different bunch length

Errors effects for short bunch case(V’’=0, 1. 2 cm) Small bunch occurs: 5. 6%. 94. 4% is the bunch center shaking

• Find the new solutions with V’=0, but V’’≠ 0, bunch length 1. 05 cm • Do errors effects again

Bunch center shaking, small bunch occurs less than 0. 01%。 Bunch center shaking less than 0. 05 rad(<48 ps)。

δacc=2. 5% solution with less momentum acceptance, the distance between inject point and bunch center is almost same as δacc=3%

Summary • With 1 st, 2 nd, and 3 rd harmonic RF systems combination(fundamental frequency 166. 6 MHz), one can get a stable RF buckets with far enough distance between injecting and circulating bunches • The distance(about 3. 06 rad/2. 9 ns) is matched with our kicker system(6 ns bottom width) • RF system does not need ramping as the 1 st and 3 rd 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.

Thanks for your attention!