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，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

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

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

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

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 4.2cm 2.1ns 2.5ns

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 nearer farther 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 4.2cm):1.2cm~4.2cm

Errors effects: 2nd 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, 1.2cm) Small bunch occurs: 5.6%. 94.4% is the bunch center shaking

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

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

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

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.

Thanks for your attention!