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### Evolution of an off-axis probe beam in EMMA

Shinji Machida

ASTeC/STFC/RAL

29 April, 2008

http://www.astec.ac.uk/intbeams/users

/machida/doc/nufact/ffag/machida20080429.pdf & ppt

Problem to be discussed (1)chromaticity

- Nonscaling FFAG has finite chromaticity.
- There is no synchrotron oscillations.
- Phase advance depends on momentum and its difference will be accumulated.

- From the figure above, average chromaticity is roughly 10.
- Take two particles with momentum difference of 1%, phase difference (/2) after 10 turns becomes 1.

Problem to be discussed (2)what does that means?

- As long as a beam is injected on-axis, the different phase advance is almost invisible.
- Once a beam is injected off-axis, smearing in phase space should appear.
- Phase difference of 2 after 10 turns means that a probe beam will be spread out completely in azimuthal direction.

y’

y’

dp>0

on-axis

off-axis

dp=0

dp<0

y

y

Simulation results (1)momentum spread

- Use the following parameters
- a=1/6, b=1/4
- Gaussian with a cut at 2 sigma
- Acceleration with 10 turns

Figure shows dp/p of a few % during acceleration.

black: on-axis

red: off-axis in vertical with 1 mm rad (normalized)

Simulation results (2)phase space evolution (z,pz)

- A beam is initially displaced in vertical direction with 1 mm rad.

- In 3 turns, a probe beam becomes a ring.
- Chromaticity is large at low momentum.

Simulation results (3)configuration space evolution (s-longitudinal,z-vertical)

- A beam is initially displaced in vertical direction with 1 mm rad.

- Correlation among longitudinal position, (momentum,) and vertical position is manifest.

Simulation results (4)summary

- A probe beam injected off-axis quickly smears out and becomes a ring with a radius of the off-set amplitude.
- This is a process which can not be restored at extraction.

Impacts on EMMA design

- With a limited aperture only for a probe beam emittance, diagnostic line after extraction just sees a fraction of a probe beam.
- We should design diagnostic line either
- for only on-axis beam with tomography measurement, or
- for off-axis large beam as well but give up tomography if it is hard to install.
- We may not have to scan the whole phase space at injection. As long as the amplitude ( m rad) of the off-set is the same, probe beams end up with the same ring in phase space.
- However, with machine errors, it is in fact difficult to know the amplitude at injection. It is useful to have some flexibilities to scan phase space.
- We also need to show that a probe beam becomes a ring in the fastest acceleration case.

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