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

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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.