R56 for a beamline with RFCA

[Aamna_Khan]

Hello Michael,

I come cross two problems while computing R56 from .mat and .sig file in the presence of RFCA: 
First, in the presence of RFCA in a beamline .mat file doesn’t give the expected output for R56: e.g. in the attached example the R56 should be around 2 m but mat output gives 0.5 m. For the individual arcs of the beamline results are fine. 

Second, R56 from .sig and .mat file is totally different for a beamline with RFCA.

Could you please look at the example if something is missing and let me know if there is any other way to calculate it? 

Thank you.
Aamna

[michael_borland]

Aamna,

The equation you use to relate S56 and R56 is not correct in the presence of acceleration. In general,

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S56 = <s delta> = <(R55*s0 + R56*delta0)*(R66*delta0 + R65*s0)>,

where '0' indicates the start of the beamline and <> indicates averaging over all particles. Expanding this gives

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S56 = <s0*delta0>*(R55*R66+R56*R65) + <s0^2>*R55*R65 + <delta0^2>*R56*R66

If there is no acceleration, R55=R66=1 and R65=0, giving

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S56 = <s0*delta0> + R56*<delta0^2>

Further assuming <s0*delta0>=0 gives the equation you used.

However, we should not expect any of these equations to work well in your example, because the bunch is long compared to the rf wavelength and it is accelerated near the crest, where the dominant term relating delta to time is second order.

--Michael