Chromaticity in Recirculation

[jpforr]

Dear all,

I am currently working on a calculation of the (horizontal) chromaticity of the recirculations of an recirculating linac. I consider only one recirculation at a time, so the beam travels only through magnets.

My first approach is an integration over all quadrupoles and sextupoles in the lattice, where I integrate betax*k1/(4*pi) or betax*r16*k2/(4*pi) respectively and sum up these contributions.

Another possibility would be to use the chromaticity parameter from the twiss_output command, but the results do not agree with my physical understanding: I get large vertical chromaticities (of several hundred), but as r36=0 for our machine, the sextupoles should not contribute, which would result in a rather low chromaticity. Also, turning off the sextupoles changes the vertical chromaticity (which would not be the case if they do not contribute anyway).

As these approaches brought different results, I tried to calculate betax from the enveloppe and emittance (via betax=s1²/ex), as our group did before in a non-elegant simulation. The result was that this betax and the one from the twiss-file differ significantly (see the attched picture, black: from twiss_output, red: calculated from enveloppe). Consequently, this resulted again in a completely different chromaticity.

(The resulting chromaticities from these three approaches were about 36, 3 and 254 respectively.)

A last attempt was to calculate the chromaticity from a calculation of the tune for different momenta (by varying an malign element, similarly to elegantExamples/Par/chromTracking2). But I encountered similar problems when I calculated the tunes. I used the tunes from twiss_output and I calculated them by an integration of 1/(2*pi*betax). For the integration I tried mathematica and sddsinteg. As the resulting tunes were already different for these methods (they ranged from 2.86 to 2.88, 1.95 to 2.09 and 2.29 to 2.52 respectively), I did not see any benefit in calculating even more different chromaticities.


My questions now are:
1. How does elegant calculate quantities like chromaticity and tune?
2. Where could the difference between the two beta-functions in the picture above come from?
3. Where could the different chromaticities and tunes come from?


I attached the .lte and .ele file I used for the chromaticity calculations.
If you are interested, I can also upload the files where I calculated the tune for different momenta.


Regards,
Jonas

[michael_borland]

Jonas,

The calculation of chromaticity is not a simple matter and long papers used to be written describing how to do it correctly. In elegant, we took a simpler approach of just using the second- and higher-order transport matrices. For simplicity, consider just the horizontal plane. The chromaticity can be computed from the chromatic derivatives of R11, R12, and R22, along with the twiss parameters. This is easiest to understand for a periodic system, as the requisite formula is easily obtained by realizing that the trace of the R matrix (R11+R22) is equal to 2*cos(nux).

Code: Select all

xchrom = -(dR11+dR22)/R12*beta0/(4*PI)

where dR11 = dR11/ddelta etc. For a transport line, the values of beta and alpha at the beginning ("0") *and* end ("1") points are needed, as is the on-momentum phase advance phi1.

Code: Select all

xchrom = ((dR12*(cos(phi1)+alpha0*sin(phi1)))/sqrt(beta0*beta1)
       - dR11*sin(phi1)*sqrt(beta0/beta1))/(2*PI)

You can derive these expressions by contemplation of the R matrix expressed in terms of twiss parameters and phase advance. (E.g., Wiedemann Vol 1, 5.135.)

To check elegant's implementation, I varied the initial momentum offset and performed fits of the resulting tunes. This gave chromaticities of x:3.320 and y:-177.59 compared to x:3.310 and y:-177.55 directly from the twiss_output command. See attached files.

--Michael