Algorithm for higher order chromaticity computations

[dpe]

I would like to understand what elegant does in order to compute the values of higher-order chromaticity, but I encountered a lot of difficulties in understanding the following manual's entry:

higher_order_chromaticity -- If nonzero, requests computation of the second- and third-order chromaticity. To obtain reliable values, the user should use concat_order=3 in this namelist and the highest available order for all beamline elements. elegant computes the higher-order chromaticity by finding the trace of off-momentum matrices obtained by concantenation of the matrix for higher_order_chromaticity_points values of $\delta$ over the full range higher_order_chromaticity_range. If quick_higher_order_chromaticity is nonzero, then a quicker concatenation method is used that gives the second-order chromaticity only.

It doesn't seem to me the same procedure that is used by MAD-X which extract the chromaticity computing the variation of betatron functions with respect to the energy. Is there a better explanation available, maybe in a paper or in a book?

Thanks a lot!

[michael_borland]

You may recall that the tune can be determined from the trace of the linear one-turn matrix, which is equal to 2*cos(2*pi*nu). If the one-turn matrix is computed for an off-momentum particle, we can get the tune vs momentum, and thus the chromaticity.

Hope this helps.

--Michael