An Algebraic Approach to the Evolution of Emittances upon Crossing the Linear Coupling Difference Resonance Page: 4 of 61
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7 Appendix B
8 Appendix C 45
List of Figures
1 Horizontal and vertical amplitudes Jx and J. and their sum
versus turn plotted in black, red and blue respectively. Here
normalized coupling matrix parameters A = 0.6 and B =
0.06. The small-amplitude high-frequency oscillations seen
on all of the curves are due to the parameter B being small
but nonzero.................................. 49
2 Amplitudes Jx and J, and their sum obtained with normal-
ized coupling matrix parameters A = 0.6 and B = 0. Here
the high-frequency oscillations are gone and the sum is constant. 49
3 Amplitudes Jx and J. and their difference plotted in black,
red and green respectively. Here normalized coupling matrix
parameters A = 0.05 and B = 0.5. The small-amplitude
high-frequency oscillations seen on all of the curves are due
to the parameter A being small but nonzero. . . . . . . . . . 50
4 Normal-mode tunes Qi and Q2 versus turn number n. Here
the black curve is Qin and the red curve is Q2n. These tunes
are given by equations (270-274) with Q = 0.25, AQmax =
0.05, AQmin = 0.005 and 2N = 4000 . . . . . . . . . . . . . . 50
5 Parameters Un, Tn, dn, en and , plotted in black, red, blue,
green and violet, respectively, for the tunes shown in Figure 4.
Note that Tn reaches zero at n = N = 2000 but does not pass
through zero. ................................ 51
6 Exact horizontal and vertical emittances ex, and eyn plotted
in black and red, respectively, for the tunes and parameters of
Figures 4 and 5. These were obtained by iteration of equation
(201) with starting parameters exo = 0.7, eyo = 0.3, co = 0
and ho = 0. The emittances are not exchanged in this case. . 51
7 Elements cn and hn of normalized beam ellipsoid matrix En
plotted in green and blue, respectively, for the tunes and pa-
rameters of Figures 4 and 5 . . . . . . . . . . . . . . . . . . . 523
43
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Gardner,C. An Algebraic Approach to the Evolution of Emittances upon Crossing the Linear Coupling Difference Resonance, report, September 1, 2008; United States. (https://digital.library.unt.edu/ark:/67531/metadc899600/m1/4/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.