Pattern Recognition: The Importance of Dispersion in Crystal Collimation Page: 3 of 15
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2 CALCULATION METHOD
2 Calculation Method
To first order, the horizontal motion of a particle in an accelerator is described by the
sum of its betatron and synchrotron oscillation displacements,
xtot x + 77, (1)
where q is the dispersion, and betatron and synchrotron oscillations are:
xp = ax cos(yx)
= nxex cos(27rQxt + 8x), (2)
3 = a cos(#5)
= ns(Qp/p) cos(27rQst + 06). (3)
The amplitudes and phases of the betatron and synchrotron oscillations are:
as = ns(ap/p), (4)
O 2 2rQxt + Ox
0s - 27rQst + Os. (5)
where nx and n8 parametrize the amplitudes of the oscillations, and ex and 0, are the
initial phases. The beam size, ax, is calculated from the emittance, and the betatron
and the synchrotron tunes are Q, and Qs. Time is represented by the turn number, t.
The emittance and other SPS beam parameters used in this calculation are listed
in Table 1. They are nominal values for CRYSTAL, except that the synchrotron tune
deviates slightly from its nominal value of Qs = 0.004, for visual clarity below. Note
the the dispersion at the crystal has a large magnitude, and is negative.
The betatron and synchrotron oscillation range of a test particle at a crystal with
negative dispersion is indicated by the black rectangle in Fig. 1. The crystal displace-
= 6o-x, (6)
is indicated with a solid blue line in Fig. 1. If 6 = 0, the particle moves on the
horizontal axis and hits the crystal only if the betatron oscillation amplitude, ax, is
greater than xe. However, if 3 -$ 0, the particle can hit the crystal even if ax is less
than x,. Similarly, if 3 is large, a particle can miss the crystal even if ax is greater than
xe because the dispersion is negative. The hit pattern of a particle is altered when
significant dispersion is present.
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Peggs,S. & Shiraishi, S. Pattern Recognition: The Importance of Dispersion in Crystal Collimation, report, September 1, 2008; United States. (digital.library.unt.edu/ark:/67531/metadc893094/m1/3/: accessed March 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.