Diagnostics of the Fermilab Tevatron using an AC dipole Page: 87 of 235
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3.2.2 Quadrupole Field Errors
In a synchrotron, quadrupole field errors produce second order perturbations
to the betatron oscillations. The sources of the quadrupole field errors are
construction errors of the magnets, errors of magnets' power supplies, and the
feeddown effects of higher order magnets. From the discussion of the second
order perturbation to a simple classical oscillator in Section 3.1.2, we can pre-
dict the quadrupole field errors change the betatron tune. In addition, because
of the correlation between the amplitude function and phase advance given in
Equation 2.31, the quadrupole field errors change the amplitude function of a
synchrotron, as well.
Tune Shift
Here, we discuss the change of the betatron tune (referred to as tune shift in
accelerator physics) due to a thin quadrupole field error. When there is a thin
quadrupole field error, the equation of motion is given by
' + Ko(s)x = -qx > (s - s2 - nC) , (3.39)
n = -
where s2 (0 < s2 < C) is location of the error and q = ABy,1t/(Bp) is the
effective strength of the quadrupole field error. From Equations 2.7 and 2.8,
the error corresponds to the transfer matrix
1 0
Mq =4 . (3.40)
-q 1
Suppose the single turn transfer matrices of the location s2 are Mo(s2+ C s2)
without this quadrupole field error and M(s2 + C s2) with this quadrupole
field error. With the transfer matrix of the quadrupole field error Mq, these75
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Miyamoto, Ryoichi. Diagnostics of the Fermilab Tevatron using an AC dipole, thesis or dissertation, August 1, 2008; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc898385/m1/87/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.