Correction of Dipole and IR Quadrupole Nonlinear Content in Large Colliders Page: 4 of 12
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F Bend Send Bend Send Send Bend Bend Bend
S S S S '
1 4 4l 14
Figure 1. A sample collider cell. The element labels are: F and D - quads, S - slots for.
correctors, C - center slot. Correction strengths on opposite sides of the thin quads
would be combined in units on either side; there are only two correctors per half-cell.
The accuracy of the solution can be understood by noting that any nonlinear effect can be
expressed in terms of integrals over the lattice. For example, one corrected octupole (b) tune
shift in a half-cell of length L can be written as:
Au, j bsa)p2 (s)da + S$ r/3.()2 + Sacfp3(L/2)2 + Sa3n/3(L)2 (2)
where SP,C,D are octupole corrector strengths. There are six first-order tune shift terms, which
include differing powers of the lattice functions j., (,L, j. The same Simpson's Rule solution
reduces all of these effects by two orders of magnitude. While initially developed for tune shift
correction, the method greatly reduces all other nonlinear effects. (For example, the Collins
sextupole distortion functions2 are exactly reduced to zero at the half-cell level by (F, C, D)
The general accuracy of correction indicates that the (F, C, D) correctors are fully equivalent
to the continuous distribution at the 1% level. A similar algorithm has been developed to
compensate varying (random) multipole content; similar cancellations are obtainable.3 However,
in this note we will emphasize the application to the correction of systematic multipole content.
The systematic effects most severely restrict linear and long-term dynamic apertures, and are
also most easily corrected.
The (F, C, D) correctors are also at optimal locations for separated-function control of
horizontal-, coupled-, and vertical-motion parameters, and these are precisely the operational
observables. The C corrector adds the ability to control coupled-motion parameters indepen-
dently of horizontal and vertical motion parameters. This tunability can be used in improving
correction from initial approximations. For instance, (F, C, D) octupoles are appropriate ele-
ments for control of all amplitude-dependent and second-order chromatic tune shifts. The (F,
C, D) elements permit exact control of the motion through 10-pole order.
The approach was described and discussed thoroughly at the second advanced ICFA beam
dynamics workshop (Lugano, Switzerland, 1988)4 and in international physics journals.5-AJ,
2.0 Application to the Large Hadron Collider (LHC)
Quasi-local (F, C, D) correction of sextupole, octupole and 10-pole components (ba, bs, b4)
is included in the current design of the LHC.8 The application is described in detail by Scandale
in these proceedings and in CERN reports by Scandale and co-workers.'0'" A graphical repre-
sentation of the effectiveness-of the correction in the LHC is shown in figure 2, from references.
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Neuffer, David. Correction of Dipole and IR Quadrupole Nonlinear Content in Large Colliders, article, January 1, 1991; [Newport News, Virginia]. (digital.library.unt.edu/ark:/67531/metadc927453/m1/4/: accessed January 19, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.