Effect of the beam-beam interactions on the dynamic aperture and amplitude growth in the LHC Page: 3 of 8
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Effect of the beam-beam interactions on the dynamic aperture and amplitude
growth in the LHC
T. Sen *, N. Gelfand, C. Johnstone, W. Wan, FNAL, Batavia, IL 60510Abstract
The dynamic aperture at collision energy is determined pri-
marily by the nonlinear fields of the IR quadrupoles but is
also influenced by the beam-beam interactions. We revisit
the choice of the crossing angle that maximizes the dynamic
aperture with an accurate modeling of the long-range inter-
actions and use of the present values of the IR quadrupole
field harmonics. A separate but related issue we address is
the amplitude growth of particles in the beam halo due to
the long-range interactions.
1 INTRODUCTION
In this note our aim is to understand two issues: a) the rela-
tive importance of triplet nonlinearities and the beam-beam
interaction in determining the dynamic aperture at collision
energy and b) the effect of the long-range beam-beam inter-
actions on amplitude growth of particles in the beam halo.
Previous studies of the first issue [1] have treated the
long-range interactions in an approximate fashion. These
assumptions have included i) neglecting the phase advance
between long-range kicks, ii) assuming the beams are round
at the locations of the long-range kicks, and iii) assuming
that the dimensionless separation between the beams stays
constant at all the kicks. All of these assumptions tend to
over emphasize the strength of the long-range kicks. We
have not made any of these approximations in our study.
As a result we hope to have a more accurate assessment
of parameters such as the optimal crossing angle given the
knowledge of the triplet error harmonics. We use the pro-
gram TEVLAT [2] to calculate the dynamic aperture when
tracking for large numbers of turns and we also use MAD
[3] for shorter term tracking to provide an independent
check.
The second issue is important because the several long-
range interactions in the LHC may scrape off particles in
the tails of a given beam when these particles are close to
the core of the other beam. This problem requires a statisti-
cal approach with a sufficiently large number of particles in
the beam distribution. It is best studied by modelling all the
beam-beam interactions accurately and ignoring the nonlin-
earities so that the effects of the beam-beam interactions on
the distribution in the tails may be followed for a large num-
ber of turns. We have written such a code and we use it here
for estimating the upper stability limit in the LHC at colli-
sion.
* email: tsen@fnal.govNormal Skew
n [(br), dbn, o(bn)] [ (an), dan, o(an)]
FNAL2/KEK2 FNAL2/KEK2
3 0, .3, .8/0, .51, 1.0 0, .3, .8/0, .51, 1.0
4 0, .2, .8/0, .29, .57 0, .2, .8/0, .29, .57
5 0, .2, .3/0, .19, .38 0, .2, .3/0, .19, .38
6 0, .6, .6/0, .5, .19 0, .05, .1/0, .10, .19
7 0, .06, .06/0, .05, .06 0, .04, .06/0, .05, .06
8 0, .05, .05/0, .02, .03 0, .03, .04/0, .02, .03
9 0, .03, .03/0, .01, .01 0, .02, .02/0, .01, .01
10 0, .03, .03/-0.25, .03, .01 0, .02, .03/0, .01, .01
Table 1: Design field harmonics, at a reference radius of
17mm, of the IR quadrupoles to be built at FNAL and KEK.
Harmonics are expressed in units of 10-4.
2 TRIPLET ERRORS AND TRACKING
DESCRIPTION
At top energy the dominant nonlinearities in the machine
are those of the IR quadrupoles. Considerable effort has
gone into the design of these magnets both at Fermilab and
KEK to ensure that the nonlinear harmonics stay within tol-
erable bounds. These errors have different sources: the
low order harmonics are primarily due to fabrication errors
and variations in coil size while the higher order harmon-
ics are mainly due to measurement errors. In the target ta-
bles specified by the two laboratories, the errors are split
into three parts: systematic errors (bn), (an), uncertainties
in the systematics dbn, dan and random variations in the er-
rors o(bn), a(an). These error tables have been refined and
updated as measurements of more model magnets (5 at Fer-
milab, 2 at KEK so far) have become available. The studies
reported here are based on error tables V2.0 (shown in Table
1) of the Fermilab and KEK magnets.
The lattice used in our tracking is derived from the
MAD lattice version 5.1. Lattice nonlinearities are the IR
quadrupole fields and chromaticity correcting sexupoles. In
most of the calculations reported here, the systematic errors
and the random errors of the body harmonics are included
but not the uncertainties in the systematic errors. Earlier
studies [4] had shown that these uncertainties reduced the
dynamic aperture by about 0.5-la. Errors in the ends of
the triplets are not included. For studies without the beam-
beam interaction we have used 100 seeds for the random er-
rors chosen from a Gaussian distribution. With the beam-
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Johnstone, C.; Wan, W.; Gelfand, N. & Sen, T. Effect of the beam-beam interactions on the dynamic aperture and amplitude growth in the LHC, article, June 17, 1999; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc718355/m1/3/: accessed April 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.