The 9th Efymposium on Accelerator Science and Technology, Tsukuba, Japan 1993

Simulation Study on Effects of Machine Imperfections m the KEK B-Factory
Haruyo Koiso and Yoshihiro Funakoshi
KEK, National Laboratory for High Energy Physics
1-1 Oho, Tsukuba, Ibaraki, 305 Japan

which amount to 62 and 92 in the horizontal and the vertical directions, respectively. The solutions of sextupoles
Effects of machine imperfections in a low beta lattice with strengths have been searched to minimize the deviations ofnoninterleaved sextupoles are studied with computer simu- the betatron tunes Vx,y and those of the Twiss parameters
lations. Misalignments of quadrupoles and sextupoles sigat the center of the RF section at 12 fitting
andnificantly degrade the dynamic aperture and the vertical- points of the momentum deviation, 6!:!pfpo = ±~61
to-horizontal emittance ratio. However, orbit corrections (i = 1, 6). The momentum bandwidth 61 is chosen to be
at sextupoles effectively recover both the dynamic aperture about 1.5 % which is required for a long Touschek lifetime.
and the emittance ratio.
We estimate the dynamic aperture by particle tracking
in the six-dimensional phase space (x,p.,,y,py,z,6). The
1. Introduction
variable z is defined as z = -vt, where t is the delay fromthe nominal particle. In this paper, the initial conditions
The KEK B-factory, an 8 GeV x 3.5 GeV two- are chosen as Pxo = Pyo = zo = 0 and Yo = axo, where a isring collider, is designed to attain a peak luminosity of a constant. We choose Yo = ~xo to check the acceptance
10 34 cm- 2 s- 1 . A small vertical beta function at the inter- for horizontal beam injection. The dynamic aperture is= 0.01m) is needed to realize the expressed by the initial values of the Courant-Snyder inaction point (IP)luminosity with reasonably small beam currents. One of variant 2Jx,yo and the momentum deviation 60 of particles
the main issues in designing such a low beta lattice is how which survive during 1000 turns without damping due toto correct a large chromaticity generated by quadrupoles the synchrotron radiation. Although this number of turns
in the interaction region while keeping a sufficient dynamic is only 1/8 of the transverse damping time, the results ofaperture for high injection efficiency and for a long beam 1000-turn tracking are almost equal to or slightly smaller
lifetime. Simulations incorporating machine errors are nec- than those of 8000 turns with the radiation damping. Allessary to estimate realistic values of the dynamic aperture. calculations concerning this study such as matching of linIn this paper, we concentrate on effects of errors in the ear optics, fitting of sextupole strengths, orbit corrections
3.5-GeV ring (LER) in which a larger dynamic aperture is and particle tracking have been done by a computer code
SAD developed at KEK [3].
required than in the 8-Ge V ring.
Abstract

f3:,fo::.f=(f3;

Dynamic Aperture with Noninterleaved Sex2.
tupoles
In order to obtain a large dynamic aperture, we plan
to adopt a chromaticity correction scheme with noninterleaved sextupole pairs. In this scheme, sextupoles areplaced in pairs each of which consists of identical sextupoles
connected by a -I transformer both in the horizontal andthe vertical planes. No other sextupoles are interleaved
between paired sextupoles. Because major transverse nonlinearities of sextupoles are canceled by the -I transformer
within a pair[1], the dynamic aperture is expected to become large in the transverse direction.
An intensive study on the noninterleaved correction
scheme has been done with computer simulations. It has
been confirmed that this scheme greatly widens the transverse dynamic aperture and that the momentum apertureis also made acceptable when we place sufficient number of
sextupole families in the ring [2].
In the present design of LER, forty pairs of sextupoles
are distributed in arcs to correct natural chromaticities

3. Effects of Errors
Performance of the noninterleaved scheme has been suspected to be sensitive to machine imperfections which
break the cancellation due to the -I transformation. Vertical emittance growth due to errors is also harmful tomaintain the required vertical-to-horizontal emittance ratio (coupling) of 1%. We discuss here effects of various
types of errors to estimate their tolerances. All of random
errors simulated below have Gaussian distributions with30' cutoff.

A)Relative Field Strength ErrorsWe introduced strength errors into quadrupoles or sextupoles. The transverse tunes were adjusted to the nominalvalues ( v., 40.12 and Vy 40.19) by varying quadrupole
strengths in each case. Figure 1 shows the dynamic aperture with random relative-strength errors of 0.1 %, 0.2%,
and 0.3% of all quadrupoles. To express the dynamic
aperture with one parameter, we introduce a parameter
A which is the area of the stable region in the 2Jx,yo and-318-==

6o plane. Average values of A and the coupling are also
written in Fig. 1, where A 0 is the area of the dynamic
aperture of an ideal lattice. The 0.2% errors are tolerable
both for the dynamic aperture and for the coupling. On
the other hand, strength errors of sextupoles are more tolerable and 0.5% errors do not affect the dynamic aperturesignificantly.

e 4.o w·5'-'0>(~2.0 JO·S

0.0 I 0~2L...OL..J-l.o.-~-'-L....1...J-.L...L....L...LJ~IIII.....Jw..~2.0

6.o w· 5 ..........-.-,.,.-r-r-r-.--.-,,....,...~,.......-,-,....,......,-,.--,

E 4.0 10'5

<fy/fxo>=0.41 +-0.22%<NAo>=O.SI+-0.06
8 examples

o~ad 0.2%'-'cr.;~2.0 !0-5of the errors (2), (3), and (4); 0.2 mm misalignments, 0.2
mrad rotations, and 0.2 mm monitor setting errors, while
we used the same rms values of the error ( 1).
We examined 12 examples. In each case, the closed orbit
distortion (COD) was measured with 364 position monitors equipped at the edge of quadrupoles and they werecorrected with 404 steering magnets. Residual values of
COD were reduced to"' 0.2 mm rms both in the horizontal
and vertical directions. After the COD correction, we carried out tracking. Figure 2 shows resultant dynamic apertures and Figure 3 shows the couplings and scaled valuesof the dynamic aperture with open circles. As are shown inthese figures, the misalignments of 0.2mm rms significantly
decrease the dynamic aperture and produce the couplingmuch larger than 1%. In order to recover these degradations, we tried to reduce COD at sextupoles by employing additional 80 monitors at the sextupoles with 0.2 mm
setting errors. In the four examples with the coupling of~ 5%, we corrected COD to"' 0.2 mm rms at monitors attached to sextupoles and quadrupoles in straight sections.
After this correction, both the dynamic aperture and thecoupling were fairly improved as indicated with the closed
circles in Fig. 3. However, further corrections must beapplied to reduce the coupling to 1%.
6.0 10" 5 ..--.--r-...-r-r-ro--r-r-r-.-.,....,,....,....,.....,,...,...........,....,

0.0 ]0~2~.0-:!--''-'-~~~.I......L-::'-::..1......1-'-....._,;~....._.__._-!

12 examples

6.o w- 5 ,.....,......,-r-r-.-.,--,,....,...~,.......-.-,....,......,-..,...,....,....,

e 4.o 1o·<~:yk,. o>= I. 60+-2. 27 •4.0 10' 5E'-'<NAo>=0.69+-0.09
8 examples

cr.;

~cr.;

~5'-'

2.0 to-5

2.0 1o-s

o.o I o~2!-.o:l--lw...-...,_..J_.L..1-ooL:9L..J_J.....L.JI~=-L....1.~2 oOo ·l%)

·Figure 1: Dynamic apertures with field strength errors ofquadrupoles. The crosses show t.he acceptance required for injection.

Figure 2: Dynamic apertures in the presence of misalignments.

1.00.9B)Misalignments of Quadrupoles and Sextupoles
In the previous study [4], we examined effects of misalignments together with strength errors. The following
errors were taken into account simultaneously: (1) 0.1%
relative strength errors of quadrupoles and 0.2% of sextupoles, (2) 0.1 mm displacements of quadrupoles and sextupoles both in the horizontal and vertical directions (3)0.1 mrad rotation errors of all magnets and ( 4) 0.1 mm setting errors of position monitors. Even with these amountsof errors, the dynamic aperture remained sufficient to clear
the injection condition and give a Touschek lifetime of~ 3hours in all of 12 examples. Although one third of theexamples brought about couplings of ~ 1.5%, we easily
reduced them to 1% by making only one or two verticalbumps of"' 1 mm height in the arcs. Misalignments of 0.1
mm rms may be difficult to be realized in the actual machine. Then in this study we have doubled the rms values••0.8<=..._< 0.70•000000.60.5002406 8 10coupling(%)

1214Figure 3: Couplings and normalized apertures of 12 examples.

C)Large Misalignments of A Single Sextupole
In the actual situation, it is likely that in addition to
the small errors studied in the previous sections there exist
small number of relatively large errors which are introduced
by some mistakes. However, it is very time consuming work-319-

80
to examine every case of such errors as is easily imagined.
In this paper, we studied an extreme case that one of the,-,.
60
strongest sextupole magnets is misaligned up to 6 mm in~'-"bl)
isit
study
thishorizontal or in vertical direction. With.s 40
expected that one has some rough idea on the situation.
-aPosition monitor::IWe gave an alignment error to a sextupole whose K2
attached to sextupoles
020
uvalue (B"l/Bp) is -6.5975. This is the only error in thelattice and the other elements are assumed to be perfect~~~~~~~~--~--~---0 -7
including the partner sextupole to the misaligned one. Fig.
0-1-2-5 -4 -3-6
Vertical misalignment (mm)4( a) shows the dynamic aperture in the cases of horizontal
misalignments of 0, -2, -4, and -6 mm. Here 0 mm means, Figure 5: Couplings due to the misalignment of a sextupole.
of course, the perfect machine. In each case except the perfect machine, COD was measured at the position monitors
shown in Fig. 5. However, the coupling is also recoveredat the locations of the quadrupoles and they were correctedby centering the orbit at the sextupoles.
using 404 steering magnets with a residual rms value of lessThrough the present study, importance of centering thethan 0.13 mm. As is seen in the figure, only one misaligned
orbit at the sextuples has become clear. This suggests that
sext.upole can affect the dynamic aperture significantly, al- we should install the position monitors at the sextupoles
though the amount of error is unreasonably large. The
rather than at the quadrupoles in the arc section. Demisalignment of 6 mm can halve the horizontal acceptance.
creasing the setting errors of the position monitors is also
However, even in this extreme case the dynamic apertureimportant.
can be almost recovered if we correct the orbit so that thebeam goes through the centers of the sextupoles. This can 4. Summary
be done by employing the position monitors attached toEffects of machine errors in a low beta lattice with nonthe sextupoles. In Fig. 4( a) the dashed line denotes thesextupoles have been studied with computerinterleaved
atcase of the 6 mm misalignment and using the monitors
the sextupoles. In the horizontal misalignment case, the simulations. This chromaticity correction scheme is not
so sensitive to field strength errors of quadrupoles andcoupling was not affected by the errors.
sextupoles. Misalignments of 0.2mm rms and rotationsof 0.2mrad rms significantly decrease the dynamic aperture and increase the vertical-to-horizontal emittance ratio.
(a) horizontal
However, we can fairly recover both the dynamic apertureand the coupling when we correct orbit distortions at sextupoles. Even with a misalignment of several millimetersof a single sextupole, we can improve the dynamic apertureand the coupling by centering the orbit at the sextupole.
These simulations show that orbit corrections at sextupoles
o.o 1o~2!-.o-:'-L...I.""""~I....L1.-'-.l..::I-::-L...J..._~~,...I......L...l...-::'2.oare important to keep good performance of the noninter6.0 1o· 5 ....-..-..,.....-,..-,-r-.r-r-r-.-..-..-,..,-,
leaved scheme.
(b) verticalAcknowledgments••.I

g 4.0 10'•5The authors wish to express their thanks to K. Oide andM. Kikuchi for many useful discussions and suggestions.

~(:::! 2.0 to-sReferences
Figure 4: Dynamic
single sextupole.

large misalignments of aIn Fig. 4(b) effects of vertical misalignments are shown.
Amounts of errors are 0, -2, -4, and -6 mm again. Thehorizontal apertures were decreased with the errors similarly to the horizontal cases. The aperture was recoveredagain by using the monitors at the sextupoles in the case of
the 6 mm misalignment as is shown with a dashed line in
the figure. The difference of this case from the horizontal
one is that the couplings were damaged by the errors as is[1] K. L. Brown, IEEE Trans. on Nucl. Sci. NS-26,3490(1979); K. L. Brown and R. Servranckx, AIP
Conf. Proc. 127, Edited by M. Month, P. F. Dahl andM. Dienes (1983), p.62.
[2] K. Oide and H. Koiso, Phys. Rev. E47, 2010 (1993).
[3] K. Oide, Nucl. Instr. Meth. A276, 427 (1989). M.
Kikuchi, Private communication.
[4] H. Koiso, To be published in Proceedings of "International Workshop on B-Factories: Accelerators andExperiments", KEK, Tsukuba, November 1992.

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