Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

LATTICE DIAGNOSTICS USING SINGLE KICK CLOSED ORBITAT KEKB
Y. Ohnishi, K. Akai, N. Akasaka, A. Enomoto, J. Flanagan, H. Fukuma, Y. Funakoshi,
K. Furukawa, S. Hiramatsu, K. Hosoyama, N. Huan, T. Ieiri, N. Iida, T. Kamitani,
S. Kato, M. Kikuchi, E. Kikutani, H. Koiso, M. Masuzawa, S. Michizono, T. Mimashi,
T. Nakamura, Y. Ogawa, K. Ohmi, S. Ohsawa, N. Ohuchi, K. Oide, D. Pestrikov,K. Satoh, M. Suetake, Y. Suetsugu, T. Suwada, M. Tawada, M. Tejima,M. Tobiyama, N. Yamamoto, M. Yoshida, and S. YoshimotoHigh Energy Accelerator Research Organization(KEK), Tsukuba, Ibaraki, JapanAbstract
We have measured beta functions using single kick closed
orbit at KEKB. The measured beta functions were compared with the model lattice and gradient errors were extracted from the result in a beta beat. The gradient errors
were also obtained by changing strength of steering magnets and measuring beam positions as the second method.
The methods of the error estimation and preliminary results
will be reported.

KEKB[1] is an asymmetric-energy, double-ring, electronpositron collider. The beam energy is 3.5 GeV forpositron(LER) and 8.0 GeV for electron(HER), respectively.
The requirement of the peak luminosity is1034 cm−2 s−1 for study of CP violation. In order to achieve
such high luminosity operations, large beam current andsmall βy∗ which is the beta function at I.P. in the verticalplane are needed. The βy∗ = 1cm optics has been done successfully for both rings. The final focusing of two beams
is provided by a pair of super conducting quadrupole magnets which are called QCS. The small βy∗ produces a large
amount of chromaticity which makes the field of sextupole
magnets strong. The nonlinear of sextupole magnets reduces the transverse dynamic aperture. It is, therefore, considered to use a pair of identical sextupole magnets which
are connected with a -I' transformer in both the horizontal and vertical planes. The nonlinear effect due to thesextupole magnets should be compensated by the -I' transformer up to the third order in Hamiltonian.
Diagnostics of the lattice has been studied since the commissioning of KEKB started in December 1998. Measurements of the closed orbit distortion(COD) give us a large
amount of information for a lattice.
The formula for the closed orbit distortion, ∆x, induced
by a single steering magnet is∆xi = Rij θj ,(2)where µ = 2πν is the betatron tune, ψ is the betatronphase advance, β is the betatron function, η is the dispersion function between two locations, α is the momentum
compaction factor, and C is the circumference of the ring.
If the last term of eq.(2) can be ignored and the location ata pair of the steering magnet and the BPM is considered,
the beta function is obtained from∆xjµβj = 2· tan(3)
θj2
When the beta functions have been measured and asignificant discrepancy between the measurement and themodel has been found, the sources of the discrepancy canbe specified. If the error sources come from a gradient error in one or more of quadrupole magnets, the differenceof measured from the model beta functions are describedas follows:∆βj =X βj βmm

2 sin µcos(µ − 2 | ψj − ψm |) · ∆Km ,(4)where ∆Km is the gradient error of the m-th quadrupole
magnet(method-1).
On the other hand, assumed that the error is the gradient error of the quadrupole magnets, the measured COD
induced by a single steering magnet is written byX∆xmeas.
= ∆xmodel+Rim ∆Km ∆xmodel,
(5)
iimm

where∆xmodeli∆xmodelm(1)where θj is a kick angle of the j-th steering magnet. The
response coefficient, Rij of the j-th steering magnet to the0-7803-5573-3/99/$10.00@1999 IEEE.

i-th BPM is calculated bypµ ηηβi βji j
Rij =− | ψi − ψj | −,µ cos
2 sin 22αC= Rij θj= Rmj θj ,(6)(7)

where Rij and Rmj are the response coefficients from the jth steering magnet to the i-th BPM or the m-th quadrupole2981

magnet, respectively. The last term of eq. (5) is the correction term due to the gradient errors of the quadrupole
magnets against the model(method-2).
Another constraint to determine the errors of the latticeis2{cot µ(cos ∆µ − 1) + sin ∆µ},βm(a)120806040(8)200where ∆µ is defined by µ−µ, and the ± signrefers to the horizontal and vertical planes, respectively. Ifthe tune changes small and not close to the half integer orinteger resonances, eq. (8) can be simplified as follows:meas.

140100010203040506070model80

no. BPMβy (m)∆Km = ±βy (m)

Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

140(b)

12010080(9)6040with taking all gradient errors into account.
In order to extract the gradient errors of the lattice, there
are two methods as described above. The beta function at
I.P, βy∗ = 1 cm optics was utilized. The horizontal-vertical
coupling has been neglected. Before the measurements of
the COD, orbit corrections with the steering magnets wereperformed. There are 462 horizontal and 456 vertical steering magnets available in LER. In this analysis, 87 horizontal and 88 vertical steering magnets were chosen to measure
the single kick closed orbit. The dispersion functions at
those steering magnets are less than 0.1m in order to avoid
the uncertainties of dispersion functions between the modeland the real machine.
There are 449 BPMs available to measure the beam positions. The beam positions are determined by an average of
8 measurements and the resolutions of the BPMs are typically less than 5µm. The closed orbit has been measured
at BPMs before and after changing every steering magnets
along the ring to minimize effects of orbit drifting. The orbit drifting at large beta function was less than 200µm inhorizontal and 1 mm in vertical plane during this measurement, respectively. Each steering magnet is set to the kickangle of 50µrad added to the original setting.
The beta functions were measured by the beam position
and a single kick angle from eq. (3). Measured transversetunes were used in eq. (3). Difference between the beta
functions at the steering magnet and those of the nearby
BPM is typically less than 10% which is estimated from themodel calculation. We used SAD program[2] to calculate
twiss parameters of the lattice model. Figure 1 shows thebeta functions at BPMs nearby the steering magnet. A significant discrepancy between measured beta functions andthose of the model without corrections can be found(fig.
1(a)). In most cases, the difference from the model beta
function will be a beta beat which is an oscillation of themeasured beta function around the design beta function at
twice the betatron frequency. The gradient errors were estimated by eq. (4) using an iterative procedure(Micado[3]).
In eq. (4) measured transverse tunes are utilized and betatron phase advances are also scaled by the ratio of measured to the model tunes. The error source is identified

20001020304050607080no. BPM

Figure 1: The β functions at BPMs nearby the steering
magnet which induces a single kick closed orbit. The measured beta functions(plots) and the model calculations for(a) no correction and (b) after corrections in the verticalplane. The solid line in fig. (b) shows only correction of
QCS magnet identified from method-1 and the dashed line
shows correction of the QCS and other quadrupole magnets
from method-2.
with QCS magnets which locate at vicinity of the interaction point. The strength of magnetic field was stronger thanthe expected value by 0.47%. Figure 1 (b) shows the corrected model for the QCS(solid line) and the beta functions
from the model lattice can well reproduce the measured
beta functions. After the correction of QCS, measured vertical tune is different from the model by 0.18, on the other
hand, the difference before the correction was 0.25.
Figure 2 represents the typical single kick closed orbit
in the vertical plane at LER. The model calculation after
the correction of QCS agrees with the measured beam positions at BPMs.
∆Y (m)

1X∆µ ' ±βm ∆Km
2 m

0.004
0.003
0.002
0.001
0-0.001-0.002-0.003-0.004
0200

400

600

800

1000

1200

1400

z (m)

Figure 2: The single kick closed orbit in the vertical plane
for a half of circumference at LER. The plots shows measured beam positions by BPMs and the solid line shows
those of the lattice model with corrections of the gradienterrors for the QCS magnet.

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Figure 3 shows the gradient errors obtained from eq.
(5) and eq. (9). The gradient error for all quadrupolemagnets are estimated by one single kick closed orbit.
In order to derive the gradient errors, the singular value
decomposition(SVD)[4] is performed to solve equations
described above. Because multiple error estimations foreach quadrupole magnet are available, the average of errors
and the standard deviations are plotted in fig. 3. Not only
the gradient error of the QCS but also a series of defocusing
quadrupole magnets(QD3P, QD5P) can be found. Those
quadrupole magnets are connected to the same power supply. The beta functions(dashed line) calculated by themodel after corrections of QCS and other quadrupole magnets are superimposed in fig. 1 (b) and also reproduce themeasured beta functions well.
The error estimations between method-1 and method-2
are consistent with each other for the QCS. However, thegradient errors of quadrupole magnets besides the QCS
were not clearly found by method-1 against method-2. The
reason is that one BPM is used for one single kick inmethod-1, on the other hand, all BPMs are used for one single kick and more information can be obtained in method2.
∆K (1/m)

0.004

(a)
0.0030.002

βy (m)

Proceedings of the 1999 Particle Accelerator Conference, New York, 1999
140120100806040200010203040506070

80

no. BPM

Figure 4: The β functions at BPMs nearby the steering
magnet which induces a single kick closed orbit after adjustment of the real QCS. The solid line shows the beta
functions calculated by the lattice model.
We have measured beta functions using the single kickclosed orbit. The gradient errors of quadrupole magnets
have been estimated using two methods. The gradient errorof the QCS could be found by both methods. The correctedmodel calculations can well reproduce the measured beta
functions and the measured closed orbit by BPMs. Theerror of the QCS was also verified from a magnetic-fieldmeasurement independent of this analysis and K value ofthe real QCS was corrected by 0.37%. Figure 4 shows themeasured beta functions compared with the model after thiscorrection. The errors of other quadrupole magnets werealso found by method-2, however, those are still under investigation. The HER should be studied and in progress.

0.001

We would like to thank the KEKB Operations staff for their
help with this experiment.

0-0.001-0.002

02550

75100

125150175200225

no. Quad.

REFERENCES∆K (1/m)

0.004[1] KEKB Design Report, KEK Report 95-7.

(b)
0.003

[2] SAD is the accelerator design tool program developed atKEK.

0.002
0.001

[3] H.G. Hereward, PS/Int. TH 59-5 (1959).
B. Autin and Y. Marti,CERN ISR-MA/73-13.

0[4] W. Press, et al., “Numerical Recipes”, Section 2.9, Cambridge University Press, Cambridge(1986).

-0.001-0.002

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275

300

325

350

375

400

425

450

no. Quad.

Figure 3: The gradient errors of quadrupole magnets predicted by method-2 at LER. Error bar shows the standard
deviation estimated from the measurements using a single
kick orbit induced by every steering magnets. The firstand last number of quadrupole magnet corresponds to theQCS magnets in question. A series of quadrupole magnets
which have small gradient errors are found systematically.

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