EFFECTS OF TUNE MODULATIONS ON THE LONG-TERM STABILITYIN ELECTRON-POSITRON STORAGE RING
Eun-San Kim and Moohyun Yoon, PAL, POSTECH, Pohang, Kyungbuk, Korea, 79-784
Abstract
the synchrotron oscillation. The change in the vertical tune
The combined effects of tune modulations and beam-beaminteraction are investigated in terms of the long-term stability in an electron-positron storage rings. The tune modulations due to power supply ripple and synchrotron oscillation are studied for the case of KEKB. The tunes aresinusoidally modulated at different amplitudes of the ripples and synchrotron oscillations. It is shown that choicesof the nominal tune greatly affect the growth of beam size
through the tune modulation and the beam-beam interaction. It is shown that the effect of a tune modulation due
to the ripple is stronger than that of tune modulation due tothe synchrotron oscillation.

1

INTRODUCTION

The long-term stability of particles in a storage ring may
be affected by various factors. To some degrees it is determined by the nominal tune of the machine and the strengthsof the nonlinearities that may exist in the machine. Factors
such as power supply ripple and ground motion are known
to increase emittance growth and particle loss rate in proton storage ring of HERA [1]. The beam-beam interaction
also imposes limitations on the performances of the colliding rings. Here we investigate the combined effects oftune modulations due to the power supply ripple and thesynchrotron oscillation on the beam-beam interaction. For
this purpose, a tracking studies are performed by the usingof KEKB parameters which are being commissioning andtracking runs of 1  10 6 turns that correspond to 10 sec inactual machine operation of the KEKB.
We model the ripple as a modulation of the tune at theripple frequency. For this work, variations of beam size arestudied by tracking particles for a million turn. Here we
investigate the effects of modulating in the vertical tune.
The change in the vertical tune is given byy = yo + Ar cos(2fr t);(1)

where yo is the nominal tune of the machine in the absenceof ripple. The ripple is represented by A r , the amplitudeof the tune modulation, f r the frequency of the modulation and t time for one turn in a ring. We study the effectat different ripple frequencies due to the presence of different harmonics. 50 Hz is considered as the main ripplefrequency. In addition to this, we study the effects at 100
Hz as another harmonic of the main ripple frequencies toconsider effect of high frequency. We also study the variation of the beam size with the amplitude of ripple at a fixedripple frequency.
Next we investiagte the effects of tune modulation due todue to the synchrotron oscillation can be written asy = yo + As sin(2s t);(2)

where yo is the nominal tune of the machine. The amplitude of tune modulation is represented by A s , s the synchrotron tune and the time t is measured in turn number.
It is shown that the tune modulations that are considered above may result in growth of beam size in the KEKBmachine. The extent of the growth depends on choice ofthe nominal tune of the machine and also on the amplitude and frequency of the modulation. In result, our simulations show that tune modulation may be a source ofemittance growth and luminosity degradation in electronpositron storage rings.
Table 1: Parameters that are used in the simulations
Parameter
UnitValue
Beam Energy
Eo8 GeVEmittance3.610 10 m
Cicumference
C3016 mEnergy SpreadÆ
6.710 4
Bunch Lengthz
5.6 mmBeam-beam tune shift
0.052
Beta function at IP
1.1 cmIP
Synchrotron tunes0.0114
Betatron tune
yo42.14

2

SIMULATION

Equations of motion for a particle is described by introducing scaled coordinates in transverse and longitudinal phasespaces. With s denoting the longitudinal coordinate, theyare given byY (s) =y (s);y0Z (s) =y 0 (s);y0
(s)E (s) =;
0Y 0 (s) =z (s);z 0IP(3)where IP represents the betatron function at the interaction point(IP) and  y0 , z0 and 0 are the nominal rms
values at the IP of the transverse beam size, the bunch
length and the relative energy spread, respectively. In theabove equation, y and y 0 are the position and the slope ofa macroparticle in a beam in the transverse direction, respectively, and  is the relative energy deviation from thenominal energy [ = (E E 0 )=E0 ]. We use

z=sct(s);(4)

where c is the speed of light, and t is the difference in thearrival time at s between the particle under considerationand the reference particle. Thus, z (s) > 0 indicates thatthe particle arrives at s earlier than the reference particle.
The evolutions of these normalized parameters in thering can be treated separately at the IP and around the arcof the ring. First, at the IP, the change in scaled coordinates
due to the beam-beam interaction is given by [2, 3]! Y R1 (Z Z )ÆP;! Y 0 + ÆP;! Z;
ÆPE ! E + R2 ÆP (Y 0 +)
2YY0Z

=

G=

A=

R1=G:

R2=

Z

=

 2n
1e A0N Y + R1 (Z Z )Y2n R1 R2 Z Ze A;822N 1 + R1 (Z Z )Y + R1 (Z Z )Y 0p;
2 (Z2(1 + R1Z )2 )
z0;
8

2 IP
020 IP(5)

1;;

z:
z 0(6)

Here,  is the nominal beam-beam parameter defined by=N re;4 0(7)where 0 is the transverse nominal emittance of the strong
beam, re is the classical electron radius and is the usual
relativistic factor. More detailed discussions of eqs. (5) and(6) can be found in ref. 2 and 3.
As mentioned above, the strong beam is divided longitudinally into several slices, and a beam-beam kick given
in eq. (6) for each slice is delivered at the barycenter Z  .
Here, Z is the longitudinal barycenter of each slice in thestrong beam and each barycenter represents n  =N of allparticles in the strong beam where N  and n are the numbers of particles in the strong beam and in each slice, respectively. In our simulation 4 slices are used.
The transformation in the transverse plane from IP to IP
around the ring is simply a rotation:

YY0




=

cos 2ysin 2ywhere y is the vertical tune.

sin 2ycos 2yY = y Y +q1

2y r^1 ; Y 0 = y Y 0 +E = 2s E +q1

2y r^2 ;1

4s r^3 ; (9)pwhere r^0 s are independent, random Gaussian variables withunit standard derivation. Here the  is damping factors defined byy = e 1=Ty ; s = e 1=Ts ;Here, the beam-beam interaction is modeled by the interaction between a macroparticle in a weak beam and one in astrong beam with several slices. The center of each slice isrepresented by the longitudinal position z  . In eq. (5), we
have definedÆP

Synchrotron radiation is given by

YY0



; (8)

(10)

where the T denotes damping times normalized by the revolution time.

3 RESULTS OF THE SIMULATION
3.1

Tune modulation due to the power supply
rippleFigure 1 shows the vertical beam sizes normalized by  yoverse the modulation amplitudes A r s at the 50 Hz ripple
frequency and  yo = 42:14. Figure 1(a) shows the casewhen there is no tune modulation. Figure 1(b),(c) and (d)correspond to the cases of A r =0.005, 0.01 and 0.015, respectively. Figure 2(a),(b),(c) and (d) show the verticalbeam sizes normalized by  yo in Ar =0, 0.005, 0.01 and0.015, respectively, at the 100 Hz ripple frequency andyo =42.14. Figure 3 shows the vertical beam sizes normalized by yo verse the nominal tune variation which areheld constant at Ar =0.02 and 50 Hz ripple frequency. Figure 3(a),(b),(c) and (d)correspond to the case of  yo =42.11,42.12, 42.13 and 42.15, repectively.
The larger tune modulation (A r ) has stronger effects.
The ripple frequency of 100 Hz shows larger growth inbeam size than that of 50 Hz. We see that the growth ofbeam size also depends on the choices of the nominal tune.

3.2

Tune modulation due to the synchrotron oscillation

Figure 4 shows the vertical beam sizes normalized by  yoover the modulation amplitudes A s and y =42.14. Figure 4(a),(b),(c) and (d) correspond to the cases of A s = 0,0.005, 0.01, 0.015 and 0.02, respectively. The result for theeffect of a tune modulation due to the synchrotron oscillation may be roughly represented byy = yo (16As )(11)

The vertical beam size is varied linearly with the amplitudeof the tune modulation.

4 RESULTS
The KEKB parameters are used to investigate the effectsof tune modulations that are generated by the power supply ripple and the synchrotron oscillation. Under current
operating conditions of the KEKB, the beam sizes may beaffected by the combined effects of the tune modulations
and beam-beam interaction, depending on the strengths ofmodulation amplitudes. Our simulations also show that theinfluence of the tune modulation on the beam size depends
on the operation tune. It is shown that tune modulation due
to power supply ripple results in larger effects in growthof the vertical beam size than tune modulation due to thesynchrotron oscillation. In our simulation, the effects ofamplitude and frequency of the modulation are also investigated. It is shown that the tune modulations may haveimportant effects on the motions of particles to nonlinearities in electron-positron storage ring.

1.3

σy

1.2

1.2

σy

1.1

1.1

1.0

1.0
0.90(a)

0.9

6

1x100

Number of Turns

6(d)

(c)

σy1x101.3

1.2

σy 1.1

1.2

1.1

1.0

1.0

0.906

Number of Turns

1x106

Number of Turns

1x10Figure 3: The vertical beam sizes normalized by  yo verse
the modulation amplitudes A r =0 :02 and 50 Hz ripple frequency. Figure 3(a),(b),(c) and (d) correspond to the caseof yo =42.11, 42.12, 42.13 and 42.15, respectively.

(b)
1.4

σy1.2

Number of Turns

1.3

0.90

1.4

(b)

(a)
1.3

σy 1.2

1.00

1.0

6

Number of Turns

1x100

Number of Turns(c)1x106(d)1.4

1.4

σy 1.2
1.0
Number of Turns

1x10

1.2

1.2

σy 1.1σy 1.11.00(b)(a)

σy 1.260

Number of Turns

1x106

1.0

1.00Figure 1: The vertical beam sizes normalized by  yo verse
the modulation amplitudes A r =0,0.005,0.01 and 0.015, respectively, at the 50 Hz ripple frequency.

1x10

1.2

σy 1.1σy 1.1

σy1.2

1.2

σy1.1

1.1

1.0

Number of Turns

1x10

60

Number of Turns

1x106

1.006

Number of Turns

01x10

Number of Turns

1x10

65

REFERENCES(d)(c)[1] O.S. Bruning and F. Willeke, Particle Accelerator, Vol.54,p.237, (1996).

1.3

1.3

σy6

Figure 4: The vertical beam sizes normalized by  yo verse
the modulation amplitudes A s =0,0.005,0.01 and 0.015, respectively, due to the synchrotron oscillation.

(b)
1.3

1x10

1.00(a)Number of Turns

(d)

(c)
1.2

1.0

1.306

σy

1.2

1.2

[2] K. Hirata, H. Moshammer and F. Ruggiero, Particle Accelerator, Vol. 40, p.205-228 (1993).

1.1

1.1
1.0

1.000

Number of Turns

1x106

Number of Turns

1x106Figure 2: The vertical beam sizes normalized by  yo verse
the modulation amplitudes A r =0,0.005,0.01 and 0.015, respectively, at the 100 Hz ripple frequency.

[3] K. Hirata, H. Moshammer, F. Ruggiero and M. Bassetti,
CERN SL-AP/90-02 (1990).