Effects of tune modulation on the long-term stability

EFFECTS OF TUNE MODULATIONS ON THE LONG-TERM STABILITY
IN 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 <strong>tune</strong>
due to the synchrotron oscillation can be written as
The combined effects <strong>of</strong> <strong>tune</strong> <strong>modulation</strong>s and beam-beam
interaction are investigated in terms <strong>of</strong> the long-term stability
in an electron-positron storage rings. The <strong>tune</strong> <strong>modulation</strong>s
due to power supply ripple and synchrotron oscillation
are studied for the case <strong>of</strong> KEKB. The <strong>tune</strong>s are
sinusoidally modulated at different amplitudes <strong>of</strong> the ripples
and synchrotron oscillations. It is shown that choices
<strong>of</strong> the nominal <strong>tune</strong> greatly affect the growth <strong>of</strong> beam size
through the <strong>tune</strong> <strong>modulation</strong> and the beam-beam interaction.
It is shown that the effect <strong>of</strong> a <strong>tune</strong> <strong>modulation</strong> due
to the ripple is stronger than that <strong>of</strong> <strong>tune</strong> <strong>modulation</strong> due to
the synchrotron oscillation.
1 INTRODUCTION
The long-term stability <strong>of</strong> particles in a storage ring may
be affected by various factors. To some degrees it is determined
by the nominal <strong>tune</strong> <strong>of</strong> the machine and the strengths
<strong>of</strong> 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 <strong>of</strong> HERA [1]. The beam-beam interaction
also imposes limitations on the performances <strong>of</strong> the colliding
rings. Here we investigate the combined effects <strong>of</strong>
<strong>tune</strong> <strong>modulation</strong>s due to the power supply ripple and the
synchrotron oscillation on the beam-beam interaction. For
this purpose, a tracking studies are performed by the using
<strong>of</strong> KEKB parameters which are being commissioning and
tracking runs <strong>of</strong> ¢ turns that correspond to 10 sec in
actual machine operation <strong>of</strong> the KEKB.
We model the ripple as a <strong>modulation</strong> <strong>of</strong> the <strong>tune</strong> at the
ripple frequency. For this work, variations <strong>of</strong> beam size are
studied by tracking particles for a million turn. Here we
investigate the effects <strong>of</strong> modulating in the vertical <strong>tune</strong>.
The change in the vertical <strong>tune</strong> is given by
Ý ÝÓ Ö Ó× ÖØ (1)
where ÝÓ is the nominal <strong>tune</strong> <strong>of</strong> the machine in the absence
<strong>of</strong> ripple. The ripple is represented by Ö, the amplitude
<strong>of</strong> the <strong>tune</strong> <strong>modulation</strong>, Ö the frequency <strong>of</strong> the <strong>modulation</strong>
and Ø time for one turn in a ring. We study the effect
at different ripple frequencies due to the presence <strong>of</strong> different
harmonics. 50 Hz is considered as the main ripple
frequency. In addition to this, we study the effects at 100
Hz as another harmonic <strong>of</strong> the main ripple frequencies to
consider effect <strong>of</strong> high frequency. We also study the variation
<strong>of</strong> the beam size with the amplitude <strong>of</strong> ripple at a fixed
ripple frequency.
Next we investiagte the effects <strong>of</strong> <strong>tune</strong> <strong>modulation</strong> due to
Ý ÝÓ ××Ò ×Ø (2)
where ÝÓ is the nominal <strong>tune</strong> <strong>of</strong> the machine. The amplitude
<strong>of</strong> <strong>tune</strong> <strong>modulation</strong> is represented by ×, × the synchrotron
<strong>tune</strong> and the time Ø is measured in turn number.
It is shown that the <strong>tune</strong> <strong>modulation</strong>s that are considered
above may result in growth <strong>of</strong> beam size in the KEKB
machine. The extent <strong>of</strong> the growth depends on choice <strong>of</strong>
the nominal <strong>tune</strong> <strong>of</strong> the machine and also on the amplitude
and frequency <strong>of</strong> the <strong>modulation</strong>. In result, our simulations
show that <strong>tune</strong> <strong>modulation</strong> may be a source <strong>of</strong>
emittance growth and luminosity degradation in electronpositron
storage rings.
Table 1: Parameters that are used in the simulations
Parameter Unit Value
Beam Energy Ó 8 GeV
Emittance ¯ 3.6¢ m
Cicumference 3016 m
Energy Spread Æ 6.7¢
Bunch Length Þ 5.6 mm
Beam-beam <strong>tune</strong> shift 0.052
Beta function at IP ¬ÁÈ 1.1 cm
Synchrotron <strong>tune</strong> × 0.0114
Betatron <strong>tune</strong> ÝÓ 42.14
2 SIMULATION
Equations <strong>of</strong> motion for a particle is described by introducing
scaled coordinates in transverse and longitudinal phase
spaces. With × denoting the longitudinal coordinate, they
are given by
Ý ×
Ý ×
× × ¬ÁÈ
Ý
Þ
Ý
Þ ×
¯ ×
× × (3)
where ¬ÁÈ represents the betatron function at the interaction
point(IP) and Ý , Þ and ¯ are the nominal rms
values at the IP <strong>of</strong> the transverse beam size, the bunch
length and the relative energy spread, respectively. In the
above equation, Ý and Ý are the position and the slope <strong>of</strong>
a macroparticle in a beam in the transverse direction, respectively,
and ¯ is the relative energy deviation from the
nominal energy [¯ ]. We use
¯
Þ × Ø × (4)

where is the speed <strong>of</strong> light, and Ø is the difference in the
arrival time at × between the particle under consideration
and the reference particle. Thus, Þ × indicates that
the particle arrives at × earlier than the reference particle.
The evolutions <strong>of</strong> these normalized parameters in the
ring can be treated separately at the IP and around the arc
<strong>of</strong> the ring. First, at the IP, the change in scaled coordinates
due to the beam-beam interaction is given by [2, 3]
Ê £ ÆÈ
ÆÈ
Ê ÆÈ
ÆÈ (5)
Here, the beam-beam interaction is modeled by the interaction
between a macroparticle in a weak beam and one in a
strong beam with several slices. The center <strong>of</strong> each slice is
represented by the longitudinal position Þ £. In eq. (5), we
have defined
ÆÈ Ò£
Ò£
Æ£ Ê £
Ê Ê £
Æ£ Ê £
Ê £
Ê £
Ô
Ê Þ
Ê
£ Þ£
Þ
¬ÁÈ
¯
¯ ¬ÁÈ
(6)
Here, is the nominal beam-beam parameter defined by
Æ£Ö
­¯
(7)
where ¯ is the transverse nominal emittance <strong>of</strong> the strong
beam, Ö is the classical electron radius and ­ is the usual
relativistic factor. More detailed discussions <strong>of</strong> 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 £.
Here, £ is the longitudinal barycenter <strong>of</strong> each slice in the
strong beam and each barycenter represents Ò £Æ£ <strong>of</strong> all
particles in the strong beam where ƣ and ң are the numbers
<strong>of</strong> 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:
Ó× Ý ×Ò Ý
where Ý is the vertical <strong>tune</strong>.
×Ò Ý Ó× Ý
(8)
Synchrotron radiation is given by
Ý
Õ Ý Ö Ý
Õ Ý Ö
× Ô ×
Ö (9)
where Ö × are independent, random Gaussian variables with
unit standard derivation. Here the is damping factors defined
by
Ý ÌÝ × Ì× (10)
where the Ì denotes damping times normalized by the revolution
time.
3 RESULTS OF THE SIMULATION
3.1 Tune <strong>modulation</strong> due to the power supply
ripple
Figure 1 shows the vertical beam sizes normalized by ÝÓ
verse the <strong>modulation</strong> amplitudes Ö× at the 50 Hz ripple
frequency and ÝÓ . Figure 1(a) shows the case
when there is no <strong>tune</strong> <strong>modulation</strong>. Figure 1(b),(c) and (d)
correspond to the cases <strong>of</strong> Ö=0.005, 0.01 and 0.015, respectively.
Figure 2(a),(b),(c) and (d) show the vertical
beam sizes normalized by ÝÓ in Ö=0, 0.005, 0.01 and
0.015, respectively, at the 100 Hz ripple frequency and
ÝÓ=42.14. Figure 3 shows the vertical beam sizes normalized
by ÝÓ verse the nominal <strong>tune</strong> variation which are
held constant at Ö=0.02 and 50 Hz ripple frequency. Figure
3(a),(b),(c) and (d)correspond to the case <strong>of</strong> ÝÓ=42.11,
42.12, 42.13 and 42.15, repectively.
The larger <strong>tune</strong> <strong>modulation</strong> ( Ö) has stronger effects.
The ripple frequency <strong>of</strong> 100 Hz shows larger growth in
beam size than that <strong>of</strong> 50 Hz. We see that the growth <strong>of</strong>
beam size also depends on the choices <strong>of</strong> the nominal <strong>tune</strong>.
3.2 Tune <strong>modulation</strong> due to the synchrotron oscillation
Figure 4 shows the vertical beam sizes normalized by ÝÓ
over the <strong>modulation</strong> amplitudes × and Ý=42.14. Figure
4(a),(b),(c) and (d) correspond to the cases <strong>of</strong> × =0,
0.005, 0.01, 0.015 and 0.02, respectively. The result for the
effect <strong>of</strong> a <strong>tune</strong> <strong>modulation</strong> due to the synchrotron oscillation
may be roughly represented by
Ý ÝÓ × (11)
The vertical beam size is varied linearly with the amplitude
<strong>of</strong> the <strong>tune</strong> <strong>modulation</strong>.
4 RESULTS
The KEKB parameters are used to investigate the effects
<strong>of</strong> <strong>tune</strong> <strong>modulation</strong>s that are generated by the power supply
ripple and the synchrotron oscillation. Under current
operating conditions <strong>of</strong> the KEKB, the beam sizes may be

affected by the combined effects <strong>of</strong> the <strong>tune</strong> <strong>modulation</strong>s
and beam-beam interaction, depending on the strengths <strong>of</strong>
<strong>modulation</strong> amplitudes. Our simulations also show that the
influence <strong>of</strong> the <strong>tune</strong> <strong>modulation</strong> on the beam size depends
on the operation <strong>tune</strong>. It is shown that <strong>tune</strong> <strong>modulation</strong> due
to power supply ripple results in larger effects in growth
<strong>of</strong> the vertical beam size than <strong>tune</strong> <strong>modulation</strong> due to the
synchrotron oscillation. In our simulation, the effects <strong>of</strong>
amplitude and frequency <strong>of</strong> the <strong>modulation</strong> are also investigated.
It is shown that the <strong>tune</strong> <strong>modulation</strong>s may have
important effects on the motions <strong>of</strong> particles to nonlinearities
in electron-positron storage ring.
σ
1.4
y1.2
1.0
1.4
1.0
(a)
0 1x10 6
Number <strong>of</strong> Turns
(c)
0 1x10 6
Number <strong>of</strong> Turns
1.4
y 1.2
1.0
1.4
σy 1.2
σy
1.2
1.0
(b)
0 1x10 6
Number <strong>of</strong> Turns
(d)
0 1x10 6
Number <strong>of</strong> Turns
Figure 1: The vertical beam sizes normalized by ÝÓ verse
the <strong>modulation</strong> amplitudes Ö=0,0.005,0.01 and 0.015, respectively,
at the 50 Hz ripple frequency.
1.3
1.2
y
1.1
1.0
1.3
1.2
y
1.1
1.0
(a)
σ σ
σ
0 1x10 6
Number <strong>of</strong> Turns
(c)
0 1x10 6
Number <strong>of</strong> Turns
1.3
1.2
y
1.1
1.0
1.3
1.2
y
1.1
1.0
(b)
0 1x10 6
Number <strong>of</strong> Turns
(d)
0 1x10 6
Number <strong>of</strong> Turns
Figure 2: The vertical beam sizes normalized by ÝÓ verse
the <strong>modulation</strong> amplitudes Ö=0,0.005,0.01 and 0.015, respectively,
at the 100 Hz ripple frequency.
σ
σ
1.3
1.2
y
1.1
1.0
(a)
σ σ
0 1x10 6
0.9
Number <strong>of</strong> Turns
σ
1.3
1.2
y
1.1
1.0
(c)
0 1x10 6
0.9
Number <strong>of</strong> Turns
1.3
1.2
y
1.1
1.0
(b)
0 1x10 6
0.9
Number <strong>of</strong> Turns
1.3
1.2
y1.1
1.0
(d)
0 1x10 6
0.9
Number <strong>of</strong> Turns
Figure 3: The vertical beam sizes normalized by ÝÓ verse
the <strong>modulation</strong> amplitudes Ö and 50 Hz ripple frequency.
Figure 3(a),(b),(c) and (d) correspond to the case
<strong>of</strong> ÝÓ=42.11, 42.12, 42.13 and 42.15, respectively.
σ
1.2
1.1
y
σ
1.0
1.2
1.1
y
1.0
(a)
0 1x10 6
(c)
0 1x10 6
Number <strong>of</strong> Turns
1.2
y 1.1
1.0
1.2
y 1.1
1.0
(b)
0 1x10 6
Number <strong>of</strong> Turns
(d)
0 1x10 6
Number <strong>of</strong> Turns
Figure 4: The vertical beam sizes normalized by ÝÓ verse
the <strong>modulation</strong> amplitudes ×=0,0.005,0.01 and 0.015, respectively,
due to the synchrotron oscillation.
5 REFERENCES
[1] O.S. Bruning and F. Willeke, Particle Accelerator, Vol.54,
p.237, (1996).
[2] K. Hirata, H. Moshammer and F. Ruggiero, Particle Accelerator,
Vol. 40, p.205-228 (1993).
[3] K. Hirata, H. Moshammer, F. Ruggiero and M. Bassetti,
CERN SL-AP/90-02 (1990).
σ
σ
σ