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Landau damping effect due to the trapped ions in a compact electron storage ring
_i!B
-H JLa
Nuclear Instrumentsand Methods in Physics Research A 370 ( 1996) 323-329
NUCLEAR
INSTRUMENTS BMETNODS IN PHYSICS
_-f
“E:%Y”
ELSEVIER
Landau damping effect due to the trapped ions in a compact electron storage ring K. Yamada*, M. Nakajima, T. Hosokawa Received 6 June 1995; revised form received 30 August 1995
Abstract
Theoretical analysis and experimental results show that the Landau damping due to trapped ions is greatly enhanced in the low-energy beams in compact electron storage rings. This enhancement originates mainly from the decrease in beam size in the low-energy region. This effect must indeed be the main reason why the recently developed compact rings can store a large current even though they have a low injection energy. In the typical energy region of compact rings, more positive utilization of this effect could make the damping rate of a lower-energy beam far larger than that of a higher-energy one. Therefore, if we could control the effect of the trapped ions, we might prefer to adopt lower energy in order to inject a large current in a compact ring.
1. Introduction
Compact storage rings for synchrotron radiation (SR) have been constructed recently [ 1,2]. At the beginning of their development, it had seemed that these machines would not store large currents because of their low injection energy (about 200 MeV) Lower-energy electrons are indeed less massive, and their motions are therefore easily disturbed by the electromagnetic fields originating from the beams themselves. This means that low-energy beams are very likely to fall into strong instabilities. Furthermore, the weaker radiation damping must aggravate the instabilities. Octupole magnets are effective in suppressing transverse instabilities, but the strong octupole field required for suppressing strong instability reduces the dynamic aperture. This effect will degrade the injection efficiency. Despite these pessimistic anticipations, recent compact storage rings can store more than 300-mA beams without serious instabilities. This means there are mechanisms stabilizing low-energy beams, and one possible mechanism is the Landau damping effect caused by the trapped ions. The biggest advantage of exploiting this mechanism is that it would not affect the dynamic aperture. Moreover, the increase in the instability growth rate with increasing stored current would be canceled out by this mechanism because the field of the trapped ions is also proportional to the stored current.
The instability suppression due to the trapped ions has been reported briefly in only few papers [ 3.41, and there is no report of detailed analysis of this effect in compact rings. In the present work, we show that the beams in a compact storage ring are strongly stabilized by the effect of the trapped ions. We do this by first analyzing the damping effect due to the trapped ions theoretically and showing various features of this mechanism, and then by presenting the results of some experiments using NIT3 electron storage ring [ 51. We, furthermore, show the guidelines for the optimum injection energy from the viewpoint of the stability mechanism including the effect of the trapped ions.
2. Theoretical analysis 2. I. Octupole component originating from the field of trapped ions The geometry we consider first is illustrated in Fig. I. The bunched electron beams have an average current of f,. and the cross section of an electron beam is elliptical. Ions are trapped along the orbit of the electron beams, and the cross section of the ion cloud is usually assumed to be the same as that of the electron beams. The line density of the ion charge is expressed by I\.
=
llle c
* Corresponding author. Tel. +81 462 40 2604. fax +81 462 40 4324. e-mail [email protected]. 0168-9002/96/$15.00 @ 1996 Elsevier Science B.V. All tights reserved SSD/Ol68-9002(95)00843-S
’
where v is the neutralization factor and c is the speed of light. The cross-sectional charge distribution K is assumed
324
K. Yamada et al./Nucl.
Instr. and Meth. in Phys. Res. A 370 (1996)
323-329
2.2. Damping rate of the transverse motions The damping rate 1/rx,dampof betatron oscillation of electron beams is the sum of the radiation damping 1/~~.,~dand Landau damping 1/7x.~o~u originating from the tune spread of betatron oscillation:
-=-+-1 7x.damp
Fig.
I. Coordinates
1
1 7rrad
7x.Lmdau
=-+- 1
WPQXl
7xsad
2rl
'
(6)
where SQXis the tune spread and 00 is the revolution frequency. The origin of SQXis the octupole focusing force. By modifying the result of the perturbation theory for nonlinear resonance [ 81, we can obtain the following rough approximation of SQ,:
for an electron beam and the trapped ions.
to be Gaussian:
C
K=K,,exp( -;{(y+
SQx N
(;)*})>
2
I
&(s)
2
kxr((s) ds,
0
where Ka = hi/2rab. The electric potential U of this twodimensional beam can be expressed by following equation r6.71.
where lXis the emittance in the x direction, C is the circumference of the ring, fix is the amplitude function, and k,.. is the octupole focusing force. The force can be separated into four terms:
(I=- a&
kz4 = kd.maa+ kd,ion + kx4,sc+ kx4,uf. dt
2EO n
.
where EC)is the dielectric constant of vacuum. For simplicity, we consider only the x motion of electrons on the y = 0 plane, and we expand the exponential term of Eq. (3) into the fourth-order polynominal. Then we obtain the following equation: 1
ab& 2EO
(a* + t)‘/*(b* + t)l/* 0 -
(a* + t)3/2(b* + t)ll*
+ 2(a* + t)s/*(b* + t)‘/*
kx4,ion
_ 1 a”4 x4’mag 6( Bp) %x3 x=o.y=a ’
’ -y2Fh7)
k;X4,0".
(10)
Here
1
dt.
(4)
The octupole component of the electric field Ex4,ionis the gradient of the last term of this equation, so we can obtain t x: octupole focusing force due to the trapped ions as eqL
k
k x4.x -
fx’
$X4
kx4,,,and kx4.W are respectively the octupole where k14,mag, focusing force of the octupole magnets, the space charge effect of the electrons, and the force due to unknown octupole components in the ring. Here kx+,,ag is expressed by
where BYis the y component of the field of the magnet and (Bp) is the momentum of the electrons divided by e. For kr+, using similar treatment to kxd,ion, we can obtain the following equation:
M
U--
(8)
(3)
(a2 + t)‘/2(b2 + t)‘/2
2a+b 3 x3 _ eEx4sion __ 49reym,c3 3(a+ b)2a3x ’ ym,c*
Fb = fiU*fRF 2c
is the bunching factor for the Gaussian bunch in which u5 iS the bunch length and fRF is the acceleration frequency. Substituting these kx4 terms into Eq. (6). we obtain 1 1 --+%,A+ rX.damp
(5)
where m, is the rest mass of an electron, e is the elemental charge, and y is the electron energy in units of the electron rest mass. This force can be one origin of the Landau damp ing of transverse motions.
(11)
7xmd
BI, + RI, (12)
where c (13)
K. Yamada er al./Nucl.
instr. and Merh. in Phys. Res. A 370 (1996)
Photo-absorber
323-329
Octupole mactnet avity
Photo-absorber
lnflector (530 MeV)
bending magnet Clearing electrode
Fig. 2. The superconducting storage ring Super-ALIS.
(14)
This leads to decreasing threshold current. For A + R < 0. however, the effects of magnetic field and ions are summed, and the threshold current increases. This condition makes the necessatyoctupole magnetic field small and enlarges the dynamic aperture.
c
e 47re,,ymrc~
B=
s
2a + b
P:3(a+
b)W
ds
(15)
.
0
2.3. Suppression of the transverse instability Here we consider the case A + R = 0, where the storage ring has no octupole components originating from magnets or the effect of the uncertain octupole components in the ring is canceled by the octupole magnets. In this case, the stability condition is as follows: 1
I
fr,damp
T.r.,xd
-=-+
where I /T,r.instis the growth rate of instabilities. Because this growth rate is proportional to the beam current, When 71satisfies the equation (17) the stability condition Eq. (16) can be always satisfied independent of I,. In other words, the beam is stable whatever may be the increase of the beam current.
2.5. Estimation of the damping rates in the super-AL/J storage ring
In this section, we show that the main stabilizing mechanism in compact rings is the effect of the trapped ions. In estimating this effect in a real machine, we take as a model the electron storage ring Super-ALIS. 2.51. Super-ALIS Super-ALE is a race-track type electron storage ring. The schematic is shown in Fig. 2. This ring has a maximum electron energy of 600 MeV. It has one octupote magnet for Landau damping, and it has two pairs of ion clearing electrodes. The photo-absorbers in the bending chambers can also be used as clearing electrodes. The injection system can be selected either 15-MeV Linac or NAR [9] ring as a synchrotron in which the extraction energy is 530 MeV. 2.5.2. Damping rates Here we estimate the damping rate under two different conditions. One is for 200-MeV beams, which is the typical injection energy of compact rings, and the other is 530 MeV Table I Design parametersof Super-ALIS 200 MeV
2.4. Cooperative e$ect between thefield and the field of the octupole magnets
of trapped ions
For the cooperation between the field of the trapped ions and that of the octupole magnets, two conditions must be considered: A + R > 0 and A + R < 0. For A + R > 0, in which the effect of magnetic field is in the direction opposite to the effect of ions, these effects are canceled.
Betatron tune
QX
w1.59 NO.55
Q? Natural emittance RF voltage RF frequency
ET [rmradl
Bunch length Circumference
0~ Iml C =2nR
kF ~RF
530 MeV
lkvl [MHz1 [ml
9.87 x 10-a 13.3 124.950
6.93 x 1o-7 35.3
2.33 x IO-* 16.8
5.92 x 10-Z
K. Yamada et al./Nucl.
326
Instr. and Meth. in Phw. Res. A 370 (1996)
323-329
3.1. Measurement of the neutralization factor and the growth rate of the instability We measured the neutralization factor using the tune shift method [ lo]. The averaged neutralization factor (T]) can be calculated by measuring the horizontal tune shift AQ.l:
In this equation lT is the transverse emittance that is the sum of the emittances of the x and y directions, and (f) is expressed by Fig.
3. Amplitude
respectively
functions
show calculated
the values measured
of Super-ALIS.
The
solid
and dotted
values in the x and y directions.
at the quadrapole
lines
(19)
o and A are
magnets.
beams, the injection energy of this machine. The machine parameters for these energies are listed in Table 1. The amplitude functions /3+ and fly are shown in Fig. 3 with solid and dotted lines, and calculated damping rates for lOO-mA, 0.5% coupled, 77= 0.001 beams are summarized in Table 2. In 200-MeV beams, half of the ions’ contribution is canceled by the space charge effect of the beams, but the damping rate due to the beams themselves is very high: about two hundred times larger than the rate of the radiation damping. The large damping rate is mainly due to a small beam size in this energy. The focusing force kr4,mapl of the octupole magnet equivalent to this ion effect is -77 rnm3. We should notice that such a strong octupole magnet limits the horizontal dynamic aperture to about f6 mm at the bending centers and makes injection difficult. In .530-MeV beams, the effect of the trapped ions decreases, but it is still comparable to the effect of the radiation damping. Consequently, we conclude that the effect of the trapped ions can be the main reason for compact rings storing large current even though their injection energy is low.
3. Observing tbe effect of trapped ions in super-ALE
where K is the coupling coefficient. We take this averaged value as 77in this work. The emittance ET can be calculated from the beam size and amplitude functions ,& and & These functions can be easily measured at quadrupole magnets by perturbing their focusing force. The measured amplitude functions shown in Fig. 3 agree well with the designed ones at quadrupoles. Therefore, the amplitude functions in the bending magnet can be expected to agree with the designed ones. We measured the beam size by numerically processing the video signals from a CCD camera monitoring the beam profile of the BL8 SR port. The measured transverse emittance in 1OOmA beams is shown in Fig. 4 as a function of the voltage on the clearing electrodes. The polarity of the voltage for clearing ions was negative. The coupling coefficient is also shown in this figure. When the clearing voltage is large, or the ions are cleared, the coupling is very small. The emittance is about 6.3 x 10-‘7r m rad which is very close to the calculated natural emittance. As shown in Fig. 5 (top), the beam profile in this condition is flat. As the ions are trapped more, the coupling increases while the emittance does not change. As shown in Fig. 5( bottom), the beam profile thus becomes round.
In this section, we confirm the effect of the trapped ions by examining the suppression of a horizontal coupled bunch instability in Super-ALE’s 530-MeV injection. This instability is excited by the TM1 10 cavity mode and would seriously limit the beam current. Table
2
Estimated
Damping
damping
Rate
rates in Super-ALIS
[s-’
1l~,.r*d I ’ /~,.,on- 1/w.c I Total
1
(7
= 0.001)
200 MeV
530 MeV
2.92
54.3
11.18 x IO3 -
6.30
x
10’1
163.2 -
I.771
= 5.50
x I02
= 61.4
5.53
102
1.16 x lo2
x
Fig. 4. Measured of the voltage
transverse emittance
on the clearing
and coupling
electrodes.
coefficient
as functions
K. Yumada et al./Nucl.
Instr. and Meth. in Phw. Rex A 370 (1996)
323-329
327
Fig. 7. Measured growth rate of the transvme the beam current. TMIIO
(The
instability
is a coupled
instability bunch
as a
instabllity
function of excited
by
mode.)
varies with the beam current. However, the factor in each current condition increases to about 0.02 when the clearing voltage goes to zero. The growth rate of the instability were measured by the following method. From Eq. ( 12) the following two threshold conditions can be obtained:
for A+ -t R > - (1 /y*Fh - q) Bf,. and
Fig. 5. Beams profile I\ and
-100
mA.
and
at the center of the bending the voltage
on clearing
magnet.
electrodes
The beam current is (top)
-1000
V
(21)
(hottom)0 V. The measured
71 is shown in Fig. 6 as a function of the voltage on the clearing electrodes. In the measurements at I, = 100 and 200 mA, the octupole magnet with kx4.mag1N -3.0 m-’ was excited for suppressing instability. The value of v can be reduced to about IO-” by applying adequate clearing voltage. The voltage giving this neutralization factor
5
R. i
& g
*015
i
i
‘._, ‘.,
’ !,
for A_ + R < -( I/$/$, - 7) St,. We assume here A+ > A_. From these equations we obtain I ,/T,.,~\,as 1
3wn~.r A+ - A_
I
-=3572+&. G.inat
(22)
A+ and A- can be calculated from the current of the octupole magnet at the threshold conditions. The measured I /TV.,,,., is shown in Fig. 7 as a function of 1,. The proportional constant (Yof the growth rate to the beam current is - 1.5 x lo3 s-’ A-‘. Substituting the measured growth rate into Eq. ( 17). WC find that a neutralization factor of 2.3 x IO-’ is needed for suppressing this instability. Because this value can easily be achieved by reducing the clearing voltage. the instability can be suppressed by the field of the trapped ions without using octupole magnets. 3.2. Observing the instability suppression To observe the suppression of the instability, we measured the threshold beam current as a function of the clearing electrodes’ voltage. In this measurement, the octupole magnet
328
Fig. 8. Threshold
K. Yamodu ef al./Nucl.
beam current ~LSa function
Insrr. and Meth. in Phys. Res. A 370 (1996)
of the voltage on the clearing
electrodes.
was not excited. As shown in Fig. 8, for clearing voltages over 200 V the threshold current was less than 100 mA. For voltage below 200 V, however, it increased to about 400 mA. This means that the condition specified by Eq. ( 17) is satisfied and the instability is suppressed regardless of the beam current. Notice, however, that the threshold current gradually decreases again as the clearing voltage decreases further. This threshold is not determined by the instability. This degradation is due to the scattering of the electron beams by the too-dense trapped ions. This scattering makes the beam lifetime short, and the equilibrium current determined by the injection rate and the lifetime thereby becomes lower. To observe the cooperative effect of the trapped ions and the octupole magnet, we measured the threshold current as a function of the octupole magnet strength. In this measurement, we made the clearing voltage as high as possible in order to keep the effect of the trapped ions from concealing the effect of the octupole magnet. Throughout this experiment the neutralization factor was about -2 x 10m3. The results are shown in Fig. 9, where the positive octupole current corresponds to the same direction as the effect of the trapped ions. The octupole current of 1.3 A, giving the minimum threshold current, corresponds to the cancellation of
323-329
the octupole magnet and the uncertain octupole components in the ring. The behavior of the threshold current in the regions above 1.3 A differs from that below this octupole current. Below an octupole current of 1.3 A the threshold current gradually increases as the octupole current decreases. Above the octupole current of 1.3 A, however, the threshold beam current jumps to more than 350 mA. Even beams over 1200 mA were stable [ 111. This extremely asymmetric behavior is quite different from the symmetric behavior observed in a similar experiment in which positrons were stored and there were no trapped ions [ 121. Therefore, our asymmetric result apparently shows the cooperative effect of the trapped ions and octupole magnets. This result means that the effect of the trapped ions must not be neglected even if the neutralization factor is as small as 0.001. This result also indicates that, if we would use octupole magnets, we should excite the octupole magnet in the same direction as that in which the effect of the trapped ions works. This policy will reduce the focusing force of octupole magnets causing the decrease of dynamic aperture.
4. Discussion Considering the effect of the trapped ions, we can show guidelines to determining the injection energy. These are important from the viewpoint of economy because a lowenergy injector results in a less expensive SR systems. For simplicity, especially to reduce the integration terms to simple functions, we use the round-beam approximation and assume that there are no octupole components originating from magnets. Substituting A + R = 0, and a = b - m into Eq. ( 12), we obtain the following relation for the damping rate: 1
1 -=-+ Tx.damp
‘bad
3woQxeL 256rr22Etjym,c3e,
II) - &I.
(23)
In this simplification, we used the following relation:
(24)
In Super-ALIS, the emittance and the radiation damping rate can be expressed as simple functions of the energy E: E+[ n-m rad] = 2.46 x lo-“E[
MeV12
(25)
and -&[
l/set]
= 3.64 x IO-‘E[MeV]s.
(26)
For the bunching factor, we assume the RF voltage as Fig. 9. Threshold magnet.
beam current as a function
of the strength of the octupole
&r[kV] = 6.67 x IO-*E[MeV].
(27)
K. Yumoda er ol./Nucl.
Instr. and Mrth.
in Phys. Res. A 370 (1996)
323-329
329
the damping rate of a lower-energy beam far larger than that of a higher-energy one. Therefore, if the neutralization factor could be controlled, we might prefer to adopt lower energy in order to inject a large current in compact storage rings.
5. Conclusion
1
,
Fig.
IO.
7 = 0.01.
Survey (b)
of
the
/
energy
1
I
dependence
/
of
I
i
the damping
rates:
(a)
The Landau damping due to the trapped ions was shown here to be very strong in compact rings. This effect is so strong that we must not neglect its contribution to the beams even when the neutralization factor is very low. This effect. furthermore, results in both strong instability suppression and large dynamic apertures at the same time. These results indicate that this effect must be the main reason why the compact storage rings store large currents in spite of their low injection energy. These results also indicate that lower energy might be preferable in injecting a large current in compact storage rings if we could control the effect of trapped ions. In such a positive utilization of trapped-ion effect, however, the following remarks must be considered; the use of this effect should be restricted only to the injection. After injection, we should use other methods for suppressing instabilities and clear up the ions in order to increase the beam lifetime.
7 = 0.001.
The energy dependence of the calculated damping rates is shown in Fig. 10. The energy range considered here is from 100 MeV to 600 MeV because the radiation damping becomes so weak below 100 MeV that the emittance will not be expressed by Eq. (25). Fig. 1Oashows the energy dependenceof the damping rate in 7 = 0.0 1case. In this case, the total damping rate increases as the energy decreases. This increase in total damping rate can be explained by the trapped ions’ effect that is much enhanced by the decrease of the transverse emittance in the low energy region. In Fig. 1Ob, showing the energy dependence of the damping rate in 77= 0.001 case, we can see the extreme decrease of the total damping rate in the region from 100 MeV to 200 MeV. This decrease correspond to the mutual cancellation of the effect of the trapped ions and space charge. In this energy region, therefore, the beams will be extremely unstable. Over 200 MeV, however, the total damping rate is nearly constant up to 600 MeV and the beams will be stable. We thus concluded that there is no problem in reducing injection energy to 200 MeV even if the neutralization factor is as low as 0.001. Furthermore, as described in the 77= 0.01 result. the more positive utilization of this effect could make
References
[I]
T. Takayama 60 (1989)
121 M.N.
Wilson,
Townsend, D.E.
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Symp.
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AS.
Bhutta
MicroProcess
and
Conf.
p. 104. S. Guiducci.
1Ith Int.
Conf.
A. Jackson,
T.
Research
Jordan, R.J. Anderson, Proc.
Biagini,
Walker, [5]
Accelerator
A.I.C.
A.R.
Andrew,
(1989) [3]
and SHI 1786.
N.
IEEE
Marks,
T.
K.
Instr. 60 ( 1989)
1783.
[6]
B. Zotter,
85-19
[7]
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Accelerators
Poole.
T.
( 1983)
Hayasaka.
Nishimura
(1989)
on High
( 1980)
G. Saxon,
Sci. NS-30
Kitayama,
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Preger, M. Serio and S. Tazzari.
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and M.
Proc.
p. 687.
V.P. Suller
and R.P
2522. S.
Ido.
Y.
Nakajima.
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A.
Rev.
SCI.
p. 253. Energy
Accelerators.
KEK.
OH0’86
(1986). (81
E.J.N.
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121 S. Sakanaka. Part. Accel.
T.
K. Nishimura
T. Kasuga, H. Y0nehara.T. Phys. 24 (1985)
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(1987)
Kinoshita
Synch<. Radiat. M.
and M. Hasumoto.
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and H.
Katoh
Kobayakawa,
-H JLa
Nuclear Instrumentsand Methods in Physics Research A 370 ( 1996) 323-329
NUCLEAR
INSTRUMENTS BMETNODS IN PHYSICS
_-f
“E:%Y”
ELSEVIER
Landau damping effect due to the trapped ions in a compact electron storage ring K. Yamada*, M. Nakajima, T. Hosokawa Received 6 June 1995; revised form received 30 August 1995
Abstract
Theoretical analysis and experimental results show that the Landau damping due to trapped ions is greatly enhanced in the low-energy beams in compact electron storage rings. This enhancement originates mainly from the decrease in beam size in the low-energy region. This effect must indeed be the main reason why the recently developed compact rings can store a large current even though they have a low injection energy. In the typical energy region of compact rings, more positive utilization of this effect could make the damping rate of a lower-energy beam far larger than that of a higher-energy one. Therefore, if we could control the effect of the trapped ions, we might prefer to adopt lower energy in order to inject a large current in a compact ring.
1. Introduction
Compact storage rings for synchrotron radiation (SR) have been constructed recently [ 1,2]. At the beginning of their development, it had seemed that these machines would not store large currents because of their low injection energy (about 200 MeV) Lower-energy electrons are indeed less massive, and their motions are therefore easily disturbed by the electromagnetic fields originating from the beams themselves. This means that low-energy beams are very likely to fall into strong instabilities. Furthermore, the weaker radiation damping must aggravate the instabilities. Octupole magnets are effective in suppressing transverse instabilities, but the strong octupole field required for suppressing strong instability reduces the dynamic aperture. This effect will degrade the injection efficiency. Despite these pessimistic anticipations, recent compact storage rings can store more than 300-mA beams without serious instabilities. This means there are mechanisms stabilizing low-energy beams, and one possible mechanism is the Landau damping effect caused by the trapped ions. The biggest advantage of exploiting this mechanism is that it would not affect the dynamic aperture. Moreover, the increase in the instability growth rate with increasing stored current would be canceled out by this mechanism because the field of the trapped ions is also proportional to the stored current.
The instability suppression due to the trapped ions has been reported briefly in only few papers [ 3.41, and there is no report of detailed analysis of this effect in compact rings. In the present work, we show that the beams in a compact storage ring are strongly stabilized by the effect of the trapped ions. We do this by first analyzing the damping effect due to the trapped ions theoretically and showing various features of this mechanism, and then by presenting the results of some experiments using NIT3 electron storage ring [ 51. We, furthermore, show the guidelines for the optimum injection energy from the viewpoint of the stability mechanism including the effect of the trapped ions.
2. Theoretical analysis 2. I. Octupole component originating from the field of trapped ions The geometry we consider first is illustrated in Fig. I. The bunched electron beams have an average current of f,. and the cross section of an electron beam is elliptical. Ions are trapped along the orbit of the electron beams, and the cross section of the ion cloud is usually assumed to be the same as that of the electron beams. The line density of the ion charge is expressed by I\.
=
llle c
* Corresponding author. Tel. +81 462 40 2604. fax +81 462 40 4324. e-mail [email protected]. 0168-9002/96/$15.00 @ 1996 Elsevier Science B.V. All tights reserved SSD/Ol68-9002(95)00843-S
’
where v is the neutralization factor and c is the speed of light. The cross-sectional charge distribution K is assumed
324
K. Yamada et al./Nucl.
Instr. and Meth. in Phys. Res. A 370 (1996)
323-329
2.2. Damping rate of the transverse motions The damping rate 1/rx,dampof betatron oscillation of electron beams is the sum of the radiation damping 1/~~.,~dand Landau damping 1/7x.~o~u originating from the tune spread of betatron oscillation:
-=-+-1 7x.damp
Fig.
I. Coordinates
1
1 7rrad
7x.Lmdau
=-+- 1
WPQXl
7xsad
2rl
'
(6)
where SQXis the tune spread and 00 is the revolution frequency. The origin of SQXis the octupole focusing force. By modifying the result of the perturbation theory for nonlinear resonance [ 81, we can obtain the following rough approximation of SQ,:
for an electron beam and the trapped ions.
to be Gaussian:
C
K=K,,exp( -;{(y+
SQx N
(;)*})>
2
I
&(s)
2
kxr((s) ds,
0
where Ka = hi/2rab. The electric potential U of this twodimensional beam can be expressed by following equation r6.71.
where lXis the emittance in the x direction, C is the circumference of the ring, fix is the amplitude function, and k,.. is the octupole focusing force. The force can be separated into four terms:
(I=- a&
kz4 = kd.maa+ kd,ion + kx4,sc+ kx4,uf. dt
2EO n
.
where EC)is the dielectric constant of vacuum. For simplicity, we consider only the x motion of electrons on the y = 0 plane, and we expand the exponential term of Eq. (3) into the fourth-order polynominal. Then we obtain the following equation: 1
ab& 2EO
(a* + t)‘/*(b* + t)l/* 0 -
(a* + t)3/2(b* + t)ll*
+ 2(a* + t)s/*(b* + t)‘/*
kx4,ion
_ 1 a”4 x4’mag 6( Bp) %x3 x=o.y=a ’
’ -y2Fh7)
k;X4,0".
(10)
Here
1
dt.
(4)
The octupole component of the electric field Ex4,ionis the gradient of the last term of this equation, so we can obtain t x: octupole focusing force due to the trapped ions as eqL
k
k x4.x -
fx’
$X4
kx4,,,and kx4.W are respectively the octupole where k14,mag, focusing force of the octupole magnets, the space charge effect of the electrons, and the force due to unknown octupole components in the ring. Here kx+,,ag is expressed by
where BYis the y component of the field of the magnet and (Bp) is the momentum of the electrons divided by e. For kr+, using similar treatment to kxd,ion, we can obtain the following equation:
M
U--
(8)
(3)
(a2 + t)‘/2(b2 + t)‘/2
2a+b 3 x3 _ eEx4sion __ 49reym,c3 3(a+ b)2a3x ’ ym,c*
Fb = fiU*fRF 2c
is the bunching factor for the Gaussian bunch in which u5 iS the bunch length and fRF is the acceleration frequency. Substituting these kx4 terms into Eq. (6). we obtain 1 1 --+%,A+ rX.damp
(5)
where m, is the rest mass of an electron, e is the elemental charge, and y is the electron energy in units of the electron rest mass. This force can be one origin of the Landau damp ing of transverse motions.
(11)
7xmd
BI, + RI, (12)
where c (13)
K. Yamada er al./Nucl.
instr. and Merh. in Phys. Res. A 370 (1996)
Photo-absorber
323-329
Octupole mactnet avity
Photo-absorber
lnflector (530 MeV)
bending magnet Clearing electrode
Fig. 2. The superconducting storage ring Super-ALIS.
(14)
This leads to decreasing threshold current. For A + R < 0. however, the effects of magnetic field and ions are summed, and the threshold current increases. This condition makes the necessatyoctupole magnetic field small and enlarges the dynamic aperture.
c
e 47re,,ymrc~
B=
s
2a + b
P:3(a+
b)W
ds
(15)
.
0
2.3. Suppression of the transverse instability Here we consider the case A + R = 0, where the storage ring has no octupole components originating from magnets or the effect of the uncertain octupole components in the ring is canceled by the octupole magnets. In this case, the stability condition is as follows: 1
I
fr,damp
T.r.,xd
-=-+
where I /T,r.instis the growth rate of instabilities. Because this growth rate is proportional to the beam current, When 71satisfies the equation (17) the stability condition Eq. (16) can be always satisfied independent of I,. In other words, the beam is stable whatever may be the increase of the beam current.
2.5. Estimation of the damping rates in the super-AL/J storage ring
In this section, we show that the main stabilizing mechanism in compact rings is the effect of the trapped ions. In estimating this effect in a real machine, we take as a model the electron storage ring Super-ALIS. 2.51. Super-ALIS Super-ALE is a race-track type electron storage ring. The schematic is shown in Fig. 2. This ring has a maximum electron energy of 600 MeV. It has one octupote magnet for Landau damping, and it has two pairs of ion clearing electrodes. The photo-absorbers in the bending chambers can also be used as clearing electrodes. The injection system can be selected either 15-MeV Linac or NAR [9] ring as a synchrotron in which the extraction energy is 530 MeV. 2.5.2. Damping rates Here we estimate the damping rate under two different conditions. One is for 200-MeV beams, which is the typical injection energy of compact rings, and the other is 530 MeV Table I Design parametersof Super-ALIS 200 MeV
2.4. Cooperative e$ect between thefield and the field of the octupole magnets
of trapped ions
For the cooperation between the field of the trapped ions and that of the octupole magnets, two conditions must be considered: A + R > 0 and A + R < 0. For A + R > 0, in which the effect of magnetic field is in the direction opposite to the effect of ions, these effects are canceled.
Betatron tune
QX
w1.59 NO.55
Q? Natural emittance RF voltage RF frequency
ET [rmradl
Bunch length Circumference
0~ Iml C =2nR
kF ~RF
530 MeV
lkvl [MHz1 [ml
9.87 x 10-a 13.3 124.950
6.93 x 1o-7 35.3
2.33 x IO-* 16.8
5.92 x 10-Z
K. Yamada et al./Nucl.
326
Instr. and Meth. in Phw. Res. A 370 (1996)
323-329
3.1. Measurement of the neutralization factor and the growth rate of the instability We measured the neutralization factor using the tune shift method [ lo]. The averaged neutralization factor (T]) can be calculated by measuring the horizontal tune shift AQ.l:
In this equation lT is the transverse emittance that is the sum of the emittances of the x and y directions, and (f) is expressed by Fig.
3. Amplitude
respectively
functions
show calculated
the values measured
of Super-ALIS.
The
solid
and dotted
values in the x and y directions.
at the quadrapole
lines
(19)
o and A are
magnets.
beams, the injection energy of this machine. The machine parameters for these energies are listed in Table 1. The amplitude functions /3+ and fly are shown in Fig. 3 with solid and dotted lines, and calculated damping rates for lOO-mA, 0.5% coupled, 77= 0.001 beams are summarized in Table 2. In 200-MeV beams, half of the ions’ contribution is canceled by the space charge effect of the beams, but the damping rate due to the beams themselves is very high: about two hundred times larger than the rate of the radiation damping. The large damping rate is mainly due to a small beam size in this energy. The focusing force kr4,mapl of the octupole magnet equivalent to this ion effect is -77 rnm3. We should notice that such a strong octupole magnet limits the horizontal dynamic aperture to about f6 mm at the bending centers and makes injection difficult. In .530-MeV beams, the effect of the trapped ions decreases, but it is still comparable to the effect of the radiation damping. Consequently, we conclude that the effect of the trapped ions can be the main reason for compact rings storing large current even though their injection energy is low.
3. Observing tbe effect of trapped ions in super-ALE
where K is the coupling coefficient. We take this averaged value as 77in this work. The emittance ET can be calculated from the beam size and amplitude functions ,& and & These functions can be easily measured at quadrupole magnets by perturbing their focusing force. The measured amplitude functions shown in Fig. 3 agree well with the designed ones at quadrupoles. Therefore, the amplitude functions in the bending magnet can be expected to agree with the designed ones. We measured the beam size by numerically processing the video signals from a CCD camera monitoring the beam profile of the BL8 SR port. The measured transverse emittance in 1OOmA beams is shown in Fig. 4 as a function of the voltage on the clearing electrodes. The polarity of the voltage for clearing ions was negative. The coupling coefficient is also shown in this figure. When the clearing voltage is large, or the ions are cleared, the coupling is very small. The emittance is about 6.3 x 10-‘7r m rad which is very close to the calculated natural emittance. As shown in Fig. 5 (top), the beam profile in this condition is flat. As the ions are trapped more, the coupling increases while the emittance does not change. As shown in Fig. 5( bottom), the beam profile thus becomes round.
In this section, we confirm the effect of the trapped ions by examining the suppression of a horizontal coupled bunch instability in Super-ALE’s 530-MeV injection. This instability is excited by the TM1 10 cavity mode and would seriously limit the beam current. Table
2
Estimated
Damping
damping
Rate
rates in Super-ALIS
[s-’
1l~,.r*d I ’ /~,.,on- 1/w.c I Total
1
(7
= 0.001)
200 MeV
530 MeV
2.92
54.3
11.18 x IO3 -
6.30
x
10’1
163.2 -
I.771
= 5.50
x I02
= 61.4
5.53
102
1.16 x lo2
x
Fig. 4. Measured of the voltage
transverse emittance
on the clearing
and coupling
electrodes.
coefficient
as functions
K. Yumada et al./Nucl.
Instr. and Meth. in Phw. Rex A 370 (1996)
323-329
327
Fig. 7. Measured growth rate of the transvme the beam current. TMIIO
(The
instability
is a coupled
instability bunch
as a
instabllity
function of excited
by
mode.)
varies with the beam current. However, the factor in each current condition increases to about 0.02 when the clearing voltage goes to zero. The growth rate of the instability were measured by the following method. From Eq. ( 12) the following two threshold conditions can be obtained:
for A+ -t R > - (1 /y*Fh - q) Bf,. and
Fig. 5. Beams profile I\ and
-100
mA.
and
at the center of the bending the voltage
on clearing
magnet.
electrodes
The beam current is (top)
-1000
V
(21)
(hottom)0 V. The measured
71 is shown in Fig. 6 as a function of the voltage on the clearing electrodes. In the measurements at I, = 100 and 200 mA, the octupole magnet with kx4.mag1N -3.0 m-’ was excited for suppressing instability. The value of v can be reduced to about IO-” by applying adequate clearing voltage. The voltage giving this neutralization factor
5
R. i
& g
*015
i
i
‘._, ‘.,
’ !,
for A_ + R < -( I/$/$, - 7) St,. We assume here A+ > A_. From these equations we obtain I ,/T,.,~\,as 1
3wn~.r A+ - A_
I
-=3572+&. G.inat
(22)
A+ and A- can be calculated from the current of the octupole magnet at the threshold conditions. The measured I /TV.,,,., is shown in Fig. 7 as a function of 1,. The proportional constant (Yof the growth rate to the beam current is - 1.5 x lo3 s-’ A-‘. Substituting the measured growth rate into Eq. ( 17). WC find that a neutralization factor of 2.3 x IO-’ is needed for suppressing this instability. Because this value can easily be achieved by reducing the clearing voltage. the instability can be suppressed by the field of the trapped ions without using octupole magnets. 3.2. Observing the instability suppression To observe the suppression of the instability, we measured the threshold beam current as a function of the clearing electrodes’ voltage. In this measurement, the octupole magnet
328
Fig. 8. Threshold
K. Yamodu ef al./Nucl.
beam current ~LSa function
Insrr. and Meth. in Phys. Res. A 370 (1996)
of the voltage on the clearing
electrodes.
was not excited. As shown in Fig. 8, for clearing voltages over 200 V the threshold current was less than 100 mA. For voltage below 200 V, however, it increased to about 400 mA. This means that the condition specified by Eq. ( 17) is satisfied and the instability is suppressed regardless of the beam current. Notice, however, that the threshold current gradually decreases again as the clearing voltage decreases further. This threshold is not determined by the instability. This degradation is due to the scattering of the electron beams by the too-dense trapped ions. This scattering makes the beam lifetime short, and the equilibrium current determined by the injection rate and the lifetime thereby becomes lower. To observe the cooperative effect of the trapped ions and the octupole magnet, we measured the threshold current as a function of the octupole magnet strength. In this measurement, we made the clearing voltage as high as possible in order to keep the effect of the trapped ions from concealing the effect of the octupole magnet. Throughout this experiment the neutralization factor was about -2 x 10m3. The results are shown in Fig. 9, where the positive octupole current corresponds to the same direction as the effect of the trapped ions. The octupole current of 1.3 A, giving the minimum threshold current, corresponds to the cancellation of
323-329
the octupole magnet and the uncertain octupole components in the ring. The behavior of the threshold current in the regions above 1.3 A differs from that below this octupole current. Below an octupole current of 1.3 A the threshold current gradually increases as the octupole current decreases. Above the octupole current of 1.3 A, however, the threshold beam current jumps to more than 350 mA. Even beams over 1200 mA were stable [ 111. This extremely asymmetric behavior is quite different from the symmetric behavior observed in a similar experiment in which positrons were stored and there were no trapped ions [ 121. Therefore, our asymmetric result apparently shows the cooperative effect of the trapped ions and octupole magnets. This result means that the effect of the trapped ions must not be neglected even if the neutralization factor is as small as 0.001. This result also indicates that, if we would use octupole magnets, we should excite the octupole magnet in the same direction as that in which the effect of the trapped ions works. This policy will reduce the focusing force of octupole magnets causing the decrease of dynamic aperture.
4. Discussion Considering the effect of the trapped ions, we can show guidelines to determining the injection energy. These are important from the viewpoint of economy because a lowenergy injector results in a less expensive SR systems. For simplicity, especially to reduce the integration terms to simple functions, we use the round-beam approximation and assume that there are no octupole components originating from magnets. Substituting A + R = 0, and a = b - m into Eq. ( 12), we obtain the following relation for the damping rate: 1
1 -=-+ Tx.damp
‘bad
3woQxeL 256rr22Etjym,c3e,
II) - &I.
(23)
In this simplification, we used the following relation:
(24)
In Super-ALIS, the emittance and the radiation damping rate can be expressed as simple functions of the energy E: E+[ n-m rad] = 2.46 x lo-“E[
MeV12
(25)
and -&[
l/set]
= 3.64 x IO-‘E[MeV]s.
(26)
For the bunching factor, we assume the RF voltage as Fig. 9. Threshold magnet.
beam current as a function
of the strength of the octupole
&r[kV] = 6.67 x IO-*E[MeV].
(27)
K. Yumoda er ol./Nucl.
Instr. and Mrth.
in Phys. Res. A 370 (1996)
323-329
329
the damping rate of a lower-energy beam far larger than that of a higher-energy one. Therefore, if the neutralization factor could be controlled, we might prefer to adopt lower energy in order to inject a large current in compact storage rings.
5. Conclusion
1
,
Fig.
IO.
7 = 0.01.
Survey (b)
of
the
/
energy
1
I
dependence
/
of
I
i
the damping
rates:
(a)
The Landau damping due to the trapped ions was shown here to be very strong in compact rings. This effect is so strong that we must not neglect its contribution to the beams even when the neutralization factor is very low. This effect. furthermore, results in both strong instability suppression and large dynamic apertures at the same time. These results indicate that this effect must be the main reason why the compact storage rings store large currents in spite of their low injection energy. These results also indicate that lower energy might be preferable in injecting a large current in compact storage rings if we could control the effect of trapped ions. In such a positive utilization of trapped-ion effect, however, the following remarks must be considered; the use of this effect should be restricted only to the injection. After injection, we should use other methods for suppressing instabilities and clear up the ions in order to increase the beam lifetime.
7 = 0.001.
The energy dependence of the calculated damping rates is shown in Fig. 10. The energy range considered here is from 100 MeV to 600 MeV because the radiation damping becomes so weak below 100 MeV that the emittance will not be expressed by Eq. (25). Fig. 1Oashows the energy dependenceof the damping rate in 7 = 0.0 1case. In this case, the total damping rate increases as the energy decreases. This increase in total damping rate can be explained by the trapped ions’ effect that is much enhanced by the decrease of the transverse emittance in the low energy region. In Fig. 1Ob, showing the energy dependence of the damping rate in 77= 0.001 case, we can see the extreme decrease of the total damping rate in the region from 100 MeV to 200 MeV. This decrease correspond to the mutual cancellation of the effect of the trapped ions and space charge. In this energy region, therefore, the beams will be extremely unstable. Over 200 MeV, however, the total damping rate is nearly constant up to 600 MeV and the beams will be stable. We thus concluded that there is no problem in reducing injection energy to 200 MeV even if the neutralization factor is as low as 0.001. Furthermore, as described in the 77= 0.01 result. the more positive utilization of this effect could make
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