- Email: [email protected]
Transverse single bunch instability study on BEPC
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 574 (2007) 3–6 www.elsevier.com/locate/nima
Transverse single bunch instability study on BEPC$ J. Gaoa, Y.P. Suna,b, a
Institute of High Energy Physics, CAS Beijing 100049, China Key Laboratory of Heavy Ion Physics, Peking University, Beijing 100871, China
b
Received 31 August 2006; received in revised form 26 December 2006; accepted 5 January 2007 Available online 12 January 2007
Abstract In recent years, a lot of experiments were done on ESRF and ELETTRA to study the single bunch transverse instability. To prevent such instabilities on BEPCII in the future, experiments were made on the single bunch transverse instability threshold current versus the chromaticity on BEPC. By analyzing the experimental data based on the theory developed in [J. Gao, Nucl. Instr. and Meth. A 416 (1998) 186 (see also PAC97, Vancouver, Canada, 1997, p. 1605).], the transverse loss factor of BEPC and the corresponding scaling law are obtained. r 2007 Elsevier B.V. All rights reserved. Keywords: Single bunch transverse instability; Chromaticity; BEPC; Transverse loss factor
1. Introduction In recent years, a lot of experiments were done on ESRF and ELETTRA to study the single bunch transverse instability [1–5]. Single bunch transverse instabilities in electron storage rings behave differently with respect to different values of chromaticity, xc [6,7]. When the chromaticity is negative it is known that the bunch is transversely unstable due to a mechanism called head–tail instability. At xc ¼ 0 the collective motion of the particles inside a bunch can benefit only little from residual Landau damping due to an equivalent residual chromaticity, xc0 , which is defined as: xc0 ¼ ð1=2pbn ÞZL, where Z is the momentum compaction factor, L is the machine circumference, and the averaged value of beta function bn is defined as 1=bn ¼ ð1=LÞ RL ðds=bðsÞÞ. The reader should not confuse the notation 0 xc0 with natural chromaticity, since its real physical cause is Z. By a rough estimate, one finds that xc0 nZ, where n is the tune shift of the machine. In fact, all the existing electron storage rings have their chromaticities compensated above zero (even if very near zero sometimes). When
xc 40, the bunch will be not only free from the head–tail instability, but also its transverse collective motion can be guaranteed by sufficient Landau damping. The instability will occur only when Landau damping effect is destroyed at a specific bunch current threshold, I th . The behavior is theoretically explained in detail in Ref. [8] (where xc0;y has been neglected). In recent years, a lot of experiments were done on ESRF and ELETTRA to study the single bunch transverse instability [1–5]. Experiments indicate a strong nonlinear dependence of I th with respect to xc . The aim of this paper is to study this nonlinear dependence and to determine the transverse loss factor of BEPC through experiments by applying the theory developed in Ref. [8]. BEPC has a 4-fold symmetrical structure, consisting of the IR/RF, the arc and the injection regions. The main parameters of the experimental mode of BEPC are shown in Table 1. In the following sections, the basic theory of this experiment, the experiment itself, the data analysis, and the conclusion will be presented. 2. Review of theory
$
Supported by NSFC (10525525).
Corresponding author. Key Laboratory of Heavy Ion Physics, Peking
University, Beijing 100871, China. E-mail address: [email protected] (Y.P. Sun). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.01.016
For a given vertical chromaticity xc;y (we will limit to our discussion on the vertical plane where the instability starts first due to stronger wake field), the threshold current for
ARTICLE IN PRESS J. Gao, Y.P. Sun / Nuclear Instruments and Methods in Physics Research A 574 (2007) 3–6
4 Table 1 Main parameters of BEPC E C V rf I b;max N sextupole
1.3 GeV 240.4 m 0.4–0.8 MV 137 mA 36
sz0 f rf ap nx =ny se
1.88–1.33 cm 199 MHz 0.016 8.72/4.75 3:432 104
the elimination of Landau damping is expressed as [8] I th ¼
4f y s0 R ðI th Þjxc;y j ehby;c iK tot ? ðsz ðI th ÞÞ
,
(1)
where f y is the vertical betatron frequency, s0 is the natural energy spread. xc;y ¼ xc;y þ xc0;y , xc0;y is the residual chromaticity in vertical plane defined above; R ðIÞ ¼ s ðIÞ=s0 , e is the electric charge of electron. K tot ? ðsz ðIÞÞ is the total transverse loss factor (vertical plane) of the machine, hby;c i is the vertical beta function in the RF cavity regime where the main contribution to K tot ? ðsz ðIÞÞ is believed to be, and sz ðIÞ is the current dependent bunch length. Defining Rz ðIÞ ¼ sz ðIÞ=sz0 , one has K tot ? ðsz ðIÞÞ ¼ Y K tot ? ðsz0 Þ=Rz ðIÞ, where Y is a constant for a given machine. We distinguish two different regimes. The first regime corresponds to very small values of xc;y . In this case I th is located in the potential well distortion dominated bunch lengthening region, and one has R 1. The second regime applies to large xc;y . I th is located in the microwave instability dominated bunch lengthening region with R Rz [9]. Taking Rz ðIÞ CI 1=3 as a rough global scaling law where C is a constant, for these two regimes Eq. (1) becomes !3=ð3YÞ 4f y s0 C Y jxc;y j I th ¼ (2) ehby;c iK tot ? ðsz0 Þ and I th ¼
4f y s0 C Yþ1 jxc;y j ehby;c iK tot ? ðsz0 Þ
!3=ð2YÞ .
(3)
3. Transverse instability experiment 3.1. Experimental process and the results The BEPC machine mode was set as the synchrotron radiation’s injection mode. The beam energy was 1.3 GeV. The beam lifetime was around 10 h. The voltage of the RF cavity was set as 300, 400, 600, 800 kV, respectively. The chromaticity was increased from 0 to 4 with the step size 1. Then one single bunch was injected into the ring until it came to the saturation current. During this process, the single bunch threshold current and the mode coupling were measured. The conclusion was that: the beam could be injected into the ring when the vertical chromaticity was between 0 and 4; the single bunch
Fig. 1.
threshold current increased at first and then decreased; the single bunch threshold current had its maximum value when the chromaticity was 1.2 approximately. Next, the step size of the chromaticity variation was set more meticulously as 0.1 when the chromaticity was increased from 0 to 4. During this process, the single bunch threshold current and the tunes of the two modes were measured. The conclusion was that: when the chromaticity was between 0 and 1, the threshold current was a power function of the vertical chromaticity; when the chromaticity was between 1 and 4, the threshold current decreased with the vertical chromaticity; for each value of the chromaticity, the bunch current at which the transverse mode coupling instability appeared was reproducible. The results were shown in Fig. 1. When the chromaticity is smaller than 1, the beam lifetime is longer than 15 h and we can see the transverse mode coupling signal on the tune monitor. It could be seen that the experimental result was abnormal when the vertical chromaticity was above 1 approximately. That was because the on-line chromaticity correction program only changed eight of the total 16 sextupoles’ strength when it did the chromaticity correction, therefore the dynamic aperture was very small due to the strong sextupole strength and the beam lifetime decreased to less than 1 h accordingly. 3.2. The scaling law From SPEAR scaling [10], one knows that the longitudinal loss factor of a storage ring scales with bunch length as sz1:21 . As for the corresponding scaling law for the transverse loss factor, one can resort to the Panofsky– Wenzel theorem [11]. Using the SPEAR impedancefrequency dependence function and applying the Panofksy– Wenzel theorem, one can prove that Y ¼ 0:7. The machine study of Super-ACO [12] shows numerically that Y ¼ 0:7. Taking Y ¼ 0:7 as a universal scaling law for transverse
ARTICLE IN PRESS J. Gao, Y.P. Sun / Nuclear Instruments and Methods in Physics Research A 574 (2007) 3–6
loss factor (equivalent to SPEAR scaling) which applies to many different machines, we conclude from Eqs. (2) and (3) that I th / xc;y 1:3 and I th / xc;y 2:3 corresponding to the two regimes, respectively. Keeping the aim of comparing only the functional dependence between I th and xc;y in mind, we have fitted the experimental results by a two parameter function I th ¼ aðxc;y Þb , where the horizontal working point ny ¼ 4:75, the momentum compaction factor Z a ¼ 0:016 [13], and xc0;y ny Z ¼ 0:076. As the experimental results are abnormal when the vertical chromaticity is above 1, we have fitted the experimental results when the vertical chromaticity is below 1 as shown in Fig. 2. When we did this experiment, the linac (injector) was not very stable. This means that the measured data of the saturation current may not be very exact, especially at the low chromaticity case. So we only fit for the whole chromaticity region (from 0 to 1) to get the power law, and let the data points distribute symmetrically on both sides of the fitting curve. The fitting results give a 17 mA, and b 1:57 with a values very close to the single bunch saturation current. We conclude that for BEPC synchrotron radiation mode, I th / xc;y 1:57 . Here as we fit the experimental results for both higher chromaticity and lower chromaticity together, we get a medium scaling law for BEPC as I th / xc;y 1:57 ,
5
Table 2 Transverse loss factor of BEPC SR mode V rf (kV)
xc;y
I saturation (mA)
sz0 (cm)
K tot ? ðsz0 Þ (V/pC/m)
300 400 600 800
0.6 0.6 0.6 0.6
10.4 7.8 9.6 11.1
2.18 1.88 1.53 1.33
160 183 216 260
which is between the two limiting dependencies I th / xc;y 1:3 and I th / xc;y 2:3 expected from the SPEAR scaling.
Fig. 3.
3.3. The transverse loss factor Using the above mentioned scaling law, the transverse loss factor can be deduced from Eq. (2), giving ! 4f y s0 xc;y tot K ? ðsz0 Þ ¼ (4) R1z;I th I 1 th ehby;c i for the low chromaticity regime.
From Eq. (4), the transverse loss factor of BEPC synchrotron radiation mode is calculated as shown in Table 2, where the natural bunch length is calculated using the machine parameters, and the bunch length after bunch lengthening is calculated according to theoretical estimation [14] which is in good agreement with the last experiment on BEPC [15]; the natural energy spread is s0 ¼ 0:000346, the horizontal beta function by;rf ¼ 12 m, and f y ¼ 5:927 MHz. From Table 2, it can be seen that the transverse loss factor of BEPC decreases with the bunch lengthening. The transverse loss factor of BEPC is 215 V/pC/m at sz ¼ 1:53 cm approximately. The transverse loss factor versus different natural bunch lengths is shown in Fig. 3. The BEPC transverse loss factor decreases with longer bunch length. 4. Conclusion
Fig. 2.
To study the single bunch transverse instability of BEPC and prevent such instabilities on BEPCII in the future, we performed an experiment on the single bunch transverse instability threshold current versus chromaticity at BEPC. After analyzing the experimental data, the bunchlength dependent transverse loss factor of BEPC was obtained.
ARTICLE IN PRESS 6
J. Gao, Y.P. Sun / Nuclear Instruments and Methods in Physics Research A 574 (2007) 3–6
Acknowledgments The author would like to thank the contributions of J.H. Yue, D.M. Zhou, Y.F. XU, and Z.Y. Guo during the experiment. References [1] J. Jacob, P. Kernel, R. Nagaoka, J.-L. Revol, A. Ropert, Experimental and theoretical studies of transverse single bunch instabilities at the ESRF, in: Proceedings of EPAC98, Stockholm, Sweden, 1998, p. 999. [2] R. Nagaoka, J.L. Revol, P. Kernel, Transverse instabilities in the ESRF storage ring: simulation, experimental results and impedance modelling, in: Proceedings of PAC99, New York, USA, 1999, p. 1192. [3] P. Kernel, R. Nagaoka, J.-L. Revol, High current single bunch transverse instabilities at the ESRF: a new approach, in: Proceedings of EPAC2000, Vienna, Austria, 2000, p. 1133. [4] J.L. Revol, R. Nagaoka, P. Kernel, Comparison of transverse single bunch instabilities between the ESRF and ELETTRA, in: Proceedings of EPAC2000, Vienna, Austria, 2000, p. 1170.
[5] J.L. Revol, R. Nagaoka, Observation, modelling and cure of transverse instabilities at the ESRF, in: Proceedings of PAC2001, Chicago, USA, 2001. [6] A.W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators, Wiley, New York, 1993. [7] R.D. Ruth, J.M. Wang, IEEE Trans. Nucl. Sci. NS-28 (1981) 2405. [8] J. Gao, Nucl. Instr. and Meth. A 416 (1998) 186 (see also PAC97, Vancouver, Canada, 1997, p. 1605). [9] J. Gao, On the single bunch longitudinal collective effects in electron storage rings, LAL/SERA 2001-152, Nucl. Instr. and Meth. A (2001), for publication. [10] H. Wiedemann, Particle Accelerator Physics, Basic Principles and Linear Beam Dynamics, Springer, Berlin, 1993, p. 395. [11] W. Panofsky, W. Wenzel, Rev. Sci. Instr. (1956) 967. [12] P. Brunelle, Etude the´orique et experimentale des faiseaux dans l’anneau VUV SUPER-ACO, The`se, Universite´ Paris 7, 1990. [13] A. Ropert, L. Farvacque, Lattice related brilliance increase at the ESRF, in: PAC97, Canada, 1997, p. 754. [14] J. Gao, Nucl. Instr. and Meth. A 491 (2002) 1. [15] Y.Y. Wei, Q. Qin, Bunch length measurement on BEPC, BEPC 2004 annual report, 2004 (in Chinese).
Nuclear Instruments and Methods in Physics Research A 574 (2007) 3–6 www.elsevier.com/locate/nima
Transverse single bunch instability study on BEPC$ J. Gaoa, Y.P. Suna,b, a
Institute of High Energy Physics, CAS Beijing 100049, China Key Laboratory of Heavy Ion Physics, Peking University, Beijing 100871, China
b
Received 31 August 2006; received in revised form 26 December 2006; accepted 5 January 2007 Available online 12 January 2007
Abstract In recent years, a lot of experiments were done on ESRF and ELETTRA to study the single bunch transverse instability. To prevent such instabilities on BEPCII in the future, experiments were made on the single bunch transverse instability threshold current versus the chromaticity on BEPC. By analyzing the experimental data based on the theory developed in [J. Gao, Nucl. Instr. and Meth. A 416 (1998) 186 (see also PAC97, Vancouver, Canada, 1997, p. 1605).], the transverse loss factor of BEPC and the corresponding scaling law are obtained. r 2007 Elsevier B.V. All rights reserved. Keywords: Single bunch transverse instability; Chromaticity; BEPC; Transverse loss factor
1. Introduction In recent years, a lot of experiments were done on ESRF and ELETTRA to study the single bunch transverse instability [1–5]. Single bunch transverse instabilities in electron storage rings behave differently with respect to different values of chromaticity, xc [6,7]. When the chromaticity is negative it is known that the bunch is transversely unstable due to a mechanism called head–tail instability. At xc ¼ 0 the collective motion of the particles inside a bunch can benefit only little from residual Landau damping due to an equivalent residual chromaticity, xc0 , which is defined as: xc0 ¼ ð1=2pbn ÞZL, where Z is the momentum compaction factor, L is the machine circumference, and the averaged value of beta function bn is defined as 1=bn ¼ ð1=LÞ RL ðds=bðsÞÞ. The reader should not confuse the notation 0 xc0 with natural chromaticity, since its real physical cause is Z. By a rough estimate, one finds that xc0 nZ, where n is the tune shift of the machine. In fact, all the existing electron storage rings have their chromaticities compensated above zero (even if very near zero sometimes). When
xc 40, the bunch will be not only free from the head–tail instability, but also its transverse collective motion can be guaranteed by sufficient Landau damping. The instability will occur only when Landau damping effect is destroyed at a specific bunch current threshold, I th . The behavior is theoretically explained in detail in Ref. [8] (where xc0;y has been neglected). In recent years, a lot of experiments were done on ESRF and ELETTRA to study the single bunch transverse instability [1–5]. Experiments indicate a strong nonlinear dependence of I th with respect to xc . The aim of this paper is to study this nonlinear dependence and to determine the transverse loss factor of BEPC through experiments by applying the theory developed in Ref. [8]. BEPC has a 4-fold symmetrical structure, consisting of the IR/RF, the arc and the injection regions. The main parameters of the experimental mode of BEPC are shown in Table 1. In the following sections, the basic theory of this experiment, the experiment itself, the data analysis, and the conclusion will be presented. 2. Review of theory
$
Supported by NSFC (10525525).
Corresponding author. Key Laboratory of Heavy Ion Physics, Peking
University, Beijing 100871, China. E-mail address: [email protected] (Y.P. Sun). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.01.016
For a given vertical chromaticity xc;y (we will limit to our discussion on the vertical plane where the instability starts first due to stronger wake field), the threshold current for
ARTICLE IN PRESS J. Gao, Y.P. Sun / Nuclear Instruments and Methods in Physics Research A 574 (2007) 3–6
4 Table 1 Main parameters of BEPC E C V rf I b;max N sextupole
1.3 GeV 240.4 m 0.4–0.8 MV 137 mA 36
sz0 f rf ap nx =ny se
1.88–1.33 cm 199 MHz 0.016 8.72/4.75 3:432 104
the elimination of Landau damping is expressed as [8] I th ¼
4f y s0 R ðI th Þjxc;y j ehby;c iK tot ? ðsz ðI th ÞÞ
,
(1)
where f y is the vertical betatron frequency, s0 is the natural energy spread. xc;y ¼ xc;y þ xc0;y , xc0;y is the residual chromaticity in vertical plane defined above; R ðIÞ ¼ s ðIÞ=s0 , e is the electric charge of electron. K tot ? ðsz ðIÞÞ is the total transverse loss factor (vertical plane) of the machine, hby;c i is the vertical beta function in the RF cavity regime where the main contribution to K tot ? ðsz ðIÞÞ is believed to be, and sz ðIÞ is the current dependent bunch length. Defining Rz ðIÞ ¼ sz ðIÞ=sz0 , one has K tot ? ðsz ðIÞÞ ¼ Y K tot ? ðsz0 Þ=Rz ðIÞ, where Y is a constant for a given machine. We distinguish two different regimes. The first regime corresponds to very small values of xc;y . In this case I th is located in the potential well distortion dominated bunch lengthening region, and one has R 1. The second regime applies to large xc;y . I th is located in the microwave instability dominated bunch lengthening region with R Rz [9]. Taking Rz ðIÞ CI 1=3 as a rough global scaling law where C is a constant, for these two regimes Eq. (1) becomes !3=ð3YÞ 4f y s0 C Y jxc;y j I th ¼ (2) ehby;c iK tot ? ðsz0 Þ and I th ¼
4f y s0 C Yþ1 jxc;y j ehby;c iK tot ? ðsz0 Þ
!3=ð2YÞ .
(3)
3. Transverse instability experiment 3.1. Experimental process and the results The BEPC machine mode was set as the synchrotron radiation’s injection mode. The beam energy was 1.3 GeV. The beam lifetime was around 10 h. The voltage of the RF cavity was set as 300, 400, 600, 800 kV, respectively. The chromaticity was increased from 0 to 4 with the step size 1. Then one single bunch was injected into the ring until it came to the saturation current. During this process, the single bunch threshold current and the mode coupling were measured. The conclusion was that: the beam could be injected into the ring when the vertical chromaticity was between 0 and 4; the single bunch
Fig. 1.
threshold current increased at first and then decreased; the single bunch threshold current had its maximum value when the chromaticity was 1.2 approximately. Next, the step size of the chromaticity variation was set more meticulously as 0.1 when the chromaticity was increased from 0 to 4. During this process, the single bunch threshold current and the tunes of the two modes were measured. The conclusion was that: when the chromaticity was between 0 and 1, the threshold current was a power function of the vertical chromaticity; when the chromaticity was between 1 and 4, the threshold current decreased with the vertical chromaticity; for each value of the chromaticity, the bunch current at which the transverse mode coupling instability appeared was reproducible. The results were shown in Fig. 1. When the chromaticity is smaller than 1, the beam lifetime is longer than 15 h and we can see the transverse mode coupling signal on the tune monitor. It could be seen that the experimental result was abnormal when the vertical chromaticity was above 1 approximately. That was because the on-line chromaticity correction program only changed eight of the total 16 sextupoles’ strength when it did the chromaticity correction, therefore the dynamic aperture was very small due to the strong sextupole strength and the beam lifetime decreased to less than 1 h accordingly. 3.2. The scaling law From SPEAR scaling [10], one knows that the longitudinal loss factor of a storage ring scales with bunch length as sz1:21 . As for the corresponding scaling law for the transverse loss factor, one can resort to the Panofsky– Wenzel theorem [11]. Using the SPEAR impedancefrequency dependence function and applying the Panofksy– Wenzel theorem, one can prove that Y ¼ 0:7. The machine study of Super-ACO [12] shows numerically that Y ¼ 0:7. Taking Y ¼ 0:7 as a universal scaling law for transverse
ARTICLE IN PRESS J. Gao, Y.P. Sun / Nuclear Instruments and Methods in Physics Research A 574 (2007) 3–6
loss factor (equivalent to SPEAR scaling) which applies to many different machines, we conclude from Eqs. (2) and (3) that I th / xc;y 1:3 and I th / xc;y 2:3 corresponding to the two regimes, respectively. Keeping the aim of comparing only the functional dependence between I th and xc;y in mind, we have fitted the experimental results by a two parameter function I th ¼ aðxc;y Þb , where the horizontal working point ny ¼ 4:75, the momentum compaction factor Z a ¼ 0:016 [13], and xc0;y ny Z ¼ 0:076. As the experimental results are abnormal when the vertical chromaticity is above 1, we have fitted the experimental results when the vertical chromaticity is below 1 as shown in Fig. 2. When we did this experiment, the linac (injector) was not very stable. This means that the measured data of the saturation current may not be very exact, especially at the low chromaticity case. So we only fit for the whole chromaticity region (from 0 to 1) to get the power law, and let the data points distribute symmetrically on both sides of the fitting curve. The fitting results give a 17 mA, and b 1:57 with a values very close to the single bunch saturation current. We conclude that for BEPC synchrotron radiation mode, I th / xc;y 1:57 . Here as we fit the experimental results for both higher chromaticity and lower chromaticity together, we get a medium scaling law for BEPC as I th / xc;y 1:57 ,
5
Table 2 Transverse loss factor of BEPC SR mode V rf (kV)
xc;y
I saturation (mA)
sz0 (cm)
K tot ? ðsz0 Þ (V/pC/m)
300 400 600 800
0.6 0.6 0.6 0.6
10.4 7.8 9.6 11.1
2.18 1.88 1.53 1.33
160 183 216 260
which is between the two limiting dependencies I th / xc;y 1:3 and I th / xc;y 2:3 expected from the SPEAR scaling.
Fig. 3.
3.3. The transverse loss factor Using the above mentioned scaling law, the transverse loss factor can be deduced from Eq. (2), giving ! 4f y s0 xc;y tot K ? ðsz0 Þ ¼ (4) R1z;I th I 1 th ehby;c i for the low chromaticity regime.
From Eq. (4), the transverse loss factor of BEPC synchrotron radiation mode is calculated as shown in Table 2, where the natural bunch length is calculated using the machine parameters, and the bunch length after bunch lengthening is calculated according to theoretical estimation [14] which is in good agreement with the last experiment on BEPC [15]; the natural energy spread is s0 ¼ 0:000346, the horizontal beta function by;rf ¼ 12 m, and f y ¼ 5:927 MHz. From Table 2, it can be seen that the transverse loss factor of BEPC decreases with the bunch lengthening. The transverse loss factor of BEPC is 215 V/pC/m at sz ¼ 1:53 cm approximately. The transverse loss factor versus different natural bunch lengths is shown in Fig. 3. The BEPC transverse loss factor decreases with longer bunch length. 4. Conclusion
Fig. 2.
To study the single bunch transverse instability of BEPC and prevent such instabilities on BEPCII in the future, we performed an experiment on the single bunch transverse instability threshold current versus chromaticity at BEPC. After analyzing the experimental data, the bunchlength dependent transverse loss factor of BEPC was obtained.
ARTICLE IN PRESS 6
J. Gao, Y.P. Sun / Nuclear Instruments and Methods in Physics Research A 574 (2007) 3–6
Acknowledgments The author would like to thank the contributions of J.H. Yue, D.M. Zhou, Y.F. XU, and Z.Y. Guo during the experiment. References [1] J. Jacob, P. Kernel, R. Nagaoka, J.-L. Revol, A. Ropert, Experimental and theoretical studies of transverse single bunch instabilities at the ESRF, in: Proceedings of EPAC98, Stockholm, Sweden, 1998, p. 999. [2] R. Nagaoka, J.L. Revol, P. Kernel, Transverse instabilities in the ESRF storage ring: simulation, experimental results and impedance modelling, in: Proceedings of PAC99, New York, USA, 1999, p. 1192. [3] P. Kernel, R. Nagaoka, J.-L. Revol, High current single bunch transverse instabilities at the ESRF: a new approach, in: Proceedings of EPAC2000, Vienna, Austria, 2000, p. 1133. [4] J.L. Revol, R. Nagaoka, P. Kernel, Comparison of transverse single bunch instabilities between the ESRF and ELETTRA, in: Proceedings of EPAC2000, Vienna, Austria, 2000, p. 1170.
[5] J.L. Revol, R. Nagaoka, Observation, modelling and cure of transverse instabilities at the ESRF, in: Proceedings of PAC2001, Chicago, USA, 2001. [6] A.W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators, Wiley, New York, 1993. [7] R.D. Ruth, J.M. Wang, IEEE Trans. Nucl. Sci. NS-28 (1981) 2405. [8] J. Gao, Nucl. Instr. and Meth. A 416 (1998) 186 (see also PAC97, Vancouver, Canada, 1997, p. 1605). [9] J. Gao, On the single bunch longitudinal collective effects in electron storage rings, LAL/SERA 2001-152, Nucl. Instr. and Meth. A (2001), for publication. [10] H. Wiedemann, Particle Accelerator Physics, Basic Principles and Linear Beam Dynamics, Springer, Berlin, 1993, p. 395. [11] W. Panofsky, W. Wenzel, Rev. Sci. Instr. (1956) 967. [12] P. Brunelle, Etude the´orique et experimentale des faiseaux dans l’anneau VUV SUPER-ACO, The`se, Universite´ Paris 7, 1990. [13] A. Ropert, L. Farvacque, Lattice related brilliance increase at the ESRF, in: PAC97, Canada, 1997, p. 754. [14] J. Gao, Nucl. Instr. and Meth. A 491 (2002) 1. [15] Y.Y. Wei, Q. Qin, Bunch length measurement on BEPC, BEPC 2004 annual report, 2004 (in Chinese).