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On the scaling law of single bunch transverse instability threshold current vs. the chromaticity in electron storage rings
Nuclear Instruments and Methods in Physics Research A 491 (2002) 346–348
On the scaling law of single bunch transverse instability threshold current vs. the chromaticity in electron storage rings J. Gao* Laboratoire de L’Acc!el!erateur Lin!eaire, IN2P3-CNRS et Universit!e de Paris-Sud, B.P. 34, 91898 Orsay Cedex, France Received 24 January 2002; received in revised form 14 May 2002; accepted 18 May 2002
Abstract Based on the single bunch transverse instability theory established by Gao (Nucl. Instr. and Meth. A 416 (1998) 186), it is shown that the functional relation between the instability threshold bunch current and the chromaticity of the machine is Ith pxdc ; with dE1:3 and dE2:3 at potential well and microwave instability dominated bunch lengthening regimes, respectively. These scaling laws are confirmed by the experimental results from ESRF. r 2002 Elsevier Science B.V. All rights reserved. PACS: 29.20.c; 29.20.Dh Keywords: Electron storage ring; Chromaticity; Transverse instability
Single bunch transverse instabilities in electron storage rings behave differently with respect to different values of chromaticity, xc [2,3]. When the chromaticity is negative it is known that the bunch is transversely unstable due to a mechanism socalled 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 R L of beta function bn is defined as 1=bn ¼ ð1=LÞ 0 ds=bðsÞ: The reader should not confuse the notation xc0 with natural chromaticity, since its real physical cause is Z: By a rough estimate, one finds that *Corresponding author. Tel.: +1-33-1-644-683-68; fax: +133-1-690-714-99. E-mail address: [email protected] (J. Gao).
xc0 EnZ; 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 > 0; 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, Ith : The behavior is explained in detail in Ref. [1] (where xc0;y has been neglected) and demonstrated clearly by the experimental results of ESRF [4–8]. Experiments indicate a strong nonlinear dependence of Ith with respect to xc : This letter aims to explain this nonlinear dependence and the corresponding scaling law by applying the theory developed in Ref. [1]. For a given vertical chromaticity xc;y (we will limit to our discussion in the vertical plane where the instability starts first due to stronger
0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 2 ) 0 1 1 3 2 - 4
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 346–348
wakefield), the threshold current for the elimination of Landau damping is expressed as [1] Ith ¼
4fy se0 Re ðIth Þjxnc;y j
ð1Þ
tot e/by;c SK> ðsz ðIth ÞÞ
where fy is the vertical betatron frequency, se0 is the natural energy spread. xnc;y ¼ xc;y þ xc0;y ; xc0;y is the residual chromaticity in vertical plane defined above; Re ðIÞ ¼ se ðIÞ=se0 ; e is the electric charge of tot electron. K> ðsz ðIÞÞ is the total transverse loss factor (vertical plane) of the machine, /by;c S is the vertical beta function in the RF cavity regime tot where the main contribution to K> ðsz ðIÞÞ is believed to be, and sz ðIÞ is the current dependent bunch length. Defining Rz ðIÞ ¼ sz ðIÞ=sz0 ; one has tot tot K> ðsz ðIÞÞ ¼ K> ðsz0 Þ=RY z ðIÞ; where Y is a constant for a given machine. We distinguish two different regimes. The first regime corresponds very small values of xc;y : In this case Ith is located in the potential well distortion dominated bunch lengthening region, and one has Re E1: The second regime applies to large xc;y : Ith is located in the microwave instability dominated bunch lengthening region with Re ERz [9]. Taking Rz ðIÞECI 1=3 as a rough global scaling law where C is a constant, for these two regimes Eq. (1) becomes !3=ð3YÞ 4fy se0 C Y jxnc;y j ð2Þ Ith ¼ tot ðs Þ e/by;c SK> z0
n 2:3 and Ith pxc;y corresponding to the two regimes, respectively. A series of machine studies have been conducted recently at ESRF (Grenoble, France) on the relation between the single bunch transverse instability and the chromaticity of the machine [4–8]. It is clearly shown experimentally in Fig. 1 that Ith depends nonlinearly on the chromaticity. Keeping the aim of comparing only the functional dependence between Ith and xnc;y in mind, we have fitted the experimental results by a two parameter function y ¼ axb ; where ny ¼ 11:39; ZEa ¼ 1:86 104 [13], and xc0;y Eny Z ¼ 2:1 103 : As indicated by the experimental results, we have fitted the first three experimental results corresponding to low xnc;y (solid line) and fitted the other experimental results corresponding to high xnc;y (dashed line) as shown in Fig. 2. The fitting results give a ¼ 15:98 (mA) and b ¼ 1:38 for solid line and a ¼ 37:17 (mA) and b ¼ 2:39 for dashed line, with b values very close to theoretical predictions. This letter shows that the transverse instability threshold current of a single bunch scales with 2:3 n xnc 1:3 and xnc 2:3 (x1:3 c and xc ; when xc is very small) at potential well distortion dominated and microwave instability dominated bunch lengthening regimes, respectively, which is confirmed by the experimental results obtained at ESRF.
ESRF Storage Ring
and 4fy se0 C Yþ1 jxnc;y j tot ðs Þ e/by;c SK> z0
20
!3=ð2YÞ
18
:
ð3Þ
From SPEAR scaling [10], one knows that the longitudinal loss factor of a storage ring scales with bunch length as s1:21 : As for the correspondz ing scaling law for the transverse loss factor, one can resort to the Panofsky–Wenzel theorem [11]. Using the SPEAR impedance-frequency 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 loss factor (equivalent to SPEAR scaling) which applies to many different machines, we conclude from Eqs. (2) and (3) that Ith pxnc;y1:3
16 14 I_th (mA)
Ith ¼
347
12 10 8 6 4 2 0
0
0.2
0.4 0.6 Vertical chromaticity
0.8
1
Fig. 1. The threshold bunch current vs. xc;y : The dots represent the experimental results [8].
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 346–348
348
ESRF Storage Ring 20 18 16
I_th (mA)
14 12 10 8 6 4 2 0
0
0.2 0.4 0.6 0.8 Vertical chromaticity+vertical residual chromaticity
1
Fig. 2. The threshold bunch current vs. xnc;y (xc0;y ¼ 0:00211): the dots and the squares represent the experimental results, and the solid and the dashed lines represent two fitting curves. The fitting formula is y ¼ axb ; and the fitting results are a ¼ 15:98 (mA), b ¼ 1:38 for solid line, and a ¼ 37:17 (mA), b ¼ 2:39 for dashed line, respectively.
References [1] J. Gao, Nucl. Instr. and Meth. A 416 (1998) 186 (see also PAC97, Vancouver, Canada, 1997, p. 1605).
[2] A.W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators, Wiley, New York, 1993. [3] R.D. Ruth, J.M. Wang, IEEE Trans. Nucl. Sci. NS-28 (1981) 2405. [4] J. Jacob, et al., Experimental and theoretical studies of transverse single bunch instabilities at the ESRF, Proceedings of EPAC98, Stockholm, Sweden, 1998, p. 999. [5] R. Nagaoka, et al., Transverse instabilities in the ESRF storage ring: simulation, experimental results and impedance modelling, Proceedings of PAC99, New York, USA, 1999, p. 1192. [6] P. Kernel, et al., High current single bunch transverse instabilities at the ESRF: a new approach, Proceedings of EPAC2000, Vienna, Austria, 2000, p. 1133. [7] J.L. Revol, et al., Comparison of transverse single bunch instabilities between the ESRF and ELETTRA, Proceedings of EPAC2000, Vienna, Austria, 2000, p. 1170. [8] J.L. Revol, R. Nagaoka, Observation, modelling and cure of transverse instabilities at the ESRF, Proceedings of PAC2001, Chicago, USA, 2001. [9] J. Gao, Nucl. Instr. and Meth. A 491 (2002), this issue. [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. 27 (1956) 967. [12] P. Brunelle, Etude th!eorique et experimentale des faiseaux dans l’anneau VUV SUPER-ACO, Th"ese, Universit!e Paris, Vol. 7, 1990. [13] A. Ropert, L. Farvacque, Lattice related brilliance increase at the ESRF, PAC97, Canada, 1997, p. 754.
On the scaling law of single bunch transverse instability threshold current vs. the chromaticity in electron storage rings J. Gao* Laboratoire de L’Acc!el!erateur Lin!eaire, IN2P3-CNRS et Universit!e de Paris-Sud, B.P. 34, 91898 Orsay Cedex, France Received 24 January 2002; received in revised form 14 May 2002; accepted 18 May 2002
Abstract Based on the single bunch transverse instability theory established by Gao (Nucl. Instr. and Meth. A 416 (1998) 186), it is shown that the functional relation between the instability threshold bunch current and the chromaticity of the machine is Ith pxdc ; with dE1:3 and dE2:3 at potential well and microwave instability dominated bunch lengthening regimes, respectively. These scaling laws are confirmed by the experimental results from ESRF. r 2002 Elsevier Science B.V. All rights reserved. PACS: 29.20.c; 29.20.Dh Keywords: Electron storage ring; Chromaticity; Transverse instability
Single bunch transverse instabilities in electron storage rings behave differently with respect to different values of chromaticity, xc [2,3]. When the chromaticity is negative it is known that the bunch is transversely unstable due to a mechanism socalled 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 R L of beta function bn is defined as 1=bn ¼ ð1=LÞ 0 ds=bðsÞ: The reader should not confuse the notation xc0 with natural chromaticity, since its real physical cause is Z: By a rough estimate, one finds that *Corresponding author. Tel.: +1-33-1-644-683-68; fax: +133-1-690-714-99. E-mail address: [email protected] (J. Gao).
xc0 EnZ; 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 > 0; 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, Ith : The behavior is explained in detail in Ref. [1] (where xc0;y has been neglected) and demonstrated clearly by the experimental results of ESRF [4–8]. Experiments indicate a strong nonlinear dependence of Ith with respect to xc : This letter aims to explain this nonlinear dependence and the corresponding scaling law by applying the theory developed in Ref. [1]. For a given vertical chromaticity xc;y (we will limit to our discussion in the vertical plane where the instability starts first due to stronger
0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 2 ) 0 1 1 3 2 - 4
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 346–348
wakefield), the threshold current for the elimination of Landau damping is expressed as [1] Ith ¼
4fy se0 Re ðIth Þjxnc;y j
ð1Þ
tot e/by;c SK> ðsz ðIth ÞÞ
where fy is the vertical betatron frequency, se0 is the natural energy spread. xnc;y ¼ xc;y þ xc0;y ; xc0;y is the residual chromaticity in vertical plane defined above; Re ðIÞ ¼ se ðIÞ=se0 ; e is the electric charge of tot electron. K> ðsz ðIÞÞ is the total transverse loss factor (vertical plane) of the machine, /by;c S is the vertical beta function in the RF cavity regime tot where the main contribution to K> ðsz ðIÞÞ is believed to be, and sz ðIÞ is the current dependent bunch length. Defining Rz ðIÞ ¼ sz ðIÞ=sz0 ; one has tot tot K> ðsz ðIÞÞ ¼ K> ðsz0 Þ=RY z ðIÞ; where Y is a constant for a given machine. We distinguish two different regimes. The first regime corresponds very small values of xc;y : In this case Ith is located in the potential well distortion dominated bunch lengthening region, and one has Re E1: The second regime applies to large xc;y : Ith is located in the microwave instability dominated bunch lengthening region with Re ERz [9]. Taking Rz ðIÞECI 1=3 as a rough global scaling law where C is a constant, for these two regimes Eq. (1) becomes !3=ð3YÞ 4fy se0 C Y jxnc;y j ð2Þ Ith ¼ tot ðs Þ e/by;c SK> z0
n 2:3 and Ith pxc;y corresponding to the two regimes, respectively. A series of machine studies have been conducted recently at ESRF (Grenoble, France) on the relation between the single bunch transverse instability and the chromaticity of the machine [4–8]. It is clearly shown experimentally in Fig. 1 that Ith depends nonlinearly on the chromaticity. Keeping the aim of comparing only the functional dependence between Ith and xnc;y in mind, we have fitted the experimental results by a two parameter function y ¼ axb ; where ny ¼ 11:39; ZEa ¼ 1:86 104 [13], and xc0;y Eny Z ¼ 2:1 103 : As indicated by the experimental results, we have fitted the first three experimental results corresponding to low xnc;y (solid line) and fitted the other experimental results corresponding to high xnc;y (dashed line) as shown in Fig. 2. The fitting results give a ¼ 15:98 (mA) and b ¼ 1:38 for solid line and a ¼ 37:17 (mA) and b ¼ 2:39 for dashed line, with b values very close to theoretical predictions. This letter shows that the transverse instability threshold current of a single bunch scales with 2:3 n xnc 1:3 and xnc 2:3 (x1:3 c and xc ; when xc is very small) at potential well distortion dominated and microwave instability dominated bunch lengthening regimes, respectively, which is confirmed by the experimental results obtained at ESRF.
ESRF Storage Ring
and 4fy se0 C Yþ1 jxnc;y j tot ðs Þ e/by;c SK> z0
20
!3=ð2YÞ
18
:
ð3Þ
From SPEAR scaling [10], one knows that the longitudinal loss factor of a storage ring scales with bunch length as s1:21 : As for the correspondz ing scaling law for the transverse loss factor, one can resort to the Panofsky–Wenzel theorem [11]. Using the SPEAR impedance-frequency 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 loss factor (equivalent to SPEAR scaling) which applies to many different machines, we conclude from Eqs. (2) and (3) that Ith pxnc;y1:3
16 14 I_th (mA)
Ith ¼
347
12 10 8 6 4 2 0
0
0.2
0.4 0.6 Vertical chromaticity
0.8
1
Fig. 1. The threshold bunch current vs. xc;y : The dots represent the experimental results [8].
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 346–348
348
ESRF Storage Ring 20 18 16
I_th (mA)
14 12 10 8 6 4 2 0
0
0.2 0.4 0.6 0.8 Vertical chromaticity+vertical residual chromaticity
1
Fig. 2. The threshold bunch current vs. xnc;y (xc0;y ¼ 0:00211): the dots and the squares represent the experimental results, and the solid and the dashed lines represent two fitting curves. The fitting formula is y ¼ axb ; and the fitting results are a ¼ 15:98 (mA), b ¼ 1:38 for solid line, and a ¼ 37:17 (mA), b ¼ 2:39 for dashed line, respectively.
References [1] J. Gao, Nucl. Instr. and Meth. A 416 (1998) 186 (see also PAC97, Vancouver, Canada, 1997, p. 1605).
[2] A.W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators, Wiley, New York, 1993. [3] R.D. Ruth, J.M. Wang, IEEE Trans. Nucl. Sci. NS-28 (1981) 2405. [4] J. Jacob, et al., Experimental and theoretical studies of transverse single bunch instabilities at the ESRF, Proceedings of EPAC98, Stockholm, Sweden, 1998, p. 999. [5] R. Nagaoka, et al., Transverse instabilities in the ESRF storage ring: simulation, experimental results and impedance modelling, Proceedings of PAC99, New York, USA, 1999, p. 1192. [6] P. Kernel, et al., High current single bunch transverse instabilities at the ESRF: a new approach, Proceedings of EPAC2000, Vienna, Austria, 2000, p. 1133. [7] J.L. Revol, et al., Comparison of transverse single bunch instabilities between the ESRF and ELETTRA, Proceedings of EPAC2000, Vienna, Austria, 2000, p. 1170. [8] J.L. Revol, R. Nagaoka, Observation, modelling and cure of transverse instabilities at the ESRF, Proceedings of PAC2001, Chicago, USA, 2001. [9] J. Gao, Nucl. Instr. and Meth. A 491 (2002), this issue. [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. 27 (1956) 967. [12] P. Brunelle, Etude th!eorique et experimentale des faiseaux dans l’anneau VUV SUPER-ACO, Th"ese, Universit!e Paris, Vol. 7, 1990. [13] A. Ropert, L. Farvacque, Lattice related brilliance increase at the ESRF, PAC97, Canada, 1997, p. 754.