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An empirical equation for bunch lengthening in electron storage rings
Nuclear Instruments and Methods in Physics Research A 432 (1999) 539}543
An empirical equation for bunch lengthening in electron storage rings J. Gao* Laboratoire de L'Acce& le& rateur Line& aire, IN2P3-CNRS, et Universite& de Paris-Sud, 91405 Orsay cedex, France Received 23 March 1999
Abstract In this paper we propose an empirical equation for the bunch lengthening in electron storage rings, where the unique dominating quantity concerning the impedance of a ring is its total longitudinal loss factor, K , for one turn. The comparisons are made between the analytical and experimental results, and the agreements are extremely well. 1999 Elsevier Science B.V. All rights reserved. PACS: 29.20.!c Keywords: Bunch lengthening; Electron storage rings; Single bunch collective instabilities
1. Introduction
where
From what we know about the single-bunch longitudinal and transverse instabilities [1,2], it is clear to see that the information about the bunch lengthening, R "p /p , with respect to the bunch X X X current is the key to open the locked chain of bunch lengthening, energy spread increasing and the fast transverse instability threshold current. In this paper an empirical bunch lengthening equation is proposed as follows:
576pe C" 55(3 c
C(R RI K ) (2CR RK I # (1) R"1# X c (R )1 c(R )1 X X
* Fax: #33-16907-1499. E-mail address: [email protected] (J. Gao)
(2)
e is the permittivity in vacuum, is Planck con stant, c is the velocity of light, I "eN c/2pR , N is the particle population inside the bunch, K is the total longitudinal loss factor at natural bunch length, c is the normalized particle energy, and R is the average radius of the ring. If SPEAR scaling law [7] is used (for example), 1+1.21 (in fact each machine has its own 1), Eq. (1) can be written as (2CR RK I C(R RI K ) # . (3) R"1# X c R cR X X In fact, the third term of Eq. (1) is due to the Collective Random Excitation e!ect revealed in
0168-9002/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 0 5 6 4 - 1
540
J. Gao / Nuclear Instruments and Methods in Physics Research A 432 (1999) 539}543
Ref. [1]. The second term, however, is obtained intuitively as explained in Section 3. The most interesting character of this empirical bunch lengthening equation is that the longitudinal loss factor, K , is the only dominating factor concern ing the impedance of a storage ring. The procedure to get the information about the bunch lengthening and the energy spread increasing is "rstly to "nd R (I ) by solving bunch X lengthening equation, i.e., Eq. (1), and then calculate energy spread increasing, R (I ) (R "p /p ), C C C C by putting R (I ) into Eq. (4) [1]: X C(R RI K ) . R"1# C cR X
(4)
Once R (I ) is found, one can use the following C formula to calculate the fast single-bunch transverse instability threshold current [2]: F f E I " e1b 2K(p ) , X
(5)
with lp F "4R "m " C C l E
(6)
where l and l are synchrotron and vertical betat ron oscillation tunes, respectively, 1b 2 is the average beta function in the RF cavity region, m is the chromaticity in the vertical plane (usually positive to control the head-tail instability), K(p ) is , X the total transverse loss factor over one turn, p C is the natural energy spread, and E is the particle energy. In practice, it is useful to express K(p ) as , X K(p )"K /RH, where K is the value at the , X , X , natural bunch length, and H is a constant depending on the machine concerned. As a Super-ACO scaling law, H can be taken as [3]. Eq. (5) is therefore expressed as F f E R X . I " e1b 2K ,
2. Comparison with experimental results In this section we look at six machines with their parameters shown in Table 1. The machine energy, natural bunch length and the corresponding longitudinal loss factor are given in Table 2. Concerning Table 1 The machine parameters Machine
R (m)
R (m)
INFN-A ACO SACO KEK-PF SPEAR BEPC
1.15 1.11 1.7 8.66 12.7 10.345
5 3.41 11.5 29.8 37.3 38.2
Table 2 The machine energy and the total loss factors Machine
c
p (cm) X
K (V/pC)
INFN-A ACO SACO KEK-PF KEK-PF SPEAR BEPC BEPC
998 467 1566 3523 4892 2935 2544 3953
3.57 21.7 2.4 1.1 1.47 1 1 2
0.39 0.525 3.1 5.4 3.7 5.2 9.6 3.82
(7)
The notation I is used with the aim of distin guishing it from the formula given by Zotter [4,5].
Fig. 1. Comparison between INFN accumulator ring (R"1.15 m and R "5 m) experimental results and the ana lytical results at 510 MeV with p "3.57 cm. X
J. Gao / Nuclear Instruments and Methods in Physics Research A 432 (1999) 539}543
541
Fig. 2. Comparison between ACO (R"1.11 m and R "3.41 m) experimental results and the analytical results at 238 MeV with p "21.7 cm. X
Fig. 4. Comparison between KEK-PF (R"8.66 m and R "29.8 m) experimental results and the analytical results at 1.8 GeV with p "1.47 cm. X
Fig. 3. Comparison between Super-ACO (R"1.7 m and R "11.5 m) experimental results and the analytical results at 800 MeV with p "2.4 cm. X
Fig. 5. Comparison between KEK-PF (2.5 GeV) (R"8.66 m and R "29.8 m) experimental results and the analytical results at 2.5 GeV with p "1.1 cm. X
the loss factors, that of INFN accumulator ring comes from Ref. [6] and the others are obtained by "tting the corresponding experimetal results with the bunch lengthening equation given in Ref. [1]. Figs. 1}8 show the comparison results between the analytical and the experimental [7}13] bunch lengthening values. It is obvious that this empirical bunch lengthening equation is quite powerful.
3. Discussion In fact Eq. (1) can be obtained from the following equation by truncating the Taylor expansion of the right-hand side of Eq. (8) up to the second order.
R"exp X
(2CR RK I . c R1 X
(8)
542
J. Gao / Nuclear Instruments and Methods in Physics Research A 432 (1999) 539}543
Fig. 6. Comparison between SPEAR (R"12.7 m and R "37.3 m) experimental results and the analytical results at 1.5 GeV with p "1 cm. X
Fig. 8. Comparison between BEPC (R"10.345 m and R "38.2 m) experimental results and the analytical results at 2.02 GeV with p "2 cm. X
results with the experimental results of six di!erent machines. The agreement between the analytical and experimental results is obvious. This empirical bunch lengthening equation challenges the existing theoretical explications for bunch lengthening e!ect.
Acknowledgements The author thanks J. Le Du! for his critical comments.
Fig. 7. Comparison between BEPC (1.3 GeV) (R"10.345 m and R "38.2 m) experimental results and the analytical results at 1.3 GeV with p "1 cm. X
From the point of view of aesthetics, Eq. (8) is more attractive (at least for the author). Even if it does not work well itself, this equation is instructive for us to establish the second term in Eq. (1).
4. Conclusion In this paper we propose an empirical bunch lengthening equation and compare the analytical
References [1] J. Gao, Nucl. Instr. and Meth. A 418 (1998) 332. [2] J. Gao, Nucl. Instr. and Meth. A 416 (1998) 186. [3] P. Brunelle, Etude theH orique et experimentale des faisceaux dans l'anneau VUV SUPER-ACO, The`se, UniversiteH Paris 7, 1990. [4] B. Zotter, Mode-coupling or transverse turbulence of electron or positron bunches in the SPS and LEP, LEP note 363, 1982. [5] B. Zotter, Current limitations in LEP due to vacuum chamber bellows, LEP note 528, 1985. [6] M. Migliorati, L. Palumbo, Wake "eld energy spread and microwave instability, Proceedings of ICFA Beam Dynamics Workshop, Frascati, October 20}25, 1997, p. 347. [7] P.B. Wilson, R. Servranckx, A.P. Sabersky, J. Gareyte, G.E. Fischer, A.W. Chao, M.H.R. Donald, IEEE Trans. Nucl. Sci. NS-24 (1977) 1211.
J. Gao / Nuclear Instruments and Methods in Physics Research A 432 (1999) 539}543 [8] R. Boni et al., Bunch lengthening and impedance measurements and analysis in DA'NE accumulator ring, Note: BM-1, Frascati, March 10, 1997. [9] Le groupe de l'anneau de collisions d'Orsay, Allongement des paquets dans ACO, Rapport technique 34}69, Orsay, le 14 novembre, 1969. [10] A. Nadji et al., Experiments with low and negative momentum compaction factor with Super-ACO, Proceedings of EPAC96, Barcelona, 1996, p. 676.
543
[11] N. Nakamura, S. Sakanaka, K. Haga, M. Izawa, T. Katsura, Collective e!ects in single bunch mode at the photon factory storage ring, Proceedings of PAC91, San Francisco, CA, 1991, p. 440. [12] SPEAR Group, IEEE Trans. Nucl. Sci. NS-22 (1975) 1366. [13] Z. Guo et al., Bunch lengthening study in BEPC, Proceedings of PAC95, Dallas TX, 1995, p. 2955.
An empirical equation for bunch lengthening in electron storage rings J. Gao* Laboratoire de L'Acce& le& rateur Line& aire, IN2P3-CNRS, et Universite& de Paris-Sud, 91405 Orsay cedex, France Received 23 March 1999
Abstract In this paper we propose an empirical equation for the bunch lengthening in electron storage rings, where the unique dominating quantity concerning the impedance of a ring is its total longitudinal loss factor, K , for one turn. The comparisons are made between the analytical and experimental results, and the agreements are extremely well. 1999 Elsevier Science B.V. All rights reserved. PACS: 29.20.!c Keywords: Bunch lengthening; Electron storage rings; Single bunch collective instabilities
1. Introduction
where
From what we know about the single-bunch longitudinal and transverse instabilities [1,2], it is clear to see that the information about the bunch lengthening, R "p /p , with respect to the bunch X X X current is the key to open the locked chain of bunch lengthening, energy spread increasing and the fast transverse instability threshold current. In this paper an empirical bunch lengthening equation is proposed as follows:
576pe C" 55(3 c
C(R RI K ) (2CR RK I # (1) R"1# X c (R )1 c(R )1 X X
* Fax: #33-16907-1499. E-mail address: [email protected] (J. Gao)
(2)
e is the permittivity in vacuum, is Planck con stant, c is the velocity of light, I "eN c/2pR , N is the particle population inside the bunch, K is the total longitudinal loss factor at natural bunch length, c is the normalized particle energy, and R is the average radius of the ring. If SPEAR scaling law [7] is used (for example), 1+1.21 (in fact each machine has its own 1), Eq. (1) can be written as (2CR RK I C(R RI K ) # . (3) R"1# X c R cR X X In fact, the third term of Eq. (1) is due to the Collective Random Excitation e!ect revealed in
0168-9002/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 0 5 6 4 - 1
540
J. Gao / Nuclear Instruments and Methods in Physics Research A 432 (1999) 539}543
Ref. [1]. The second term, however, is obtained intuitively as explained in Section 3. The most interesting character of this empirical bunch lengthening equation is that the longitudinal loss factor, K , is the only dominating factor concern ing the impedance of a storage ring. The procedure to get the information about the bunch lengthening and the energy spread increasing is "rstly to "nd R (I ) by solving bunch X lengthening equation, i.e., Eq. (1), and then calculate energy spread increasing, R (I ) (R "p /p ), C C C C by putting R (I ) into Eq. (4) [1]: X C(R RI K ) . R"1# C cR X
(4)
Once R (I ) is found, one can use the following C formula to calculate the fast single-bunch transverse instability threshold current [2]: F f E I " e1b 2K(p ) , X
(5)
with lp F "4R "m " C C l E
(6)
where l and l are synchrotron and vertical betat ron oscillation tunes, respectively, 1b 2 is the average beta function in the RF cavity region, m is the chromaticity in the vertical plane (usually positive to control the head-tail instability), K(p ) is , X the total transverse loss factor over one turn, p C is the natural energy spread, and E is the particle energy. In practice, it is useful to express K(p ) as , X K(p )"K /RH, where K is the value at the , X , X , natural bunch length, and H is a constant depending on the machine concerned. As a Super-ACO scaling law, H can be taken as [3]. Eq. (5) is therefore expressed as F f E R X . I " e1b 2K ,
2. Comparison with experimental results In this section we look at six machines with their parameters shown in Table 1. The machine energy, natural bunch length and the corresponding longitudinal loss factor are given in Table 2. Concerning Table 1 The machine parameters Machine
R (m)
R (m)
INFN-A ACO SACO KEK-PF SPEAR BEPC
1.15 1.11 1.7 8.66 12.7 10.345
5 3.41 11.5 29.8 37.3 38.2
Table 2 The machine energy and the total loss factors Machine
c
p (cm) X
K (V/pC)
INFN-A ACO SACO KEK-PF KEK-PF SPEAR BEPC BEPC
998 467 1566 3523 4892 2935 2544 3953
3.57 21.7 2.4 1.1 1.47 1 1 2
0.39 0.525 3.1 5.4 3.7 5.2 9.6 3.82
(7)
The notation I is used with the aim of distin guishing it from the formula given by Zotter [4,5].
Fig. 1. Comparison between INFN accumulator ring (R"1.15 m and R "5 m) experimental results and the ana lytical results at 510 MeV with p "3.57 cm. X
J. Gao / Nuclear Instruments and Methods in Physics Research A 432 (1999) 539}543
541
Fig. 2. Comparison between ACO (R"1.11 m and R "3.41 m) experimental results and the analytical results at 238 MeV with p "21.7 cm. X
Fig. 4. Comparison between KEK-PF (R"8.66 m and R "29.8 m) experimental results and the analytical results at 1.8 GeV with p "1.47 cm. X
Fig. 3. Comparison between Super-ACO (R"1.7 m and R "11.5 m) experimental results and the analytical results at 800 MeV with p "2.4 cm. X
Fig. 5. Comparison between KEK-PF (2.5 GeV) (R"8.66 m and R "29.8 m) experimental results and the analytical results at 2.5 GeV with p "1.1 cm. X
the loss factors, that of INFN accumulator ring comes from Ref. [6] and the others are obtained by "tting the corresponding experimetal results with the bunch lengthening equation given in Ref. [1]. Figs. 1}8 show the comparison results between the analytical and the experimental [7}13] bunch lengthening values. It is obvious that this empirical bunch lengthening equation is quite powerful.
3. Discussion In fact Eq. (1) can be obtained from the following equation by truncating the Taylor expansion of the right-hand side of Eq. (8) up to the second order.
R"exp X
(2CR RK I . c R1 X
(8)
542
J. Gao / Nuclear Instruments and Methods in Physics Research A 432 (1999) 539}543
Fig. 6. Comparison between SPEAR (R"12.7 m and R "37.3 m) experimental results and the analytical results at 1.5 GeV with p "1 cm. X
Fig. 8. Comparison between BEPC (R"10.345 m and R "38.2 m) experimental results and the analytical results at 2.02 GeV with p "2 cm. X
results with the experimental results of six di!erent machines. The agreement between the analytical and experimental results is obvious. This empirical bunch lengthening equation challenges the existing theoretical explications for bunch lengthening e!ect.
Acknowledgements The author thanks J. Le Du! for his critical comments.
Fig. 7. Comparison between BEPC (1.3 GeV) (R"10.345 m and R "38.2 m) experimental results and the analytical results at 1.3 GeV with p "1 cm. X
From the point of view of aesthetics, Eq. (8) is more attractive (at least for the author). Even if it does not work well itself, this equation is instructive for us to establish the second term in Eq. (1).
4. Conclusion In this paper we propose an empirical bunch lengthening equation and compare the analytical
References [1] J. Gao, Nucl. Instr. and Meth. A 418 (1998) 332. [2] J. Gao, Nucl. Instr. and Meth. A 416 (1998) 186. [3] P. Brunelle, Etude theH orique et experimentale des faisceaux dans l'anneau VUV SUPER-ACO, The`se, UniversiteH Paris 7, 1990. [4] B. Zotter, Mode-coupling or transverse turbulence of electron or positron bunches in the SPS and LEP, LEP note 363, 1982. [5] B. Zotter, Current limitations in LEP due to vacuum chamber bellows, LEP note 528, 1985. [6] M. Migliorati, L. Palumbo, Wake "eld energy spread and microwave instability, Proceedings of ICFA Beam Dynamics Workshop, Frascati, October 20}25, 1997, p. 347. [7] P.B. Wilson, R. Servranckx, A.P. Sabersky, J. Gareyte, G.E. Fischer, A.W. Chao, M.H.R. Donald, IEEE Trans. Nucl. Sci. NS-24 (1977) 1211.
J. Gao / Nuclear Instruments and Methods in Physics Research A 432 (1999) 539}543 [8] R. Boni et al., Bunch lengthening and impedance measurements and analysis in DA'NE accumulator ring, Note: BM-1, Frascati, March 10, 1997. [9] Le groupe de l'anneau de collisions d'Orsay, Allongement des paquets dans ACO, Rapport technique 34}69, Orsay, le 14 novembre, 1969. [10] A. Nadji et al., Experiments with low and negative momentum compaction factor with Super-ACO, Proceedings of EPAC96, Barcelona, 1996, p. 676.
543
[11] N. Nakamura, S. Sakanaka, K. Haga, M. Izawa, T. Katsura, Collective e!ects in single bunch mode at the photon factory storage ring, Proceedings of PAC91, San Francisco, CA, 1991, p. 440. [12] SPEAR Group, IEEE Trans. Nucl. Sci. NS-22 (1975) 1366. [13] Z. Guo et al., Bunch lengthening study in BEPC, Proceedings of PAC95, Dallas TX, 1995, p. 2955.