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On the single bunch longitudinal collective effects in electron storage rings
Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8
On the single bunch longitudinal collective effects 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 29 January 2002; received in revised form 14 May 2002; accepted 18 May 2002
Abstract After giving an analytical expression for the single bunch short range wake potential of a storage ring, we have discussed separately the roles of linear and nonlinear terms of the Taylor expansion of the wake potential on the bunch longitudinal motion. The equations describing bunch lengthening and increase in energy spread are established. Applications to different operating machines are made. r 2002 Elsevier Science B.V. All rights reserved. PACS: 29.20.c; 29.20.Dh Keywords: Electron storage ring; Bunch lengthening; Single bunch instability
1. Introduction The bunch lengthening phenomenon in an electron storage ring was first observed in ACO [1] and later in other machines, where accompanying bunch lengthening, one finds an increase in the single bunch energy spread with a more or less appreciable threshold current. The first empirical bunch lengthening formula [1] found in ACO is expressed as Ib ðmAÞ 2 2 3 st ðnsÞ ¼ st0 ðnsÞ 1 þ 2 10 ð1Þ E 4 ðGeVÞst ðnsÞ where st ðnsÞ is the bunch rms duration measured in nano-second, st0 ðnsÞ corresponds to st ðnsÞ at zero bunch current, Ib is the bunch current, and *Tel.: +1-33-1-644-683-68; fax: +1-33-1-690-714-99. E-mail address: [email protected] (J. Gao).
EðGeVÞ is the particle energy (the E 4 ðGeVÞ dependence in Eq. (1) was soon corrected to be E 3 ðGeVÞ in the later theoretical works, such as Refs. [2,3]). Since then, understanding these single bunch longitudinal collective phenomena has become one of the main battle fields for accelerator physicists. Reviewing a list (incomplete) of these efforts, Refs. [2–20], one may ask questions, such as ‘‘is this battle finished and is the field cleaned up?’’. Fortunately, for the theorists, and unfortunately, for the experimentalists the answer is negative. As shown in Ref. [19], where the discrepancy between experimental and theoretical results manifests itself to an untolerable degree, it is ever clearer that more theoretical works should be devoted in this classical problem. This paper presents a theoretical framework to explain the physical processes and to establish a
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 1 - 2
2
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8
bunch lengthening equation and, for the first time, the equation for the increase in the energy spread of the bunch due to wake potential. In Section 2, an analytical expression for the wake potential of the total machine is proposed. In Section 3, the theoretical framework is established to explain and calculate bunch lengthening and increase in bunch energy spread. Applications to different operating storage rings are given in Section 4. What one should bear in mind is that understanding the longitudinal collective effects is the starting point to treat properly the single bunch transverse instability [21,22].
2. Wake potential expression of a storage ring We start with finding an analytical expression that describes the wake potential of a storage ring. For the convenience of our theoretical treatment coming later, we will use a function of three parameters, i.e., bunch length sz ; total loss factor kðsz ), and the total inductance Lðsz ), to describe the total wake potential of the machine. As an Ansatz, we propose the following analytical expression: 2z2 Wz ðzÞ ¼ akðsz Þ exp 2 7sz 2 Zi z pffiffiffi cos 1 þ atan atan p 2Zr 3sz Zi þatan ð2Þ 2Zr where a ¼ 2:23; Zi ¼ 2pL=T0 ; Zr ¼ kðsz ÞTb2 =T0 ; T0 ¼ 2pRav =c; Tb ¼ 3sz =c; Rav is the average radius of the ring, sz is the bunch length, c is the velocity of light, and z ¼ 0 corresponds to the center of the bunch. The effectiveness of the wake potential expression shown in Eq. (2) will be demonstrated in Section 4. The formulae through out the paper are in MKS units. Rigorous derivation of an expression similar to Eq. (2) is not the point of this paper. The reader may find a more accurate expression using a broadband impedance model.
3. Theory According to electron storage ring theory [23] we know that the single particle ‘‘bunch length’’ sz0 can be expressed as ca 2 2 2 sz0 ¼ sE0 ð3Þ Os0 E0 where a is the momentum compaction factor, sE0 is the energy spread, E0 is the particle’s energy, and Os0 is the angular frequency of the synchrotron oscillation which can be obtained from the following differential equations [24]: dW ¼ eV ðsin f sin fs0 Þ dt
ð4Þ
df 1 hZos ¼ W dt 2pps Rav
ð5Þ
where W ¼ 2pðE E0 Þ=os ; os is the angular revolution frequency of the synchronous particle, V is the peak rf voltage, fs0 the synchronous phase, h is harmonic number, ps is the synchronous particle’s momentum, Z ¼ ð1=g2 Þ a; g is the normalized particle’s energy, and Rav is the average radius of the machine. If Eq. (4) is linearized as dW ¼ eV cos fs0 Df dt
ð6Þ
where Df ¼ f fs0 ; one gets O2s0 ¼
eV cos fs0 hZos : 2pRav ps
ð7Þ
Eq. (3) implies that, for a machine of given a and E0 ; there are only two possible parameters through which the bunch length can be changed, i.e., Os0 and sE0 : In the following, we will show how the single-bunch, short-range wake potential perturbs Os0 ; sE0 ; and finally sz0 : Firstly, we expand the analytical wake potential expression shown in Eq. (2) in a Taylor series: 2 z z Wz ðzÞ ¼ Akðsz Þ þ Bkðsz Þ þ Ckðsz Þ sz sz þ Oðz3 Þ
ð8Þ
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8
where a A ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi Zi 1 þ 2Z r
ð9Þ
1 Zi B0:289aZi 1 þ 0:637 atan atan 2Zr C C r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi B¼B @ A 2 Zi Zr 1 þ 2Zr ð10Þ a C ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2
Zi 1 þ 2Z r 0 2 1 Zi 1 þ 0:637 atan atan 2Zr B2 C @ þ A: 7 6
ð11Þ Equipped with the analytical expression of the wake potential of the ring, we turn now our attention to the dynamics of the particles inside a bunch. We consider then only the dominating linear part of Wz ðzÞ; denoted as Wlz ðzÞ and expressed as: z Wlz ðzÞ ¼ Bkðsz Þ : ð12Þ sz To treat the Ne different particles in the bunch on the same footing and including the effect of wake potential, Eq. (6) should be modified as follows: dW l lrf 2 ¼ eV cos fs0 Df þ e Ne Wz Df dt 2p Bkðsz Þlrf ¼ 1 e2 N e 2peV cos fs0 sz0 eV cos fs0 Df: The resultant Os is obtained: e2 Ne Bkðsz Þc O2s ¼ O2s0 1 : sz0 hos eV cos fs0
potential, the linear longitudinal motion (Df) can be derived from the following Hamiltonian: H0 ¼
0
ð13Þ
ð14Þ
The discussion made above treats only the static aspect, or the so-called potential well distortion effect [3]. To facilitate studying the role of the nonlinear part of the wake potential, we will use the Hamiltonian formalism. Without the wake
3
p2f 2
þ
O2s0 2 Df 2
ð15Þ
where pf ¼ dDf=dt: To facilitate our study, we introduce the nonlinear wake potential term into the unperturbed Hamiltonian in the form of series of delta functions separated by a revolution period. The perturbed Hamiltonian is expressed as p2f O2s0 2 e2 Ne Ckðsz Þ lrf 2 H¼ þ Df þ 3 2 2 2psz N X hZos dðt nT0 Þ ð16Þ Df3 T0 2pps Rav n¼N where we have ignored the perturbation from the linear wake potential term. At this point recall the general expression for the dynamic aperture in a circular storage ring due to delta multipoles derived in Ref. [25], which has been successfully applied in the beam-beam interaction studies in a lepton circular collider [26]. The perturbed Hamiltonian is expressed as H¼
N p2x Kx ðsÞ 2 b2 L 3 X x þ x þ dðs iLÞ 2 3r i¼N 2
ð17Þ
where px ¼ dx=ds; Kx ðsÞ represents linear lattice focusing, b2 is the coefficient of the delta function sextupole magnetic field, and r is the local bending radius. According to Ref. [25] the corresponding one dimensional dynamic aperture, Adyna;x ; can be calculated analytically: pffiffiffiffiffiffiffiffiffiffiffiffiffi 2b ðsÞ r Adyna;x ¼ pffiffiffi x 3=2 ð18Þ jb2 jL 3bx ðs1 Þ where bx ðsÞ is the beta function of the storage ring and s1 is the location where the delta function sextupole is located. By analogy between Eqs. (16) and (17), one gets the maximum value, ADf ; of Df beyond which Df will undergo phase instability and move in a stochastic way. ADf is expressed as rffiffiffi 2 V cosðfs0 Þ 2psz 2 ADf ¼ : ð19Þ 3 eNe Ckðsz ÞT0 Os0 lrf
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8
4
Defining sDf ¼ 2psz =lrf and dividing ADf by sDf ; one gets rffiffiffi ADf 2 sz V cosðfs0 Þ S¼ ¼ ð20Þ 3 eNe Ckðsz ÞT0 fs0 lrf sDf where fs0 is the synchrotron oscillation frequency. If we take SX1 as a rough criterion for the particles inside a bunch to undergo longitudinal phase stochastic motions, one can get a threshold of particle population inside the bunch, Ne;th ; expressed as follows: rffiffiffi 2 sz0 V cosðfs0 Þ Ne;th ¼ ð21Þ 3 eCkðsz0 ÞT0 fs0 lrf where sz has been replaced by sz0 for simplicity. We consider first the case of Ne XNe;th : As stated above, the phases of the particles inside the bunch will execute stochastic motions (phase instability as we call it). We have to explore the kinetic aspect of Wlz ðzÞ; which is closely related with energy spread increasing. Expressing Wlz ðzÞ as z0 sin Os t þ fi l Wz ðzÞ ¼ Bkðsz Þ ð22Þ sz where fi is a random phase due to the above mentioned stochastic motions, one gets the increment of the random energy fluctuation, w; due to the short range wake potential over one turn (T0 ) expressed as T0 Dw ¼ e2 Ne Wlz ðzÞ exp tE T0 þ 1 exp w ð23Þ tE where tE is the synchrotron radiation damping time. Assuming that Dw can be regarded as a Markov random variable, one knows that the distribution function of Dw is governed by Fokker–Planck equation @F ðt; DwÞ @ ¼ ðGF ðt; DwÞÞ @t @Dw 1 @2 þ ðDF ðt; DwÞÞ 2 @ðDwÞ2
ð24Þ
//ðDwÞ2 SS T0
ð26Þ
where //SS denotes the average over all possible value of z0 and fi : Putting Eq. (23) into Eqs. (25) and (26), one gets e2 N e T0 //ðWlz ðzÞÞSS exp G¼ T0 tE T0 w þ 1 exp ð27Þ T0 tE 2 2 T0 w ðe2 Ne Þ2 D ¼ 1 exp þ tE T0 T0 2T 0 //ðWlz ðzÞÞ2 SS exp tE
ð28Þ
where //Wlz ðzÞSS ¼ 0 and //ðWlz ðzÞÞ2 SS ¼ ðBkðsz ÞÞ2 =2: Inserting Eqs. (27) and (28) into Eq. (24), one gets T0
@F ðt; DwÞ @t
T0 @ ðwF ðt; DwÞÞ ¼ 1 exp @Dw tE 2 1 T0 @2 þ 1 exp ðw2 F ðt; DwÞÞ 2 tE @ðDwÞ2 1 ðe2 Ne Bkðsz ÞÞ2 2T0 exp þ 2 2 tE 2 @ ðF ðt; DwÞÞ: ð29Þ @ðDwÞ2
Now, multiplying both sides of Eq. (29) by w2 and integrating over w; one has d/w2 S 2T0 ¼ 1 exp T0 /w2 S dt tE ðe2 Ne Bkðsz ÞÞ2 2T0 exp þ : ð30Þ 2 tE Since T0 =tE 51; Eq. (30) can be reduced to
with G¼
D¼
//DwSS T0
ð25Þ
T0
d/w2 S 2T0 2 ðe2 Ne Bkðsz ÞÞ2 ¼ : /w S þ dt tE 2
ð31Þ
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8
Solving Eq. (31), one gets ðe2 Ne Bkðsz ÞÞ2 tE s 1 exp /w2 S ¼ tE 2 2T0 s þ exp ð32Þ w2 tE 0 where w0 is the initial value. Obviously, when s-N; one obtains the additional energy spread due to the kinetic motions of the particles s2E;w ¼ /w2 SN
ðBe2 Ne kðsz ÞÞ2 ¼ tE : 4T0
where
ð33Þ
Defining Rz ¼ sz =sz0 and RE ¼ sE =sE0 ; we can rewrite for an isomagnetic ring. Eq. (34) can be expressed explicitly: ! CðRRav Ib Bkðsz0 ÞÞ2 2 RE ¼ 1 þ ð35Þ g7 R2B z where 576p2 e0 C ¼ pffiffiffi : 55 3_c3
For an isomagnetic ring, Eq. (37) can be simplified as eBkðsz0 ÞIb JE Rsz0 1 2 Rz ¼ 1 m0 c3 aCq g3 RBz ! CðRRav Ib Bkðsz0 ÞÞ2 ð38Þ 1þ g7 R2B z
Cq ¼
The total resultant energy spread can be expressed as ! ðBe2 Ne kðsz ÞÞ2 2 2 2 2 sE ¼ sE;0 þ sE;w ¼ sE;0 1 þ tE : ð34Þ 4T0 s2E;0
5
55_ pffiffiffi ¼ 3:84 1013 m 32 3m0 c
ð39Þ
JE ¼ 2 þ aRav =R: JE is called the energy damping partition number. To avoid the unphysical solution, Eq. (38) should be simplified as eBkðsz0 ÞIb JE Rsz0 R2z ¼ 1 þ m0 c3 aCq g3 RBz þ
CðRRav Ib Bkðsz0 ÞÞ2 : g7 R2B z
ð40Þ
For the case of Ne oNe;th ; Eqs. (40) and (35) are reasonably reduced to eBkðsz0 ÞIb JE Rsz0 R2z ¼ 1 þ ð41Þ m0 c3 aCq g3 RBz and
ð36Þ
Ib ¼ eNe c=T0 ; Ne is the particle population inside the bunch, R is the local bending radius, e0 is the permittivity of vacuum, _ is Planck constant, and B is a constant for a given machine (B ¼ 1:21 if one uses SPEAR scaling [6]). Well prepared with Eqs. (14) and (35), one can calculate the resultant bunch length under the influence of the single bunch short range wake potential; ca 2 2 2 sz ¼ sE Os E0 1 e2 Ne Bkðsz0 Þc ¼ s2z0 1 sz0 RBz hos eV cos fs0 ! CðRRav Ib Bkðsz0 ÞÞ2 1þ : ð37Þ g7 R2B z
R2E ¼ 1:
ð42Þ
Having established the bunch lengthening and the energy spread equations for different bunch particle population regions, Ne oNe;th and Ne XNe;th ; respectively, we can apply the results to specific examples.
4. Applications We take first the low energy ring of KEK BFactory as an example with E ¼ 3:5 GeV; R ¼ 16:3 m; Rav ¼ 480 m; V ¼ 4:2 MV; fs ¼ 1:14 kHz; frf ¼ 509 MHz; fs0 ¼ 0:365; sz0 ¼ 0:004 m; L ¼ 22 nH; kðsz0 Þ ¼ 42 V=pC; and Ne;th ¼ 4:3 107 : Fig. 1 shows that the analytical calculation of the total longitudinal wake potential of the ring fits well with that from the numerical simulation [27], where the Gaussian charge distribution
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8 100
5
80
4
60
3
40
2
20
Wz (V/pC)
Wz (V/pC)
6
0 −20 −40
1 0 −1 −2
−60
−3
−80
−4
−100 −0.02
−0.01
0
0.01
0.02
−5 −0.05
z (m)
Fig. 1. KEKB low energy ring: the dots and the solid line represent the wake potentials calculated numerically [27] and analytically by using Eq. (2), respectively, with sz0 ¼ 0:004 m; L ¼ 22 nH; and kðsz0 Þ ¼ 42 V=pC: The dashed line shows the Gaussian bunch shape with arbitrary units.
−0.03
−0.01
0.01
0.03
0.05
z (m)
Fig. 3. The solid line represents the total longitudinal wake potential of PEP-II low energy ring with sz0 ¼ 0:01 m; L ¼ 83:3 nH; and kðsz0 Þ ¼ 2:9 V=pC: The dashed line shows the Gaussian bunch shape with arbitrary units.
3
2 1.8
2.5
1.6 1.4
2
Rz, Re
Rz, Re
1.2 1.5
1 0.8
1
0.6 0.4
0.5
0.2 0
0
0.5
1
1.52
2.5
3
3.5
4
Ne (10^10)
0
0
1
2
3 4 Ne (10^10)
5
6
7
Fig. 2. KEKB low energy ring: the solid line and the dashed line are the bunch lengthening and the energy spread increasing vs. the particle population inside the bunch, respectively (sz0 ¼ 0:004 m).
Fig. 4. PEP-II low energy ring: the solid line and the dashed line are the bunch lengthening and the energy spread increasing vs. the particle population inside the bunch, respectively (sz0 ¼ 0:01 m).
represented by dashed line has arbitrary units and it is true for all following figures. The analytical estimations of bunch lengthening (solid line) and energy spread increasing (dashed line) vs. the particle population inside a bunch are shown in Fig. 2 where a ¼ 0:0002 [28]. As for the low energy ring of PEP-II [16] with E ¼ 3:1 GeV;
R ¼ 13:751 m; Rav ¼ 350 m; V ¼ 3:2 MV; fs ¼ 4:6 kHz; frf ¼ 476 MHz; fs0 ¼ 0:236; sz0 ¼ 0:01 m; L ¼ 83:3 nH; kðsz0 Þ ¼ 2:9 V=pC; a ¼ 0:00131; and Ne;th ¼ 9:3 108 : We show in Fig. 3 the analytical calculation of the total longitudinal wake potential of the machine, and in Fig. 4, the analytical estimations of bunch lengthening (solid
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8 10 8 6
Wz (V/pC)
4 2 0 −2 −4 −6 −8 −10 −0.04
-0.02
0
0.02
0.04
z (m)
Fig. 5. The solid line corresponds to the total longitudinal wake potential of ATF damping ring with sz0 ¼ 0:0068 m; L ¼ 14 nH; and kðsz0 Þ ¼ 4:5 V=pC: The dashed line shows the Gaussian bunch shape with arbitrary units.
7
lated analytically, and in Fig. 6 the analytical estimations of bunch lengthening (solid line) and energy spread increasing (dashed line) vs. Ne : To compare with the experimental data, in Fig. 6, we have drawn some experimentally obtained results (dots) from the experimental fitting curve [29] which is more close to the case of sz0 ¼ 0:0068 m: It is obvious that by using the machine impedance L ¼ 14 nH and kðsz0 Þ ¼ 4:5 V=pC at sz0 ¼ 0:0068; one has the theoretically predicted bunch lengthening curve very close to the experimentally observed results. We conclude that the observation in Ref. [19] ‘‘The impedance values implied by our analysis are much larger than expected: L is about a factor of 3 larger than earlier calculations, and R is about a factor of 10 larger, a result that is especially puzzling’’ is due to the fact that the experimental results are fitted with a theoretical model which is not totally correct.
2
5. Conclusion
1.75 1.5
In this paper, we have established a theoretical framework to describe the single bunch longitudinal collective effects, such as bunch lengthening and energy spread increasing through corresponding equations. Applications to different operating machines are made.
Rz, Re
1.25 1 0.75 0.5 0.25 0
Acknowledgements 0
1
2
3
4
5
6
7
8
9
10
Ne (10^9)
Fig. 6. ATF damping ring: the solid line and the dashed line are the bunch lengthening and the energy spread increasing vs. the particle population inside the bunch, respectively (sz0 ¼ 0:0068 m). The dots are experimental results.
The author appreciates very much the useful comments and constant interests from J. Le Duff and J. Haissinski.
References line) and energy spread increasing (dashed line) vs. the particle population, Ne : Finally, looking at ATF damping ring [29] with E ¼ 1:54 GeV; R ¼ 5:73 m; Rav ¼ 22 m; V ¼ 200 KV; fs ¼ 8:95 kHz; frf ¼ 714 MHz; fs0 ¼ 0:811; sz0 ¼ 0:0068 m; L ¼ 14 nH; kðsz0 Þ ¼ 4:5 V=pC; a ¼ 0:00197; and Ne;th ¼ 4:36 107 : we show in Fig. 5 the total longitudinal wake potential of the machine calcu-
[1] R. Belbeoch, et al., Recent experiments with ACO, Proceedings of the National Conference on Particle Accelerators, Vol. 1, Moscow, 1968, p. 129 (in Russian). [2] C. Pellegrini, A.M. Sessler, The equilibrium length of highcurrent bunches in electron storage rings, Nuovo Cimento, A 3 (1971) 116. [3] J. Haissinski, Nuovo Cimento B 18 (1) (1973) 72. [4] A. Piwinski, Bunch lengthening and power losses due to the vacuum chamber walls, DESY 72/72, 1972.
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J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8
[5] D. Boussard, CERN LabII/RF/int./75-2, 1975. [6] P. Wilson, et al., IEEE Trans. Nucl. Sci. NS-24 (1977) 1211. [7] F.J. Sacherer, IEEE Trans. Nucl. Sci. NS-24 (1977) 1393; F.J. Sacherer, IEEE Trans. Nucl. Sci. NS-51 (1995) 43. [8] K. Bane, P.B. Wilson, Bunch lengthening due to potential well distortion from cylindrical cavities with beam ports, SLAC-PUB-1909, PEP-241, 1977. [9] R. Siemann, Nucl. Instr. and Meth. A 203 (1982) 57. [10] S. Krinsky, J.M. Wang, Particle Accelerators 12 (1982) 107. [11] J. Gareyte, Nucl. Instr. and Meth. A 239 (1985) 72. [12] J. Le Duff, Nucl. Instr. and Meth. A 239 (1985) 83. [13] B. Chen, A. Chao, Bunch lengthening effect and localized impedance, Proceedings of the PAC93, Vol. 5, 1993, p. 3348. [14] A.W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators, Wiley, New York, 1993. [15] K. Oide, Particle Accelerators 51 (1995) 43. [16] S. Heifets, G. Sabbi, Single bunch stability in LER of PEPII SLAC-AP-104, 1996, p. 12. [17] J. Gao, Nucl. Instr. and Meth. A 418 (1998) 332.
[18] J. Gao, Nucl. Instr. and Meth. A 432 (1999) 539. [19] K.L.F. Bane et al., Impedance analysis of bunch length measurements at the ATF damping ring, SLAC-PUB8846, May 2001. [20] R. Holtzapple, et al., Phys. Rev. ST Accel. Beams 3 (2000) 034 401. [21] J. Gao, Nucl. Instr. and Meth. A 416 (1998) 186 (see also PAC97, Vancouver, Canada, 1997, p. 1605). [22] J. Gao, Nucl. Instr. and Meth. A 491 (2002), this issue. [23] M. Sands, The physics of electron storage rings, an introduction, SLAC-121, November 1970. [24] J. Le Duff, Longitudinal beam dynamics in circular accelerators, CERN 85-19, 1985, p. 125. [25] J. Gao, Nucl. Instr. and Meth. A 451 (2000) 545. [26] J. Gao, Nucl. Instr. and Meth. A 463 (2001) 50. [27] Y.H. Chin, K. Satoh, Impedance budget for the KEK B-Factory, PAC95, p. 3082. [28] T. Ieiri, Measurement of bunch length based on beam spectrum in the KEKB, EPAC2000, Vienna, Austria, 2000, p. 1735. [29] H. Hayano, Impedance measurement of ATF DR, EPAC98, Stockholm, Sweden, May 1998, p. 481.
On the single bunch longitudinal collective effects 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 29 January 2002; received in revised form 14 May 2002; accepted 18 May 2002
Abstract After giving an analytical expression for the single bunch short range wake potential of a storage ring, we have discussed separately the roles of linear and nonlinear terms of the Taylor expansion of the wake potential on the bunch longitudinal motion. The equations describing bunch lengthening and increase in energy spread are established. Applications to different operating machines are made. r 2002 Elsevier Science B.V. All rights reserved. PACS: 29.20.c; 29.20.Dh Keywords: Electron storage ring; Bunch lengthening; Single bunch instability
1. Introduction The bunch lengthening phenomenon in an electron storage ring was first observed in ACO [1] and later in other machines, where accompanying bunch lengthening, one finds an increase in the single bunch energy spread with a more or less appreciable threshold current. The first empirical bunch lengthening formula [1] found in ACO is expressed as Ib ðmAÞ 2 2 3 st ðnsÞ ¼ st0 ðnsÞ 1 þ 2 10 ð1Þ E 4 ðGeVÞst ðnsÞ where st ðnsÞ is the bunch rms duration measured in nano-second, st0 ðnsÞ corresponds to st ðnsÞ at zero bunch current, Ib is the bunch current, and *Tel.: +1-33-1-644-683-68; fax: +1-33-1-690-714-99. E-mail address: [email protected] (J. Gao).
EðGeVÞ is the particle energy (the E 4 ðGeVÞ dependence in Eq. (1) was soon corrected to be E 3 ðGeVÞ in the later theoretical works, such as Refs. [2,3]). Since then, understanding these single bunch longitudinal collective phenomena has become one of the main battle fields for accelerator physicists. Reviewing a list (incomplete) of these efforts, Refs. [2–20], one may ask questions, such as ‘‘is this battle finished and is the field cleaned up?’’. Fortunately, for the theorists, and unfortunately, for the experimentalists the answer is negative. As shown in Ref. [19], where the discrepancy between experimental and theoretical results manifests itself to an untolerable degree, it is ever clearer that more theoretical works should be devoted in this classical problem. This paper presents a theoretical framework to explain the physical processes and to establish a
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 1 - 2
2
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8
bunch lengthening equation and, for the first time, the equation for the increase in the energy spread of the bunch due to wake potential. In Section 2, an analytical expression for the wake potential of the total machine is proposed. In Section 3, the theoretical framework is established to explain and calculate bunch lengthening and increase in bunch energy spread. Applications to different operating storage rings are given in Section 4. What one should bear in mind is that understanding the longitudinal collective effects is the starting point to treat properly the single bunch transverse instability [21,22].
2. Wake potential expression of a storage ring We start with finding an analytical expression that describes the wake potential of a storage ring. For the convenience of our theoretical treatment coming later, we will use a function of three parameters, i.e., bunch length sz ; total loss factor kðsz ), and the total inductance Lðsz ), to describe the total wake potential of the machine. As an Ansatz, we propose the following analytical expression: 2z2 Wz ðzÞ ¼ akðsz Þ exp 2 7sz 2 Zi z pffiffiffi cos 1 þ atan atan p 2Zr 3sz Zi þatan ð2Þ 2Zr where a ¼ 2:23; Zi ¼ 2pL=T0 ; Zr ¼ kðsz ÞTb2 =T0 ; T0 ¼ 2pRav =c; Tb ¼ 3sz =c; Rav is the average radius of the ring, sz is the bunch length, c is the velocity of light, and z ¼ 0 corresponds to the center of the bunch. The effectiveness of the wake potential expression shown in Eq. (2) will be demonstrated in Section 4. The formulae through out the paper are in MKS units. Rigorous derivation of an expression similar to Eq. (2) is not the point of this paper. The reader may find a more accurate expression using a broadband impedance model.
3. Theory According to electron storage ring theory [23] we know that the single particle ‘‘bunch length’’ sz0 can be expressed as ca 2 2 2 sz0 ¼ sE0 ð3Þ Os0 E0 where a is the momentum compaction factor, sE0 is the energy spread, E0 is the particle’s energy, and Os0 is the angular frequency of the synchrotron oscillation which can be obtained from the following differential equations [24]: dW ¼ eV ðsin f sin fs0 Þ dt
ð4Þ
df 1 hZos ¼ W dt 2pps Rav
ð5Þ
where W ¼ 2pðE E0 Þ=os ; os is the angular revolution frequency of the synchronous particle, V is the peak rf voltage, fs0 the synchronous phase, h is harmonic number, ps is the synchronous particle’s momentum, Z ¼ ð1=g2 Þ a; g is the normalized particle’s energy, and Rav is the average radius of the machine. If Eq. (4) is linearized as dW ¼ eV cos fs0 Df dt
ð6Þ
where Df ¼ f fs0 ; one gets O2s0 ¼
eV cos fs0 hZos : 2pRav ps
ð7Þ
Eq. (3) implies that, for a machine of given a and E0 ; there are only two possible parameters through which the bunch length can be changed, i.e., Os0 and sE0 : In the following, we will show how the single-bunch, short-range wake potential perturbs Os0 ; sE0 ; and finally sz0 : Firstly, we expand the analytical wake potential expression shown in Eq. (2) in a Taylor series: 2 z z Wz ðzÞ ¼ Akðsz Þ þ Bkðsz Þ þ Ckðsz Þ sz sz þ Oðz3 Þ
ð8Þ
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8
where a A ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi Zi 1 þ 2Z r
ð9Þ
1 Zi B0:289aZi 1 þ 0:637 atan atan 2Zr C C r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi B¼B @ A 2 Zi Zr 1 þ 2Zr ð10Þ a C ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2
Zi 1 þ 2Z r 0 2 1 Zi 1 þ 0:637 atan atan 2Zr B2 C @ þ A: 7 6
ð11Þ Equipped with the analytical expression of the wake potential of the ring, we turn now our attention to the dynamics of the particles inside a bunch. We consider then only the dominating linear part of Wz ðzÞ; denoted as Wlz ðzÞ and expressed as: z Wlz ðzÞ ¼ Bkðsz Þ : ð12Þ sz To treat the Ne different particles in the bunch on the same footing and including the effect of wake potential, Eq. (6) should be modified as follows: dW l lrf 2 ¼ eV cos fs0 Df þ e Ne Wz Df dt 2p Bkðsz Þlrf ¼ 1 e2 N e 2peV cos fs0 sz0 eV cos fs0 Df: The resultant Os is obtained: e2 Ne Bkðsz Þc O2s ¼ O2s0 1 : sz0 hos eV cos fs0
potential, the linear longitudinal motion (Df) can be derived from the following Hamiltonian: H0 ¼
0
ð13Þ
ð14Þ
The discussion made above treats only the static aspect, or the so-called potential well distortion effect [3]. To facilitate studying the role of the nonlinear part of the wake potential, we will use the Hamiltonian formalism. Without the wake
3
p2f 2
þ
O2s0 2 Df 2
ð15Þ
where pf ¼ dDf=dt: To facilitate our study, we introduce the nonlinear wake potential term into the unperturbed Hamiltonian in the form of series of delta functions separated by a revolution period. The perturbed Hamiltonian is expressed as p2f O2s0 2 e2 Ne Ckðsz Þ lrf 2 H¼ þ Df þ 3 2 2 2psz N X hZos dðt nT0 Þ ð16Þ Df3 T0 2pps Rav n¼N where we have ignored the perturbation from the linear wake potential term. At this point recall the general expression for the dynamic aperture in a circular storage ring due to delta multipoles derived in Ref. [25], which has been successfully applied in the beam-beam interaction studies in a lepton circular collider [26]. The perturbed Hamiltonian is expressed as H¼
N p2x Kx ðsÞ 2 b2 L 3 X x þ x þ dðs iLÞ 2 3r i¼N 2
ð17Þ
where px ¼ dx=ds; Kx ðsÞ represents linear lattice focusing, b2 is the coefficient of the delta function sextupole magnetic field, and r is the local bending radius. According to Ref. [25] the corresponding one dimensional dynamic aperture, Adyna;x ; can be calculated analytically: pffiffiffiffiffiffiffiffiffiffiffiffiffi 2b ðsÞ r Adyna;x ¼ pffiffiffi x 3=2 ð18Þ jb2 jL 3bx ðs1 Þ where bx ðsÞ is the beta function of the storage ring and s1 is the location where the delta function sextupole is located. By analogy between Eqs. (16) and (17), one gets the maximum value, ADf ; of Df beyond which Df will undergo phase instability and move in a stochastic way. ADf is expressed as rffiffiffi 2 V cosðfs0 Þ 2psz 2 ADf ¼ : ð19Þ 3 eNe Ckðsz ÞT0 Os0 lrf
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8
4
Defining sDf ¼ 2psz =lrf and dividing ADf by sDf ; one gets rffiffiffi ADf 2 sz V cosðfs0 Þ S¼ ¼ ð20Þ 3 eNe Ckðsz ÞT0 fs0 lrf sDf where fs0 is the synchrotron oscillation frequency. If we take SX1 as a rough criterion for the particles inside a bunch to undergo longitudinal phase stochastic motions, one can get a threshold of particle population inside the bunch, Ne;th ; expressed as follows: rffiffiffi 2 sz0 V cosðfs0 Þ Ne;th ¼ ð21Þ 3 eCkðsz0 ÞT0 fs0 lrf where sz has been replaced by sz0 for simplicity. We consider first the case of Ne XNe;th : As stated above, the phases of the particles inside the bunch will execute stochastic motions (phase instability as we call it). We have to explore the kinetic aspect of Wlz ðzÞ; which is closely related with energy spread increasing. Expressing Wlz ðzÞ as z0 sin Os t þ fi l Wz ðzÞ ¼ Bkðsz Þ ð22Þ sz where fi is a random phase due to the above mentioned stochastic motions, one gets the increment of the random energy fluctuation, w; due to the short range wake potential over one turn (T0 ) expressed as T0 Dw ¼ e2 Ne Wlz ðzÞ exp tE T0 þ 1 exp w ð23Þ tE where tE is the synchrotron radiation damping time. Assuming that Dw can be regarded as a Markov random variable, one knows that the distribution function of Dw is governed by Fokker–Planck equation @F ðt; DwÞ @ ¼ ðGF ðt; DwÞÞ @t @Dw 1 @2 þ ðDF ðt; DwÞÞ 2 @ðDwÞ2
ð24Þ
//ðDwÞ2 SS T0
ð26Þ
where //SS denotes the average over all possible value of z0 and fi : Putting Eq. (23) into Eqs. (25) and (26), one gets e2 N e T0 //ðWlz ðzÞÞSS exp G¼ T0 tE T0 w þ 1 exp ð27Þ T0 tE 2 2 T0 w ðe2 Ne Þ2 D ¼ 1 exp þ tE T0 T0 2T 0 //ðWlz ðzÞÞ2 SS exp tE
ð28Þ
where //Wlz ðzÞSS ¼ 0 and //ðWlz ðzÞÞ2 SS ¼ ðBkðsz ÞÞ2 =2: Inserting Eqs. (27) and (28) into Eq. (24), one gets T0
@F ðt; DwÞ @t
T0 @ ðwF ðt; DwÞÞ ¼ 1 exp @Dw tE 2 1 T0 @2 þ 1 exp ðw2 F ðt; DwÞÞ 2 tE @ðDwÞ2 1 ðe2 Ne Bkðsz ÞÞ2 2T0 exp þ 2 2 tE 2 @ ðF ðt; DwÞÞ: ð29Þ @ðDwÞ2
Now, multiplying both sides of Eq. (29) by w2 and integrating over w; one has d/w2 S 2T0 ¼ 1 exp T0 /w2 S dt tE ðe2 Ne Bkðsz ÞÞ2 2T0 exp þ : ð30Þ 2 tE Since T0 =tE 51; Eq. (30) can be reduced to
with G¼
D¼
//DwSS T0
ð25Þ
T0
d/w2 S 2T0 2 ðe2 Ne Bkðsz ÞÞ2 ¼ : /w S þ dt tE 2
ð31Þ
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8
Solving Eq. (31), one gets ðe2 Ne Bkðsz ÞÞ2 tE s 1 exp /w2 S ¼ tE 2 2T0 s þ exp ð32Þ w2 tE 0 where w0 is the initial value. Obviously, when s-N; one obtains the additional energy spread due to the kinetic motions of the particles s2E;w ¼ /w2 SN
ðBe2 Ne kðsz ÞÞ2 ¼ tE : 4T0
where
ð33Þ
Defining Rz ¼ sz =sz0 and RE ¼ sE =sE0 ; we can rewrite for an isomagnetic ring. Eq. (34) can be expressed explicitly: ! CðRRav Ib Bkðsz0 ÞÞ2 2 RE ¼ 1 þ ð35Þ g7 R2B z where 576p2 e0 C ¼ pffiffiffi : 55 3_c3
For an isomagnetic ring, Eq. (37) can be simplified as eBkðsz0 ÞIb JE Rsz0 1 2 Rz ¼ 1 m0 c3 aCq g3 RBz ! CðRRav Ib Bkðsz0 ÞÞ2 ð38Þ 1þ g7 R2B z
Cq ¼
The total resultant energy spread can be expressed as ! ðBe2 Ne kðsz ÞÞ2 2 2 2 2 sE ¼ sE;0 þ sE;w ¼ sE;0 1 þ tE : ð34Þ 4T0 s2E;0
5
55_ pffiffiffi ¼ 3:84 1013 m 32 3m0 c
ð39Þ
JE ¼ 2 þ aRav =R: JE is called the energy damping partition number. To avoid the unphysical solution, Eq. (38) should be simplified as eBkðsz0 ÞIb JE Rsz0 R2z ¼ 1 þ m0 c3 aCq g3 RBz þ
CðRRav Ib Bkðsz0 ÞÞ2 : g7 R2B z
ð40Þ
For the case of Ne oNe;th ; Eqs. (40) and (35) are reasonably reduced to eBkðsz0 ÞIb JE Rsz0 R2z ¼ 1 þ ð41Þ m0 c3 aCq g3 RBz and
ð36Þ
Ib ¼ eNe c=T0 ; Ne is the particle population inside the bunch, R is the local bending radius, e0 is the permittivity of vacuum, _ is Planck constant, and B is a constant for a given machine (B ¼ 1:21 if one uses SPEAR scaling [6]). Well prepared with Eqs. (14) and (35), one can calculate the resultant bunch length under the influence of the single bunch short range wake potential; ca 2 2 2 sz ¼ sE Os E0 1 e2 Ne Bkðsz0 Þc ¼ s2z0 1 sz0 RBz hos eV cos fs0 ! CðRRav Ib Bkðsz0 ÞÞ2 1þ : ð37Þ g7 R2B z
R2E ¼ 1:
ð42Þ
Having established the bunch lengthening and the energy spread equations for different bunch particle population regions, Ne oNe;th and Ne XNe;th ; respectively, we can apply the results to specific examples.
4. Applications We take first the low energy ring of KEK BFactory as an example with E ¼ 3:5 GeV; R ¼ 16:3 m; Rav ¼ 480 m; V ¼ 4:2 MV; fs ¼ 1:14 kHz; frf ¼ 509 MHz; fs0 ¼ 0:365; sz0 ¼ 0:004 m; L ¼ 22 nH; kðsz0 Þ ¼ 42 V=pC; and Ne;th ¼ 4:3 107 : Fig. 1 shows that the analytical calculation of the total longitudinal wake potential of the ring fits well with that from the numerical simulation [27], where the Gaussian charge distribution
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8 100
5
80
4
60
3
40
2
20
Wz (V/pC)
Wz (V/pC)
6
0 −20 −40
1 0 −1 −2
−60
−3
−80
−4
−100 −0.02
−0.01
0
0.01
0.02
−5 −0.05
z (m)
Fig. 1. KEKB low energy ring: the dots and the solid line represent the wake potentials calculated numerically [27] and analytically by using Eq. (2), respectively, with sz0 ¼ 0:004 m; L ¼ 22 nH; and kðsz0 Þ ¼ 42 V=pC: The dashed line shows the Gaussian bunch shape with arbitrary units.
−0.03
−0.01
0.01
0.03
0.05
z (m)
Fig. 3. The solid line represents the total longitudinal wake potential of PEP-II low energy ring with sz0 ¼ 0:01 m; L ¼ 83:3 nH; and kðsz0 Þ ¼ 2:9 V=pC: The dashed line shows the Gaussian bunch shape with arbitrary units.
3
2 1.8
2.5
1.6 1.4
2
Rz, Re
Rz, Re
1.2 1.5
1 0.8
1
0.6 0.4
0.5
0.2 0
0
0.5
1
1.52
2.5
3
3.5
4
Ne (10^10)
0
0
1
2
3 4 Ne (10^10)
5
6
7
Fig. 2. KEKB low energy ring: the solid line and the dashed line are the bunch lengthening and the energy spread increasing vs. the particle population inside the bunch, respectively (sz0 ¼ 0:004 m).
Fig. 4. PEP-II low energy ring: the solid line and the dashed line are the bunch lengthening and the energy spread increasing vs. the particle population inside the bunch, respectively (sz0 ¼ 0:01 m).
represented by dashed line has arbitrary units and it is true for all following figures. The analytical estimations of bunch lengthening (solid line) and energy spread increasing (dashed line) vs. the particle population inside a bunch are shown in Fig. 2 where a ¼ 0:0002 [28]. As for the low energy ring of PEP-II [16] with E ¼ 3:1 GeV;
R ¼ 13:751 m; Rav ¼ 350 m; V ¼ 3:2 MV; fs ¼ 4:6 kHz; frf ¼ 476 MHz; fs0 ¼ 0:236; sz0 ¼ 0:01 m; L ¼ 83:3 nH; kðsz0 Þ ¼ 2:9 V=pC; a ¼ 0:00131; and Ne;th ¼ 9:3 108 : We show in Fig. 3 the analytical calculation of the total longitudinal wake potential of the machine, and in Fig. 4, the analytical estimations of bunch lengthening (solid
J. Gao / Nuclear Instruments and Methods in Physics Research A 491 (2002) 1–8 10 8 6
Wz (V/pC)
4 2 0 −2 −4 −6 −8 −10 −0.04
-0.02
0
0.02
0.04
z (m)
Fig. 5. The solid line corresponds to the total longitudinal wake potential of ATF damping ring with sz0 ¼ 0:0068 m; L ¼ 14 nH; and kðsz0 Þ ¼ 4:5 V=pC: The dashed line shows the Gaussian bunch shape with arbitrary units.
7
lated analytically, and in Fig. 6 the analytical estimations of bunch lengthening (solid line) and energy spread increasing (dashed line) vs. Ne : To compare with the experimental data, in Fig. 6, we have drawn some experimentally obtained results (dots) from the experimental fitting curve [29] which is more close to the case of sz0 ¼ 0:0068 m: It is obvious that by using the machine impedance L ¼ 14 nH and kðsz0 Þ ¼ 4:5 V=pC at sz0 ¼ 0:0068; one has the theoretically predicted bunch lengthening curve very close to the experimentally observed results. We conclude that the observation in Ref. [19] ‘‘The impedance values implied by our analysis are much larger than expected: L is about a factor of 3 larger than earlier calculations, and R is about a factor of 10 larger, a result that is especially puzzling’’ is due to the fact that the experimental results are fitted with a theoretical model which is not totally correct.
2
5. Conclusion
1.75 1.5
In this paper, we have established a theoretical framework to describe the single bunch longitudinal collective effects, such as bunch lengthening and energy spread increasing through corresponding equations. Applications to different operating machines are made.
Rz, Re
1.25 1 0.75 0.5 0.25 0
Acknowledgements 0
1
2
3
4
5
6
7
8
9
10
Ne (10^9)
Fig. 6. ATF damping ring: the solid line and the dashed line are the bunch lengthening and the energy spread increasing vs. the particle population inside the bunch, respectively (sz0 ¼ 0:0068 m). The dots are experimental results.
The author appreciates very much the useful comments and constant interests from J. Le Duff and J. Haissinski.
References line) and energy spread increasing (dashed line) vs. the particle population, Ne : Finally, looking at ATF damping ring [29] with E ¼ 1:54 GeV; R ¼ 5:73 m; Rav ¼ 22 m; V ¼ 200 KV; fs ¼ 8:95 kHz; frf ¼ 714 MHz; fs0 ¼ 0:811; sz0 ¼ 0:0068 m; L ¼ 14 nH; kðsz0 Þ ¼ 4:5 V=pC; a ¼ 0:00197; and Ne;th ¼ 4:36 107 : we show in Fig. 5 the total longitudinal wake potential of the machine calcu-
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8
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[5] D. Boussard, CERN LabII/RF/int./75-2, 1975. [6] P. Wilson, et al., IEEE Trans. Nucl. Sci. NS-24 (1977) 1211. [7] F.J. Sacherer, IEEE Trans. Nucl. Sci. NS-24 (1977) 1393; F.J. Sacherer, IEEE Trans. Nucl. Sci. NS-51 (1995) 43. [8] K. Bane, P.B. Wilson, Bunch lengthening due to potential well distortion from cylindrical cavities with beam ports, SLAC-PUB-1909, PEP-241, 1977. [9] R. Siemann, Nucl. Instr. and Meth. A 203 (1982) 57. [10] S. Krinsky, J.M. Wang, Particle Accelerators 12 (1982) 107. [11] J. Gareyte, Nucl. Instr. and Meth. A 239 (1985) 72. [12] J. Le Duff, Nucl. Instr. and Meth. A 239 (1985) 83. [13] B. Chen, A. Chao, Bunch lengthening effect and localized impedance, Proceedings of the PAC93, Vol. 5, 1993, p. 3348. [14] A.W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators, Wiley, New York, 1993. [15] K. Oide, Particle Accelerators 51 (1995) 43. [16] S. Heifets, G. Sabbi, Single bunch stability in LER of PEPII SLAC-AP-104, 1996, p. 12. [17] J. Gao, Nucl. Instr. and Meth. A 418 (1998) 332.
[18] J. Gao, Nucl. Instr. and Meth. A 432 (1999) 539. [19] K.L.F. Bane et al., Impedance analysis of bunch length measurements at the ATF damping ring, SLAC-PUB8846, May 2001. [20] R. Holtzapple, et al., Phys. Rev. ST Accel. Beams 3 (2000) 034 401. [21] J. Gao, Nucl. Instr. and Meth. A 416 (1998) 186 (see also PAC97, Vancouver, Canada, 1997, p. 1605). [22] J. Gao, Nucl. Instr. and Meth. A 491 (2002), this issue. [23] M. Sands, The physics of electron storage rings, an introduction, SLAC-121, November 1970. [24] J. Le Duff, Longitudinal beam dynamics in circular accelerators, CERN 85-19, 1985, p. 125. [25] J. Gao, Nucl. Instr. and Meth. A 451 (2000) 545. [26] J. Gao, Nucl. Instr. and Meth. A 463 (2001) 50. [27] Y.H. Chin, K. Satoh, Impedance budget for the KEK B-Factory, PAC95, p. 3082. [28] T. Ieiri, Measurement of bunch length based on beam spectrum in the KEKB, EPAC2000, Vienna, Austria, 2000, p. 1735. [29] H. Hayano, Impedance measurement of ATF DR, EPAC98, Stockholm, Sweden, May 1998, p. 481.