Burov, Alexey V

Zholents, A A

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Particle Accelerators, 1990, Vol. 28, pp. 53-58Reprints available directly from the publisherPhotocopying permitted by license only© 1990 Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaSTABILIZATION OF THE BUNCH LENGTHENING IN STORAGE RINGA.V. BUROV, A.A. ZHOLENTSInstitute of Nuclear Physics, 630090, Novosibirsk, USSRAbstract. The method of amplifying the longitudinal focusing is considered as ameans against the lengthening effect of an intense bunch in the storage ring.Main limitations of this approach are analyzed. A conclusion for having a bigmomentum compaction factor to obtain a small longitudinal emittance is made.PROPOSALTo provide a high luminosity on the colliding beam facility it is important to have thebunch length smaller than the value of p-function at the collision point, which in itsturn is also advantage to be made as small as possible. The obtaining of a shortlength for intense bunch is hindered by the bunch lengthening effect, when the lengthis determined from a well known equation I :a3 = Ne2Rc I~I. (1)-j2;x nHere and below CGS system and the following designations are used: (J is r.m.s. longi-tudinal particle deviation from the bunch mass centre; N is a number of particle in abunch; R-average machine radius, R= n/2Jt, where n is the circumference; Z -broad-band impedance of the vacuum chamber; n - azimuthal harmonic number, n = R/a;x - rigidity of RF focusing, determined so, that a particle with a deviation x from theequilibrium receives an energy xx per one revolution. For NN cavities with a length 'A/2and an amplitude of the electric field on the axis ~ Rx=2e~RNR • (2)Synchrotron oscillation frequencyE is the equilibrium particle energy in a storage ring; T is the revolution period; ex isthe momentum compaction factor.Substituting (2) into (I) one can get the number of cavities required to providethe bunch length of a.(3)The data obtained from measurements of vacuum chamber impedance for a number of[531]/5354/[532] A. V. BUROV AND A. A. ZHOLENTSstorage rings indicate that for the case of a carefully smoothed vacuum chamber onecan expect a value of the given impedance2I~ I=(1-3) Q .Assuming in (3) I~ I=1.5Q, N=6.1011 • 'if R =6MV/m, n=600m, to provide 0=1 emone should haveNR =140.In Reference3 a class of broadband impedance is found which with the growth of Ndoes not correspond to the swelling of the bunch, but, on the contrary, to its short-ening. However, the uncertainty in the real utilization of such a shortening broadbandsystem makes one to search for new means to obtain a short bunch length.The only means left as it seems is an increase on the focusing rigidity x. Afterattaining the maximum of ~ R it should be done by increasing on the number of cavi-ties. That does not mean that all cavities should hr obligatory active. A high-qualitypassive cavity with a right tuned eigenfrequency is also known to focus a bunch. Sinceboth the effects - the bunch lengthening (I) and the focusing by passive cavities - areproportional to the number of particles in the hUJlch, one can suspect that the bunchlength selfstabilization will take place at the ifJt(,ll~itil\~ ovrr the lengthening threshold.The present paper gives a detailed consideration of tlli~ proposal and related problems.LONGITUDINAL DYNAMICSSynchrotron oscillations of electrons ill tilt' storage ring with M bunches, each con-taining N particles, are determined by thr accelerating voltage of activr cavities andthe total field, induced by all bunches in passive cavities. The equations of synchrotronoscillations have the form:X= -ace,. w; N e'2 <x' - [ x Je=-x-- L W kT\- -.ac ET ck= - ,Xl(4)Here x is the longitudinal displacenlent of electron, e is the relative energy shift, W(I)is the wake function of the cavity series, T1 is the bunch pass period and (t)., is thesynchrotron oscillations frequency in the absense of passive cavities.From Eq. (4) the total frequency of synchrotron oscillations is determined:Here2 2 • Ne'2a M <X' -Qs=Ws-l --2- \' (PWI) Z(pwt).ET Lp= - 00Z(W) = 1W(t) eiwidto(5)(6)is the narrow-band impedance, WI =Mwo is the bunch pass frequency.In the case of a narrow-band impedance with a small tune shift of the eigenfre-quency with respect to integer multiple of WI one only consider in sum (5) the terms inthe vicinity of pole Z: p=±po, POWI=Q; Q~wR"The wake function of a cylindrical single-mode cavity with a radius b and alength g has the form:STABILIZATION OF THE BUNCH LENGTHENINGW(t) = WOe-VR'cosooRt, CW=NRW) ,2.4cooR=-"b-'The corresponding narrow-band impedance is equal to:Z(Q) -- W iQ-- 0Q2~ 2+2· (,.wR · lYR"'''Substituting (11) into (5), one can obtain:Q2 = 002+ a Ne2 WO N8 M .!!!JLS S ET2 L\Q 'where L\Q=Q-wR"From here it follows, that the condition:L\Q>O[533]/55(7)(8)(9)( 10)(11)(12)(13)satisfies the focusing action of passive cavities. The amplitude of the longitudinal elect-ric field on the cavity axis is7! = Ne WoM =0.8 MNe 101 ~R Tg 181 L\~~ Rb!\n (14)and it is convenient to express the synchrotron oscillation frequency in terms of thisamplitude(15)( 16)Let us now consider the multibunch coherent oscillations. The motion of short bunches,as compared to the cavity length, is described by the system of equations:XJl (qT) = -acEJl (qT) ,e~(qT) = :; x~(qT) - ~~2 I W[(q-k) T+ I1,MV T- x~(qT) -;xv(qT)].v.1?Here XJl is the longitudinal shift of the electron in the J.t-bunch (.... =0, 1, ... M -1), EJl isits relative energy shift, q is the revolution number. From here the eigenfrequencies ofthe multibunch motion are determined( 17)Here 1 is the number of the multibunch mode (1=0, 1, ... ,M"-I). The mode 1=0 isstable when condition (13) is satisfied. The modes 1=1=0 in correspondence with (17)are unstable. The instability increment quickly decreases with the number of I.56/[534] A. V. BUROV AND A. A. ZHOLENTS( 18)For a quantitative estimation of the increment let us calculate the tune shift ~Q/ooRand the synchrotron frequency using formulae (14), (15). Assuming the electric fieldamplitude for a superconductive cavity to be 6MV/m and N=6.10 11 , M=20,g=30cm, b=23cm (A==60cm), TI=600m, E=5GeV, NR=140, a=5.10- 3, one canfindQ s =0.16.<UoAssuming 'VR/ooR =5 .10- 10, one can obtain1m [00 (/=1)]=0.11 S-1 .Such a small value of the increment indicates the real absence of corresponding cohe-rent effects.According to (16) the power of coherent losses in the narrowband system is( 19)For the values of the storage ring parameters mentioned Clhovt', it isPR = 1.1 kWLIMITATIONS OF THE CAVITIES NUMBERBy increasing the nUITlber of cavities in the storage ring we simultaneously increase thebroadband longitudinal impedance. If the impedance introduced does not exceed theparasitic impedance of the machine, the bunch length can be stabilized. Otherwise thelengthening will occur already at the introduced impedance. This limits the maximumnumber of cavities:NR<N t=±Ill"I' (20)Zoo is the broadband longitudinal impedance of the cavity at the harmonic n=R/a 4:IZ",I=50~ Q.aHere a is the radius of the tube connecting the cavities. At IZI/n=1.5Q, g=30cm,a = 7.5 em and a = 1 em, one can obtainN t =400.The second limitation for the number of cavities is connected to the power of coherentlosses at their broadband impedance. Since the energy loss of a single cavity is equalt04 :pt=2M(Ne)2 _lgnTa V (J (21)then, with the RF power P available for compensating these losses, we come to the fol-STABILIZATION OF THE BUNCH LENGTHENINGlowing limitations:pNR <N2=-.PIAssigning P=2.5 MW, one can find[535]/57(22)(23)The RF power, required for providing the length a of a bunch with N particles, isdetermined by (3) and (21).2M (Ne)3c21 ~ ,,[;P,£ = (2n)5/2 (J7/2 ~R a .The third limitation for the number of cavities follows from the analysis of stability oftransverse coherent oscillations. The distortion of the longitudinal potential well resultsin a large frequency spreadAQs~Qs .Due to this fact the slow coherent oscillations compared to the synchrotron frequencyare stable. An approximated condition for the absence of fast instabilities can be ob-tained, for example, in the model with two rnacroparticles;):2 2Ne Rwo G N ~28wl)QsE l.- R~ , (24)where Gl.- is the average value of the single cavity transverse wake function Gl.- (s), w~is the betatron oscillations frequency. According to the results of Reference4:G ()=~~l.- S Jt a3 .Assuming Gl.-~G.l(2a), Qs/wo=0.16, w~/wo=20 and for the same values of N, R, Eand a as presented above, it follows from (24) that to provide the transverse stability,the following condition should be satisfiedNR<N3= 104• (25)Of all the limitations for the number of cavities considered above, condition (22) seemsmost important.LONGITUDINAL EMITTANCEIt often occurs that not only a short bunch, but also a small longitudinal emittance inthe whole is required. For instance, this problem is significant for damping rings, pro-ducing beams for linear colliders and storage ring factories worked on the narrowresonances. The longitudinal emittance ao£, of a bunch with the number of particlesover the lengthening threshold is calculated from (1) - (3):_ 1 n 1/6 2 Z 2/3 100.- 2n ( 2e ~R NR) (Ne 1-;-1 c) -.;;;i' (26)It follows from here that the choice of the storage ring optics with a big momentum58/[536] A. V. BUROV AND A. A. ZHOLENTScompaction factor promotes the achievment of the ultimately small longitudinal emit-tance. The above formula gives the value of emittance, when the bunch lengthening isdetermined entirely by a microwave instability. If the lengthening (I) is also caused bythe distortion of the potential well, which is not accompanied by an increase in theenergy spread, then the longitudinal emittance will be lower.REFERENCESl. L.J. Laslett, V.K. Neil and A.M. Sessler. Rev. of Sci. Inst., N 3, 32 (1961);D. Boussard. CERN Lab./RFlINT175-2, 1975. -2. L. Rivkin et al. Bunch Lengthening in the SLC Damping Rings.SLAC-PUB-4656. EPAC ConL Rome, 1988;K.L.F. Bane. The calculated Longitudinal Impedance of the SLC Damping Rings.SLAC-PUB-4618, EPAC ConL Rome, 1988.3. A.V. Burov. Bunch Lengthening- is it Inevitable? INP Preprint 89-12 (Novosi-birsk, 1989); Proc. of this ConL4. A.V. Novokhatski. On the Estimation of the Wake Potential for an Ultrarelativis-tic Charge in an Accelerating Structure. INP Preprint 88-39 (Novosibirsk, 1988).5. A. Chao. In Physics of High Energy Particle Accelerators. AlP ConL Proc.,N 105 (New York, 1983).

Stabilization Of The Bunch Lengthening In Storage Ring

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