Possible performance limitations coming fromcollective effects in the CLIC damping rings are the subject of this paper. In particular, the consequences of space charge, due to the very high beam brilliance, and of the resistive wall impedance, due to the locally very small beam pipe, are considered potentially dangerous in spite of the high beam energy. Space charge has been studied in detail with the HEADTAIL code, which was modified in order to take into account a finer lattice structure. This study also includes requirements on the broad band impedance of the damping rings and ion effects in the electron ring (electron cloud in the positron ring is treated in a companion paper). Its goal is to identify all the potential design constraints determined by these phenomena

Jeanneret, J B

Papaphilippou, Y

Quatraro, D

Rumolo, G

CERN Document Server

CERN-AB-2008-026COLLECTIVE EFFECTS IN THE CLIC DAMPING RINGS*Possible performance limitations coming from collective effects in the CLIC damping rings are the subjectof this paper. In particular, the consequences of space charge, due to the very high beam brilliance, and ofthe resistive wall impedance, due to the locally very small beam pipe, are considered potentially dangerousin spite of the high beam energy. Space charge has been studied in detail with the HEADTAIL code, whichwas modified in order to take into account a finer lattice structure. This study also includes requirements onthe broad band impedance of the damping rings and ion effects in the electron ring (electron cloud in thepositron ring is treated in a companion paper). Its goal is to identify all the potential design constraintsdetermined by these phenomena.EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN   -   AB DepartmentGeneva, SwitzerlandAugust 2008AbstractCERN-AB-2008-026CLIC-Note -762EUROTeV-Report-2008-038*Work supported by the Commission of the European Communities under the 6th Framework Programme“Structuring the European Research Area”, contract number RIDS-011899.G. Rumolo, J.-B. Jeanneret, Y. Papaphilippou, D. QuatraroCERN, Geneva, SwitzerlandPresented atEPAC'08, 11th European Particle Accelerator Conference, Genoa, Italy - June 23-27, 2008COLLECTIVE EFFECTS IN THE CLIC DAMPING RINGS∗G. Rumolo, J.B. Jeanneret, Y. Papaphilippou, D. Quatraro, CERN, GenevaAbstractPossible performance limitations coming from collectiveeffects in the CLIC damping rings are the subject of thispaper. In particular, the consequences of space charge, dueto the very high beam brilliance, and of the resistive wallimpedance, due to the locally very small beam pipe, areconsidered potentially dangerous in spite of the high beamenergy. Space charge has been studied in detail with theHEADTAIL code, which was modified in order to take intoaccount a finer lattice structure. This study also includesrequirements on the broad band impedance of the dampingrings and ion effects in the electron ring (electron cloud inthe positron ring is treated in a companion paper).Its goalis to identify all the potential design constraints determinedby these phenomena.INTRODUCTIONCollective effects can be the bottle-neck in the perfor-mances of any accelerator and usually pose limitations toits ultimate performances. In the design stage of an accel-erator, it is necessary to budget a total impedance that isconsistent with the beam intensity requirements. Besides,the intensity requirement must be checked not to clash withother possible multi-particle limitations coming from spacecharge, electron/ion effects, Intra-Beam Scattering.The impedance of an accelerator is usually thought of asmade of three main contributions: resistive wall, severalnarrow band resonators modeling cavity-like objects, andone broad-band resonator that models the rest of the ring.The broad-band resonator, which models the global effectof all discontinuities of the beam pipe and several non-resonating objects (like pick-ups, kickers, etc.), is shortrange and mostly affects the particle dynamics within onesingle bunch. Both resistive wall and narrow band res-onators usually produce severe multi-bunch effects, sincethey are associated to slowly decaying wake fields caus-ing mainly the motion of one bunch to affect subsequentbunches.The aim of this paper is to review the impact of collectiveeffects for the CLIC Damping Rings, whose parameters aresummarized in Table 1.∗This work is supported by the Commission of the European Com-munities under the 6th Framework Programme Structuring the EuropeanResearch Area, contract number RIDS-011899.Table 1: Damping Ring parameters used in our studyEnergy p0 (GeV) 2.424Norm. transv. emitt. ǫxn,yn (nm) 381, 4.1Bunch length σz (mm) 1.53Momentum spread σδ 1.43 ×10−3Bunch spacing ∆Tb (ns) 0.5Bunch population Nb 4.1× 109Circumference C (m) 365.2Coupling (%) 0.13Mom. compact. α 8× 10−5Number of bunches nb 312Tunes Qx,y,s (m) 69.82, 33.80Store time/train Tst (ms) 20Energy loss ∆E (MeV/turn) 3.857Damping times τx,y,z (ms) 1.5, 1.5, 0.74RF frequency frf (GHz) 2RF voltage Vrf (MV) 4.115Bend length Lbend (m) 0.545Bend chamber rad. Rbend (cm) 2Number of bends Nbend (m) 96Wiggler length Lw (m) 2Wiggler field Bw (T) 2.5Number of wigglers Nw (m) 76Wiggler radius rw (mm) 9SINGLE BUNCH EFFECTSSpace ChargeThe incoherent (direct) space charge tune spread is givenby:∆Qy =NbreC(2π)3/2γ3σz√1ǫy··〈√βy√βxǫx +D2xσ2δ +√βyǫy〉 ≃ 0.188 (1)In this formula, ǫx,y are the beam geometric emittances,which can be obtained from the normalized ones dividingby γ. The incoherent tune shift is certainly more criticalin the vertical plane. The horizontal one, due to the largeremittance, is in the order of 0.01. Furthermore, the coher-ent tune shift coming from image charges is much belowthis value (in both planes), because it is inversely propor-tional to the square of the mean chamber radius, rather thanto the product of the beam rms sizes.A possible emittance growth due to space charge has beenstudied with macroparticle simulation by tracking the beamin the DR over 1000 turns with a variable number of spacecharge kick locations chosen randomly in the lattice (andaveraging over different seeds). It is clear that, with a highenough number of kicks the growth levels at about 10%(see Fig. 1), which could be still intolerable if extrapolatedto the 16000 turns of residence time of the beams in thestorage rings. However, this might still be a numerical ef-fect and besides, it could quickly saturate as the emittancegrows and the space charge effect is reduced. Simulationcampaigns with higher numbers of space charge kicks andscans in the parameter space are foreseen in the future. Itis also planned to track the beams with space charge andat least one nonlinearity in the machine to study whetherpossible phenomena of periodic resonance crossing couldbe important for the CLIC-DRs. 10 20 30 40 50 60 70 80 20  30  40  50  60ε y# of S. C. kicks per turn% of εy growthFigure 1: Space charge induced emittance growth as a functionof the number of space charge kicks given to the bunch around thering.Longitudinal and Transverse Broad-bandImpedanceThe broad-band resonator model gives a description ofthe impedance at frequencies ∼ ωr = c/b, where b is thebeam pipe half-height. In the higher frequency range, themore detailed diffraction model gives better results, but itneeds to be applied only on a scale |z| ≪ b (where z isa longitudinal coordinate along the bunch). In the lowerfrequency range, the broad band model does not take intoaccount possible resonant responses to sharply defined fre-quencies (cavity modes). These cavity modes occur at fre-quencies below the cut-off frequency c/b (since they haveto be trapped) and give rise to wake functions which ringfor long periods of time. Neglecting these long-range con-tributions, we certainly do not get accurate results from themodel in the range |z| ≫ b. Taking into account that theCLIC-DR vacuum chamber radius ranges between 8 and20 mm, the frequency ωr lies between 2.4 and 6 GHz. Asufficiently smooth vacuum chamber design, as is in mod-ern accelerators, can easily allow for an impedance of about1Ω, or even lower. The transverse impedance is roughlylinked to the longitudinal one by:Z⊥1 (ω) =Cπb2Z||0 (ω)n, (2)where C is the ring circumference. Assuming the longi-tudinal impedance to be 1 Ω, we then easily get to a trans-verse impedance of about 0.3− 2MΩ/m.A broad band impedance is expected to cause microwaveinstability above a certain threshold, which we can roughlyevaluate using the Boussard criterion (coasting beam cri-terion extended to bunched beams). However, even belowthreshold, there could be enough potential well distortionas to cause bunch lengthening, which can be associated tobunch shape deformation or coherent undamped dipole mo-tion of the bunch within the bucket. The Boussard criterionfor longitudinal microwave instability (or turbulent bunchlengthening) is [1]:∣∣∣∣∣Z||0n∣∣∣∣∣ < 1.7 ln(2)Z0 |η|γNbr0 σ2δσz (3)Plugging the CLIC-DR values into the above formula,we obtain the expected threshold of microwave instabilityto sit around 44 mΩ. This value is very low, but it be-comes reasonably high (1–7 Ω) if we include in the aboveformula the suppression factor (b/σz)2 [2], which for theDRs ranges between 30 and 170.In the transverse plane, head-tail interaction through awake field can drive the bunch unstable. Regular head-tail instability can be avoided by running the ring at someslightly positive value of chromaticity. Strong head-tail in-stability, also known as Transverse Mode Coupling Insta-bility (TMCI), can nevertheless affect the DR bunch andcause rapid beam loss if the transverse impedance is strongenough. For a round beam in a round chamber, a criterionto find the threshold of TMCI, obtained from the applica-tion of the kinetic theory and the mode coupling condition,applicable in the two limits of short or very long bunch,reads [3]:ξ <Qsωrσtif ωrσt ≤ 1 (4)ξ <√2QQs(ωrσt)2 if ωrσt ≫ 1 (5)whereξ =ωr/2π < βy > RTNbe3.75QE/eIf we consider ωr ranging between 2π × 2.4 and 6 GHz,the discriminating number ωrσt remains below 1. Conse-quently, the DR bunch is in the regime of short bunch andEq. (4) should be applied to predict the threshold for theTMCI. In terms of broad-band resonator parameters and inconvenient units, we can simply recast this equation in thefollowing form:RT [kΩ/m] f2r [GHz]Q≤ 0.6 E [GeV] Qs< βy > [m] Qb [C] σt [ps],(6)where Qb = Nbe represents the bunch charge in Coulomband fr = ωr/2π.Using the DR parameters from Table 1, Eq. (6) yieldsRT [MΩ/m] f2r [GHz]Q≤ 5.5× 102 ,which translates into a threshold of 15 MΩ/m for the RTparameter of the transverse broad band resonator in the fre-quency range of interest (6 GHz) and for a quality factorQ = 1. As said above, the present design of storage ringsshould allow for an impedance well below this value.MULTI-BUNCH EFFECTSCoupled Bunch InstabilitiesThe rise time of the coupled bunch modes caused by re-sistive wall are calculated from the imaginary part of theformula:∆ωx,yµ,m = −i2Γ(m+ 1/2)2mm!Nbr0c2 <βx,y>γCσz··∞∑p=−∞Zx,y1(ωp)hm(ωp − ωξx,y)∞∑p=−∞hm(ωp − ωξx,y), (7)withhm(ω) =(ωσzc)2mexp(−ω2σ2zc2)ωp = (pM + µ+Qx,y +mνs)ω0µ = 0, 1, ...,M − 1m = 0,±1,±2, ....We consider the resistive wall impedance (inversely pro-portional to the square root of the conductivity and to thethird power of the beam radius) and use the pipe verticalsize from the wigglers (flat structure). M is assumed to bethe harmonic number of the radiofrequency. The numbersobtained are pessimistic because 1) the wigglers only coverhalf of the ring, therefore the worst growth time would stillbe about twice larger, and 2) they assume a ring fully filledwith bunches, which would approximately scale by anothernb/M factor the real growth rate of the instability. Thegrowth/damping rates for m = 0 are plotted in Fig. 2. Theminimum growth time of 1 ms corresponds to about 1000turns (and is also in the order of the vertical damping time)and can be easily damped with a transverse feedback. 0.001 0.01 0.1 1 10 100 1000 0  500  1000  1500  2000  2500τ (s)Mode numberGrowth/decay timesHorizontalVerticalFigure 2: Growth/damping times of the m = 0 coherent modesas a function of the µ mode number for a coupled bunch resistivewall instability in the CLIC-DRs.Ion EffectsIn the electron DR, the ion oscillation frequency insidethe bunch train during the store is in the range of 300 MHz(horizontal plane) to about 1 GHz (vertical plane), to bedivided by the the square root of the mass number of theion√A. However, not all ion types will be trapped in thebunch train, and the critical mass for trapping of a singlycharged ion is:Acrit =Nb∆Tbcrp2σy(σx + σy)≃ 13 (8)This means that ions like CO+, N+2 or H2O+ will betrapped and will accumulate around the electron beam, po-tentially becoming a source of fast ion instability. The in-duced tune shift introduced by the ion cloud at the end ofthe train is given by∆Qion ≃ NbnbreCπγ√ǫxǫy(σionpkBT)≃ 0.05 (9)where a standard pressure p=1 nTorr and 2 MBarn ofionization cross section were assumed. The exponentialrise time of the fast ion instability is:τinst ≃ 0.1 · γσxσyNbnbcreβyσion(kBTp)√8π(10)and it amounts to about 1.1 µs, which is about one revo-lution time and would therefore require a very demandingmulti-bunch feedback system to be controlled. Therefore,a lower vacuum would be desirable.CONCLUSIONSIn conclusion, the requirements in terms of longitudinaland transverse broad band impedance (low frequency partof the total machine impedance) are not too stringent be-cause a few Ω in longitudinal and apparently a few MΩ/min the transverse plane would still be acceptable to guaran-tee the beam stability against single bunch effects. Spacecharge causes a very large incoherent tune spread, danger-ous in terms of resonance crossing, and can be potentiallya source of emittance growth. However, this specific studyneeds further investigation and cleaning of possible numer-ical effects.Resistive wall multi bunch effects do not seem to be criticalbecause of the large rise time of the associated instabilities(in the order of the damping times). The fast ion instabilityposes a constraint on the acceptable vacuum level, whichmust be much smaller than 1 nTorr.REFERENCES[1] A. W. Chao, “Physics of Collective Beam Instabilities inHigh Energy Accelerators”, John Wiley & Sons, Inc. 1993[2] F. Zimmermann et al., in Proc. of PAC05, Knoxville, Ten-nessee, US(2005)[3] B. Zotter, CERN/ISR-TH/82-10 (1982)

Collective Effects in the CLIC Damping Rings

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