SINGLE AND MULTIBUNCH DYNAMICS STUDY DURING THE DAΦNEMAIN RING COMMISSIONING

A. Gallo for the DAΦNE Commissioning Team1

LNF-INFN, Frascati, Italy

Abstract

The Φ-factory DAΦNE is a high luminosity, 510 MeVdouble ring collider presently in operation at the Frascatilaboratories of INFN. The longitudinal and transversebeam dynamics in DAΦNE is a crucial issue, since thestrategy chosen to get the required luminosity calls forhigh values of both single and multibunch currents. Beammeasurements (such as bunch lengths in various regimes,beam transfer functions, coupled bunch oscillation growthrates, microwave and head-tail instability thresholds) arereported and the feedback systems aimed to cure or preventthe beam instabilities described. The agreement betweenthe experimental observations and the expected behavioursis emphasized, while the still open problems arediscussed.

1 INTRODUCTION

The Frascati Φ-factory DAΦNE [1] is a 510 MeV e+e-

collider consisting of two separated rings intersecting eachother with an horizontal crossing angle in two interactionregions (IRs). The collision energy is tuned on theΦ meson resonance to study the CP violation occurringin the Φ decay into neutral kaons [2]. The statisticsneeded for an accurate measurement of the CP violationasks for a high luminosity value (of the order of 1032 cm-

2s-1). The main machine design parameters are sum-marized in Table I.

The strategy adopted to get the required luminosityrelies on high values of both single and multibunchcurrents.

1 C. Biscari, R. Boni, M. Boscolo, A. Clozza, G. DelleMonache, G. Di Pirro, A. Drago, A. Gallo, A. Ghigo,S. Guiducci, F. Marcellini, G. Mazzitelli, C. Milardi,L. Pellegrino, M.A. Preger, R. Ricci, C. Sanelli,F. Sannibale, M. Serio, F. Sgamma, A. Stecchi,A. Stella, C. Vaccarezza, G. Vignola, M. Zobov,INFN-LNF; J. Fox, S. Prabhakar, D. Teytelman,A. Young, SLAC.

A design single bunch luminosity of ≈ 4 1030 cm-2s-1

can be obtained with a single bunch current of ≈ 44 mA.Since up to 120 bunches can be stored in each ring, thelimit luminosity is of the order of ≈5 1032

cm-2s-1 witha stored multibunch current as high as 5 A. From theseconsiderations it is clear that single and multibunch lon-gitudinal and transverse dynamics are crucial issues for aneffective machine operation.

Table I: DAΦNE design parameters

Energy E 510.0 MeV

Emittance εx / εy 1/0.01 mm.mrad

Beam-beam tune shift ξx / ξy 0.04/0.04

Betatron tune νx / νy 5.09/5.07

RF frequency f rf 368.263 MHz

Harmonic number h 120

Revolution frequency f0 3.0688 MHz

Bunch average current I0 43.7 mA

Particles per bunch N 9.0 1010

Momentum compaction αc 0.017

Natural energy spread σε 0 / E 0.000396

Bunch length σz 2.0 ÷ 3.0 cm

Synchr. radiation loss U0 9.3 keV/turn

Damping time τε / τx 17.8/36.0 ms

RF voltage Vrf 100 ÷ 250 kV

Synchrotron tune νs 0.011

Beta functions at IP βx

* / βy* 450/4.5 cm

Max. luminosity L 5.3 1032 cm-2 s-1

The commissioning of DAΦNE has begun in autumn1997 and has been stopped in November 1998 to allowthe KLOE [2] detector roll-in. The commissioninginteraction regions (the so called "day one" IRs), equipped

with standard electro-magnetic quadrupoles and abundantdiagnostics, have been replaced by the new ones (KLOEand DEAR IRs) and the DAΦNE commissioning with theKLOE detector has restarted in the early spring 1999. Themachine is presently in operation, most of the time todeliver luminosity to the experiment. However, someshifts are still dedicated to luminosity tune-up andmachine performance optimization.

2 SINGLE BUNCH DYNAMICS

The single bunch dynamics is dominated by the shortrange wakefields i.e. by the broadband machineimpedance. An extensive study aimed to reduce thecontribution of every vacuum chamber element to theDAΦNE longitudinal impedance has been carried out atthe machine design stage [3]. The overall short-rangewake function has been calculated by adding up the con-tributions of almost all the vacuum chamber disconti-nuities, that were estimated by analytical calculations ornumerical 2D and 3D code (ABCI [4], MAFIA [5])simulations assuming a 2.5 mm gaussian distribution.The resulting wake function and broadband impedance areshown in Fig. 1.

1.5 101 4

-1 101 4

-5 101 3

0

5 101 3

1 101 4

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0 0.02 0.04 0.06 0.08 0.1

w(z), V/C

z, m

0

0.5

1

1.5

2

2.5

3

3.5

0 3 6 9 12 15

|Z/n| [Ω]

f [GHz]

Figure 1: DAΦNE wake potential and broadbandimpedance of a 2.5 mm Gaussian bunch.

It is noticeable that the |Z/n| value is about 0.6 Ω forfrequencies higher than 1.5 GHz. The wake function andimpedance of Fig. 1 have been assumed as the basicinformation to estimate the bunch lengthening process inDAΦNE as well as to evaluate the microwave instabilitythresholds and analyze the bunch behaviour in theturbolent regime. The results have been compared to thebeam measurements.

2.1 Bunch lengthening: simulations

The DAΦNE bunch lengthening has been simulated bymeans of a tracking code similar to those successfullyused to study the same process in the SLC damping rings,SPEAR, PETRA and LEP. This method consists intracking the motion of N superparticles in the longitu-dinal phase space over 4 damping times. The turn-by-turnequation of each macroparticle motion includes latticedispersion, radiation damping, stochastic quantum excita-tion, interactions with the RF field and with the wake ofall the macroparticles ahead [6]. In order to avoid numeri-cal growth of the bunch energy spread, a number ofmacroparticles larger than 105 has to be used to model thebunch.

The bunch physical properties of interest (such as rmslength, rms energy spread, bunch centroid shift) arecalculated by averaging over the last damping time.

The rms bunch length and energy spread calculated forbunch current from 0 to the nominal value are shown inFigs. 2a) and 2b), while the bunch current distributionsfor RF voltages of 100 and 250 kV are shown in Fig. 2c.

The distribution centroid in both Fig. 2c plots isdelayed by the effect of the resistive part of the impedance.

0.5

1

1.5

2

2.5

0 10 20 30 40

I, mA

sigma, cm

250 kV

100 kVa)

Figure 2a): rms bunch length

3.5

4

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5

5.5

6

0 10 20 30 40

I, mA

σE/E, 10-4

250 kV

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c)

Figure 2b): rms energy spread

0

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-60 -40 -20 0 20 40 60

λ, a.u.

z, mm

250 kV

100 kV

d)

Figure 2c): Current distribution @Ib = 44 mA

It is evident from Fig. 2 that the estimated rms bunchlength is always lower than the nominal value of 3 cm,even for RF voltages as low as 100 kV. This result hasbeen confirmed by experimental measurements.

2.2 Bunch lengthening: experimental results

The DAΦNE bunch length has been measured at variousRF voltages by acquiring the beam induced signal in abroad band button electrode with a sampling oscilloscopeTEKTRONIX 11801A through a low attenuation cable(ANDREW FSJB-50A) about 8 m long.

The oscilloscope was equipped with a sampling headSD-24 with a rise time of 17.5 ps and an equivalentbandwidth of 20 GHz and the acquired waveforms havebeen stored on a PC through a GP-IB interface for off-lineanalysis.

The off-line elaboration of the stored waveforms wasnecessary to take into account the contribution of thebutton electrode transfer impedance and the cable frequencyresponse. In fact the Fourier transform V(ω) of the signal

v(t) acquired by the oscilloscope is related to the bunchspectrum I(ω) by:

V(ω) = I(ω) Zt(ω) α(ω)

where Zt(ω) and α(ω) are the button transfer impedanceand the cable frequency response that are both known fromprevious calibration measurements. Once V(ω) has beenacquired it is easy to reconstruct the bunch currentdistribution i(t) as the inverse Fourier transform of I(ω).

The comparison between bunch length measurements(circles) and simulations (solid line) at 100 kV of RFvoltage for the positron ring are reported in Fig. 3. Themeasurements were taken before the KLOE detector roll-in, that means in presence of the temporary day-oneinteraction regions. The agreement between measurementsand simulation is really good for bunch currents largerthan 5 mA. The low current disagreement is attributed tothe frequency response of the button, that peaks and rollsabove the cut-off frequency of the TE11 mode (≈ 5 GHz);short bunches, having a wide spectral contents, producedistorted pulses at the button output, and their originalshapes can be hardly recovered.

1

1.5

2

2.5

3

3.5

0 10 20 30 40

FWHM/2.35, cm

I, mA

Figure 3: Bunch lengthening (FWHM) at 100 kV RFvoltage. Solid line: numerical calculations; circles: mea-surement results.

The bunch length measurement has been performed alsoon the electron ring, and has been repeated on both ringsafter the the detector roll-in, in presence of the new KLOEinteraction region.

The measurements have shown that electron bunches,in same conditions, are few percent longer than positronones. This is probably due to the extra contribution givento the electron ring impedance by several ion clearingelectrodes placed along the vacuum chamber.

The measurements taken after the detector roll-in arevery similar to those taken before, with the exception thatin this case they have been stopped at ≈ 20 mA due to the

harmful presence of a quadrupole single-bunch instabilitiythreshold in both ring. This last effect will be discussed inthe microwave instability paragraph.

As a conclusion, the very good agreement between theDAΦNE bunch length simulations and measurementsshows that the efforts to design a low impedance vacuumchamber and to accurately evaluate the overall machinewake function were successful. Neverthless, it seems thatthe installation of the new interaction region has given anunexpected contribution to the machine impedance whichdoes not affect significantly the bunch length but which isanyway capable to modify the microwave thresholds by alarge amount.

2.3 Microwave instability: analytical evaluations

The coupling between bunch longitudinal coherent modesis the driving source of the microwave instability.Different azimuthal modes may couple if their natural fre-quencies are shifted by amounts comparable to the syn-chrotron frequency ("strong" instability), while radialmodes having the same azimuthal number may couplealready for much smaller frequency shift ("weak" insta-bility). The coherent frequency shift is due to theinteraction between the bunch distribution and themachine impedance.

To study the microwave instability for the DAΦNEbunch we assumed again the machine wake function andbroadband impedance shown in Fig. 1 and we used thedouble water-bag analytical method [7]. This consists inapproximating the real bunch distribution in the longitu-dinal phase space by a double water bag. This is one ofthe simplest methods to take into account the splitting ofthe azimuthal modes with different radial numbers.

The approximated double water bag bunch distributionis put in the Vlasov equation to obtain, with a normalmode expansion of the distribution perturbation, aneigenvalue system with an infinite number of variables(the angular frequencies Ωm,k of the coherent modes) andequations, labelled by the azimuthal and radial indexes mand k (m=1÷∞; k=0,1). The system can be approxi-mately solved by truncating it at some mmax; we limitedour analysis to mmax = 9.

The plot of the calculated coherent mode frequenciesfm,k (m = 1 ÷ 9; k = 0,1) as a function of the bunchcurrent for 4 different RF voltages (100, 150, 200 and 250kV) are reported in Fig. 4. The frequencies of the radialmodes with the same azimuthal number m are equal atzero current (fm,k = m fs), as shown in Fig. 4, whilesplit with increasing current leading to various couplingof modes.

0

5 104

1 105

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2 105

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f, H

z

I, mA

a)

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5 104

1 105

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I, mA

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2 105

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I, mA

c)

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5 104

1 105

1.5 10 5

2 105

2.5 10 5

3 105

3.5 10 5

0 10 20 30 40

f, H

z

I, mA

d)

Figure 4: Longitudinal coherent mode coupling at different RF voltagesa) - 100 kV; b) - 150 kV; c) - 200 kV; d) - 250 kV

Accordingly to these results, the microwave instabilityat low RF voltages is driven by the coupling of radialmodes with low azimuthal index. For instance, atVRF = 100 kV the lowest thresholds are given by thecoupling of the radial quadrupole modes (@ Ib ≈ 24 mA)and sextupole modes (@ Ib ≈ 28 mA). At higher RFvoltages the microwave instability is driven by thecoupling of modes with high azimuthal index.

Since the excitation of coherent longitudinal modesproduce a bunch length modulation, which can in turnexcite beam-beam resonances, the turbulent regime mayaffect seriously the collider luminosity. From this pointof view a high RF voltage operation seems to bepreferable since unstable high order coherent modes drivehigh order beam-beam resonances which are much lessharmful for the beam-beam interaction.

2.4 Microwave instability: experimentalobservations

In spite of the fact that the nominal DAΦNE singlebunch current of ≈44 mA is already a quite large value, inthe first commissioning stage before the detector roll-inwe managed to store routinely single bunch currents of110 mA in both rings, about a factor 2.5 above thenominal value, with no evidence of destructive singlebunch instabilities.

Indeed, we also got a good agreement with theanalytical calculation since we observed, during the bunchlength measurements at VRF=100 kV, the presence of

pure quadrupole and dipole thresholds at Ib ≈ 26 mA and Ib≈ 35 mA respectively. By increasing the RF voltage to150 kV, the quadrupole threshold was shifted to ≈ 38 mAand the dipole one was pushed beyond the nominal singlebunch value. The observed thresholds correspond preciselyto those predicted for the low azimuthal mode coupling bythe double water bag method and shown in the Fig. 4plots.

At higher RF voltage we got no evidence of instability,even for single bunch currents much larger than nominalvalue. This probably means that high azimuthal modecoupling is Landau damped, or, more likely, that themode coupling analytical predictions are much lessaccurate for high azimuthal indexes as a consequence ofthe eigenvalue equation system truncation.

Unfortunately, the well consistent scenario describedabove appeared to be modified after the insertion of thedetector interaction region. Although we did not expectsizeable contributions to the machine wake function andimpedance by the new IR, an harmful quadrupoleinstability has appeared in both rings. The typicaldependence of the single bunch threshold on the RFvoltage for both e+ and e- beams is shown in Fig. 5. Thefirst noticeable effect is the threshold reduction at higherRF voltage. This behaviuor, on one hand, is opposite tothat observed in the early commissioning stage, and, onthe other hand, forced us to operate the RF system atlower voltage (≈100 kV) which is certainly not beneficialfor the beam lifetime and the RF beam loading.

5

10

15

20

25

30

35

75 100 125 150 175 200

electron threshold [mA]positron threshold [mA]

VRF

[kV]

Figure 5: Single bunch current thresholds of the quadrupole mode for the DAΦNE e+ and e- beams.

The threshold dependence on the RF voltage can beexplained by the bunch lengthening effect which reducesthe effective impedance at low RF voltages. Thisinterpretation has been confirmed by measuring thetreshold tendency with the ring momentum compactionαc at fixed RF voltage: the results show that, bylengthening the bunch through an αc increase, thequadrupole thresholds shift up. This can also explain whythe measured thresholds of Fig. 5 were different for thetwo rings: in fact we know that the αc values of the twolattices were not equal during that measurements.

Being the quadrupole thresholds so sensitive to thebunch length, we can conclude that something has beenmodified in the high frequency part of both ringimpedances during the machine stop of last winter. Thesource of this modification is situated with highprobability in one common part of the rings, that meansin one of the two IRs, where some delicate devices arelocated (some shielded bellows, the KLOE shieldedvacuum chamber, ...). At this time we are not able toformulate a more accurate hypothesis.

Anyway, since the bunch length control has beendemonstrated to be effective in pushing up the instabiltythresholds, and since the measured bunch length is stilllower than the nominal value, we are planning to installan high harmonic RF cavity to contro the bunch length.This should allow us not only to freely control the bunchlength, but also to add substantial Landau damping.

2.5 Transverse single bunch dynamics

So far we did not carry out measurements especiallydedicated to single bunch transverse dynamics, but someobservations have shown that the DAΦNE broadbandtransverse impedance is small.

First, an head-tail instability threshold as high as13 mA with sextupoles off has been achieved after anaccurate orbit correction; second, the observed verticaltune shift is a small fraction of the synchrotron tune inthe whole current range from 0 to the nominal value,indicating that the DAΦNE operation is quite far from thetransverse mode coupling threshold.

3 MULTI-BUNCH DYNAMICSThe multibunch dynamics is dominated by the long rangemachine wakefields, that means by the resonant fieldstrapped in various regions of the machine vacuumchamber.

The strategy adopted to optimize the machine design formultibunch operations was based on three guidelines:

- reduction of the number of trapped modes bysmoothing the geometry of the vacuum chamber compo-nents and shielding the resonant volumes;

- damping of the residual trapped modes by means ofbroadband waveguides (RF cavities, longitudinal feedbackkickers) or dedicated probes (transverse kickers, injectionkickers);

- development of fast and efficient feedback systems(bunch-by-bunch longitudinal feedback, 0-mode feedback)to actively cure the coupled bunch instabilities.

3.1 Longitudinal Multi-bunch dynamics

The bunch-by-bunch longitudinal feedback systems [8]have been succesfully commissioned on both rings (July'98 for electrons, September '98 for positrons). With thetemporary day-one IRs, a maximum current of ≈ 550 mAin 30 bunches configuration has been obtained in bothrings, limited by the poor IR vacuum. Recently, about≈ 700 mA have been stored in 120 bunches in both ringswith the longitudinal feedback turned off; this indicatesthat the machine vacuum has been substantially improvedby the new IRs.

The bunch-by-bunch longitudinal feedback systemshave been developed in the frame of an international col-laboration involving the PEP II (SLAC), ALS (LBNL)and DAΦNE (INFN-LNF) groups. Many other machinesin the world are presently implementing similar systems.

The basic architecture of this systems is shown inFig. 6. The longitudinal position of each bunch is trackedby the front-end electronics which consists in a broadbandphase detector followed by a fast digitizer. The digitizedsignal is downampled and sent to a DSP farm where,through a wide choice of possible digital filters, thelongitudinal oscillation of each bunch is shifted by π/2 togenerate a proper out-of-phase kick. The correction signalsare recombined in a single channel by a hold buffer, andthe individual kicks are transferred to the bunches bybroadband kickers driven by pure class A solid stateamplifiers, providing damping of the oscillations. Thelongitudinal kickers are waveguide overloaded cavities thathave been designed for DAΦNE to optimize the shuntimpedance and reduce the HOM content [9].

A plot of the beam spectrum around a revolutionharmonic with feedback off and on is reported in Fig. 7giving a glance of the system effectiveness in dampingdipole coupled bunch oscillations. A damping timeshorter than ≈ 200 µs has been measured under variousconditions.

The feedback system is also a very powerful diagnosticstool when used to perform the so called "grow-damp"measurements [10]. In this modality the feedback gain isinverted or set to 0 for a time long enough to let theinstability amplitudes grow to a measurable level, thenturned again in the standard operation mode restoringbeam control.

Σ

P ickup

Bunched Beam

$ $Power

Amplifier

LongitudinalKicker Cavity

AmplitudeModulator

CombGe nerator

PhaseDetector

LP Filter

MasterOscillator6 (4) * fRF

Timing/Control

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C/D

ownS

ampl

er

DA

C/H

old

Buf

fer

CarrierOscillator

VMEDSP Farm

2.25 (3.25)* fRFQPSK

Modulator

Figure 6: Basic architecture of the bunch-by-bunch longitudinal feedback system

a) feedback off b) feedback on

Figure 7: Longitudinal pick-up signal spectrum of 30 bunches

The output data of the system front-end during both"grow" and "damp" phases are acquired and post-processed.The amplitude vs. time of any possible coupled-bunchmode can be singled out allowing to determine the mostdangerous coupled bunch modes, their growth rates, wherethey are probably located in the frequency spectrum, howmuch they are damped by the feedback system. Anexample of grow-damp measurements on the DAΦNEelectron ring with 30 bunches is shown in Fig. 8. Thetime evolution of the bunch oscillation amplitudes, thetime evolutions of the coupled bunch mode amplitudes,the mode growth rates and damping rates pre and postbreakpoint are reported.

The modes #6 and #11 are the most unstable ones (withrise times of ≈ .8 ms and ≈ 1.3 ms respectively), andthey are efficiently damped by the feedback system(damping times ≈ 200 µs and ≈ 300 µs respectively).

As explained in section 1.4 we have been recentlyforced to operate the RF systems at a lower voltage

(VRF ≈ 100 kV). The negative shift of the synchrotronfrequency of the beam barycentric oscillation (the coupledbunch mode "0") with beam loading is emphasized underthis conditions.

The longitudinal feedback system got in some troublewhen dealing with low synchrotron frequency lines locatedout of the digital filter band. To avoid such kind ofmalfunctioning we decide to increase the damping of the0-mode by adding a dedicated analog feedback acting onthe RF low-level controls. It consists of an RF phasemodulation π/2 out of phase with respect to thelongitudinal barycentric oscillation detected by the beatingbetween a sample of the cavity voltage and the beam-induced signal on a longitudinal monitor [11]. The π/2phase shift is produced by an analog AC coupledintegrator circuit simply based on OP-AMPs. Thecalculated damping time provided by this system isroughly proportional to the integrator bandwith, and is≈ 50 µs for a 6 kHz bandwidth.

Figure 8: Example of multibunch "grow-damp" measurement on the electron beam

The implementation of the 0-mode feedback systemswas successful in preventing the anomalies in thelongitudinal feedback system operations. At present, themulti-bunch current limits are set only by the quadrupolesingle bunch threshold or by the machine vacuum, whichis still improving due to the beam conditioning.

3.2 Transverse Multi-bunch dynamics

During the early commissioning phase we observed thepresence of a coupled-bunch vertical instability limitingthe total positron current. At beginning the transverseinstability has been studied by simply analyzing thespectrum of the signal induced by the unstable beam on avertical position monitor. The frequency location of thehighest betatron lines gave us information on what werethe most excited coupled bunch modes. We also noticedthat the vertical oscillations would start as soon as thelongitudinal feedback was turned on, while almost only

longitudinal oscillations were present with the feedbackturned off. Moreover, the vertical instability could bedamped by increasing the strength of the magneticsextupoles, that means by overcorrecting the ringchromaticity.

More accurate measurements have been made on thepositron beam by digitizing the signal from a transverseBPM using the front-end electronics of the longitudinalfeedback system of the electron ring. The transversemotion of the bunches has been acquired and off-lineprocessed to extract information on the dynamics.

The most unstable coupled bunch mode was found to bethe number 55 in 120 bunch operation. The instabilitysource has been identified as a 1304 MHz mode trapped inthe injection kicker that was found by bench impedancemeasurements performed before the machine assembling.Its resonant frequency is located very near to an unstablesideband of the transverse coupled bunch mode 55;

moreover, the instability threshold is clearly correlated tothe vertical orbit in the injection kickers, as it has beenexperimentally demonstrated by setting local orbit bumpsin all the vacuum chamber regions considered potentiallydangerous for the transverse dynamics.

Once the betatron tunes have been changed (from5.11/5.08, close to the design values, to 5.15/5.21) tooperate the machine at a less critic, farther from theinteger working point[12], the driving mode has beendetuned with respect to the unstable line, and the verticalinstability almost disappeared. Anyway, since the HOMstrapped in the injection kickers are potential source ofboth longitudinal and transverse multibunch instabilities,we have designed special HOM dampers based oncapacitive coupling of the resonant fields[13]. A full setof new injection kickers equipped with HOM dampers ispresently in fabrication, and we plan to install them onthe rings in the next machine shutdown.

4 CONCLUSIONSThe dynamics measurements made during the DAΦNEcommissioning are globally in good agreement with theexpected beam behaviour. The very good agreementbetween the bunch length measurements and simulationsindicates that the computed machine wake function andimpedance are quite realistic, and the measured values ofthe single bunch thresholds (microwave, head-tail, ..)show that the design efforts aimed to reduce thelongitudinal and transverse broadband impedances weresuccessful.

The longitudinal multibunch dynamics is kept undercontrol by the feedback systems, which means that thenatural growth rates due to the resonant modes of thevacuum chamber are not too large. The installation of thenew damped injection kickers should improve thetransverse multibunch dynamics, which however has notlimited the machine performances so far.

Presently, the only beam dynamics serious issue isrepresented by a pure quadrupole longitudinal instabilitylimiting the single bunch current to ≈15 mA. This isprobably a microwave threshold caused by an unexpectedcontribution to the high frequency part of the machinebroadband impedance given by some element in the newinteraction regions. Although the efforts to overcome thisproblem were ineffective so far, we are quite confident thatthe installation of an high harmonic RF system is theright solution to the problem since we have experimentalevidence that the instability threshold raises with thebunch length; moreover we can also expect in this case abeneficial contribution from the Landau damping comingfrom the double RF system operating in the lengtheningregime.

REFERENCES

[1] S. Guiducci and DAΦNE Team "DAΦNE OperatingExperience", PAC 99, New York, March-April1999.

[2] The KLOE collaboration "KLOE a General PurposeDetector for DAΦNE", Frascati Internal NoteLNF-92/109, April 1992.

[3] M. Zobov et al. "Collective effects and impedancestudy for the DAΦNE Φ-Factory" Internationalworkshop on Collective Effects and Impedance for B-factories, Tsukuba, Japan, June 1995, KEKProceedings 96-6, August 1996 (A), pp. 110 - 155.

[4] Y.H. Chin "User's guide for ABCI version 8.8(Azimuthal Beam Cavity Interaction)", LBL-35258 ,UC-414, February 1994.

[5] R. Klatt et al. "MAFIA - A three-dimensional elec-tromagnetic CAD system for magnets, RF structuresand transient wake field calculations", SLACreport 303, 1986.

[6] M. Zobov et al. "Bunch lengthening and microwaveinstability in the DAΦNE Positron Ring", DAΦNETechnical Note BM-3, Frascati, (June 98).

[7] A. Chao, B. Chen and K. Oide, “A Weak Mi-crowave Instability with Potential Well Distortionand Radial Mode Coupling”, in Proceedings of the

16th IEEE Particle Accelerator Conference (PAC95)and International Conference on High EnergyAccelerators (IUPAP), Dallas, Texas, May 1995, pp.3040 - 3042.

[8] M. Serio et al. "Multibunch Instabilities & Cures"Proc. of EPAC '96, Sitges, Spain, June 1996,p. 148.

[9] R. Boni et al., "A waveguide overloaded cavity aslongitudinal kicker for the DAΦNE bunch-by-bunchfeedback system", Particle Accelerators, Vol. 52,p. 95.

[10] J.D. Fox et al., "Observation, control and modalanalysis of longitudinal coupled bunch instabilitiesin the ALS via a digital feedback system", Proc. ofEPAC '96, Sitges, Spain, June 1996, p. 346.

[11] D. Boussard "Design of a ring RF system", RFEngineering for Particle Accelerators, CERNAccelerator School, CERN 92-03 p. 474.

[12] M. Zobov et al. "Beam-beam interactions inDAΦNE: numerical simulations and experimentalresults", this Workshop.

[13] D. Alesini et al. "DAΦNE injection kicker: elec-tromagnetic analysis of trapped modes and dampingantenna design", this Workshop.