A beam-tracking simulation has been done for study of the electron- cooling bunching of ion beams around a few hundred MeV/u. In the simulation, the field due to the space charge of a beam in a round vacuum chamber is calculated from the charge distribution of macro- particles and the image particles due to the macro-particles. The simulation results show that the bunched beams meet a transverse instability more easily than coasting beams. (2 refs)

Katayama, T

Takanaka, M

CERN Document Server

TRANSVERSE INSTABILITY DUE TO THE SPACE CHARGE DURINGTHE ELECTRON-COOLING BUNCHING OF ION BEAMSM. Takanaka, T. Katayama, RIKEN, Wako, JapanAbstractA beam-tracking simulation has been done for study of theelectron-cooling bunching of ion beams around a few hun-dred MeV/u. In the simulation, the field due to the spacecharge of a beam in a round vacuum chamber is calculatedfrom the charge distribution of macro-particles and the im-age particles due to the macro-particles. The simulationresults show that the bunched beams meet a transverse in-stability more easily than coasting beams.1 INTRODUCTIONCollision experiments with ion-electron beams and ion-ion beams are planned at the Double Storage Ring ofRIKEN RI-beam factory project. Ion-beam bunching un-der electron cooling has been investigated by using a beam-tracking simulation for realization of the bunched beamsrequired for the experiments. The previous-simulation re-sults for the case of a 150-MeV/u U92+238 -ion beam of 3.4mA[1] have shown that 1) the beam has wide incoherent-tune spreads longitudinally and transversely, 2) the beamis not trapped into resonances due to the nonlinear fieldsof magnets, and 3) the beam is stable longitudinally underthe longitudinal broad-band impedance of jZbbL =nj = 5 Ωbased on the broad-band impedance model.The transverse space-charge impedance for a circle typeof dipole charge distribution is described as follows;ZscT = iZ0R2γ21a2− 1b2; (1)where Z0 is the free space impedance, R the mean radiusof a ring, ap2rms beam size, and b the inner radius ofa round vacuum chamber. Such a type of impedance be-comes no cause of instability, as the force induced throughthe impedance is proportional to the transverse beam posi-tion from the chamber center. There is a question whetherthe description is applicable to the cases of the space-charge-dominated beams or not, or a question whether atransverse space-charge force induced through the vacuumchamber becomes a cause of instability or not. For the sakeof answering the question, the field due to the space chargein a round vacuum chamber is calculated from the chargedistribution of the macro-particles and the image particlesdue to the macro-particles in the simulation on the assump-tion that the longitudinal component of the field is negligi-ble.Besides the above calculation of the field, in the simula-tion where coupled-bunch phenomena are not dealt, forcesinduced through the transverse broad-band impedance ZbbTand the transverse resistive wall impedance have newlybeen taken into account as forces acting on ions. The nota-tion ZbbLT =n is used just for parameterizing ZbbT ;ZbbT =2Rb2ZbbLTn: [2] (2)The transverse broad-band impedance of jZbbLT =nj =50 Ω has been used on the assumption of the broad-bandimpedance model. The ring and the beam input parametersto the simulation have been listed in[1]. The longitudinaland transverse cooling times for a zero-current beam are 5ms and about 20 ms in the simulation, respectively. Thefollowing simplification has been done in the simulation.1) The vacuum chamber has a round cross section of in-ner radius b around the ring. 2) The  functions along thering are equal to the average ones, and  functions equalto 0. 3) The transverse position displacement of an ion dueto the momentum difference is neglected in order that theinstability described later is seen to have no relation withthe displacement. 4) The effects of nonlinear fields of themagnets on beams are neglected because of the authors’ fo-cus just on the instability. 5) The transverse field due to thespace charge is calculated from the transverse charge distri-bution that is made vertically symmetry by counting half amacro-particle at the coordinates (x, y) located at (x, −y).Otherwise, the instability can occur in the horizontal andthe vertical directions, and the analysis of the beam behav-ior becomes more complicated.2 SMALL VACUUM CHAMBERIn the vacuum chamber of b = 40 mm, a 150-MeV/u U92+238 -ion coasting beam of 3.4 mA is initially electron-cooled to6rms momentum spread of 510−4, and is bunched byapplying a fundamental RF (harmonics no., or h =87) volt-age. Every RF bucket is filled with the beam. The bunchspacing is 3.0 m. The RF voltage is increased graduallywith the momentum-spread kept constant in order that thebunch length is made short.Such a process was simulated with results as shown inFig 1. The amplitude of the horizontal beam position be-comes increased a synchrotron period after applying the RFvoltage. After then, the horizontal emittance is blown upten-times large within two synchrotron periods. The maxi-mum instantaneous current at 5.1 ms is 10 mA at the bunchcenter. The horizontal phase-space distribution at 5.1 ms isshown in Fig 2. The instability is seen to be of the trans-verse dipole mode. The betatron-oscillation-phase differ-0-7803-5573-3/99/$10.00@1999 IEEE. 1719Proceedings of the 1999 Particle Accelerator Conference, New York, 1999ence from the head to the tail of the dense bunch 1.5 m longis about 1=2. The difference b is understood in termsof the slippage factor , the chromaticity , and the bunchlength lb;b =lbR: (3)01234564 10-45 10-46 10-47 10-48 10-49 10-41 10-31.1 10-3RV voltage [kV]Momentum spread (6*rms of Dp/p)Bunch lengthD p/pVRFBunch length (6*rms) [m]10-710-610-510-40.0010.010.11101000 0.8 1.6 2.4 3.2 4 4.8 5.6Transverse emittance (rms) [p m*rad]|Horizontal position| [mm]Time [ms]|x|e hevFigure 1: Instability in the vacuum chamber of 40 mm innerradius during the electron-cooling bunching.Figure 2: Horizontal phase-space distribution. The ellipseis for a reference.3 COHERENT TRANSVERSEOSCILLATIONThe dipole moment of a coherent transverse oscillation(transverse displacementI  I1) is Fourier-expanded asfollows;I1 (t; s) =XΩ;nA(Ω; n) exp−iΩt + insR; (4)where A is the component amplitude, t the time, s the lon-gitudinal position, Ω the coherent frequency, and the inte-ger n the oscillation number around a ring. !0, describedlater, is the revolution frequency.Figure 3 shows the spectrum of the horizontal dipole mo-ment around 5.2 ms. There is no remarkable different spec-trum between the fast waves with n > 0 and the slow waveswith n < 0. A single peak of the major structure of the sig-nal along Ω=!0 − nh  p + 0:38 ( the integer p=fixed,the integer n=variable ) means that the head-tail mode ofthe instability is the so-called rigid-bunch mode. The sig-nal has synchrotron satellites on the both sides, as shownin two lower of Fig.3. The fractional coherent tune is read0.3733 and 0.3867 from peak’s coordinates of the signalswith p  0 and p < 0, respectively, while the fractionalincoherent tune is estimated at 0.380 0.005 from an FFTanalysis.0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.710-710-610-59.3 9.35 9.4 9.45 9.5 9.55 9.6 9.65 9.710-7Figure 3: Spectrum of the dipole mode of coherent trans-verse oscillation which has low-level signals cut away.4 LARGE VACUUM CHAMBERThe simulation was done on the same condition as de-scribed in Section 2 except for the double-sized vacuumchamber of b = 80 mm. The simulation results show thatthe beam is as stable as in the previous case[1] up to onthe way. After the beam is bunched by applying the fun-damental RF voltage 20 kV and becomes shorter than 1 mto the degree that the third harmonic RF voltage can be ap-plied, the bunching is prompted by applying the third har-monic RF voltage. The beam meets the instability whenthe 6rms bunch length reaches 0.6 m. Figure 4 shows theemittance blowing-up on two impedance conditions; theone being the broad-band impedance of jZbbLT =nj = 50 Ωand the resistive wall impedance, and the other no broad-band impedance nor the resistive wall impedance. Themaximum instantaneous currents at 38.5 ms and 39.5 msare 37 mA and 39 mA at the bunch center, respectively.The instability threshold is seen to be dependent not only1720Proceedings of the 1999 Particle Accelerator Conference, New York, 199910-710-610-510-40.0010.010.111010034 35 36 37 38 39 40Transverse emittance (rms) [p m*rad]|Horizontal position| [mm]Time [ms]|x|e hev00.511.51 10-41.2 10-41.4 10-43rd harmonic RF voltage [20 kV]Momentum spread (6*rms of Dp/p)Bunch length (6*rms) [m]V3*RFBunch lengthD p/pFigure 4: Instability in the vacuum chamber of inner radiusb=80 mm during the electron-cooling bunching. The Thickcurves are for the high impedance condition, and the thincurves for no-impedance condition.on the inner radius of the vacuum chamber but also on thetransverse broad-band impedance.It is preferable from the viewpoint of the instability curethat the inner sizes of the vacuum chamber surroundingsaround the ring are designed to be as wide as possible tothe degree that the broad-band impedance of the ring doesnot increase so high.5 COASTING BEAMSFor the sake of checking whether the instability describedin Section 2 is merely due to the increase of the instan-taneous beam current or not, the simulation was done ofhigher-current coasting beams under the electron coolingin the small chamber of b = 40 mm.The simulation results, as shown in Fig. 5, show that thecoasting beam is stable even in the case where the coasting-beam current of 70 mA is 7 times higher than the maximuminstantaneous current of the bunched beam. This meansthat the increase of the current is the necessary conditionfor the instability of the bunched beams but not the suffi-cient condition.6 COHERENCY OF TRANSVERSEOSCILLATIONIn the case of a coasting beam, even if the beam has co-herent transverse oscillation on some account, when theelectromagnetic interaction between the beam and the vac-uum chamber surroundings are negligibly small, the coher-ent oscillation is diluted to disappear for the following rea-son. Localized ions spread around the ring and spread az-imuthally 2 on the transverse phase planes in time owingto the momentum difference.0 1002 10-44 10-46 10-48 10-41 10-30.0010.010.11101003.5 4 4.5 5 5.5Horizontal emittance (rms) [p mm*rad]|Horizontal position| [mm]Time [ms]Momentum spread (6*rms of Dp/p)e hD p/p|x|Figure 5: 70 mA coasting beam electron-cooled in the vac-uum chamber of inner radius b=40 mm.In the case of a bunched beam, the coherent transverseoscillation is not diluted for the following reason. Local-ized ions synchrotron-oscillate and remain localized in theRF separatrix, although ions localized near the boundaryof the separatrix spread along the boundary in time owingto the non-linearity of the synchrotron frequency. There isa limitation on the betatron-oscilation-phase difference be-tween an ion and the synchronous particle. The variationof the difference is described in the term of the longitudi-nal distance zi to the ion from the synchronous particle byEq. (3) in which lb is replaced with −zi. Therefore, thedistribution of the phase difference of the beam is periodicat the synchrotron frequency.It can be said from the above that coherency of the trans-verse oscillation is very low for a coasting beam and highfor a bunched beam. Such high coherency is consideredto be one of causes that make the bunched beams meet theinstability more easily than coasting beams.7 CONCLUSIONSA beam-tracking simulation of the electron-cooling bunch-ing of ion beams showed the following results. The space-charge dominated beams meet a transverse instability whenthey are bunched by using an RF system and an electroncooler. The instability is due to the transverse field inducedthrough the vacuum chamber by the space charge of thebeam, or through the transverse space-charge impedance.The instability has the dipole mode as the transverse modeand the rigid-bunch mode as the head-tail mode. Thebunched beams meet the instability more easily than coast-ing beams at least by 7 times in the regard of the instanta-neous beam current.Coherency of the transverse oscillation is higher forthe bunched beams than for coasting beams. Such highcoherency is considered to be one of causes that makebunched beams meet the instability more easily than coast-ing beams.8 REFERENCES[1] M. Takanaka, T. Katayama: Proc. of PAC’97, 1807(1997).M. Takanaka, T. Katayama: Proc. of EPAC’98, 1014(1998).[2] B. Zotter, F. Sacherer: CERN 77-13, 175(1973).1721Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

Transverse instability due to the space charge during the electron-cooling bunching of ion beams

Abstract

Similar works

Full text

Available Versions

CERN Document Server