Hübner, K

Keil, Eberhard

Zotter, Bruno W

CERN Document Server

COHERENT SPACE CHARGE PHENOMENA IN THE ISRK. HUbner, E. Keil, B. ZotterCERN, Geneva, Switzerland.I for l"O!:oh =0,2[A]150-"II"Qcoh././ 800 A.///././600A400A----- ------~.//././3 /////~////// ~///50 ///~","-100+--~----10Coherent instabilities of bunched andcoasting beams in the ISR are compared to theore-tical predictions. The present phase space densityis below the threshold for the longitudinal cohe-rent instability which has not yet been observedin the ISR, while transverse instabilities havebeen seen and eliminated by the application ofadditional sextupole fields.Abstract1. IntroductionThe theory of coherent oscillations of aparticle beam in a metallic vacuum chamber usuallyneeds a number of simplifYing assumptions to allowmathematical treatment. In general we have assumedcylindrical or rectangular geometry, small pertur-bations, and uniform particle beams. Longitudinaland transverse oscillations are treated separately,and only the dipole mode has been considered.Fig.l. Current provoking a given coherent Q-shiftCurve 1: a.c.fields penetrate vacuum chamber2: a.c.fields not penetrating3: a.c.fields not penetrating incl.finite resistivity of walls and plates.The dots are experimental values.3. Coherent Transverse Instability of coastingbeams2. Coherent Q-shiftThe betatron frequency of a transverselyoscillating beam is influenced by the fields ofthe oscillating charges, and by the boundary condi-tions on the vacuum chamber and/or the magneticpole faces. The coherent Q-shift can be expressed1) bywhere the last two terms have £ or ~ when the magne-tic field of the oscillating beam does or does notpenetrate the vacuum chamber wall. For the ISR,we find for the two cases -8,9Y A and -23y A, wherethe larger figure is probably closer to reality asthe thickness of the vacuum chamber is about twoskindepths. Corrections for the influence of thefinite conductivity of the chamber walls and forthe clearing electrodes are rather small. Measu-rements at low current levels 2) gave a 6I/6Q of-7 Y A. The results are shown in Fig. 1. Thediscrepancy between the calculated and measuredvalues indicates that elements apart from thesmooth resistive wall contribute to the Q-shift.(4)y... NN ap/p1> --vlnmQI =: a > 1.3ap/p~ap/pI(n-Q)~ - ~ n I> -.L Y... -1L (2)ap/p ap/p v~ N op/pIn addition to the shift of the real part ofthe betatron frequency, a finite wall resistivitycan cause an imaginary part. This will lead toan instability of modes with n>Q unless sufficientfrequency spread is provided to cause Landau dam-ping. The stability criterion can be written 4)where v~ is the normalised stability limit varyingbetween close to zero (U»V) and 2/n (V»U), butalmost independent of the distribution functionassumed. For mode numbers close to Q the Q-spreadcontributes most to the LHS, and we getFor the lowest mode (n = 9) in a rectangular cham-ber corresponding to ISR dimensions we find(1)The incoherent 6I/6Qi has also been measured3) and was found to be about 400A at Y = 16,4, ingood agreement with calculated values.for a small stack (~lA) with the actual phase spacedensity presently obtained in the ISR. For largerstacks the criterion becomes less stringent due tothe variation of the geometrical factor F35).- 296 -In the earlier running-in tests of the ISH,sudden "brick-wall" losses were limiting the maxi-mum intensity achievable. Application of sextupolefields to bring Q' above unity made stacking to higherintensities possible. The fact that sextupoles arean efficient cure of this transverse instabilityshows that it occurs at low mode numbers. It istherefore believed to be mainly due to the resisti-vity of the vacuum chamber. Figs.2 shows stacks 6)made with various amounts of Q'. The beam loss at5,5A is due to the tripping of the clearing powersupplies.6A4. Transverse bunched beaminstabilityValues of Q' > 1.4 are required to damp cohe-rent rigid bunch oscillations, similar to the coas-ting beam case. However, synchrotron oscillationsof the particles in the RF-bucket can provide acoupling between the front and rear end of thebunch, and lead to instable oscillation for largervalues of Q'. An estimate of the growth rate dueto the wall resistivity (without Landau damping)is given by 8,9)is the phase shift between front and rear end ofthe bunch. For typical ISH parameters, X = 4,6 Q'and only higher modes (m >2) should be unstablefor a positive Q'. As the maximum growth rate isonly 2 sec -1, Landau damping should be effective.For a negative Q' also the lowesu mode (m = 0) canbecome instable with a growth rate of about 13 sec-I.4wherex = - 2AwnQ'Q (6)8 min 4 oTransverse instability at injection has beenseen in the ISH, and the beam blows up horizontally.Q' > 1 suppresses the instability, while Q' = -2does not. The growth-rate does not vanish forsmall Q', but is of order of 100 sec -1 contraryto the head-tail effect theory.5. Evolution of the Mode SpectrumA large number of signals have been seen on aspectrum analyser with a frequency response froma - 110 MHz connected to pick-Up plates, both du-ring injection and with a circulating beam. Duringinjection, sidebands of the multiples of the revo-lution frequency are usually seen at (n-Q), butalso at other frequencies corresponding to (Q-n)and (n-2Q) modes.Fig. 2. Stacking with di~~erent amounts o~sextupole ~ield. The beam loss at 5.5 Ais due to the tripping o~ the clearingpower supplies.At higher mode numbers, resonating elementsof the vacuum envelope can cause large increases ofV/N. An evaluation for the clearing electrodesshows, however, that the LHS of Eq. (2) increaseseven more due to the contribution o~ the spread inrevolution frequency, and safety factors of over 10are found. Also the influence of resonant cross-section variations is much smaller for the trans-verse than for the longitudinal case 7).With circulating beams o~ a ~ew amperes, singlelines or clusters of lines corresponding to (n-Q)modes have been observed between 30 and 70 MHz, andsometimes as high as the upper ~requency limit ofthe analyser. Some lines appear stationary, remai-ning for minutes, other drift downwards in frequen-cy, covering as much as 10 MHz in a few minutes.There are no obvious resonant elements at thosefrequencies ; this explanation would not fit thedrifting lines in any case. One tentative expla-nation which has been proposed is by electronsoscillaring in the potential well of the protonbeam 10 , which may render transverse proton beam- 297 -modes unstable. Their frequency would lie roughlyin the observed range.2Nc 2r e'1TRb(b+a) ~ 100 MHz (7)where ~ lies between zero and one and expresses thecorrection due to the side-hole, and X > 1 thefrequency correction. A more detailed analysis,taking the finite resistivity of the side tUbesinto account, uses matrices evaluated ntmlericallyby computer 13).Fig. 3 shows a typical spectrtml analyser output.Fig. 3. Frequency spectrtml extending from Q-100MHzpicked up by a plate at a current level of3,5A. The negative spike is instrtmlental.6. Coherent longitudinal instabilitiesMeasurements of the frequency response ofvarious elements have shown high Q-resonances 1,+)in the lower microwave region, but no unambiguousmicrowave signals have been picked up from the beam.7. ConclusionsThe agreement of theory and experiment ofthe transverse resistive wall instability is quitesatisfactory, and the application of the calculatedamount of sextupole component led to its disappea-rance. The situation is less clear for the longi-tudinal case, where theory predicts too largecoupling impedances in the microwave region due toresonant elements, but no direct evidence for aninstability has been obtained so far. At present,the beam intensity does not seem to be limited bycoherent effects.The criterion for stability of coherent lon-gitudinal oscillations of the beam can be written1ijReferenceswhere Z/n is the coupling impedance divided by themode ntmlber. For the first few injected pulses,the limit on Z/n at design phase space density isof the order of 10 n, and much larger (~lkn) fora full stack with not too steep flanks. Becausethe design density has not yet been obtained, thepresent limit should be about lOon.At even higher frequencies, many differentelements of the vacuum envelope become resonant.The coupling impedance then equals roughly theproduct of the low frequency value and the qualityfactor. For example, the large number (~300) ofvariations from elliptical to circular cross-sec-tion have been calculated to contribute about 3.5nper pair, yielding close to 1 kn if they all addin phase. Large tanks can yield even higherimpedances given roughly by 12).B. Zotter ISR Running-in report (Run 84),12.7.1971 (unpublished)W. Schnell ; CERN/ISR-RF/70-7B. Zotter ; CERN/ISR-TH/71-13K. HUbner, B. Zotter ; CERN/ISR-TH/71-18M. Sands ; SLAC-TN-69/8 and 10B. Zotter; CERN/ISR-TH/69-6010) H.G. Hereward ; CERN-71, to be published11) E. Keil, W. Schnell; CERN/ISR-RF-TH/69-4812) E. Keil, B. Zotter ; CERN/ISR-TH/70-30 and 33B. Zotter ; CERN/ISR-TH/70-4713) E. Keil, B. Zotter ; Particle Accelerators(to be published)14) P. Bramham ; BEIC document No. 64, 5.7.1971(unpublished)7)1) L.J. Laslett, L. Resegotti; 6th Int. Conf. onHigh Energy Acc., 1967, p.1502) K. Hubner; ISR Running-in report (Run 89)26.7.1971 (unpublished)ISR Running-in report (Run 105, 107,110), 14.9.1971 (unpublished)3) J.P. Gourber and H. Laeger; ISR Running-inreport (Run 105), 8.9.1971 (unpublished)4) K. Hubner, P. Strolin, V. Vaccaro, B. ZotterCERN/ISR-RF-TH/70-25) B. Zotter CERN/ISR-TH/68-516)8)9)(8)( t>p/m c)2o[33 cr1/2200 Rg(~)n resAt the lowest mode (n =1) the resistive wallcontributes about 5 n, while most other equipment(clearing electrodes, guard plates of pick-upstations, inflection kicker screen, cross-sectionvariations, RF cavities) contribute inductivelyof the order of 10 to 20 i n. For increasing modenumbe~s, the resistive wall contribution decreasesas n- 2 • At the resonant frequency of the RFcavities, the coupling impedance has been reducedto a few n by feedback.

Coherent space charge phenomena in the ISR

Abstract

Similar works

Full text

Available Versions

CERN Document Server