In this paper we review calculations done recently to demonstrate the stability of beam bunches in the AGS Booster for both cases of protons and heavy ions. 1 ref., 2 figs., 2 tabs

Ruggiero, A G

Ruggiero, Alessandro G.

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Impedances and instabilities of the AGS Booster

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Particle Accelerators, 1990, Vol. 28, pp. 65-70Reprints available directly from the publisherPhotocopying permitted by license only© 1990 Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaIMPEDANCES AND INSTABILITIES OF THE AGS BOOSTER*A.G. RUGGIEROBrookhaven National Laboratory, Upton, NY 11973, USAAbstract In this paper we review calculations done recently to demonstratethe stability of beam bunches in the AGS Booster for both cases of protonsand heavy ions.INTRODUCTIONThe AGS Booster project,1 presently under construction at Brookhaven NationalLaboratory, and expected to be ready for commissioning during 1991, has twomajor objectives:(i) Increase the proton intensity in the AGS by a factor of four (to 6 X 1013 ppp).The Booster will operate at the repetition rate of 10 Hz and produce 3 bunchesper cycle each of 5 x 1012 protons.(ii) Accelerate heavy ions up to gold in the Booster for AGS and eventually forRHIC. In this mode of operation the injection energy is very low, ({J '" 0.05).Because of the large proton intensity and the low velocity at injection of theheavy ions and their large charge state, it is important to demonstrate that beambunches are stable against collective effects that can cause coherent oscillationsboth longitudinal as well as transverse.In the following we shall make first an estimate of the longitudinal and trans-verse coupling impedance and then evaluate the beam stability.LONGITUDINAL COUPLING IMPEDANCEThe energy range of the AGS Booster is limited. The energy does not exceed 1.5GeV for protons and it is only a fraction of a GeV/nucleon for heavy ions. Theinjection energy though is quite low; thus we expect that the space charge providesthe largest contribution to the total longitudinal coupling impedance which is largeand essentially capacitive.We wrote a small program (ZOVERN) which calculates and adds the con-tributions from several wall components, like: wall resistivity (stainless steel),bellows, beam position monitors and variations of the vacuum chamber cross-section. An example of the total result is shown in Fig. 1 for "y = 2.6. As one* Work performed under the auspices of the U.S. Department of Energy.[543]/6566/[544] A. G. RUGGIEROcan see, the imaginary part is very large. The real part is constant and below 10ohm. To observe also that the cut-off harmonic number is relatively low, of onlyfew thousand.Independent estimates of the impedance due to injection/extraction kickermagnets and to the vacuum ports have also been made. They seem to lead to asmall value which add insignificantly to the results shown in Fig. 1.The contribution of the rf cavities is not taken into account here.Table I gives the space charge contribution alone, because it is so dominant,for different ion species. As one can see, it can be as large as 4 kohm in somecases.TRANSVERSE COUPLING IMPEDANCEThere are four major contributions to the transverse coupling impedance in theAGS Booster.The space charge. The expression for the contribution isiRZo (1 1)Zl. = 132'12 . a2 - b2where Zo = 3770, a is the average beam radius and b is the vacuum chamberradius. For protonsZ1.. ~ i 53 MO/ m at injection~ i 14 MO/m at extractionThe resistivity of the vacuum chamberZ _ (1 - i)R ( 2RZOP) 1/21.. - b3 p(n _ II)where R is the average radius of the closed orbit, P the wall resistivity, and II thebetatron tune number. The contribution is small compared to the space-chargeeffect. For protons again, for (n - II) = 5 - 4.8,Z..L = (1 - i)0.058 MO/m at injection0.045 MO/ m at extractionIn the circular approximation, by virtue of the deflection theorem, the longitudi-nal coupling impedance estimate can be translated into an equivalent transversecoupling impedance2R ~Zl.= fJb2 n-llIMPEDANCES AND INSTABILITIES OF AGS BOOSTERIf we take IZlI/nJ ~ 100 as previously estimated, thenIZJ.I ~ 0.2 MO/m[545]/67Finally there are transverse modes proper due to several resonanting struc-tures, like rf cavities, but they are difficult to estimate and can usually be observedwith the beam itself.LONGITUDINAL SINGLE BUNCH INSTABILITYTo estimate the longitudinal stability of individual bunches in the AGS Boosterone calculates the following complex quantityU' - iV' = -i 2eIpp2(Z/n) Q1r1"IE(~EIE)FWHM Awhere e = charge of the electron; Q = particle charge state; A = particle atomicmass number; {J = ratio of particle velocity to speed of light; Z = the complexbeam- environment coupling impedance; n = the harmonic number of the insta-bility; E = the total energy per nucleon of the particle; (~EIE) FWH M = thefull-width half-maximum relative bunch energy spread.Alsowhere 'Y = E / Eo; Eo = the particle rest energy per nucleon; and 'YT is evaluatedat the Booster transition energy. ("IT = 4.88).Finally I p is the bunch peak currentI_ NQe{Jcp- _to=V 21r0'N = the number of particles in a bunch; u = rms bunch length. The bunches areassumed to have a bi-gaussian distribution.In our notation Z = X + iY with X > 0 a resistance and Y is positive for acapacitive reactance and negative for an inductive reactance.If there is no resistance, the reactance being positive (capacitive) and the ac-celerating cycle always below the Booster transition energy, the individual bunchesare always stable.Only the presence of a resistance in the coupling impedance can cause thebunches to be unstable. We can calculate the tolerances on Xln. The results aregiven in Table II. We show the values of U' with space charge at injection and68/[546] A. G. RUGGIEROextraction for each case. Based on the stability diagram shown in Fig. 2 we canthen infer the maximum allowed values for V'. Since we are below the transitionenergy, sign (Ko) > o.The choice of V' depends critically on the shape of the energy distribution.The range of U'for the proton beam during the acceleration cycle is shown inFig. 2. With the exception of a truncated cosine distribution (8) and a first-orderparabolic distribution (9), the beam bunch is always stable provided V' < 0.4, thelimit being set by a second-order parabolic distribution (7) at top energy. Thiscorresponds to the resistive impedance limit X/n < 60 O.TRANSVERSE SINGLE BUNCHIn the worst case of no Landau damping, the growth rate of an instability withinan individual bunch in the transverse plane is given by-1 Ipro R (z )T =-- e 1.ev'YZoThis formula is valld in the case that the growth rate is larger than a synchrotronoscillation period (fast head-tail instability). For a slower rate, one would recoverthe conventional head-tail instability that can be controlled by letting the chro-maticity take a slightly negative value. (We are below the transition energy at-all times!) Only the real part of the transverse coupling impedance Zl. gives acontribution to the instability growth rate.We could identify only the resistivity of the wall as the major contributor tothe real part of Zl., especially in the low frequency range when n ~ v. Insertingthe values, we obtain for 'Y '"oJ1, at injection, T ~ 1 msec.The imaginary part enters in the stability criterion that can be written as(EO) 1rv{l'Y /::i.pIZ.d < --;: IpR I(n - 11)'1 + el pwhere /::i.p/p is the bunch momentum spread, ethe chromaticity. Because of thelarge contribution of space charge to the coupling impedance, the stability criterioncannot be easily satisfied. Nevertheless, in analogy to the longitudinal microwaveinstability, we expect that only perturbations with wavelengths shorter than thebunch length can cause instability. This corresponds to a harmonic number largeenough when the coupling impedance is greatly reduced and, the growth rate issubstantially smaller.IMPEDANCES AND INSTABILITIES OF AGS BOOSTERCOUPLED BUNCHED INSTABILITIES[547]/69Using codes like ZAP and with analytical calculations, we found that the beam inthe booster is stable against longitudinal bunch-to-bunch coherent motion. Thisis true for both the smooth wall components and the parasitic resonating modes.The beam in the booster is unstable against transverse bunch-to-bunch mo-tion. The instability is induced by the wall resistivity, with n - II = 0.2 beingthe predominant mode. There is not enough Landau damping to make the beamstable, especially at low energy, because of the very large space-charge forces. Theestimated growth time is 3 IDS. There is a need of a transverse damper acting onany bunch-to-bunch mode providing a damping rate of at least 300 s-l.Higher order parasitic modes do not seem again to cause any harm to thebeam transverse stability.REFERENCES1. W.T. Weng, invited talk to this conference.TABLE I Space Charge Z / n for Heavy Ion Beams in kOCarbon Sulfur Copper Iodine GoldA 12 32 63 127 197Q 6 14 21 29 33Nxl09 22 6.7 4.7 3.2 2.2Pinjection 0.1262 0.1002 0.0782 0.0595 0.0478Pextraction 0.8714 0.8716 0.8534 0.7900 0.6868Z/ n at injection 1.47 1.86 2.40 3.16 3.93at extraction 0.23 0.23 0.27 0.40 0.65TABLE II Bunch Stability Requirement in the BoosterU'Injection Extraction V' X/nProton 0.67 0.90 0.4 600Carbon 0.11 0.066 0.5 5.30Sulfur 0.066 0.041 0.5 8.5Copper 0.052 0.033 0.5 10.6Iodine 0.034 0.021 0.5 16.7Gold 0.019 0.012 0.5 29.270/[548]IlealA. G. RUGGIEROIrnaginary (--)(+) "'I == 2.6~ "" ••••• .if 11 ..•••••••• - 4 ••••100 1---------- 4--- - " ....&-L..oI~........-"";-.7.. ~:"""..-~_ -r-- _--~• • • .... w •••.. . .. ........-.10FIGURE 1 Z/n versus n, '1 == 2.6.-[5100n1000(2)1.5/(1)10,000FIGURE 2 Stability diagram. Distributions: (1) Lorentzian, (2) Gaus-sian, (3) 5th-order parabolic, (4) 4th...order parabolic; (5)3rd-order p,arabolic,(6) squared cosine, (7) 2nd-order parabolic, (8) truncated cosine, (9) 1st-orderparabolic.

Impedances And Instabilities Of The Ags Booster

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Impedances And Instabilities Of The Ags Booster

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