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Envelope Instability in the Fermilab BoosterP. Zhou, J.B. Rosenzweigand S. Stahl

Fermi National Accelerator Laboratory!P.O. Box 500, Batavia, IL 60510

AbstractThe transverse emittance in the FNAL Booster grows athigh beam intensity, in a time less than 100 turns. Weexamine the possible contributions to the fast emittancegrowth from coherent instability of the beam envelope.Theoretical analysis, through the use of envelope equa-tions, is employed to predict the dependence of envelopeinstability on the peak current the Booster beam. Thesepredictions are compared to the results of multi-particletracking calculations.

IntroductionThe intensity limit for operation of low energy protonsynchrotrons is often determined by the space charge tuneshift[l], which can drive individual particle tunes from theoperating point onto a significant machine resonance. Inthis way a relatively small tune shift (usually a few percent)can cause the emittance of a synchrotron beam to growsignificantly. Experimentally, this effect manifests itselfby a linear asymptotic growth of the emittance with peakbeam current. This type of behavior has been observed inexperimental studies of the Fermilab Booste r[2].In contrast to scenario described above, linear transportallows much higher intensity beams to propagate withoutdegradation of the beam emittance, as the beam travelsmuch shorter distances in general, and does not encounterthe periodic resonance-driving defects found in a circularmachine. The limits of stable beam propagation in a lin-ear focusing channel have been demonstrated experimen-

tally by Tiefenback, et al. [3] at LBL. It was found that aslong as the bare phase advance per cell uc did not exceedapproximately 90 degrees, the beam emittance was sta-ble up to very high intensities. At phase advances above90 degrees, however, it was verified that instability of thebeam envelope caused emittance growth in the transmit-ted beam. This result is in agreement with the theoreticalpredictions of Boffman, Laslett, Smith and Haber[4]. AsPresent address: UCLA Dept. of Physics, 405 Hilgard Ave., Los

Angeles, CA 90024tOpmated by the Universities Research Association under con-

tract with the U. S. Department of Energy

many synchrotrons are designed with bare phase advanceper cell near 90 degrees to minimize the necessary aperturein the machine, it is reasonab le to ask whether this effectis important for weaker intensity beams found in circularmachines. In particular, this question was raised with re-gards to the Fermilab Booster, both in its present stateand after the upgrade in injected energy from the linac[5].We review below the standard derivation of rms trans-verse beam envelope equation, and extend it to includethe effects of design orbit curvature, momentum spreadand dispersion. The envelope equation obtained is thenanalyzed for its stability properties under perturbation.This technique is used to examine the Fermilab Booster todetermine possible problems with envelope instability.

The Envelope Equation andInstability

The first envelope equations for describing beam propa-gation including the effects of space charge self-forces werederived by Kapchinskij and Vladimirskij[6] using K-V dis-tribution function.Sacherer generalized the treatment by concentrating onthe evolution of the rms envelopes[7]. This approach isequivalent to following the second moments of the Vlasovequation for the four-dimensional transverse phase space ofthe beam. Assuming no z-y coupling, the second momentsin the (z,p=) plane, where p, f z, satisfy the followingequations:

;T = 2w (1)Zg= -x(%)22 + tFz + z (2)-Y=P -2K(z)zp, + 2pz3.z (3)

where rz is the beams space charge self force divided byP2ymc2. The term p,Fz is related to rms emittance byZrm* = &qqg$ (4)

In addition, if the beam spatial density has elliptical sym-metry, the term zF, depends only on the rms beam di-0-7803-0135-8/!?1$01.00@IEEE 1779

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mensions X E &? and & z J- 3,XQxF,= x+y (5)

where Q = 1/(1ep3r3) with 10 N 31 MA for protons. andthe rms envelope equations areX + K(%)X = $ + y&y (6)

2Y - K(%)Y = 5 + A--x+Y (7)The K-V envelope equations in rms form are contained inEqs. 6 - 7 as a special case.The eigenvalues obtained by perturbation analysis onEqs. 6 - 7 tell the stability of envelope mismatches. When-ever eigenvalues move off of the unit circle on the complexplane, envelope instability occurs. The instabilities startwhen two eigenvalues collide on the unit circle. Necessarycondition for envelope instability is that at least one of thebare phase advances per cell 00 > 90 degrees.Circular Machines

The major changes to beam behavior in circular ma-chines which have relevance to deriving envelope equationsare due to curvature focusing, and to the presence of mo-mentum dispersion[8].Following Sacherers analysis and ignoring the chromaticnature of the focusing we arrive at the equation fo:r the rmshorizontal beam size X in a circular machine --X + [I

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0 300 600 900 1200 1500 1800Beam Current (mA)Figure 2: Instability Growth Rate vs Peak Beam Current

dieted by the non-linear field energy theory[9] is negligi-ble compared to the growth observed in the simulation,thus not countable. However, the current at which emit-tance growth occurs corresponds to when the maximumtune shift pushes the individual particles phase advanceto below 90. This seems to suggest that it is the fourthorder resonance driven by the space charge force, which isexactly what drives the envelope instability. The fact thatthe emittance can change a noticeable amount in just onecell makes us believe that the simple perturbation methodused in the calculation of envelope instability is not ap-propriate. Nonetheless, it does predict the point where asignificant emittance growth occurs. The contribution ofthe above to the emittance growth is however very smallfor the Fermilab booster even at maximum operating cur-rent, thus cannot be the cause of the observed explosiveemittance growth.

In the case including momentum spread, nothing signif-icantly different was observed. Chromaticity was seen toplay little role. We have to point out however that the sim-ulation does not include any external impedance or fielderrors. The failure in this case again points to the faultof the way perturbation was done, i.e. keeping the emit-tance as a known function while perturbing the envelopeequation. The effect of momentum spread, if any, is to in-troduce more incoherence and seen to reduce the emittancegrowth slightly.

ConclusionsThe treatment of the envelope instability in circular ma-chines is a limited success. While it does not predict theexact onset of emittance growth, it does point to the in-tensity of significant emittance growth. The former seems,as suggested by simulation, to be determined by when theparticles with maximum tune shift, instead of the coherentmoment, hit the fourth order resonance. In the case of Fer-milab booster, the emittance growth due to this process is

negligible even at the maximum operating beam intensityand could not have been responsible for the fast emittancegrowth observed. The treatment to include momentumspread failed to give reasonable description and simulationresults suggest minimal effect of momentum spread.

ReferencesPI

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L.J. Laslett, On Intensity Limitations Imposed byTransverse Space-Charge Effec ts in Circular ParticleAccelerators, in Proceedings of the 1963 SummerStudy on Storage Rings, Accelerators, and Experi-mentation at Super-High Energies, 324, BrookhavenNational Laboratory, 1963 (BNL 7534).Y. Chao, J. Crisp, S. Holmes, J. Lackey, W. Merz,Improving the Fermilab Booster Emittance , Pro-ceedings of the 1988 European Particle AcceleratorConference, 663 (World Scientific, London, 1989).M.G. Tiefenback and D. Keefe, Measurement of Sta-bility Limits of a Space-Charge-Dominated Ion Beamin a Long A.G. Transport Channel, IEEE Tkans.Nucl. Sci. NS-32,2483 (1985). Also M.G. Tiefenback,Space-Charge Limits on the Transport of Ions Beamin a Long Alternating Gradient System, LBL-22465,PhD. Thesis, UC-Berkeley (1986).I. Hoffman, L.J. Laslett, L. Smith, and I. Haber, Sta-bility of the Kapchinskij-Vladimirskij (K-V) Distribu-tion in a Long Periodic Transport Channel, ParticleAccelerators 13, 145 (1983).F. Mills and J. MacLachlan, private communication,and C. Celata and L.J. Laslett, Envelope InstabilityStatus Report, unpublished.I.M. Kapchinskij and V.V. Vladimirskij, Limitationsof Proton Beam Current in a Strong Focusing LinearAccelerator Associated with the Beam Space Charge,in Proc. of the Second int. Conf. on High Energy Act.and Inst., 274, CERN (1959).F.J. Sacherer, RMS Envelope Equation With SpaceCharge, IEEE Trans. Nucl. Sci. NS-18, 1105 (1971).P. Zhou and J.B. Rosenzweig, Envelope instability inlow-energy proton synchrotrons, Nucl. Znstr. Meth.A297, 24 (1990).M. Reiser, Theory and Design of Particle Beams,1989 U.S. Particle Accelerator Summer School notes(unpublished book draft).E.L. Hubbard, Ed. Booster Synchrotron, FermilabPublication FN-405 (Fermilab, 1973).L. Schachinger and R. Talman, TEAPOT: A Thin-Element Accelerator Program for Optics and Track-ing, Particle Accelerators, 22, 35 (1987)

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