Balbekov, V I

Gertsev, K F

Kopylov, L I

Mikheev, M S

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Particle Accelerators, 1990, Vol. 27, pp. 215-220Reprints available directly from the publisherPhotocopying permitted by license only© 1990 Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaTHE EFFECT OF MAGNETIC FIELD NONLINEARITIESON THE BUNCHED BEAM TRANSFER FUNCTIONV.I.Balbekov, K.F.Gertsev, L.I.Kopylov, M.S.MikheevInstitute for High Energy Physics, 142284, Serpukhov,Moscow region, USSRAbstract The bunched beam transfer response has been found withaccount of a nonlinear character of oscillations. The signal processingtechnique was proposed for reconstructing the function of particle dis-tribution over betatron amplitudes and for evaluating the octupole fieldnonlinearity. These parameters have been measured at the U-70 machinewith the help of this technique.INTRODUCTIONThe beam transfer function technique (BTF) is applied at acceleratorsto measure various parameters, in particula~ to extimate the stabilityof the beam and its behaviour near betatron resonances. However, thistechnique has been developed only for a continuous beam and the caseof linear oscillations1 . The form of the debunched beam transfer func-tion in the presence of a pronounced octupole field nonlinearity hasbeen considered in 2 but, to our best knowledge, there are no experimen-tal works available on this subject. The RTF technique, applied for abunched beam, i.e. for the majority of real situations, encounterstheoretical and experimental problems.The form of the bunched beam transfer function (BBTF) has beenobtained in this work with account of the octupole nonlinearity effectfor the case when synchrotron satellites don't overlap in the BBTFspectrum. The results on measuring the BBTF at the U-70 machine arepresented. The technique allowing the reconstruction from the data ofthe particle distribution over betatron amplitudes has been offered.2. THEORET ICAL ASPECTSLet us consider for definiteness beam oscillations along axis z. Thestrength of the exciting electric field E and the relevant linear den-sity of a dipole moment D will be presented as a superposition of wa-ves e ik6 - i )! t, where 8 is the azimuth in the coordinate systemco-moving with the beam. A linear relationship exists between the am-plitudes of harmonics:CDL ~. (0) .~. (0)k' =-00(1)where GJ s and /3 s are the angular and reduced velocities of a synchro-nous particle. The explicit form of the conductivity matrix 11 kk (52)will be given later and in the meanwhile we note the following twospecific features:i) the singularities of function Jl kk (~l) are in the lower half-plane of the complex variable 52 ;[461]/215216/[462] V. I. BALBEKOV ET AL.ii) on the real axis these functions differ noticeably from 0within narrow ranges of Q .~ ±LUsQz ' where Qz is the betatron tune ofvertical oscillations.Let field E cos (wt) be produced by a short-length pusher havingan angular extent of ae , which is placed on the azimuth B=0 (in thelab. coordinate system). The field harmonics then take the form~,(n)E'llS1--2 'Jtn + k'ws(n + k'w )2 - ifs(2)To reconstruct the space-time pattern it is necessary to make theinverse Laplace time transform and Fourier azimuth transform. Due toproperty (a) the signal shape will be determined for large t by thepoles of function (2) only. Therefore at point Bg , where the pick-upis placed (in the lab. system) we haveD(t)00 :Ute 00 1(k'W -kW -W)• ~ e g. ~ {~.(-k'Ws+W) • e S S +k=~ k'=-oo1(k'W -lEU> +W) }+~,(-k'ws-w).e s s. (3)In our further calculations, we confine ourselves with the mostinteresting case when the excitation frequency is small, (f~\< LVs /2)and the pick-Up bandwidtQ does not exceed Ws /2 either. Taking intoaccount property (ii) one may get convinced that in the observed frac-tion of the signal only harmonics k=k'=~ko' where ko is the integerclosest to Qz' are left:D(t) N(4)The sign of ~ is now specified in such a way thatThe conductivity matrix can be found by solving the linearizedkinetic equation. The result takes the following form:EFFECT OF MAGNETIC FIELD NONLINEARITIES ON BBTF~ (0)1(k'-~)eJ1(C+nw Q )t-1(k-~).e (e.u.t).e ' e S Z .dt ..o[463]1217(5)Here J is the mean average beam current, Qzz is the betatron tune, Psis the synchronous particle momentum, Ix z are the betatron amplitudessquared, u is the deviation of the momentUm from the synchronous one.The beam distribution function F(Ix ' I z , 8 , u) is normalized by unity.The parameter de has the form(6)where ~ z is the vertical chromaticity, ~ is ~he momentum compactionfactor, y is the reduced energy. Function f)rc)(I} , u, t) is the initialazimuth of the particle being at the instant of time t at point (B, u).For the solution of the problem formulated above the most conve-·nient is the operating mode of the machine for which 'de =0, i.e.t =(a-1-2 ),(Q -K )~z zs -iQ 0In this case, the matrix elements of expression (4) have the form(7 )(8)with the "transverse" distribution function F.ol (Ix, I z ) being normali-zed by 1. It is taken into account here that definition (5) refers tothe upper half-plane, whereas definition (4) requires to be extendedanalytically onto the real axis. For this to be done, the integrand of(5) has to be multipliplied by e- bvsEt and further £ has to approach +0.Putting (8) into (4) we obtainD(t)=D .cos(wt-k 8 )+D .s1n(wt-k0 8g )cOg s (9 )D+1Dc s(10)218/[464] V. I. BALBEKOV ET AL.In Qzthe dependence on Ix' Izshould be taken into account; themomentum-dependence was taken into account in a special way by intro-ducing chromaticity. If Qzis a linear function of its arguments, thenmeasuring Dg one may find the projection of F~ onto a straight linein the plane Ix' I z . Let us consider the most interesting cases:Dse E J fle • [~FIZ]4Q P wf la I u 1 =1zs s s zz z z w1I = - <kc-Q - w/w )(&) azzzs s(11)(lla)where Fz{Iz ) is the projection of the distribution function onto axis I z .Dswhere Fx{I ) is the projection of FJ., onto Ix; I W is determined fromformula (11a) with ~zz replaced by d... zx.3. EXPERIMENTAL RESULTSThe BBTF was measured during the U-70 injection flattop for the fol-lowing beam p~ram~ters: Ekin = 1.5 GeV, N = 2-1011 p/bunch, S2. c ==2.1t·1.8 A 103 s 1, LU s = 2·5L·1.83·105 s-l, I z = 4.9 cm2 , Qz = 9.825,Qr = 9.88, Ap/p = 3-10- 3 . During the measurements the establishedchromaticity was such that be equal to O. In this case, only thecentral peak (m=O) is left in the BBTF, with the amplitudes of theremaining satellites being equal to zero.The equipment used in the measurements contained a conventionalset of units 3 : a pickup, amplifier, filter, noise generator, wide-bandpusher and a two-channel spectrum analyser. The measurements were madeat the lowest-frequency betatron oscillation harmonic (n=10), thefrequency resolution was OQ = 2.5 10- 4 .The correction system for the octupole nonlinearity makes it pos-sible to establish the required dependenceby an independent tuning of coefficients O<zz and c(zx.Figure 1 shows the BBTF's measured for oLzx=O and two values ofcX.zz: o(zz==2.2-10- 3 cm- 2 (curve A) and o(.zz=1.2.10- 3 cm- 2 (curve B).The width of the distribution over level 0.15 from the maximum is~Ql = 6.5-10- 3 , OQ2 == 1.1-10- 2 , respectively, i.e. 8'Q2- bQ1 == 4.5-10- 3 . The relevant calculated increase of the width of the distri-bution over frequencies due to the octupole nonlinearity isEFFECT OF MAGNETIC FIELD NONLINEARITIES ON BBTFand is in a good agreement with the measurements.1.0, J, ~I n~\B .....I q ~_ A0.81 r ~ IO.6~ Iit \II I II If \ \0.4l II \ \I ),( I ......."0.2-11 1ft \ \L....'1~,.,A\~I ... ..,","", .,~"'" .\ ~'.':J,..c\ ...-"-.>.0.0 ~:(?~.~\( 'T- -r ."... T ... •......- aA i9.81 9.82 9.83 9.84 9.85 Qz[465]/219FIGURE 1 The beam transfer function for two different operationalmodes of the octupole nonlinearity correction system:_I -3 -2 ...I -3 -2~zzA = 2.2-10 cm , ~zzB = 1.2-10 cm .Using formula (II) one may reconstruct the particle distributionover betatron amplitudes from the results on measuring the BBTF. Thereconstruction procedure is as follows. First, the point Qzs corres-ponding to particles with a small betatron amplitude is found, with thedistribution edge corresponding to ~s remaining stationary with thevalue of oC zz varying. Then the frequency spectrum is transformed tovariable I z with the known value of ~zz taken into account. Dividingthe result by I z and integrating it one obtains the form of functionF(Iz ). In the region of small amplitudes, where the noise-to-signalratio is larger than unity, the inaccuracy increases essentially. There-fore the experimental points were fit by a smooth second-order depen-dence. Figure 2 presents the azimuthal distribution functions calcula-ted with the help of this technique. In this figure, curve A shows theinitial beam, curve B shows the one cut vertically with a scraper.Figure 3 shows the form of the BBrF for the case 01.- z =0, 0( zx= 2-10- 3 cm- 2 directly corresponding, in accordance with f12), tofunction Fx .One should bear in mind that the nonlinearity of synchrotronoscillations and that of space charge effect contribute into the BBTF.The value of this contribution, Qo' is estimated from the maximum peakwidth with the natural octupole nonlinearity compensated by the cor-t' -3rection system. In our case, oQo = 2-10220/[466] V. I. BALBEKOV ET AL.1.0- _.0.80.6:::J \,~o. 0 ~r-T""""1"--'--ir-r--~T"-r-,--,.-.......-r-I~"""'--."""'~-'·_"-'··'·~·~·~I·---····0.5 1.0 1.5 2.0 2.5 3.0i r3.5I4.0 A~FIGURE 2 The distribution function from the amplitudes of verti-cal oscillations: A - initial beam, B -the one scrapedvertically.9.849.839.811.00.80.6M~ ~"'f.,I \I \0.4 . 10.2 ~ I \J L/,\1 ~ 'y>"',,/,,~0.0 -bo~ ,I~ .- I .. --_ .. 1-' _. \.......M.... ~:':::J...9.82FIGURE 3 -3 2The beam transfer function forol. zz=o,ot zz =2·10 cm- '.4. CONCLUS IONSThe have showen that the BBTF reflects the octupole nonlinearity effecton the spread of betatron frequencies in the beam and may be used to re-construct the distribution of particles in the beam from betatron ampli-tudes. But, in our opinion, the most important point is that the measu-red spread in betatron frequencies is directly responsible for Landaudamping and allows one to estimate the beam stability.REFERENCES1. J.Rorer, R.Tung. CERN 84-15, p. 385, references therein.2. L.Vos. CERN SPS/86-3.3. K.F.Gertsev et ale Preprint IHEP 88-157, Serpukhov, 1988.

The Effect Of Magnetic Field Nonlinearities On The Bunched beam Transfer function

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