Gertsev, K F

Kopylov, L I

Mikheev, M S

CERN Document Server

Particle Accelerators, 1990, Vol. 26, pp. 241-246Reprints available directly from the publisherPhotocopying permitted by license only© 1990 Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaTHE MEASUREMENT OF BETATRON: AND SYNCHROTRON TUNESHIFTS IN AN INTENSE BUNCHED BEAMK.F.GERTSEV, L.I.KOPYLOV, M.S.MIKHEEVInstitute for High Energy Physics, 142284, Serpukhov,Moscow region, USSRAbstract1. INTRODUCTIONThe transverse transfer function of a bunched beam hasbeen measured in a wide intensity range at the U-70injection flat bottom. The separation of synchrotron mo-des of oscillation allows one to determine the synchrot-ron and betatron tune shifts and to estimate theacting impedances from the well known theoretical models.The applicability of this method for prediction of beaminstabilities and for adjusting the active feedback loopsis discussed.The form of the transverse transfer function of a bUllched beam (BBT}')has been obtained in 1 without the sp~ce charge effects taken into ac-count. In this case, the BBTF is a number of equidistant spectral lineswith the amplitude(])and the distanc~ between the neighbouring lines equals the synchrotronfrequencyuU s . Here Nb is the number of particles in the bunch, the nor-malization coefficient ?(A) is dependent ~rt the amplitudes of betatronoscillations and the formfactor Gm is determined ase'OGm = JJm(AB)Jm(~B - kB) F,,(B)B dB.In this in~egral, Jmis a Bessel function, ?t = Q -~ ~ is chroma-o{- 3""2' lticity, F,,(B) is the function of distribution over the amplitudes ofsynchrotron oscillations, Q is the betatron tune, ~ is the orbit compac-tion factor, ¥ is the particle energy Note that in this case the sa-tellites possessing the same number but opposite signs have equal ampli-tudes.The space charge effects vary the form of the transfer function, fre-quency shifts depending on the number of the mode of syncrotron oscil-lation occur. The theoretical aspect of this shift has been treated ina number of papers 2 . As was shown in3 , not only mutual location of thesyncrotron modes of oscillations but also their amplitudes vary withintensity.This paper presents the results on measuring the BBTF within a widerange of the U-70 intensities. The separation of synchrotron modes ofoscillations made it possible to determine the synchrotron and betatronfrequency shifts. The resqlts are compared with these obtained using theknown theoretical models.241242 K. F. GERTSEV, L. I. KOPYLOV AND M. S. MIKHEEV2. MEASUREMENT RESULTSThe BBTF has been measured at the U-70 injection flat bottom for E == 1.4 GeV (Y=2.5). Here the revolution frequency is LOo=2jt'41.83 fJ lb5 s-~the betatron tunes are. ~=9.88, Qz=9.81, the synchrotron frequency isQs=9 a 10- 3 and the acceleration harmonic number is q=30. The octupolenonlinearity correction system was adjusted in such a way that the na-tural octupole nonlinearity could be compensated for and the maximumseparation of synchrotron satellites maintained. The sextupole nonline-arity correction system allowed one to obtain the chromaticity f =15which, in accordance with formula (1~ yields the following relationshipbetween the satellites amplitudes for a bunch 80 ns long and zero beamcurrent:The measurements were carried out at the lowest-frequency harmonic (k=10)of the revolution frequency with the help of standard hardware 1 . The-4frequency resolution of a two-channel spectrum analyzer was 2.5 10 WOoNote also that the feedback transverse system was switched on.'1~I29.8229.8'o9.80-+===;:==>=;pae::;::::~~~~""""-"'~:...:;.....,~~~...,..=:;::~-'-QH9.81 9.82 9.831.00.80.60.40.20.09.19 9.801.00.80.00.40.20.09.19FIGURE 1 The beam transfer function for Nb=2~109 p/b (fig.1a)and for Nb =3·1011 p/b (fig.1b).Figure 1 presents the BBTF measured in the Z-direction for differentbeam intensities: a) Nb =2'109 p/b, b) Nb =3-10 11 p/b. The numerals inBETATRON AND SYNCHROTRON TUNE SHIFTS 243the lower part of the figure correspond to the numbers of synchrotronmodes. With this chromaticity, the modes m=-1,0,1,2 are identifiedwithout difficulty. Figure 2 shows the tune location of these modesversus the beam intensity within the range from Nb=1.5~109 p/b toNb=3~1011 p/b while fig.3 gives the relative amplitude of synchrotronsatellites within the same intensity range.100201020-190.82Q, ~¥--JW----"-'-------"-'---"""0.83FIGURE 2. The tunes of synchrotron modes of oscillationsversus bunch current.Let us note some qualitative features of these dependences:a) the satellites numbered +1, -1 are shifted almost concurrently;b) the relative amplitude of these satellites varies essentially withintensity;c) the shift of mode tune increases with its number.It was also noted that with the intensity varying from 7-109 p/b to3'1011 p/b the bunch length also varies from 80 to 120 ns. This is ac-companied by the variation of the momentum bunch spread from (~P/P)min==3.65"10- 3 to (6p/p)max=5.5-10- 3 . This means that an increse of the in-tensity leads to a 2.3 factor of growth of the phase beam volume ratherthan to a deformation of the potential well due to the space charge ef-fects. In our opinion, this is caused by thp coherent synchrotron oscil-lations at the injection flat bottom.In addition, the frequency spectrum of the longitudinal signal in theneighbourhood of the RF harmonic n=3 has also been measured (see fig.4).The numerals in the figure show the numbers of synchrotron harmonics. Asis seen from these measurements, the coherent shift of synchrotron fre-quency does not exceed 6Qs < 10-3, which corresponds to a longitudinalimpedance of Zn/n < 100i Ohm. Taking into account the known from thetheory2 relation between coherent and incoherent shifts one may concludethat in the betatron spectrum the convergence of the satellites havingthe same number but opposite signs should not exceed ~Qc=2010-3, whichis just confirmed in fig.2.2441.51.00.5K. F. GERTSEV, L. I. KOPYLOV AND M. S. MIKHEEV2 10 20 100 200 ~/O-gFIGURE 3. The relative amplitudes of synchrotron satellites versusbunch intensity.To estimate the coherent betatron shift let us used the well-knownformula obtained by F.Sacherer for a wide-band low-frequency impedance 2 :(Wcm - m~) = '}(fuel Cm(iZ (W~»4 oC~QB l' (2)Herp Wcm is the coherent frequency of the m-th synchrotron mode, Cm isthe formfactor of the m-th mode, Z(w~) is the value of the transverse wide-band impedance at the chromatic shift frequency~ = dQdp/p'where ~o is the revolution frequency, B is the bunching factor. Thewall impedance of an elliptic vacuum chamber with semiaxes 5xl0 cm isZ=2.94i MOhm/m (inductive wide-band impedance) at a chromatic shiftfrequency of £.Or =21t·1. 3 .107 s-1 For a beam intensity of Nb =3 -lOll p/bthe coherent zero-mode tune shift caused by this impedance isAQoc=(6-8)IIO- 3 , i.e. it is in a good agreemept with the experimentalvalue.The following values of th~ coefficients Cm have been obtained inpaper2 :CO=1.1,which means a decrease of the shift with number m as (Iml+1)-1. However,our experimental data contradict this model. The shift of the 1st mode isBETATRON AND SYNCHROTRON TUNE SHIFTS 245hQ1c=1.9· ~QOc' that of the 2nd one is AQ2c=2.6 -ll QOc' Such a shift canbe interpreted within the frames of the above model only by assuming thepresence of a frequency-dependent inductive impedance having the follo-wing values:Z ] =]2 .] MOhm/m atZ2=24.1 MOhm/m atHowever, this does not seem realistic.0.80.60.40.20.0 ....L-o-.oo..,...-s----r-....o-.0,...10--~~O.O'-T1-S...--..--- ....0-.0T"'20~~....,..:0.=;;02;;....5..----0-.0+-aoQsFIGURE 4. The longitudinal signal spectrum in the neighbourhoodof the n=3 RF harmonic.In our opinion, to interpret this effect it is necessary to take intoaccount the incoherent space charge shift effect. The following formulafor the synchrotron satellite frequency has been obtained in paper4 :Here ~ Qio is the incoherent space charge betatron tune shift}?e f'" ]is the normalization factor,'X is the formfactor of distribution overthe amplitudes of synchrotron oWcillationa According to our calculations,~Qio=3.4.]O-2, X o =O.3, X]=o.82. Unfortunately, the coefficient X 2has not been calculated in4 , neither have the amplitudes of satellites.We have obtained the following tune shifts:This indicates that the model under discussion yields the results whichagree qualitatively with the experimental ones.The intensity-dependent behaviour of the amplitudes of synchrotronmodes can be explained only with their mutual influence taken into ac-count. This follows from the theory of coupled mode instability (see.for example3 ). The amplitudes of synchrotron modes are enhanced mutuallywith the distance between them decreased due to the coupling beam impe-dances. The experimental data availablp do not contradict this theoryqualitatively though it does not take into account the incoherent spacecharge shift.246 K. F. GERTSEV, L. I. KOPYLOV AND M. S. MIKHEEVTo the best of our knowledge, the experimental results presented aboveare the first measurements which cannot be explained within the framesof the presently availabl'e models of transverse bunched beam instabili-ty and therefore this problem requires a further study.3. OONCLUSIONThe authors have failed to find a fairly complptp model explaining theexperimental data. But still, we hope that these data will be helpfulfor theorists displaying during the recent years a keen interest tothese problems, in particular, to the ones related to the developmentof the theory of coupled modes instability.Simultaneously we see that the obtained results can find direct ap-plications for choosing the working point of the accelerator and fordamping transverse instabilities. For example, as is evident from theBBTF shown in fig.lb, the description of this function with thp help ofonly one parameter, betatron tune, is not only conditional, butalso in many cases illegal. Therefore it is necessary to take into ac-count, firstly, its complicated nature and, secondly, the degree ofexcitation of higher oscillation modes when measuring betatron frequen-cies.Since the beam stability is directly related to the BBTF it is ne-cessary to construct a sound technique of adjusting the damping mode(for sextupoles, octupoles, feedback loops).To conclude, the authors would like to express a deep gratitude totheir notewprthly collegues V.I.Balbekov for a number of valuable re-marks, O.P.Lebedev and A.D.Yermolayev for their kind assistance in car-rying out the experiments.REFERENCES1. V.I.ijalbekov et ale Report at this conference.2. J.L.Laclare. CERN 87-03. Geneva 1987.3. Young-Ho Chin. CERN/SPS/85-2. Geneva 1985.

The Measurement Of Betatron And Synchrotron Tune Shifts In An intense bunched Beam

Abstract

Similar works

Full text

Available Versions

CERN Document Server