Arakawa, D

Hidaka, S

Ikegami, M

Machida, S

Mochihashi, A

Mori, Y

Ninomiya, S

Noda, K

Oki, T

Toyama, T

Uesugi, T

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OBSERVATION OF QUADRUPOLE MODE FREQUENCYAND ITS CONNECTION WITH BEAM LOSST.Uesugi, S.Machida, Y.Mori, KEK-Tanashi, Midori-cho, Tanashi-shi, Tokyo 188-8501, JapanM.Ikegami, JAERI, Tokai-mura, Naka-gun, Ibaraki-ken 319-1195, JapanS.Ninomiya, A.Mochihashi, T.Oki, S.Hidaka, RCNP, Ibaraki, Osaka 567-0047, JapanD.Arakawa, T.Toyama, KEK 1-1 Oho, Tsukuba-shi, Ibaraki-ken 305-0801, Japanand K.Noda, NIRS, Anagawa, Inage-ku, Chiba-ken 263-8555, JapanAbstractRecent Simulation results imply that the number of parti-cles in a high intensity synchrotron is limited by the res-onance crossing of coherent mode oscillations, not by theincoherent one. In order to verify it, we measured the tuneshift of quadrupole mode oscillations in HIMAC at Na-tional Institute of Radiological Sciences (NIRS) to showthe connection with beam loss. In this paper, we discusssignals observed in a coasting beam. We also presents thenumerical calculation of envelope motion for a bunchedbeam.1 INTRODUCTIONIt is believed that space charge effects limit the number ofparticles in a high intensity synchrotron. However, the de-tailed mechanism of beam loss is not clear. One modeltells that the incoherent tune of an individual partcle is re-duced by the space charge field to a major resonance line,and resonate with lattice field errors. Another model, pro-posed by Sacherer in 1960's[1], is that the coherent modeoscillations of a beam resonate with lattice field errors. Theincoherent model is not self-consistent because it neglectsthe time evolution of distribution due to the space chargeforce.Recent simulation results seem to support the lattermodel[2]. In order to verify this, we planned to measurethe tune shifts of coherent mode oscillations and their con-nection with beam loss. We observed coherent mode sig-nals in a coasting beam in HIMAC synchrotron. To startwith, we discuss the three dimensional envelope equations,where it is shown that the envelope oscillation frequenciesin a bunched beam do not differ much from that in a coast-ing beam. Then, we report our measurements.2 ENVELOPE OSCILLATIONS IN ABUNCHED BEAM2.1 3D envelope equationsFor a bunched beam, envelope oscillations occur in threedimensions. Sacherer studied three dimensional envelopeequations and found that they depend only on rms beamsize, as far as the distribution has ellipsoidal symmetry [3].They ared2~xd2+20x Nr0R2223g(~x; ~y; ~z)~x =(Rx)2~x3d2~yd2+20y Nr0R2223g(~y; ~z; ~x)~y =(Ry)2~y3d2~zd2+20z Nr0R2223g(~z; ~x; ~y)~z =(Rz)2~z3(1)where  is the azimuth along the circumference of a syn-chrotron, R is the ring average radius, 0xis bare tune, andthe geometric factor g is elliptic integral,g(a; b; c) =32Z10ds(a2+ s)3=2(b2+ s)1=2(c2+ s)1=2(2)and 3is about 1=5p5, which very weakly depends on de-tails of distribution. In the parenthesis of the left hand sideof eq.1 corresponds to a square of depressed tune 2xwhen(~x,~y,~z) is the matched beam size. Differentiating eq.1 anduse x= x~x20=R, we haved2d2~x+ 42x~x =3Nr0R222~x0gj~x0;~y0;~z0d2d2~y + 42y~y =3Nr0R222~y0gj~y0;~z0;~x0d2d2~z + 42z~z =3Nr0R222~z0gj~z0;~x0;~y0(3)where the subscript “0” denotes matched solutions. Theequations above can not be solved analytically because ofelliptic integral. One way to solve them is to take someapproximations[4] and substitute arithmetic function for g.Here, we employ numerical integration.2.2 Numerical solutionWe solved eigen equations of eq.3 numerically, for a givenfocusing force, emittance and beam intensity. It is expectedthat, in the limit of long bunch and low synchrotron fre-quency, the behavior of envelope motion will be similar tothat of a coasting beam.In solving eq.3, matched beam size (~x0; ~y0; ~z0) wascalculated by eq.1 with d2=d2 = 0 and elliptic integral0-7803-5573-3/99/$10.00@1999 IEEE. 1821Proceedings of the 1999 Particle Accelerator Conference, New York, 199955.566.577.5810 2 10 3 104rms bunch length(mm)mode tune data2D formulaFigure 1: Bunch length dependence of eigen mode frequen-cies with the parameters of table1.is evaluated with a numerical package available [5]. Wechose parameters in Table1. In order to study bunch lengthdependence, we varied longitudinal focusing force, and ad-justed beam intensity simultaneously so that transverse in-coherent tune remains constant.At first, we show in Fig.1 the eigen frequencies of enve-lope oscillations, together with the one calculated by a 2Dformula.2=~x0+ ~y02~y0x0xq(220x  220y)2  (0xx)2= 7:012; 6:392: (4)In each region except ~z0 250mm, two of calcu-lated modes agree with the values estimated by eq.4.Eigen vectors, shown in Fig.2, prove that these modesare pure horizontal, vertical and longitudinal one. Ataround 0z 0x; 0y, the oscillations form two coupledmodes, ie. breathing(even) mode with higher frequencyand quadrupole(odd) mode with lower frequency. This be-havior is similar to the coupled phenomena of a single par-ticle.In conclusion, the coupling of transverse and longitudi-nal envelope oscillations due to space charge force seemsnegligible for a synchrotron where the synchrotron oscilla-tion frequency is much less than betatron oscillation. Moredetailed discussion could be found in Ref.[6]Table 1: Parameters used in envelope mode calculation.particle 6MeV/u He2+ ( = 1:006)bare tune (0x; 0y)=(3.6,3.3)0z=0.00978 @~z=5000mm100% emittance =(10,10,55000)mm-mradbeam intensity 11010ppp @~z=5000mm-1-0.500.51240 250 260 270 280mode 1amplitude (arbitrary)-1-0.500.51240 250 260 270 280mode 2-1-0.500.51240 250 260 270 280mode 3rms bunch length(mm)Figure 2: Amplitude ratio of eigen modes in x(solid line),y(broken line) and z(dot line). Mode 1, 2 and 3 is of thehighest, middle and the lowest frequency.3 MEASUREMENTS IN HIMAC3.1 The HIMAC tune shiftsWe chose a flatbase operation (no acceleration) with aHe2+ beam to obtain the largest tune shift. Machineparameters are same as Table 1 except horizontal emit-tance of 264mm-mrad and maximum beam intensityof 1  1011ppp. That is due to multi-turn injection.The incoherent tune shift of vertical oscillation is cal-culated as -0.0446/1011ppp. The coherent ones are -0.00216/1011ppp for dipole mode and -0.0590/1011ppp forquadrupole mode.3.2 Quadrupole monitorAmong coherent mode oscillations, the tune shifts ofquadrupole mode is dominant. Therefore, we installed aquadrupole pickup monitor in HIMAC in the summer of1998. It is an electrostatic type pickup with four elec-trodes(Fig.3). The amplitude of quadrupole mode can beseen in a quadrupole channel output, which is subtractingthe sum of vertical two electrodes signal from that of hori-zontal. The difference between opposite two electrodes areused to observe dipole mode oscillations. The estimatedmagnitude of output signal for a 6MeV/u He2+ beam isaround 15mV/1011ppp in each channel, and a quadrupolecomponent is about the order of (x=a)2 lower than that,where a is the radius of pickup electrodes.3.3 Spectrum and tune shiftsWe observed beam spectrum of a coasting beam of 3 1010ppp with a real time spectrum analyzer. We usedRFQ[7] in order to excite quadrupole mode oscillations.Fig.4 shows the beam spectrogram around 4 times of revo-1822Proceedings of the 1999 Particle Accelerator Conference, New York, 1999Figure 3: Quadrupole monitor in HIMAC.y 2x Qx x Qy QyFigure 4: Spectrogram of coasting beam.lution frequency. Quadrupole mode and dipole mode oscil-lations in horizontal and vertical space can be seen. Someof them are shifting in frequency as the beam current de-creases at 1000msec after injection.We estimated the space charge tune shift of each modeby plotting frequency versus beam current seen by aslow current transformer. Fitted value of tune shifts forquadrupole mode is about -0.082/1011ppp, which is not ingood agreement with theoretical value. A part of the dis-crepancy can be attributed to uncertainty of emittance. Weare continuing measurements at other operationg points.117711781179118011811182118311841185118611870 0.5 1 1.5 2 2.5 3beam current (mA)frequency (kHz)Figure 5: Tune shift vs circulating current.3.4 Resonance crossingFinally we show in Fig.6, where the operating poiont ismoving gradually by varying defocusing field. As thecurrent of defocusing magnet goes down, the quadrupolemode frequency approaches to revolution frequency. At2500msec after beam injection, it finaly meets 2y=7.0 res-onance line and beam disappears rapidly. The tune shift ofcoherent quadrupole mode oscillations at the beam currentjust before the resonance is so small that it is difficult todetermine if the resonance occurs by coherent mode oscil-lation or incoherent one.Figure 6: Half integer resonance. Frequency span is200kHz centered at 1.64MHz. Time scale is 3000msecfrom top (beam injection) to bottom.4 SUMMARYFirst, we solved 3D envelope equations numerically, andfound that the coupling between transverse and longitudi-nal motion due to space charge force is negligible in syn-chrotrons. Therefore, the mode frequencies in a coastingbeam are fairly good approximations of that in a bunchedone except one longitudinal mode.Secondly, we presented our experiment in HIMAC syn-chrotron. We identified the coherent mode signals. Thetune shift is not so good agreement with the expected value,at this moment. However, a part of the discrepancy canbe attributed to uncertainty of emittance. We are carryingon the experiments to reveal the connection between tuneshifts and beam loss.5 ACKNOWLEDGEMENTWe would like to thank the staff of Accelerator EngineeringCompany for the help of our experiment in HIMAC.6 REFERENCES[1] F.J.Sacherer, Ph.D.Thesis(1968).[2] S.Machida, Proceeding of EPAC' 98(1998).[3] F.J.Sacherer, IEEE Trans.Nucl.Sci, 18(3)(1971).[4] M.Pabst and K.Bongardt, Proceeding of EPAC' 98(1998).[5] For example, Numerical Recipe.[6] T.Uesugi et.al., in preparation.[7] K.Sato, Proceeding of EPAC' 98(1998).1823Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

Observation Of Quadrupole Mode Frequency And Its Connection With Beam Loss

Observation Of Quadrupole Mode Frequency And Its Connection With Beam Loss

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