Temnykh, A B

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CBN 01-17Some aspects of the use of Vibrating WireTechnique for a wiggler magnetic eldmeasurement.1Alexander B. TemnykhLaboratory of Nuclear Studies, Ithaca NY 14853, USASeptember 21, 2001AbstractNonlinear perturbation in beam dynamics caused by wiggler's mag-netic eld may signicantly degrade storage ring performance. Thecommonly used magnetic measurement techniques such as Hall probe,long ipping coil measurement, scan with a small searching coil cannot revile all causes of the perturbation. Since the beam trajectoryis wiggling, the eld seen by particles moving through the wiggler isdierent from the eld measured along strait lines with above tech-niques.In contrast, the modied vibrating wire (VW) technique describedhere may provide a simple way to measure magnetic eld along thepath imitating wiggling beam trajectory. The VW technique uses thetaut wire stretched through the testing eld region as a magnetic eldprobe. Application of DC current to the wire will cause its displace-ment imitating the beam trajectory2, then exciting the wire vibratingmodes with AC current one can measure magnetic eld along the wire,i.e., along the path imitating beam trajectory.The paper presents results of the wiggler magnetic eld measure-ments using "wiggling" and strait vibrating wire. Comparison revealseects caused by the trajectory wiggling. Good agreement with theo-retical model was observed.1Work supported by the National Science Foundation2This eect is well known and sometimes is used for magnetic eld measurement11 Introduction and TheoryEquation for the displacement of the taut wire with DC current stretchedthrough the magnetic eld and equation for a particle motion in the magneticeld are very similar:T@2Xw@z2=  IdcBy(z) (1)@2Xp@z2=qPBy(z) (2)Here (1) describes the wire displacement and (2) is for particles motion. Inthese equations z is a longitudinal coordinate, By(z) is vertical magnetic eldas function of z,Xwand Xpare the wire and particle horizontal displacementsresprectively, T is the wire tension, Idcis the DC current applied to the wire,q and P is a particle charge and momentum. The equations become identicalifIdcT=  qP(3)In many cases the vertical magnetic eld in wigglers can be approximated asBy' Bwsin(2z=d) (4)where d is a wiggler period, Bwmaximumwiggler eld. For this type of eldthe wire horizontal displacement ( and beam trajectory) will be sinus-likefunctionXw(z) = Awsin(2z=d) (5)with the wiggling amplitudeAw=IdcBwT(d2)2(6)For Bw= 0:780T and d = 12cm (used in experiments described below) thelater gives:Aw[mm] = 0:28Idc[A]T [N ](7)Let's consider eects caused by the wiggling of beam trajectory usingtheories described in [2] and [3].One eect is a vertical focusing. Because of the trajectory wiggling inhorizontal plane, longitudinal wiggler eld component in coordinate system2related to the trajectory will produce horizontal eld. In linear approxima-tion and under assumption that the wiggler eld is a sinus-like function (4),the dependence of the horizontal eld integrated along the wiggling path (5)on vertical position at x = 0 will be:dIx(x = 0; y) =ZL0BzX0w(z)dz =  BwAwL2(2d)2y (8)Here L is wiggler length. This eld causes a beam deection in vertical planewhich results in vertical focusing.1For Bw= 0:780T, d = 12cm, L = 3m andAw= 0:060 mm (parameters used in our experiments) the above expressiongives:dIx(x = 0; y)[Gm] =  1:92  y[mm] (9)Note that at x = 0 the wiggling does not eect horizontal eld integral, i.e.,dIy(x = 0; y) = 0 (10)Another eect caused by the trajectory wiggling relates to the single poleseld roll-o. Let Bp(x) be the dependence of vertical eld on horizontalposition across a single pole. Consider trajectories at y = 0. Because of thedierent horizontal position in "positive" and "negative" wiggler poles thevertical eld integral over the wiggling trajectory will depend on wigglingamplitude, Bp(x) and horizontal position x as:dIy(x; y = 0) =  AwLw2@Bp(x)@x(11)If wiggler pole design is symmetric in x, i.e., Bp(x) is even function of x, then@Bp(x)=@x and dIy(x; y = 0) should be odd function and will produce termslike quadrupole, octupole and etc. multipoles. The Hall probe measurementof the vertical magnetic eld across of single pole of the wiggler used in ourexperiments indicated dependence:Bp(x)[G] = 7:80  103  6:53  10 8 x6[mm] (12)1It should be mentioned that while particles moving trough the wiggler are alwaysexperiencing focusing eect, the taut wire with DC current, independent of the currentdirection, is always pushed o the magnet center indicating defocusing eect.3Length [m] 3.0Number of poles 50Period [cm] 12.0Peak magnetic eld [T] 0.780Gap [cm] 4.0Pole width [cm] 11.0Table 1: General parameters the wiggler used in measurements.For Aw= 0:060 mm and L = 3m (experimental condition) equation (11)gives the dependence of vertical eld integral on x :dIy(x; y = 0)[Gm] = 3:56  10 8 x5[mm] (13)Note that at y = 0 horizontal eld integral should not depend on the trajec-tory wiggling:dIx(x; y = 0) = 0 (14)2 Measurement SetupIn the measurements the wiggler recently built for Cornell High Energy Syn-chrotron Source was used. The wiggler's general parameters are given inTable 1.The measurement setup schematically shown on Figure 1. It consisted ofcopper-beryllium 100 micron diameter wire 383:2cm in length (1), stretchedthrough the wiggler(2). The wire mid point has been placed exactly in themiddle of wiggler. The wire ends were xed on stages movable in horizontaland vertical directions. At one end of the wire was a tension mechanism(3),while optical wire motion detectors(4) were placed close to the opposite end.A \LAB-NB" board installed in the Macintosh and programs based on \Lab-View" software provided all needed signal analysis. Power supply driving DCcurrent used to "wiggle" the wire was connected to the wire ends in serieswith transformer. AC current in wire was excited by wave form generator\HP33120A" connected to the other transformer input. The wire sag esti-mated from fundamental mode frequency ( 18Hz) was 0.9mm.4(3)DC currentPower supplyMacintosh Quadra 800with Lab-NB Board(1)Wave FormGenerator(2)(4)Figure1:Measurementsetup.(1)-100microncopper-berylliumwire,(2)-wiggler,(3)-tensionmechanism,(4)-horizontalandverticalwirepositiondetectors.3Results3.1WigglingamplitudecalibrationTherstexperimentwastocalibratewirewigglingamplitude.Thewirexpositioninthe7adjacentpoleswasmeasuredbyopticalmeansasfunc-tionofDCcurrentowingthoughthewireandtension.TheresultsofthemeasurementshowninFigure2indicatedthelineardependenceofwigglingamplitudeAwonratioIdc=T.Fitwithlinearfunctionofthedatayields:Aw[mm]=(0:2550:006)Idc[A]T[N](15)ThisisingoodagreementwithcalculationinSection1.3.2Thehorizontaleldappearenceduetowiggling.Inthisexperiment,horizontalmagneticeldalongstraitandwigglingwirewasmeasuredusingVWtechniqueatthreeverticalpositions(y=5mm,y=0andy= 5mm)atx=0.The1:68Ntensionand0:8ADCcurrentappliedtothewireresulted,accordingtocalibration(15),inAw=0:124mm500.050.10.150.20.250.30 0.2 0.4 0.6 0.8 1 1.2Aw[mm]Idc[A]/T[N]Figure 2: Measured dependence of the wire wiggling amplitude on ratioIdc=T . Idcis a current owing through the wire, T - the wire tension. Fitwith linear function gives Aw[mm] = (0:255  0:006)  Idc[A]=T [N ]of the wire wiggling amplitude. Horizontal eld distribution along the wirewas measured using 9 odd (1th,3d... 17th) modes of vertical wire vibration.Only odd order modes were used because modes of even orders were stronglyaected by systematic eld perturbation at the wiggler ends. Results areshown in Figure 3. The plots depict the dierence between data obtainedwith strait and wiggling wire, i.e., the horizontal eld appeared due to thewire wiggling. At y = 5mm and y =  5mm the eld has approximatelyequal amplitude but opposite sign. At y = 0, where longitudinal eld is zero,the horizontal eld did not appear. This is consisted with a wiggler model.It should be mentioned that for 12cm wiggler period, 5mm vertical oset and0.125mm amplitude wiggling, the model calculation gives the horizontal eldaveraged over wiggler period of 0:8  10 3of maximum wiggler eld. Thissuggests that 1[r.u.] of the vertical scale corresponds to 410 4of maximumwiggler eld or 3:3G.Another remark is regarding eld distribution along z. Because the lengthof the wire standing waves used in this measurement was longer than thewiggler period, the horizontal component of magnetic eld was averaged6-3-2-101230 50 100 150 200 250 300y  = 5mmy = 0y = -5mmdBx [r.u.]z[cm]Figure 3: Horizontal eld at dierent vertical position caused by wiggling.1[r.u] in vertical scale ' 4  10 4of maximum wiggler eld or 3.3G, see intext. Magnetic structure (wiggler) located between z = 0 and z = 300cm.over few the wiggler periods. The result was that the measured distributionappeared to be uniform along the magnet.3.3 Eect of the wiggling on vertical and horizontaleld integrals.If the function describing distribution of the eld along magnet is known,one can use only one wire vibrating mode to measure the eld integral. Let'sprove this statement for a wiggler eld. Suppose the distribution of the elddistortion caused by the wiggling along integrating path (wire) is:ÆB(s) = ÆB0sin2(2s=d); s1 s  (s1+ L) (16)Here s1is the position of wiggler's end, L is the wiggler's length, L = Nd.B0is the distortion amplitude, which is not known. The eld integral overthe path along the wire, 0  s  lw, lwis the wire length, will be:dI =Zlw0ÆB(s)ds =ÆB0L2(17)7From other side, the rst longitudinal eld harmonic, H1, which can bedirectly measured with fundamental wire vibrating mode can be written as:H1=2lwZlw0ÆB(s) sin(slw)ds = ÆB02sin(L2lw) (18)It was assumed that the wire middle point is placed exactly in the middle ofwiggler. Comparing expressions (17) and (19) one can nddI = H1lw2L2lwsin 1(L2lw) (19)i.e., the eld integral is equal to the rst longitudinal harmonic multipliedby well dened coeÆcient related to the geometry of setup. Note that if thewire is much longer than the magnetic structure, lw L, the coeÆcient doesnot depend on the length of the structure and the expression for eld integralbecomes:dI = H1lw2(20)In experiments described below the vibrating wire measurement was cali-brated with small shim placed in the middle of magnet. Previously, theeld integral change due to the shim was measured with a long ipping coil.Geometric factor was calculated using formula (19).In the rst experiment, there was measured the dependence of the changein horizontal and vertical eld integrals due to wiggling versus vertical po-sition at x = 0, i.e., dIx;y(x = 0; z). To create the wire wiggling 0:42ADC current was applied. At 1:68N tension, according to calibration (15),it resulted in 0:060mm of wire wiggling amplitude. The experimental dataplotted on Figure 4. One can see evident variation of dIxwith vertical po-sition and much smaller variation of the dIy. Fit with liner function of dIxgives:dIx(x = 0; y)[Gm] = ( 1:99 0:17)  y[mm] (21)This is in a perfect agreement with a model calculation made in Section 1.Note that according to model vertical eld integral should not depend onwiggling,i.e., dIy(x = 0; y) = 0. The observed dependence of dIyon y canbe explained by a small tilt of horizontal wire position detector around z. Inresult the sensor may parasitically detect signal from vertical wire vibration.In second experiment dIxand dIywere measured versus horizontal wireposition at y = 0. The wire wiggling amplitude was 0:060mm, the same as8-20-1001020-10 -5 0 5 10dIydIxdI x, dIy [Gm]y[mm], x=0Figure 4: Dierence in horizontal and vertical eld integrals measured withstrait and wiggling wire as function of vertical wire position at x=0. Fitwith linear function gives: dIx[Gm] = ( 1:99  0:17)  y[mm], dIy[Gm] =( 0:21 0:08)  y[mm]-20-1001020-60 -40 -20 0 20 40 60dIydIxdIx, dIy [Gm]x[mm], y=0Figure 5: The dierence in the eld integrals measured with strait and wig-gling wire as a function of horizonal wire position at y=0. Dependence ofvertical eld integral on x was tted with function dIy= C x5. The t givesC = 3:60  0:12 [Gm/mm5]9in previous experiment. Results are showed in Figure 5. Fit of the measureddependence of dIyon x yieldsdIy(x; y = 0)[Gm] = (3:60  0:12)  10 8 x5[mm] (22)This is also in a very good agreement with the calculations made in Sec-tion [1]. Variation of dIxwith x is negligible as it should be according to themodel.4 ConclusionVibrating Wire technique was used to measure magnetic eld distributionand eld integrals along a wiggling path imitating the beam trajectory in awiggler's magnetic eld. To wiggle the wire, which ws used as a eld probe,DC current was applied to it. Obtained data are in a good agreement withmodel calculation.The Vibrating Wire technique has unique features which can be eectivelyexploited for the measuring of wiggler-like magnetic structures.References[1] A. Temnykh, Vibrating wire eld-measuring technique, Nuc. Inst., A399 (1997) 185-194[2] H. Wiedemann, Particle Accelerator Physics, Berlin: Springer, 1999[3] J. Safranek et, Nonliner Dynamics in SPEAR Wigglers, Proceedings ofthe 1999 Particle Accelerator Conference, New York, 1999, pp. 157-161.10

Some Aspects of the Use of Vibrating Wire Technique for a Wiggler Magnetic Field Measurement

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