ADVANCED FFAG OPTICS, DESIGN AND EXPERIMENT

Transcription

1 ADVANCED FFAG OPTICS, DESIGN AND EXPERIMENT J.-B. Lagrange, Y. Mori, Kyoto Univerity Reearch Reactor Intitute, Oaka, Japan Abtract Much progre ha been made in the FFAG deign with novel idea, for example, FFAG traight line, FFAG with race track hape, FFAG with vertical orbit excurion, etc. Some of thee were demontrated experimentally. The talk will review the recent progre around the world. INTRODUCTION FFAG accelerator have been deigned only in a circular hape, with horizontal excurion until recently, and zerochromaticity i obtained with the vertical magnetic field B in the mid-plane following the o-called caling law [1] k r B = B 0 F, 1 r 0 withr the radiu,r 0 the reference radiu,k the contant geometrical field index, F an arbitrary azimuthal dependent function andb 0 = Br 0. It contrain the tranvere motion, ince the machine can only be in a circular hape, and the excurion in the horizontal direction. Recent development give the poibility to drive particle with no overall bend in FFAG while keeping zero-chromaticity [2], and vertical excurion FFAG are invetigated thee day [3, 4]. Fixed frequency acceleration ha alo been experimentally demontrated. Thi paper mainly report on the traight FFAG line, theoretically and experimentally. Other recent development are then briefly mentioned. ZERO-CHROMATIC FFAG STRAIGHT LINE Theory The motion of a charged particle with a given momentum i decribed by the following linearized equation for mall amplitude { d 2 x d + 1 n 2 ρ x = 0, 2 d 2 z d + n 2 ρ z = 0, 2 in the local curvilinear coordinate ytem x,, z. The field index n i defined a: n = ρ B, 3 B with B the vertical component of the magnetic field. The ymbol mean that the partial derivative i done in the work upported by Japan Science and Technology Agency lagrange@rri.kyoto-u.ac.jp machine mid-plane perpendicularly to the cloed orbit, and evaluated at x = 0. A neceary and ufficient condition of the invariance of the betatron ocillation along the curve = cont. i p = 0, n p = 0. 4 From Eq. 4, ince the curvature radiu ρ doe not depend on momentum, the imilarity of the reference trajectorie i a neceary condition for the invariance of the betatron ocillation. In the carteian coordinate X, Y, z, it lead to X co p, = χp+g, 5 where g i independent of momentum and can be een a the hape term, common of all the reference trajectorie, while χ i the tranlation term. We now focu on the invariance of the field indexn with momentum. If we derive in the median plane the equation p = qbρ, for a particle of momentumpand chargeq, with repect to x and at x = 0, we have: B +ρ B = 1 q p. 6 The field indexncan thu be rewritten a: n = ρ p. 7 p The field index n can then be expreed in the coordinate χ,,z: χ [ ] n = ρm +, 8 χ χ with m a parameter defined by: mχ, = 1 dp pdχ. 9 The parameter m can be rewritten with the magnetic field B a the normalized field gradient m = 1 B db dχ. 10 The normalized field gradient m i independent of, ince the momentum i independent of. Since ρ doe not depend on momentum, it doe not depend on χ either: χ = 0, χ = dρ d. 11 FM cyclotron and caling FFAG No Sub Cla

2 Geometrical conideration [2] give that the field index can be rewritten a n = ρcoϕ+inϕtan +ϕ m+ dρ d tanϕ+, 12 with ϕ the angle between the horizontal carteian vector u X and the vector x, and the angle between u X and the line = cont. Since n i independent of momentum along the curve = cont., we can expre the normalized field gradient mχ, with Eq. 12: mχ = m 1 +m 2 tan, 13 with m 1 andm 2 parameter independent of momentum: m 1 = 1 dρ ρ d inϕ ncoϕ m 2 = 1 dρ ρ d coϕ+ninϕ. 14 Concerning the curve = cont., they mut be choen continuou and at leat two time differentiable. Their longitudinal variation mut alo fulfill the condition that the cloed orbit never cro each other. Once the curve = 0 i choen, the imilarity provide that every curve = cont. follow the ame law but for a contant. The angle i choen independent of X, i.e. the curve = cont. follow a tilted traight line: Y =cont. X = X X 0 tan +Y =cont.,0, 15 with Y =cont.,0 = Y =cont. X 0. The particular cae where the angle i null lead to the rectangular cae, ince the curve = cont. are perpendicular line ee Fig. 1. With the angle independent of momentum,m = m 1 + m 2 tan i contant, and the momentum follow: p = p 0 e mχ χ0. 16 Furthermore, it would be intereting to write thi expreion in carteian coordinate. Since we have from Eq. 5 we can conclude that χ χ 0 = X co,p X co,p 0, 17 X co,p = X co,p m ln p p The magnetic field ditribution follow in carteian coordinate BX,Y = B 0 e mx X0 F Y X X 0 tan. 19 Experiment The experiment aim to verify the zero-chromaticity of the traight caling FFAG law by meauring the horizontal phae advance of two different energie. The traight caling FFAG experiment i performed in the accelerator complex of Kyoto Univerity Reearch Reactor Intitute, in Oaka, Japan. The main component of thi experiment are FM cyclotron and caling FFAG No Sub Cla X Y Figure 1: Tilted cell with contant angle top. The magnet 1 and 3 are not identical to keep the ame effective path for the particle. The rectangular cae bottom i the particular cae where the tilt angle i nul. 1. the H Linac injector of the 150 MeV FFAG complex, delivering 7 MeV and 11 MeV kinetic energy H particle collimated with 2 collimator, 2. a Courant-Snyder parameter meaurement ytem, 3. the prototype of the traight caling FFAG cell. A chematic view of the etup i preented in Fig. 2. In thi figure, the reference trajectory of the particle i plotted in red, and the traight FFAG cell in blue. Two collimator are intalled before the traight cell to et the Courant-Snyder parameter. The Courant-Snyder parameter meaurement ytem i added after the traight cell to determine the beam parameter at the exit. The ize of the collimator and the ditance between them determine the Courant-Snyder parameter and the emittance at the entrance. The emittance ǫ i a function of the collimator ize and the ditance between the two collimator. If we backtrack the beam at the center of the collimator ytem, the beta-function i half of the ditance between the two collimator and α = 0. Size of the collimator ha however to be mall compared with the incident beam ize. The collimator aperture ha been et at 2 mm and the ditance between the two collimator i et to 1530 mm. The 100% emittance i aumed to be 9 time the rm emittance. We then have at the entrance of the ytem ǫ 100% = 1.3π mm.mrad, ǫ rm = ǫ 100% 9 = 0.14π mm.mrad, β = 0.77 m, α = The Courant-Snyder parameter meaurement ytem ued in thi experiment i imilar to the emittance meaurement lit-grid method [5], but with the grid replaced by a fluorecent creen. The lit i ued to meaure the horizontal angle of the beam, and the lope of the line x v. x. When the lit i removed, the ize of the beam on the creen fitted with a gauian give acce to the beta function value, and the center of the gauian i taken a the poition of the beam. Since the width of the lit i crucial for the accuracy of the meaurement, a lit of 0.1 mm width ha been manufactured and ued in the ytem. 121

3 H- beam from Linac Bellow Downtream Collimator Vacuum Chamber Bellow Movable lit Uptream Collimator Collimator ytem Movable traight Scaling FFAG cell Camera Fluorecent creen Courant-Snyder parameter meaurement ytem Figure 2: Schematic view of the traight caling FFAG experiment. Since the cell mut be able to move in X direction to match the different trajectorie without changing the magnetic field, it i intalled on a ball-bearing ytem. Bellow before and after the cell allow the vacuum chamber to move along with the magnet without breaking the vacuum. The prototype for the traight caling FFAG cell i a FDF triplet. Since the traight cell mut move to match the different reference trajectorie, one mut limit the maximum neceary diplacement. T o that end, one would prefer to have a mall diperion, and m-value of the cell ha been choen to be 11 m 1. The diplacement between 11 MeV reference trajectory and 7 MeV reference trajectory i then around 2 cm. The parameter of the cell are ummarized in T a b le 1. T a b l e 1: Parameter of the Straight Cell Type FDF m-value 11 m 1 T o t a l length 4.68 m Length of F magnet 15 cm Length of D magnet 30 cm Max. B Field D magnet 0.3 T Max. B Field F magnet 0.2 T Horizontal phae advance 87.7 deg. A C type magnet ha been choen to have an eaier acce to the pole for field meaurement. The minimum full gap ize ha been choen to be 2.3 cm, to limit the ampere turn. Field clamp with 12 mm thickne have been adjuted at 60 mm from the pole to match the fringe field of the field model. The pole hape ha been deigned firt in 2 dimenion with the finite element computation oftware POISSON [6], then it ha been optimized in 3 dimenion with TOSCA code [7]. Concerning the coil, the current denity i 2.7 A/mm 2 in the F magnet, and 4.4 A/mm 2 in the D magnet, and indirect water cooling ha been choen. The total power conumption of the three magnet i around 1 kw. T o check the manufacturing preciion and have tracking in meaured field map, meaurement of the field ha been done. A 3-dimenional magnetic probe i controlled by the ytem YAMAHA RCX 142 erie. The three magnet are 122 Figure 3: Picture of the traight caling FFAG magnetprototype. firt aligned with a theodolite, then the probe i aligned according to the magnet ytem. Each magnet i meaured eparately. The field i meaured on 3 vertical layer: in the midplane z = 0,z =+5 mm andz = 5mm. Each plane i meaured in the good field region, with a horizontal tep ize of 5 mm, and a longitudinal tep ize of 10 mm. Drift paceof20cmfrombothide ofeachmagnetareaddedto the effective field boundarie to meaure the fringe field. Particle tracking in field model ha been firt done with a code baed on Runge-Kuttaintegration developedat Kyoto Univerity. Field follow the traight caling law in the rectangular cae, and fringe field have linear fall-off. The horizontal phae advance i contant and i equal to 87.7 deg. Tracking ha then been done in field map from TOSCA. The interpolation of the field between the map node i done linearly in 3 dimenion. The local horizontal phae advancein the Straight Cell are plotted in Fig. 4. The local normalized field gradient in the cell or local m- FM cyclotron and caling FFAG No Sub Cla

4 Snyder parameter a well a poition and angle of the beam i meaured at the exit of the Straight Cell when it i launched off it reference trajectory. The linear phae advanceψ can be calculated from: tanψ = α 1 β 1x 1 x Field model TOSCA map Meaured map Figure 4: Horizontal local phae advance veru kinetic energy in the Straight FFAG line with field model red plain line, with TOSCA field map black dahed line and with meaured field map blue mixed line. value i defined from Eq. 18 a [ mx i = ln B zx i,ydy B zx ref,ydy mx ref = mx ref 1+mX ref+1 2, ] 1 X i X ref, 21 with B z X i,ydy the integral of the vertical magnetic field of the traight caling FFAG cell along a traight line longitudinally at a contant horizontal abcia X i in the mid-plane. The reulting curve i preented in Fig. 5. The variation in the local m-value i o mall that the variation in the horizontal phae advance i negligible. Particle tracking ha finally been done in the meaured field map. The horizontal local phae advance and local m-value in meaured field map are hown in Fig. 4 and in Fig. 5, repectively. The agreement with the TOSCA tracking i within 1% for them-value and for the horizontal phae advance. The goal of thi experiment i to meaure the horizontal phae advance for 7 MeV and 11 MeV in the traight caling FFAG cell prototype. T o achieve thi, the Courant- Field model TOSCA map Meaured map Figure 5: Localm-value veru horizontal abcia in the Straight FFAG line in the good field region with field model red plain line, with TOSCA field map blackdahedline and with meaured field map blue mixed line. FM cyclotron and caling FFAG No Sub Cla with x 1 and x 1 the ditance and the angle to the reference trajectory at the exit of the cell, repectively, while α 1, β 1 are the Courant-Snyder parameter at the exit of the cell, i.e. at the creen. The reult of the experiment conit of 6 erie of data, taken at a different magnet poition. The firt poition magnet correpond to the reference trajectory poition, from which the Courant-Snyder parameter are extracted. Then the poition +10 mm and the poition -10 mm give the angle and poition of the beam off the reference trajectory. The econd order of the dependance in amplitude of the phae advance i canceled by adding the 2 reult. T o have more tatitical data, Courant-Snyder parameter are extracted from all magnet poition. Data conit of picture of the fluorecent creen with different lit poition. The alignmentof the creen with the collimator i done with H 0 particle. The picture of the creen with the lit at the central poition aligned with a theodolite i ued to determine the reference point of the picture. Since an H 0 beam i preent at the creen, the collimator ytem i intalled off the mid-plane to eparate the two beam at the creen, uing the verticalbetatronocillation. Iti then poible to detect the H beam at the creen without being diturbed by the H 0 beam. The angle of the beam x can be obtained from the poition at the creen and the poition of the lit. The picture without lit fitted by a gauian can give the rm beam ize at the creen. Auming the geometrical emittance from the collimator ytem, we can thu get the beta function at the creen. The lope c of the line x x 0 = cx lit x lit0 i the ratio of the Courant Snyder parameterα/β at the lit. By tracking in the drift pace between the lit and the creen, we can get the alpha parameter at the creen. The reulting phae advance are preented in T a b le 2. W e then have a difference of 1.6% between the two energie. There i no difference of phae advance with TOSCA tracking in the 11 MeV cae, and the difference i 1.7% for the 7 MeV cae. Thi good agreement clarifie the traight caling law. Theerroraretatitical andindependent. Theyareevaluated from the beta function error due to the fluctuation of the beam ize. It affect both the Courant Snyder pa- T a b l e 2: Phae Advance ψ exp. at the Screen in the Straight Scaling FFAG Experiment and Tracking Phae Advance ψ track in TOSCA Field Map ψ exp. deg ψ track deg 11 MeV 87.5± MeV 86.1±

5 rameter β 1 and α 1 at the lit, ince α 1 i evaluated from the value of β 1. It i 3.7% for 11 MeV, and 11.1% for 7 MeV. The incident beam fluctuate, epecially for the 7 MeV cae, degrading the emittance quality. It affect the beam ize at the creen, with an error three time bigger than in the 11 MeV cae. Since the caue of the error i from the incident beam, it i inevitable to have a large error for 7 MeV. FFAG INSERTIONS Combination of Scaling FFAG Cell T w o different FFAG cell can be combined by matching the diperion [8]. The periodic beta-function of the cell have alo to be matched to limit the amplitude of the betatron ocillation. If a correct matching i not achievable, then a tranparent inertion with a phae advance multiple of 180 deg. can be done for one of the two different type of cell. Diperion Suppreor A principle of a diperion uppreor in caling FFAG i preented in Fig. 6. The component of thi cheme are three type of caling FFAG cell. The area 1 contain FFAG cell with a diperion D 1 at the border, the area 2, contituting the diperion uppreor itelf, contain FFAG cell with a diperion D 2 at the border, and the area 3 contain FFAG cell with a diperion D 3 at the border. The condition to have a diperion uppreor are a phae advance of 180 deg. for the cell of the area 2 and the diperion D 2 i half of the um of D 1 and D 3. Thi principle i baed on the linear theory, o i valid a long a the effect of non-linearitie i negligible. The amplitude detuning can break the zero-chromaticity if the difference of diperion i too large regarding the momentum range. One olution to overcome thi problem i to reduce the diperion with everal diperion uppreor. F o r n diperion uppreor in cacade to go from an initial diperion D ini to a final diperiond fin, the periodic diperiond i of the diperion uppreor numberimut then follow D ini + 1 n+1 D fin = 2 n 1 i+1 D i. 23 i=1 Figure 6: Principle of a diperion uppreor with caling FFAG cell. 124 VERTICAL FFAG Excurion in FFAG can be alo done in the vertical direction. It ha been firt propoed by T. Ohkawa [9] in 1955 a an electron cyclotron. It ha been redicovered recently [3, 4]. In thi cae, the zero-chromaticity condition can be obtained with an exponential increae in the vertical direction. Vertical FFAG could be an aet when it come to accelerateultra-relativitic particle, becaue of it quai-iochronicity. FIXED FREQUENCY ACCELERATION Some development have alo been done in FFAG acceleration. Serpentine acceleration in caling FFAG [10] and harmonic number jump acceleration [11] allow a fixed RF frequency acceleration in caling FFAG. Serpentine acceleration ha been demontrated theoretically and experimentally. CONCLUSION There ha been a trong revival in FFAG in the lat year. Recent development in optic for zero-chromatic FFAG, with traight line, vertical excurion lead to new ring hape and the introduction of inertion. Fixed RF frequency acceleration become alo poible in zerochromatic FFAG. It open a new range of application for thi type of machine. AKNOWLEDGEMENTS The author would like to thank Dr. Y. Ihi, Dr. T. Ueugi and Dr. Y. Kuriyama, KURRI, for their valuable help in the traight experiment. REFERENCES [1] K. R. Symon, D. W. Kert, L. W. Jone, L. J. Lalett, K. M. Terwilliger, Fixed-Field Alternating-Gradient Particle Accelerator, Phy. Rev [2] J.-B. Lagrange et al, Straight caling FFAG beam line, Nucl. Intr. Meth. A, vol. 691, pp , [3] S. Brook, Proc. of IPAC10, [4] S. Brook, Proc. of IPAC11, [5] P. Fork, Lecture note on beam intrumentation and diagnotic, Joint Univeritie Accelerator School, [6] L. A. A. C. Group, POISSON/SUPERFISH reference manual, Lo Alamo National Laboratory, [7] Vector Field Ltd., TOSCA OPERA-3D reference manual. [8] J.-B. Lagrange, Study of zero-chromaticity in FFAG accelerator, doctoral thei, [9] T. Ohkawa, JPS meeting, [10] E. Yamakawa et al, Serpentine acceleration in zerochromatic FFAG accelerator, Nucl. Intr. Meth. A, vol. 716, pp , [11] T. Planche et al, Harmonic number jump acceleration of muon beam in zero-chromatic FFAG ring, Nucl. Intr. Meth. A, vol. 632, pp. 7 17, FM cyclotron and caling FFAG No Sub Cla