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I : . .- SLAC-PUB-6058 SLAC/SSRL-0008 February 1993 (SSRL-ACD) Permanent Magnet Edge-Field Quadrupoles As Compact Focussing Elements For Single- Pass Particle Accelerators * ROMAN Stanford Stanford Stanford . TATCHYN Center Laboratory CA 94309 Linear Accelerator Radiation Stanford, Synchrotron University, Abstract A previously proposed orthogonally asymmetric arrangement of permanent magnet - (PM) material can be made to generate a highly approximate quadrupole field distribution in the vicinity of its symmetry axis. If a small (X 1 cm) device gap is permitted, a relatively small amount, of conmlerically available PM material can generate focussing gradients in excess of 100 T/m. In this article some of the general characteristics of this configuration are examined and applications to the design and construction of ultra-long undulators on single-pass nlachines are considered. S&mitted to Review of Scientific Instruments * Work supported by the US Departjment of Energy under contract DE-AC03-76SF00515 I ., _ ._ .- - 1. Introduction The properties of axisymmetric in linear quadrupole field distributions are well on is as focussing known a elements or recirculating magnetic be machines [1,2]. The current-driven yoke the has proven to quadrupole configured steel particularly element important, and perhaps preponderant optical machines. employed in present-day there are areas can to as accelerators of field arise iron), pure and circular in which Nevertheless, or design design disadvantages material based complications referred from utilizing and permeable methods would (henceforth on either be flux alternative magnets pure currents to could or be be permanent generally useful such considered densities are preferable, attained. relevantbroad devices has _- provided One area comparably in which considerations particularly design and currently A of is that of insertion exists without defining devices, in the iron, the undulators. construction amount of base of experience with done and on both and a large study which are been circumstances to undulator under construction alternative suitable In technological approaches [3,4,5]. recent work, iron-based been at SLAC a 60 m long pure-PM undulator with a - superimposed lattice has quadrupole focussing/defocussing Free-Electron 1). Analytical facets lattice need Laser studies to be (FODO) @EL) have kept the of designed [6] for applications (see Figure shown [7] that the quadrupole yoke e significantly far away from the PM undulator axis in order to prevent surrounding modifications undesirable 2 the on-axis a field amplitude. large power At the minimum consumption the required attention out of full by aperture (12 cm), is significantly in order In of the guadrupoles necessary (115 T/m). to attain of focussing gradients has been given to the view this, possible lattice us8 could PM structures which an alternative‘FOD0 be configured. paper the us8 of a previously-disclos8d for this purpose is PM structure As In this [8] as a possible in Figure candidate examined. pieces indicated 2, it consists in the of four identical which, a if distributed quadrupole the bore lies the in evenly azimuthal equivalent direction, to that would generate field 8888ntially generated along of an ordinary the Contiguity lower iron-based of the structure. The basic difference in to the pieces which the (and their intuitively B Y that field edge-fields) can upper and the PM-pairs, of be be seen in a underlie. maximization It should gradient x-direction. rearrangement be _ineffectual; by evident attempting similar would field one the a in for of: _ of the pole it would, to poles facets in in an iron-based fact, the the d8Cr8aS8 flux quadrupole the On-axis gradients of the helping channel into lines directly In be the from contiguous PM of other. will SeqU81, to exhibit proposed number principle certain the e compact 1) edge-field properties configuration and potential shown advantages iron-based examples element axes of of that could make it useful In of for replacing particular, the guadrupoles be given applications. possible focussing and us8 will for proposed the ways configuring (iron/PM) the economy lattices 2) some along possible hybrid undulators, improving 3 . : and 1. performance of the pure-PM undulator structure shown in Fig. 2. Field distributions in the PM edge-field quadrupole Expressions structure shown for the B,(O,y,O) 2 and By(x,O,O) components derived: of the in Fig. (for L --) -) are easily By(X) s > [,tan -l[c) - 2tan-lIh t*s/2) - tac'[ lgif ] + tan -'[~++,;zl - tan-1(xgi2V) + taG1[~~g~J]; (1) ._ Bx(y) - Br n (y + g/a2 X w2 + + (y - (h + g/W2 (h + g/W2 x w (y + g/212) (y + 2 + (y (h + g/212 + (h + g/2) I2 - In (Y - II (2) l In the vicinity sufficiently derived s =x(O,~,O) 3Y from large, the of the axis, the (1xl,lyl Q( g/2), and for w Y approximate formula8 by gradients of Bx and B can b8 above 4 I .. _ . and aB (x,0,0) ax It is 8Vid8nt are that both to 0th order the gradients as the in the holds PM above for an approximation ordinary hOW8V8r1 away from constant From the this and identical, asymmetry condition of quadrupole. it is clear structure, too far the that cannot the persist the and axis. In order to indicate extent to which vertical away from horizontal axis for gradients quadrupole term8 w. vs. remain sufficiently of similar practical the dimensions interest, for the higher-order values the of in 8qU'S. w 2 (3-4) mUSt g/2, be examined studies equal practical that For nUm8riCal indicate to within region or vertical horizontal gradients remain 54% across an axis-centered homog8n8ity1' diameter of S O.lg. with typical If this sizes of "gradient is associated beams occupation it diameter8 b8COm88 and of particle evident effective from in linear (3) or Circular (4) could To that be machines, _extremely prepared th8 8qU'S. and compact from foCUSSing PM 818m8ntS commercially 8ff8ctiv8ness Set available of the materials. indicate if we we be remarkable edge-field w=8cm, of quadrupole, L=4Ocm, 100 choose find the parameter that focussing in the (g=lCm, h=3mm, in B,=lT), T/m can gradients 8~~888 generated center of the device. to make at of this the point pieces magnet is with an in at A natural comparison * orthogonally symmetrized Fig. 2, i.e., with each arrangement rectangular depicted placed permanent 5 . : 9o" azimuthal increments For about the z axis, placed as the PO188 of an ordinary to and 8aCh quadrupole. other the magnet8 the is field as Cl088 viz., shown as possible with that given g-w, the by to .maximize h*g, it gradients, assuming straightforwardly are SymTtk8triZ8d PM 8Brh/ng2, (4)) l qUadrUpO18 about half gradients that approximately edge-field quad i.e., of the (cf. eq. 3. PM edge-field quadrupole fOCUSSing 1attiC88 It is of interest edge-field To do this Fig. quadrupole both 3), it to examine (quad) the focussing performance of the vis-a-vi8 and a conventional lattice to structure. arrangement thin-lens advance _ individually, wi11 be in a FODO to (see convenient refer the approximation per unit lattice _in cell [2]. to the quadrupole and the associated induced phase of the betatron oscillation by the depicted Designating equ's. (3) quad by the and absolute (4) by value G[T/ml, of either th8 of the two length gradients f of the is focal edge-field given in the thin-lens (and near-axis) approximation simply 3.35E[GeV] fbl L [ml G [T/ml . 2E(g2 t 2gh) (5) 3BrLh The phase advance per unit cell is given approximately by 6 I -; _ ;. _ $ [radl -D/f, wavelength with defined th8 total phase advance for the entire betatron to be 2n rad. If we now note denominator made than l-2 for Of orders in that the rang8 eq. (5) can by of values simple for of the term G-in the be parameter the selection of magnitude greater with PM edge-field quad apertures, in we iron-based recognize and economy the quads the restricted immediately flexibility example, for corresponding by the use increase of the design For implied PM option. that referenced quads FEL design, we can estimate the 40 cm long with 15 T/m in Fig. 1 could in principle be replaced volume 6 cm long 100 T/m edge-field quads utilizing of 1888 than 10 a total cm3. of magnetic requiring example, the material shorter (per quad) In designs for [g]), would could _ betatron wavelengths (as would at lower unit be typical, energies that quad for shorter-period on the FELs operated adVanC8 per limitation from be the phase cell result also restricted by focal parameters factors any of an iron with iron the extended MOr8 that with corresponding for edge-field it is be per structure. _apparent configured betatron attractive insertion tabulation g8n8rally, an giV8n lattice, could alternative f8W8r the PM structure easily Cell8 either or Significantly latter of option the more Unit wavelength, when device the being especially along reference, lengths for variation j3 parameter For the a a - length n88d8 t0 be minimized. field gradients is given 3, we can and of attainable of quad again design focal 1. broad range s Referring mechanical dimensions to Fig. in Table also by note the a significant field advantage permitted symmetric 7 distribution the ability about to in the axis all of the the edge-field PM quad, with namely, all the arrange two lattice sections magnet-pairs access to the parallel d8ViC8 planes, gap. providing maximum lateral inS8rtiOn 4. Selected applications In this section for some we examine both hybrid some and alternative pure-PM focussing lattice configurations and discuss We first 3 into the undulator structures of their implications. of imbedding (i.e., a given iron/PM) D the the lattice of Fig. It is the the a the consider gap of a the notion hybrid for be pole order as undulator. gap intuitively individual proximity non-matching imbedded evident period evident quads of the period. that would iron In fields. of by with of aperiodically faces to of an the modulated undulator regularity it in thus 8nhanC8 as lattice that 2D is an fields optimal much possible, is one appears the of FODO the arrangement to 4) on the some and the PM which multiple made Au identical (888 Fig. Based that integral the PM undulator over the period PO18 it is quads are of centered prior us of faces. 8Vid8nt obs8rvations quad structure and the section, with the provides rang8 fr88dOm of matching implying be D to a wide that continuous any two undulator s structures complement periods, could virtually relatively selected t0 optimized independently one another.' 8 The faces pole remaining quad are Taking issue fields. .flat, Wn2w, concerns For the actual effect of the that far pole the from on the faces simplicity, and we will assume horizontal, TnL, of sufficiently saturation. for the and w%gu, an approximate gradients is easily expression G of the modification quad by the the free-space set of images edge-field viz., first derived, G- 161Brlh n 1 t g@ (g;(g/2) + g2/8 + hg/4 t 2, (g$ W (g/2) 1 2, (6) (g2 +2gh) Obviously, pole faces be for undulator Paladin but structures the with wedged expressions symmetry or curved will in (e.g., [lo]), if the derived general the iron different, the appropriate quad exists in _ geometry, loading of the also fields should perturb to their fr88-SpaC8 their the distributions symmetrically, as helping maintain usefulness, magnitude to first order, focussing elements. under To examine _th8 eq. For be abOv8 of a typical if image-loading (gu=5Cm, effect g=2Cm, apprOXimatiOn8, we choose h=Smm), field. (6) predicts a concavely possible We to a 15% change curved make in pole in the face, PM quad's free-space such of as on Paladin, the loading it should the magnitude the significantly of being in it smaller. able to note, passing, evident with the importance edge-field attain small dimensions quad reducing the analytical and * to a given hybrid d8ViC8. We now consider mechanical complexity of matching soine of the implications of replacing the 9 iron two quad lattice of in Fig. Fig. 1 with PM edge-field two optics. In the top of structures 5 we show alternative other OUtSid while has the plaC8m8ntS the quad magnet lattice, array one inside, and th8 The latter the undulator requiring advantage a of gap. larger option, substantially generating field, large quantity of material, larger a prOpOrtiOnat8ly required "h0m0g8n80u8" or transport quadrUpOl8 systems formulas, with we and as may be dynamic for machines apertures. verify even that Based on the abOV8 general both the attainable quad can readily gradients materials favorable H-lcm) . volume for for the out-of-gap FEL are still of Fig. 1 extremely (g,=l.Scm; the referenced structure A potentially using . tuning either interesting two possibility in the that top could of B. accrue from of the structures on-axis image field Fig. 5 is the the of the undulator by induced this the are amplitude (and h8nC8 K-parameter) . _ for doing fields. on Two possible the bottom and of configurations Fig. 5. Again schematized one for choosing _wide left we simplicity assuming sufficiently magnets, field readily d8riV8 an approximation Bi, as a function to the modified gap g,, gap 8 on-axis the amplitude magnet of the undulator and the vertical permanent the block height H, between iron plates: -2n(s e Bb B. It8 - ('3, + WI/Au t ..... I . (7) 10 Assuming for example, could either method fields should lattices this enable tuning will 8 is closed.down that using to its minimum size, gt2H, W8 .see, undulator PM blocks with H=lcm and +u-8cm 40%. the modulationof configuration, also B. by up to approximately it is evident modify that the the image In tuning lattice one simultaneo~usly to eq.. (6).... For quadrupole according of course a particular the combined application, tuning attemptto optimize of both Should or for to facilitate prove Of the r8gUir8d unrealizable, fr88dOm either undulator performance. strategy additional n88d quad t0 or be (electrical introduced mechanical) independently fields. Apart degree8 Will the adjusting from the88 undulator lattice that be observations, 5 could, it should b8 apparent conditions, the tuning implemented methods of Fig. under suitable significantly movement to be 1888 8Xp8nSiV8ly than alternative for of or methods.involving example, ultra-long less. prove of the undulator jaws. This mightK in of the an case octave particularly requiring relevant ranges undulators tuning 5. Discussion A number edge-field briefly on * additional for further of general issues related In to the this introduction section, field W8 of the quad field merit attention. radiation fOCU8 some quality, damage, and tuning, possible study. applications, 8818Ct8d r8comm8ndations 11 It has generally to attain somewhat than with b88n aCC8pt8d that r8C8nt quality magnets. t8ChnOlOgy with The i8 able higher pure field permanent iron-based quality of structures commercially of ongoing available improvement still-evolving PM material, [ll], and hOW8V8r, it can is still be argued in a stag8 that based be made with on to special, ultraprecise deliver Thus, could fabrication and sorting, Comparable t8ChnigU88 PM8 could measurements quality the field at least to iron-based depicted designs. in Fig. 2 for example, in principle pieces. four monolithic from magnets sets be aSS8mbl8d Her8 of much smaller, the sorted rectangular simple more such it al80 of the seems likely quad that extremely to be of planar conducive structure to edge-field could and prove enhancing the it with range more the implementation control techniques than complicated noted orthogonally that the symmetric geometry. Furthermore, even is to be rather 8ff8ctiv8n888 possible the for of the PM quad, . _ low values of Br, makes further and the use of a wide that of PM materials, t8ChnigU88 fields enhancing components likelihood _engineering construction sufficiently high-quality of PM8 can be found. applications damage. can be is A Well-known the sensitivity machines, of limitation of their in accelerator to radiation energies, field quality at high Since intense cases to be linear especially radiation, minimum its that generators th8 the product beam noise it is 8Vid8nt that in many of the axis distance maximum the small FEL of approach operating heights of a PM quad lifetime will times we note in our limited. in Here of the quad poles, particular referenced structure, imply 8XpOSUr88 12 not significantly (dipole) S), mOV8d we worse that those In experienced by the StrUCtUr8'8 (e.g., PO188 primary see could Fig. b8 magnets. have to al80 alternative that the configurations edge-field from the quad axis It seen large rather distances or without is also seriously reasonable' techniques radiation Perhaps edge-field Equ's. compromising to will damage one expect economy that to performance. effective to help increasingly be developed in a given engineering minimize the continue or its effects of configuration. with regard to the the more relates that gap an of obvious to of is questions potential three StrUCtUr8 (3-4) G, show its the easily for tunability. parameters primary For determining we may also G 0ni.y th8 that size varied. tuning of completeness, (both for note the additional the region parameter gradient magnet and varying could apart . - and hOmOg8n8ity) poles evenly be introduced in the by pulling the contiguous Both this horizontal direction. which g are obviously - mechanical in _terms tuning of motions, and could be 8Xp8Ct8d to be ad8gUat8 number of precision An response time for a large of mechanical applications. discussed possibly smaller fields value, currents e response alternative is faster the method tuning, could abOV8, attain mass of Only image-field times approach, due to 5). the which response element by potentially PM quad - its tuning to be (888 Fig. small If the need then varied amounts about a constant "GC~S88W faster an arrangement in than Using 818CtriCally be able to controlled attain even would, times principle, an ordinary (current-controlled) perhaps iron quad. natural In considering further applications, the mOSt 13 question is Whether machines apart the th8 proposed PM quad rings). would be usable regard, on it recirculating is clear be given that, to (e.g., from the of storage issue In this of tuning, of consideration must size the region gradient that or into homogeneity. to change the Regarding their mandatory tuning (as on during the machines injection lattice could n88d lattice of parameters separating ramping), possibility pure-PM, 8COnOmiCal would energy. dynamic upper quad. be quad tunable-iron, lead to more and tunable-PM designs. attainable Regarding aperture components the possibly Ostensibly, most' dramatic machines that economization inject at full the on fixed-energy the region of field gradient homogeneity, set a 'of recirculating Small8gt limited horizontal not8 machines gap could size by strong limit This to the may in p8rmiSSibl8 even vs. of an edge-field the higher-order distributions. large regions (see _ be the further vertical asymmetries We might, . - field hOW8V8rl that even and quad at Substantially of gradient l), the hOmOg8n8ity edge-field economical warranting the effects lattice in of correspondingly can Still be larger designed An gaps to Table _- deliver impressive possibility, mitigating th8 quads and focussing further field performance. study and could additional for employ - analysis, be to asymmetries Using to alternative as Configuration the of alt8rnat81y"rOtat8d Fig. 3. Assuming that described r8solutions caption above in satisfactory the issues, the it is 8Vid8nt of an intriguing future e machines, nam8ly the rings based on direction d8V8lOpm8nt evolution circular or '*mini" be of ultra-compact, t8ChnOlOgy non-superconducting could 14 facilitated. Some that final possible applications 1) the into of the PM edge-field easy insertion quad of the the d888rV8 mention are: potentially storage Sp8Cially-Configured lattice momentum amplitude us8 of th8 functions compaction at desired sections rings ,e.g., to modulate to the [13]; change p for special factor applications or to [12], modulate ring function 2) the mini-f) for locations around the and quad's either intense on fOCUSSing or gradients to construct machines, e.g., sections, accomodating undulators With analytical Optical for linear recirculating such as, small-gap [14]. regard' and to devices, micropole SUbS8gU8nt study t0 be of r888arCh, the a more detailed as an numerical would with 888m edge-field W8 quad 818m8nt -quads warranted. field recall that 8V8n symmetric doublet distributions, a high degree a of _ focussing/defocussing . _ astigmatism distributions -advantage [2]. In can exhibit the this regard, quad could asymmetric be utilized field to in the edge-field for these and perhaps Other certain if the The natural multi-element rotated” to extended the - configurations, configuration lattice8 vertical other in particularly could be 8mplOy8d. "alternately same applies machines featuring emittance aSymm8tri88 In the88 b8tW88n vs. horizontal cases, the functions. in the prove and perhaps field asymmetries edge-field quad higher-order components e interest. of the could to be of practical 15 6. ACknOWl8dg8m8ntS . Useful research Department of comments was by Sherman at Winick which are ,acknowledged. Operated Sciences, of by .This the 'performed SSRL is of Energy, Sciences: Office'.of -That Basic Office's Energy Division Materials Chemical, Division research. Sciences has provided support for this 16 TableA 3818Ct8d edge-field fOCUSSing gradients G and focal length8 f attainSbl8 in quadrupol8s~~compos8d _. of PM material with B,=lT.E-5GeV s[cml 0.2 0.2 0.2q* h[cml 2w [cm1 L[cmL G[T/mL fCm1 2.3 1.5 0.04 0.04 0.04 0.08 0.4' 0.4 2 2 364 566 0.5 0.5 0.10 0.10 0.20 1.0 1.0 5 5 146 226 2.3 1.5 0.10 1.0 1.0 0.20 0.20 0.20 0.40 2.0 2.0 10 10 73 113 2.3 1.5 2.0 2.0 0.40 020 0.40 0.80 4.0 4.0 20 20 36 56 2.3 1.5 - 4.0 4.0 0.80 0.80 0.80 1.60 8.0 8.0 40 40 18 28 2.3 1.5 4* 8.0 8.0 1.60 1.60 1.60 3.20 16.0 16.0 80 80 9 14 2.3 1.5 - * diameter over which equal. vertical vs. horizontal gradients remain -approximately 17 7. Reference8 [II M. S. Livingston McGraw-Hill, New and J. P. B18W8ttt York, 1962. Particle Acc818rat0r8, 121 E. Regenstreif, Triplets," Academic "Focusing with Quadrupoles, Particles, 353-410. Doublets, and ed., in Focusing Press, of Charted 1967. A. Septier, New.YOrk, pp. [31 R. Tatchyn Insertion No. 582, and I. Lindau, D8ViC8S 1986'. eds., International Sources, Conference on for Svnchrotron SPIE PrOC88dingS 141 R. Tatchyn, Workshop "Soft X-Ray Applications," in PrOC88dingS Source, 88/02, of the on PEP as a Synchrotron eds., Radiation No. R. Coisson p.97. and H. Winick, SSRL Publication [51 P. J. Viccaro, Insertion Workshop R. Tatchyn, Working and R. Coisson, Group Discussions "Summary at the of the SSRL in D8ViC8 on 4th Generation of the Workshop Synchrotron on Fourth Light Sources," Light Proceedings S0urc88, No. Generation eds., SSRL M. Cornacchia p. 86. and H. Winick, Publication - 92/02, [61 s C. Pellegrini, Tatchyn, Seeman, J. ROS8nZW8ig, K. Bane, H.-D. Nuhn, P. Pianetta, R. J. H. Winick, P. Morton, T. Raub8nh8im8r1 K. Halbach,, K.-J. Kim, and J. Kin, "A 2 to 4 nm High 18 1 . Power FEL on th8 SLAC Linac," 1992, Kobe, presented Japan; at the to appear 1992 FEL in Nucl. Conference, Instr. August and Meth. 1993; . . [7] R. Tatchyn, unpublished. .,’ : VI R. Tatchyn, Tanh-Function' Undulator Workshop wPreliminary-Thoughts ,_ Insertion,D8ViC8: Filter," on a 'Transverse Its Characteristics in Proceedings Radiation No. as an and EmittanC8 on PEP of the R. Coisson as a Synchrotron Source, 88/02, and H. Winick, eds., SSRL Publication p. 244. [9] R. Tatchyn, Operation," Generation SSRL "Optimal Insertion of Device Parameters for SASE FEL in Proceedings Light Sources, No. the Workshop on Fourth eds., M. Cornacchia p. 605. and H. Winick, Publication 92/02, [lo] G. A. Deis, Rego, A. R, HarVey,,C. D. Parkinson, D. Prosnitz, "A Long J. E. T. Scharlemann, Wiggler PrOC88dingS and K. Halbach, for the Paladin of El8CtrOmagn8tiC Experiment, " Free-Electron Inernational 23-26, Laser the Tenth Conference LLNL/UC - on Magnet Preprint Technology, . Boston, September 1987, [ll] K. Halbach, m - private communication. [12] A. Amiry and C. Pellegrini, "Design of a Quasi-Isochronous 19 . : Light Source," in PrOC88dingS Sources, No. of the Workshop on Fourth eds., Generation Light M. Cornacchia p. 195. and H. Winick, SSRL Publication 92/02, [13] M. H. R. Donald, PrOC8edingS S0urc88, "Low-Remittance Lattice for PEP," in Light of the Workshop on Fourth Generation eds., M. Cornacchia p. 122. and H. Winick, SSRL Publication No. 92/U, [13] R. Tatchyn, micropole Rev. Sci. P. Csonka, undulators Ins&m. and A. Toor, "Perspectives radiation on in synchrotron 60(7), 1796(1989). technology," 20 Fiqure Captions ‘. Figure 1. Pure PM undulator field design with a superimposed quadrupole lattice generated by iron-based quadrupoles. FigUr8 2. Schematic All axis of the permanent have their magnet easy edge-field quadrupole. to the y field four pieces axes parallel with the and are identically directed magnetized, vectors as shown. Figure 3. Parameters composed of a focussing/defocussing of PM edge-field would quadrupoles. (FODO) lattice An alternative lattice arrangement in which oriented, . _ the z-axis. the be to first in each set up a similar section second were quad segments identically by 90' about and then rotate every _- Figure 4. Arrangement of an edge-field guadrupole with FODO lattice poles. in By the gap of an insertion centering th8 the quads d8ViC8 permeable faces over the iron pole and keeping (Or some of each - same periodicity as the undulator identical image StrUCtUr8 modulation multiple quad's of it), virtually field by the induced fields is ensured. m - Figure 5. Front views of pure-PM undulator structures. primary The large dipole rectangles represent the undulators' 21 arrays. In th8 upper tW0 figUr88, pole alternative pieces placements of an edge-field vicinity bottom field quadrupole's dipole in the In the of the primary left figure; arrays is shown. the a method undulator is shown. for tuning with on-axis. pure-PM of a pure-PM lattice the a superimposed guadrupole 8 b8tW88n By changing on-axis induced the gap height field is modified iron plates, image the by the equivalent An alternative in the bottom charges in the plates. is schematized realization right figure. of this method 22 J. , I 8cm Fig. 1 I : . Fig. 2 r Fig. 3 tY Fig. 5

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