WEPEA003 - work is supported by the French "Agence Nationale de la Recherche" as part of the program "investing in the future" under reference ANR-10-EQPX-51, and also by grants from Region Ile-de-France.International audienceThom-X is a 50MeV compact ring based on the Compton back-scattering which is being built in LAL, France. With a very short bend radius of 0.352 m, the nonlinear effects of the dipole fringe fields become critical to the beam dynamic . This paper compares the modelings of the dipole fringe field using four popular codes: MadX, Elegant, BETA, and Tracy3, and then discuss the proper model to have consistent results between the analytical calculation and the symplectic tracking of the Thom-X ring

Loulergue, A.

Zhang, J.F.

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Dipole Fringe Field Effects in the ThomX RingJ.F. Zhang, A. LoulergueTo cite this version:J.F. Zhang, A. Loulergue. Dipole Fringe Field Effects in the ThomX Ring. IPAC 13 - 4thInternational Particle Accelerator Conference, May 2013, Shanghai, China. Joint AcceleratorConferences Website, pp.2504-2506, 2013. <in2p3-00823304>HAL Id: in2p3-00823304http://hal.in2p3.fr/in2p3-00823304Submitted on 30 Jul 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.DIPOLE FRINGE FIELD EFFECTS IN THE ThomX RING∗Jianfeng Zhang† , LAL, Universite Paris 11, IN2P3/CNRS, 91898 Orsay, FranceAlexandre Loulergue, Synchrotron SOLEIL, St-Aubin, 91192 Gif-sur-Yvette, FranceAbstractThomX is a compact machine based on the Comptonback-scattering, which is being built by LAL in France.The ThomX ring has a short circumference of 16.8 m anda low design energy of 50 MeV, and the dipole bendingradius of 0.352 m. With such a short bending radius, thenonlinear effects due to the second order dipole fringe fieldin the ring become non-trivial. In this paper, we discuss theappropriate models of the dipole fringe field in the particletracking, and then compare the vertical chromaticity of theThomX ring calculated using four different codes: Tracy3,BETA, MADX and ELEGANT.INTRODUCTIONA compact machine ThomX is being built in LAL,France [1]. This machine aims to produce hard X-rays withan average flux of 1011 - 1013 ph/s, which will be dedicatedto the medical applications, imagining, and especially thecultural heritage recovery in the museums such as the Lou-vre museum in Paris.In the Thom-X machine, the electrons are generated bya laser electron gun, then accelerated up to 50 MeV by aLinac, and finally transferred and injected into the ring tocollide with a laser to produce X-ray. The storage durationis planned to be 20 ms. Table 1 shows the main parametersof the ThomX ring.Table 1: Main Parameters of the ThomX RingParameters valuesCircumference [m] 16.80Nominal energy [MeV] 50Betatron tunes νx / νy 3.175 / 1.64Natural chromaticities -10.522 / -11.255Momentum compaction factor 0.0148RF frequency [MHz] 500RF harmonic 28Beta / dispersion @ IP 0.1 / 0.1 / 0Number of dipoles 8Number of quadrupoles 24Number of sextupoles 12∗work is supported by the French “Agence Nationale de la Recherche”as part of the program “investing in the future” under reference ANR-10-EQPX-51, and also by grants from Re´gion Ile-de-France.† zhangj@lal.in2p3.frVERTICAL CHROMATICITY ANDDIPOLE FRINGE FIELDNormally the dipole fringe field is modeled using the K.Brown’s second order transfer matrix [2] and the verticalchromaticity is calculated using an analytical formula, suchas in the simulation codes BETA [3], MADX [4] and EL-EGANT [5]. The ThomX ring is designed using the codeBETA.Although it is important to correctly calculate the beamproperties using an analytical method, the beam dynamicsneed to be further studied using the particle tracking, suchas to investigate the details of the dynamic aperture usingthe Frequency Map Analysis.In the particle tracking code Tracy3 [6], the initial kickmap of the dipole fringe field was the first order K. Browntransfer matrix [2] with the correction of 1/1+δi [7],pfx = pix +tan θρxi (1)pfz = piz −11 + δitan(θ − ψ)ρzi (2)where i and f respectively denote the particle enters andexits the fringe field region; px,y are the canonical momen-tums; the horizontal coordinate x, the vertical coordinate z,the longitudinal coordinate −ct, and the momentum offsetδ are unchanged after the fringe field region.In eqn. 1 and 2, ρ is the bending radius of the dipole; θis the entrance or exit angle of the dipole, ψ is a parameterdue to the dipole fringe field,ψ =1ρKg1 + sin θ2cos θ, (3)where g is the full magnet gap, andK =∫ +∞−∞Bz(s)[B0 −Bz(s)]gB20ds, (4)where B0 is the vertical design dipole field, Bz(s) is themagnitude of the fringing field on the magnetic middleplane at a position s. The value of K is chosen as 0.5 inTracy3.With the first order kick maps of the dipole fringe field inEqn. 1 and 2, the vertical chromaticity from Tracy3 is closeto the value from BETA and also the experimental valuein the storage ring of the Synchrotron Soleil, which has acircumference 354 m, the beam energy 2.739 GeV, and thedipole bending radius 5.36 m. However, the natrual verticalchromaticity from Tracy3 is -5.072 which has a great dis-crepancy from the value -11.255 calculated using BETA,for the ThomX ring which has a circumference 16.8 m, theWEPEA003 Proceedings of IPAC2013, Shanghai, ChinaISBN 978-3-95450-122-92504Copyrightc ○2013byJACoW—ccCreativeCommonsAttribution3.0(CC-BY-3.0)05 Beam Dynamics and Electromagnetic FieldsD02 Non-linear Dynamics - Resonances, Tracking, Higher Orderbeam energy 50 MeV, and the dipole bending radius 0.352m. This discrepancy is in the expectation, since it is wellknown that the second order term of the dipole fringe fieldis non-trivial to the calculation of the vertical chromaticityin a ring with a small bending radius [8].The K. Brown’s second order matrix of dipole fringefield is an accurate model, but it can’t be used directly inthe particle tracking codes like Tracy3, because an accu-rate and symplectic model is necessary to have a correctprediction to the beam dynamics using particle tracking.IMPORTANCE OF THE SYMPLECTICITYTO THE PARTICLE TRACKINGIn the particle tracking, the particle coordinates before(Xi) and after (Xf ) the lattice element are connected usinga map M , That is Xf = M(Xi). To have a correct predic-tion of the particle motion using the particle tracking, Mmust be both accurate and symplectic, otherwise a wrongpicture of the beam dynamics will be obtained.Figure 1 compares the phase space tracking during 1000turns in the ThomX ring with a symplectic model (eqn. 1and 2) and a non symplectic model (K. Brown’s secondorder transfer matrix) of the dipole fringe field. To keepthe simplicity of the tracking, the sextupoles are turned off,and only dipoles, quadrupoles are included in the lattice.With a symplectic model, the phase space is a closed ellipse(Fig. 1 a) which is in the expectation; while with a nonsymplectic model, the phase space spirals out (Fig. 1 b)which is the wrong physical picture, since the system is alinear dynamic system.−0.015 −0.01 −0.005 0 0.005 0.01 0.015−4−2024x 10−3x [m]xp [rad](a)−0.015 −0.01 −0.005 0 0.005 0.01 0.015−4−2024x 10−3x [m]xp [rad](b)Figure 1: Phase space tracking in 1000 turns using a sym-plectic (a) and non symplectic model (b) in the ThomX ringwithout sextupoles.GEOMETRIC CORRECTION OF THELINEAR DIPOLE FRINGE FIELDThe field distribution of the dipoles in the ThomX ring isshown in Fig. 2. The position s = 0 is the physical edge ofthe dipole. The effective dipole field Bz begins to decreasefrom the design value B0 at the position −ε and disappearsat the position ε.0Figure 2: Sketch of the dipole field in the ThomX ring.From the Maxwell equation  × B = 0, it is easy toget the longitudinal field Bs = ∂Bz∂s z. From the Lorentzequation, the particle is vertically kicked by a force dPz/dt= −qVxBs with Pz the vertical momentum, q the particlecharge, Vx the horizontal speed. Here the horizontal dipolefringe field Bx is omitted since the magnet width is muchlarger than the horizontal coordinate x.The design particle entres/exits the dipole with its closedorbit normal to the dipole face, so its Vx = Vs tan θ with Vsthe longitudinal speed, and the vertical focusing along thefringe field at the entrance/exit of the dipole isdpzds=− tan θBρ∂Bz∂sz = Kzz (5)where Kz is the usual notation of the normalizedquadrupole strength, ds = Vsdt, pz = Pz/P0 and Bρ =P0/q. Furthermore, integrating Kz over the dipole fringefield region leads to the total focusing effect (1/fz) in thevertical plane:1fz=∫ +ε−εKzds =− tan θρ, (6)which is the linear kick map of eqn. 2 without the correc-tion term 1/(1 + δi) and ψ.However, the closed orbit of the off momentum particleis not normal to the dipole pole faces when it entres/exitsthe dipole, and its horizontal divergence x′ or px at the ex-trance/exit of the dipole contributes to the absolute speedProceedings of IPAC2013, Shanghai, China WEPEA00305 Beam Dynamics and Electromagnetic FieldsD02 Non-linear Dynamics - Resonances, Tracking, Higher OrderISBN 978-3-95450-122-92505 Copyrightc ○2013byJACoW—ccCreativeCommonsAttribution3.0(CC-BY-3.0)Vx, that is Vx = Vs tan(θ ± px/(1 + δ)) with “+” at theentrance while “-” at the exit of the dipole. Consequently,the modified first order kick map of eqn. 1 and 2 with thisgeometric correction ispfx = pix +tan θρxi, (7)pfz = piz −11 + δitan(θ − ψ ± pix1+δi )ρzi. (8)In Eqn. 7 and 8, θ is due to the pole face rotation fromthe Cartesian coordinate system in the straight section tothe curvilinear coordinate system in the dipole, θ = 0 forthe dipoles in the ThomX ring; ψ is due to the dipole fringefield as shown in eqn. 3 and 4; pix/1 + δi is due to theclosed orbit of the off momentum particle is not normal tothe dipole pole face, and this term has a non-trivial contri-bution to the vertical chromaticity.SYMPLECTIC MAP OF THE DIPOLEFRINGE FIELDIn a conservative accelerator system without the energyloss (radiation damping) and the energy gain (RF cavitiesoff), the particle motion can be described by a Hamiltonian,H(x, px; z, pz;−ct, δ; s)= −(1 + xρ)√(1 + δ)2 − (px − AxBρ)2 − (pz − AzBρ)2+xρ+x22ρ2− AsBρ+ δ,where Ax and Az , are respectively the horizontal, vertical,vector potential of the magnet. Normally Az is omittedsince the magnet width is much larger than the horizontalcoordinate x; Ax is non zero when the longitudinal fringefield is considered. x/ρ+x2/2ρ2 is the longitudinal vectorpotential of the dipole, and As is the longitudinal vectorpotential of the other higher order magnets (quadrupoles,sextupoles, etc).With the Hamitonian H and following the proceduresproposed by E. Forest [9], the symplectic map of the dipolefringe field up to second order iszf =2zi1 +√1− 2 ∂φ∂piz zi(9)xf = x+12∂φ∂px(zf )2 (10)pfz = piz − φzf (11)(−ct)f = (−ct)i − 12∂φ∂δ(zf)2 (12)whereφ(px, pz, δ) =pxρ(1 + p2z)− gKρ2{(1 + δ)2 − p2zp3s+p2xp2s(1 + δ)2 − p2xp3s}From Eqn. 9 to 12, it is clear that if zi = 0, the particlecoordinates are changed after the dipole fringe field region.Repectively with the linear kick map (eqn. 1 and 2), thegeometric correction of the linear kick map(eqn. 7 and 8)and the Forest’s symplectic map (eqn. 9 to 12), the betatrontunes and natural chromaticities of the ThomX ring werecalculated using the code Tracy3; and then compared to thevalues from BEATA, MADX and ELEGANT which usethe K. Brown’s second order transfer matrix of the dipolefringe field. The results are shown in Table 2. From this ta-ble, the second order term of the dipole fringe field has noobvious effects on the betatron tunes and horizontal chro-maticity, while has a strong effect on the vertical chromatic-ity of the small machine with a small bending radius. Thegeometric correction to the modified linear kick map of thedipole fringe field is a simple and appropriate model and itgives the values close to the Forest’s symplectic model.Table 2: Tunes and Natural Chromaticities of ThomX Ringνx νz ξx ξzTracy 3 (Corr.) 3.175 1.64 -10.663 -10.860Tracy 3 (Forest) 3.175 1.64 -10.663 -10.860Tracy 3 (no corr.) 3.175 1.64 -10.663 -5.072BETA 3.175 1.64 -10.522 -11.255MADX 3.175 1.64 -10.522 -11.255ELEGANT 3.175 1.64 -10.523 -10.735CONCLUSIONThe importance of the symplecticity in the particle track-ing is shown in this paper. A simple geometric correctionto the K. Brown’s modified first order transfer matrix of thedipole fringe field is proposed, and compared to the sym-plectic model and also K. Brown’s second order transfermatrix. The results show that this geometric correction cangive the correct prediction of the vertical chromaticity ofthe small ring like the ThomX ring.REFERENCES[1] A. Variola, “The ThomX project”, IPAC’11, p. 1903-1905.[2] K. Brown, “A First and Second Order Matrix Theory forthe Design of Beam Transport Systems and Charged ParticleSpectrometers”, SLAC-75.[3] J. Payet, Beta Code, CEA, SACLAY.[4] H. Grote and F. Schmidt. PAC 2003, p. 3497.[5] M.Borland, ANL Advanced Photon Source Report No. LS-287, 2000.[6] H. Nishimura, “TRACY,A Tool for Accelerator Design andAnalysis”, EPAC 88, p.803,1989.[7] E. Forest, SSC-141(October 1987).[8] A. J. Dragt, “Exact Numerical Calculation of Chromaticity inSmall Rings”, Nuclear Science, 1981.[9] E. Forest, et al. “Second order fringe in MAD-X for the mod-ule PTC”.WEPEA003 Proceedings of IPAC2013, Shanghai, ChinaISBN 978-3-95450-122-92506Copyrightc ○2013byJACoW—ccCreativeCommonsAttribution3.0(CC-BY-3.0)05 Beam Dynamics and Electromagnetic FieldsD02 Non-linear Dynamics - Resonances, Tracking, Higher Order

Dipole Fringe Field Effects in the ThomX Ring

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