Lattice Design of a Carbon-Ion Synchrotron based on Double-Bend Achromat Lens
LLattice Design of a Carbon-Ion Synchrotron based onDouble-Bend Achromat Lens
Xuanhao
Zhang
School of Physics, University of Melbourne, [email protected]
Abstract
A normal-conducting carbon-ion synchrotron design for cancer therapy is proposed. The lattice design is based onthe double-bend achromat lens with additional quadrupole magnets. The proposed synchrotron design is compactwith a circumference of 55 meters. The lattice is tuned for slow extraction at the third integer resonance and theoptical functions are optimised with considerations for reducing the power consumption during operation. i. introduction
Irradiation of tumour using carbon-ions is a non-invasivemethod of cancer therapy. There are several advantagesfor using using carbon-ions over the more widely avail-able proton therapy, such as greater relative biologicaleffectiveness (RBE) and better depth precision [1]. How-ever, there is only a handful of carbon-ion therapy fa-cilities around the world partially due to the high costof building a dedicated facility. The size of the typi-cal synchrotron required to accelerate carbon-ions totherapeutic energies presents a significant cost barrier.In addition, once built the ongoing electricity bill ofrunning the facility could cost more than 1 M A C per year[2]. This paper presents a novel synchrotron design forcarbon-ion therapy purposes, the design is optimisedfor both the overall size of the machine and the ver-tical beam aperture that could potentially reduce theelectricity consumption. ii. design requirements
The maximum penetration depth required for clinicaltreatment is 38 cm equivalent depth in water, this cor-responds to an energy of 430 MeV/u for carbon ions.The maximum dipole strength is set to 1.5 T and thelength of each dipole element should be kept between2-3 m for most efficient manufacturing costs [3]. Themomentum compaction factor γ t of the machine shouldbe kept above 1.46 to ensure longitudinal beam stability.It is preferable to have long dispersion-free sections inthe ring to accomodate RF-cavities and septum magnetsto simplify the accelerator cycle.Design of the synchrotron is constrained by the re-quirement for a scanning pencil beam treatment methodwhere a smooth beam extraction over the period of 1 s is needed, this is typically achieved through resonanceextraction at the third integer tune. The chromaticity ofthe machine ξ x should be negative for maximum trans-verse stability of the waiting beam during extraction[4]. Optics of the ring is kept constant during extractionwhile a sextupole is used to excite the third-order res-onance. Particles are smoothly accelerated toward theresonance by either an RF-cavity or betatron core. It isdesirable to place the resonance sextupole (SXR) in adispersion-free section of the ring to allow independentresonance and chromaticity control.The particle motion in phase space in the plane ofextraction is given by the Kobayashi Hamiltonian: H = 6 πδQ X + X (cid:48) ) + S n XX (cid:48) − X ) , (1)where δQ is the distance to the third order resonancetune n ± / , ( X, X (cid:48) ) are the normalised phase spacecoordinates and S n is the normalised SXR strength. Par-ticles leave the stable region along separatrices describedby the Kobayashi Hamiltonian and they are extractedfrom the ring through a combination of electrostatic(ES) and magnetic septa (MS). The size of the stableregion in phase space is slowly reduced as particles areextracted to provide a constant beam spill. Aligningthe separatrices of particles with different momenta atthe ES will minimise losses and reduce the size of theseptum aperture, this is known as the Hardt condition[4]: D n cos( π − ∆ µ ) + D (cid:48) n sin( π − ∆ µ ) = − πξ x /S n , (2)where D n , D (cid:48) n is the normalised dispersion function andits derivative at the ES, ∆ µ is the betatron phase ad-vance between SXR and ES, and ξ x is the chromaticityin the plane of extraction.1 a r X i v : . [ phy s i c s . acc - ph ] J u l able 1: Summary of synchrotron parameters at major operational carbon therapy facilities and the proposed DBA lattice.
PIMMS GSI Hitachi NIRS Siemens DBAProton • - • - • -Carbon • • • • • • No. built 2 2 1 5 1 -Circumference (m) 75 65 60 62 65 55Cell type FODOF FD doublet FODO FODO FODO DBANo. of cells 8 6 6 6 6 2No. of dipoles 16 6 12 18 12 12No. of quads 24 12 12 12 12 14 E max (MeV/u) 400 430 480 400 430 430Max. β y (m) 15 22 13 13.4 15.5 11.8Tune ( Q x /Q y ) 1.6666/1.72 1.72/1.13 1.72/1.43 1.672/1.72 1.7/1.8 1.672/1.72The power consumption of the lattice can be reducedby minimising the size of the vertical beam aperture.The total current of a normal conducting dipole is givenby: N I = B gap hµ η , (3)where B gap is the dipole field strength between the poles, h is the pole gap height, µ is the vacuum permeabilityand η ≈ for non-saturated dipole with iron yoke. Inthe case where both beam injection and beam extractionare performed in the radial plane, the vertical chamberaperture is given by [5]:aperture = ± (cid:104) n (cid:112) β(cid:15)/π + collimator margin (cid:105) , (4)the first term in the brackets is the vertical betatronenvelope for a given RMS emittance, it corresponds tothe ‘good’-field region of the magnets where a goodfield uniformity is required, the collimator margin cor-responds to the ‘poor’-field region which is typicallyreserved as a safety margin that is half the size of the‘good’-field region. Therefore the current required fornormal conducting dipoles is approximately proportionalto square-root of the vertical betatron function. iii. existing designs Currently there are 12 facilities around the world thatprovide cancer therapy treatment using carbon ions,they are located in Japan (6), China (2), Germany (2),Austria (1) and Italy (1). All 12 facilities rely on nor-mal conducting synchrotrons as the main accelerator.The majority of these synchrotron lattice designs canbe traced back to one of five projects lead by the follow-ing institutions: CERN (PIMMS)[5], GSI [6], Hitachi[7], NIRS [8] and Siemens [9]. Table 1 shows a latticeparameter summary of these five designs.Most of these designs have separate function, alter-nating focusing FODO type lattices. The FODO lattice is a relatively simple lattice that offers a high degreeof periodicity, however there are several tradeoff factorsto the design considerations. There exists an inverserelationship between the size of the main synchrotronand the amplitude of betatron oscillation, this relation-ship means any lattice design has to carefully balancecompactness of the machine and beam parameters. Inaddition, the basic FODO lattice does not have anydispersion free sections and often requires three or morefamilies of quadrupoles to tune the lattice for slow ex-traction. The basic FODO lattice can be modified toaccomodate dispersion free drifts but at the cost of ad-ditional quadrupoles and overall size. In the case of thePIMMS design, the number of quadrupoles magnets inthe main ring is double the number in that of a typicalFODO lattice. iv. lattice design
The motivation for using the Double-Bend Achromat(DBA) [10] is mainly driven by the preference for disper-sion free drift sections. In comparison with a compactPIMMS type FODOF lattice [11], the DBA requiresfewer number of quadrupole magnets and independentpower supplies.Figure 1 shows the layout of the original DBA. TheDBA was initially designed as a candidate for electronstorage rings where small beam divergences and verylow emittance in long drift sections are required to acco-modate insertion devices. It was shown that the originalDBA can be optimised for minimum betatron functionin the long drifts by changing the distance between thebending dipoles and the central quadrupole [12] as il-lustrated in Figure 1. This property is exploited in theproposed DBA design in this paper to minimise thevertical betatron envelope.The lattice design was performed using MADX [13].First, two 180 ◦ DBA were joined together to form aclosed ring with two dispersion-free long drifts. The2 igure 1:
Original double-bend achromat arc consist of twodipoles and five quadrupoles arranged symmetri-cally. [10]
A. JACKSON
AchromatInsertionSymmetry PointInsertionStraightSection I Symmetry Point A
0" F ' 0 / I "
IsperSlon unction, // J ;' " FIGURE 1 Basic DFA structure.
AchromatInsertionSymmetry PointStraightSection Symmetry PointAchromat
FIGURE 2 Basic TBA structure.
This paper details the difficulties with the DFA structure and show how theyare overcome in the TBA design. The principles are illustrated in two lattices thathave been optimized for a 1.5-GeV synchrotron light source.2. THE CHASMAN-GREEN LATTICEDFA structures are utilized in the VUV and XRAY rings (both operational) atthe National Synchrotron Light Source and are the basis of the conceptualdesigns in proposals for higher-energy synchrotron radiation sources both in theU.S. and in Europe. However, as beam emittance is pushed ever smaller andthe requirement to optimize the new sources for insertion devices is emphasized,so the relative inflexibility of this type of structure becomes apparent.First we demonstrate that the radial phase advance between the centers of thedipoles forming the achromat is approximately 180 (This should be intuitivelyobvious by noting the form of the beta function or by noting that an electrontrajectory starting with some angle and zero displacement in the middle of thefirst bend is focused back to the axis in the middle of the second bend.)Consider a DFA lattice with the minimum possible emittance. Sommer hasshown that for this condition the beta value has a minimum given by = ·lb[l + · - · · .J (1) at a distance into the magnet given by S = i b[1- + · · .J, (2) Figure 2:
Betatron amplitude and dispersion functions ofthe proposed DBA lattice at Q x = 1 . , Q y =1 . . s (m) D x ( m ) , β x ( m ) , β y ( m ) D x β x β y total bending length is adjusted to 27.72 m correspond-ing to a maximum dipole field of 1.5 T. Each dipolesection was subdivided into three dipoles that are each2.31 m in length. An additional quadrupole was addedsymmetrically on both sides of the central quadrupoleto achieve a tune that is close to the third integer. Itwas necessary to shift the additional quadrupoles alongthe ring by one dipole element to provide the requiredphase advance between the SXR, ES and MS for slowextraction scheme. The SXR and a placeholder RF-cavity is located at one of the dispersion-free section,while both injection and extraction magnetic septa arelocated at the other dispersion-free section. The choiceof septa arrangement potentially limits the orientationof injection and extraction to opposite sides of the ring.It is possible to swap the injection septum with theRF cavity by increasing the length of the dispersion-free drift length. The ES is placed upstream of theextraction MS that satisfies the Hardt condition, where D n = 2 . , D (cid:48) n = − . , ∆ µ ( SXR, ES ) = 226 ◦ and ∆ µ ( ES, M S ) = 70 . ◦ . Space is reserved in dispersiveregions along the ring for a pair of chromaticity sex-tupoles. As such, the chromaticity of the machine andthe resonance conditions can be controlled separately.Additional work is required to include bump magnetsfor injection and extraction.The vertical betatron envelope was optimised by vary-ing the drift lengths and quadrupole strengths. Theminimum drift length between any element was limitedarbitrarily to 0.2 m as an engineering margin. Figures2 and 3 show the lattice functions and the schematiclayout of the final optimised design. Specifications ofthe proposed DBA lattice is included in Table 1 forcomparison with exisiting designs. Detailed parametersof the final design can be found in Table 2. Figure 3:
Schematics of the synchrotron, the designed dipolemagnets are 30 ◦ sector dipoles. QF: focusingquadrupole, QD: de-focusing quadrupole. QF1
QF2
QD2
QD1
QD2
QF1
QD1 BM RF cavity
SXR ES MS Inj.
MS Ext.
Table 2:
Detailed parameters of the proposed DBA lattice.
Transition gamma, γ t ρB β x , β y D x ξ x , ξ y -1.093, -1.276Dipole length 2.31 mDipole strength 1.5 TQuadrupole length 0.4 mQF1 strength 1.049 m − QF2 strength 0.800 m − QD1 strength -1.299 m − QD2 strength -0.650 m − Typically, ES is placed in a region with relatively largehorizontal betatron amplitude to enhance the ‘kick’ from3he electric field and prevent the extracted particles fromcolliding with the MS wire. The horizontal displacementat the MS, ∆ x MS , due to a deflection of θ ES,x from theES upstream is given by: ∆ x MS = θ ES,x (cid:112) β ES,x · β MS,x · sin(∆ µ ) , where ∆ µ is the betatron phase difference between MSand ES, ( β ES,x , β
MS,x ) are the horizontal betatron func-tions at ES and MS respectively. The deflection θ ES,x of a charged particle beam in an electric field is givenby: θ ES,x = tan − (cid:20) E x l ES qp β × (cid:21) , where E x is the electric field strength, l ES is the effectivefield length of ES, p is the momentum in GeV/c and β is the relativistic velocity. In the current design, anelectric field of 61.1 kV/cm will result in a 0.6 mmdisplacement at maximum extraction energy. The fieldrequired is within feasible limits of ES field strength andthe resultant deflection is enough to clear a ‘thin’ MSsimilar to that used in the PIMMS design. v. conclusions A normal conducting synchrotron was designed to reducethe size of carbon-ion therapy facilities and potentiallylessen the electricity cost of operating the machine. Thelattice has been designed to satisfy the Hardt conditionfor third-order resonant extraction. acknowledgement
This work is supported by the Australian GovernmentResearch Training Program Scholarship. references [1] D. Schardt, Nuclear Physics A , 633 (2007),Proceedings of the 9th International Conference onNucleus-Nucleus Collisions.[2] U. Amaldi et al. , A facility for tumour therapy andbiomedical research in South-Eastern Europe
CERNYellow Reports: Monographs (CERN, 2019).[3] E. Benedetto, Personal communication.[4] M. Pullia, Hardt condition for superposition ofseparatrices, in
Slow extraction from synchrotronsfor cancer therapy , 1996.[5] Accelerator Complex Study Group, P. J. Bryant et al. , Proton-Ion Medical Machine Study (PIMMS),2
CERN-PS-2000-007-DR (CERN, 2000). [6] A. Dolinskii, H. Eickhoff, and B. Franczak, Thesynchrotron of the dedicated ion beam facility forcancer therapy, proposed for the clinic in heidelberg,in ,2000.[7] F. Noda et al. , Conceptual design of carbon/protonsynchrotron for particle beam therapy, in , pp. 1300–1302, 2009.[8] T. Furukawa et al. , Design of synchrotron at NIRSfor carbon therapy facility, in , 2004.[9] V. Lazarev et al. , Technical overview of the siemensparticle therapy accelerator, in , 2011.[10] A. Jackson, Particle Accelerators , 111 (1986).[11] H.-S. Kang, J. Huang, and J. Choi, J KOREANPHYS SOC (2008).[12] M. Sommer, Optimization of the emittance of elec-trons (positrons) storage rings (Universite de Paris-Sud, 1983).[13] CERN - Accelerator Beam Physics Group, MAD- Methodical Accelerator Design, 2020, https://mad.web.cern.ch/mad/https://mad.web.cern.ch/mad/