In view of the CERN Proton Synchrotron proposed replacement with a new ring (PS2), a detailed optics design has been undertaken following the evaluation of several lattice options. The basic arc module consists of cells providing negative momentum compaction. The straight section is formed with a combination of FODO and quadrupole triplet cells, to accommodate the injection and extraction systems, in particular the H- injection elements. The arc is matched to the straight section with a dispersion suppressor and matching module. Different lattices are compared with respect to their linear optics functions, tuning flexibility and geometrical acceptance properties

Barranco, J

Bartmann, W

Benedikt, M

Carli, C

de Maria, R

Goddard, B

Papaphilippou, Y

Peggs, S

Trbojevic, D

In view of the CERN Proton Synchrotron proposed replacement with a new ring (PS2), a detailed optics design has been undertaken following the evaluation of several lattice options. The basic arc module consists of cells providing negative momentum compaction. The straight section is formed with a combination of FODO and quadrupole triplet cells, to accommodate the injection and extraction systems, in particular the H{sup -} injection elements. The arc is matched to the straight section with a dispersion suppressor and matching module. Different lattices are compared with respect to their linear optics functions, tuning flexibility and geometrical acceptance properties

Papaphilippou,Y.

De Maria, R.

Barranco, J.

Bartmann, W.

Benedikt, M.

Carli, C.

Goddard, B.

Peggs, S.

Trbojevic, D.

UNT Digital Library

BRDD·KHAVENN /tTI0 1';J :ptL LABORi\TOR''{Linear optics design ofnegative momentumcompaction lattices for PS2Y. Papaphilippou, J. Barranco, Wo Hartmann,Mo Benedikt, c. CarH~ s. GoddardCERJ.~, Geneva, SwitzerlandR.. de Maria, S. Peggs, D. TrbojevicBrookhaven National Laboratory, Upton, NY, USAPresented at the Particle Accelerator Conference (PAC09) .Vancouver, B.C., CanadaMay 4-8, 2009Coltider-Acce lerator DepartmentBrookhaven National LaboratoryP.O. Box 5000Upton, NY 11973-5000vvww'.bnl.govNotice: This manuscript has bee n auth ored by emplo yees of Brookhaven Science Associates , LLC underContr act No. DE-AC02-98CHl 0886 with the U.S. Department of Energy. The publisher by accepting themanuscrip t for publication acknowledges that the United States Government retains a non-exclusive, paid-up,irrevocable, world -wide license to publish or reproduce the publi shed form of this manuscript, or allow othersto do so, for United States Government purpos es.This preprint is intended for publication in a journal or proceedings. Since changes may be made beforepubilca tlon , it may not be cited or reproduced without the autho r's permiss ion.DISCLAIMERThis report was prepared as an account of work sponsored by an agency of theUnited States GoVeI1l--111ent. Neither the United States Government nor anyagency thereof, nor any of their employees , nor any of their contractors,subcontractors, or their employees, makes any warranty, express or implied, orassumes any legal liability or responsibility for the accuracy, completeness, or anythird party's use or the results of such use of any information, apparatus, product ,or process disclosed, or represents that its use would not infringe privately ownedrights. Reference herein to any specific commercial product, process, or serviceby trade name, trademark, manufacturer, or otherwise, does not necessarilyconstitute or imply its endorsement, recommendation, or favoring by the UnitedStates Government ot any agency thereof or its contractors or subcontractors.The views and opinions of authors expressed herein do not necessarily state orreflect those of the United States Government or any agency thereofLINEAR OPTICS DESIGN OF NEGA1"lVE I\10MENTUM COMPACTIONLATTICES FOR PS2Y Papaphilippou, 1. Barranco, W. Bartmann, M. Benedikt, C. Cadi, B. GoddardCERN, Geneva, Switzerland,R. de Maria, S. Peggs, D. Trbojev ic, Brookhaven National Laboratory, Upton, N--Y~ USAAbstractTable 1: Lattice constraints for the PS2.In view of the CE~r..r Proton Synchrotron proposed re-placement with a new ring (PS2), a detailed optics designhas been undertaken following the evaluation of several lat-tice options. The basic arc module consists of cells provid-ing negative momentum compaction . The straight sectionis formed with a combination of FODO and quadrupoletriplet cells, to accommodate the injection and extractionsystems, in particular the H- injection elements. The arc ismatched to the straight section with a dispersion suppres-sor and matching module. Different lattices are comparedwith respect to their linear optics functions, tuning flexibil-ity and geometrical acceptance properties.Parameter [unit]Injection kinetic energy [GeV]Extraction kinetic energy [GeV ]Circumference [rn]Transition energy [GeV]Maximum bending field [T]Maximum quadrupole gradient [Tim]Maximum beta functions [rn]Maximum dispersion function [m]Minimum drift spac e for dipoles [rn]Minimum drift space for quads [rn]value4501346.4-Imaginary1.7176060.61.2,~ -2.5PS2 RING OPTICSFigure 1: Horizontal (blue), vertical (red) beta functionsand dispersion (green) of the !'l'MC module3 /):!:75 c2.0i 5, -!.o20. ' 3'0. " 40, ' 50, ' 60. ' 70, i~. .solso.~I 50.4IJ.JO.20,10.The backbone of the NMC ring is the arc module whoseoptics is shown in Fig. 1. The total length is 80m and fiveof these cells form the arc. The design is based on a com-bination of FODG cells and quadrupole doublets (in total4 quadrupole families). The bending strength is varied be-tween these two structures, in order to create a dispersionoscillation. Imposing a negative dispersion at the entranceof the module, drives the momentum compaction factor U cbelow 1.2 T, at a radius of 70 mm. The beta functions anddispersion should be kept below 60 and 6 m respectivelyto assure conformable geometrical aperture. Finally, andtaking into account space constraints for correctors and in-strumentation, coil ends and vacuum equipments, the driftspace around quadrupoles is set to at least 1.2m and dipolesare separated with a drift of O.6m.LATTICE DESIGN CONSTRAINTSThe upgrade strategy of the Large Hadron Collider(LHC) (1] is heavily based on the renewal of the CERNinjector complex including the replacement of the old Pro-ton Synchrotron (PS) with a new ring (PS2). Some basicparameters guiding the optics design of PS2 are displayedin Table 1. A proton beam with kinetic energy of 4 GeVwill be injected directly from the Super-Conducting ProtonLinac (SPL) (2], accelerated up to 50 GeV and extractedtowards either the SPS or a dedicated experimental area.The LHC ion beams should be injected from the existingion complex. The injection and extraction energy choice isdriven from space charge tune-shift and instability consid-erations in the PS2 and SPS [3]. The ring circumferenceis fixed to 3OOOr.17m, i.e. 15/77 of the SPS circumfer-ence (2200r.m) for better filling and synchronization pur-poses (4]. After a detailed analysis and review process [5J,a lattice that avoids transition was considered as the pre-ferred option. Several versions of Negative MomentumCompaction (NMC) lattices aiming to low imaginary tran-sition values were first considered (7, 8]. This was basedon the need for fast synchrotron motion to achieve the highenergy longitudinal manipulations necessary for the LHCbunch pattern production, with an RF system similar tothe one of the actual PS. Considering a pre-choped beamfrom the SPL with the required bunch spacing and accel-erated with a 40MHz RF in the PS, relaxes the constrainton the transition energy which has still to be flexible allow-ing fast RF gymnastics. This option imposes a challenginglarge tunability RF system for accelerating the ion beams aswell, with RF frequency range from below 20MHz up to 40MHz. The NMC arcs are filled with conventional dipolesnot exceeding 1.7 T and quadrupoles with pole tip fieldsto negative values, Three dipoles in the FODO part of theNMC is the best choice for reducing dipole lengths to be-low 4m , but the module gets slightly longer. The inclusi onof central bends in the middle of the cell has the advantageof incre asing the dipole filling factor but also has an impacton the final n c value, as this point is associated with hig hpositive dispersion. The best compromise was found by in-eluding a single central bend . Th is module alone providesa relatively low transition energy of ""ft; = 26i . In orderto achieve low imaginary transition energies, focusing hasto be increased resulting to high phase advances especiallyin the horizontal plane with /Lx = 2670 and /l.y = 1580 .The maximum betas and dispers ion are kept below 5Smand 3m, respectively.Figure 2: Layout of the injection-extraction LSS (top) andoptics (bottom) for accommodating an injection schemewith foil (left) or laser stripping (right).The two long straight sections (LSS) will be filled withRF cavities, injection and extraction elements. The generallayout of the future LHC injector chain implies to insta llall PS2 injection and ejection elements in the same longstraight section. In consequence, the LSS has to be longenough to accommodate all beam transfer systems and pro-tection equipment like dumps or even collimators [9]. Onthe top of Fig. 2, the layout of the injection and extrac-tion LSS is displ ayed . The different systems enable the H-charge exchange injection , fast ion injection and fast , slowand multitum eject ions. The LSS is around i 45m long ,form ed by a series of FODO cells with 90 ° phase advanceand a centra l quadrupole triplet with a long drift to housethe H- injection system. Two option s are considered forstripping (foil and laser) [10] with two optics alternatives,presented in the bottom part of Fig . 2. In the usual strip -ping by foil option , the width of the foil can be minimizedby a mismatch of the optics functions between the injectedand circulating beam, imposing (3x,y ~ 22.5m at the injec-tion point (Fig. 2 bottom, left) and injected beam beta of,8x ,y > 10 m. Con sidering laser stripping, the laser peakpower is proportional to the vertic al beam size if the beam-laser interaction is horizo nta l and an optics with minimumvertical beam size at the waist is necessary (Fig. 2 bottom ,right). Th is latest optics is limited by the available aper-ture at the last two triplet quadrupoles which in any casehave to be enlarged to accom modate the injected/extractedtrajectories .. 2.40.0.0-3:.Figure 3: Horizontal (blue), vertical (red) beta func tionsand dispersion (green) of the dispersion suppressor.Figure 4: Horizontal (blue), vertical (red) beta functionsand dispersion (green) of one quarter of the PS2 ring .The link between the LSS and the NMC arcs is providedby dispersion suppressor and matching cells. The disper-sion suppression can be also achieved by fixing the totalphase advance of the arc to a multiple of 21r [11] as in theJ-PARC main ring [12]. Although this option provides acompact arc and partial resonant compensation, the addi-tion al dispersion oscillation s induced lead to larger max-imum dispersion and aperture requirements. This optionalso WOUld necessitate wide tuning flexibility of the LSSwhich is limited in the PS2 case by the beam transfer con-straints. The optics of the tuned cell can be found in Fig. 3.The first half of the suppressor is similar to a 'N'MC halfmodule for zeroing dispersion and matching to the straightsection optics. Six independent quadrupole families areneeded to achieve the matching constraints, while keepingthe maximum beta functions to reasonable levels.The optics func tions for a quarter of the li ng are plot-ted in Fig . 4 for tunes of Qx = 13.25, Qy = 8.21 andtransition energy of "'(t = 36-i. The ring is composed of166, 3.78m-long dipoles with 1.7T field at top energy, 132quadrupoles in 17 families of 6 types with maximum gra-dients of 18T/m. The natural chromaticities are not high(~x, ~y) = (-22, - 13) consideri ng that dispersion is largeenough allowing efficient correction with low chromaticf ~~r ]-"'r ~J}I I..'tillt~;,f~;~ -t s!. .ro~~~~,..c.....~_-:±l:~~!MC:..l!~ .m!Wl sa nnr1fll Ht!~! :~tp .< ::;jjl}!':,f9 JJ_r.i. ~Figu re 5: Tune space area with resonance lines up to 3rdorder (left ) and zoom around the chosen working point withresonances up to 4th order (right) .sextupole strengths,REFERENCES(1] R. Garoby, EPAC08, Gen ova, Italy, p.3734, 2008.[2] F. Gerigk (ed.), Concept ual des ign of the SPL II, CERN-200 6-006 .[3] M. Benedikt, these proceedings; E. Shaposhnikova,HB2008, Nashville, TN, US, 2008 .[4·] R. Garob y, CERN internal AB -Note- 2007-020.[5] http ://indico.cern .ch/conferenceDisplay.py?confId=31855[6] W. Bartmann, et a1. PAC 2007, Albuquerque, NM , USA,p.739,2007.(7J Y. Papaphilippou et aI, BB2008, Nashville, TN, USA, 2008.[8] D. Trbojevic et al, EPAC 2008, Genova, Italy, p. 370 , 2008 .[9] J. Barranco, the se proceedings.[1O} J. Uythoven et a1. these proceedings.[11J Yu. Senichev, A. Chechenin, J. Exp. Th. Phys., 105-5, 988,2007.[12] Accelerator Technical Design Report for JPARC, KEK Re-port 2002-13 .(13] Y. Papaphilippou et aI., these proceedings.dipole and quad (1 .1 instead of 1.2m), thus reducing thetot al leng th to 77.5m. Each arc is formed by 3 such mod-ules. This module achieves I t = 25i for similar phaseadvances (270° and 163.r). The minimum dispersion isslightly larger and the beta func tions the same as for thenom inal ring. The dispersion suppressor is shortened byreducing drift space between magnets . The final ring hasa '"'It = 38i and is tuned for (Qx ,Qy) = (14.25, 9.3)The number of dipoles is slightly increased (172) and theirlength reduced. The number of quads rem ains identical.The slightly bigger chrornaticities of -22 .8 and -15.7 shouldnot affect the design. The final choice of this alternativewill be driven by non-linear dynamics simulations includ-ing space-charge and space constraints for the beam trans-fer elements.Figure 7 : Optics of a 3-periodic PS2 module (top left), LSS(top right), suppressor (bottom left ) and ring (bottom right).·24-21 ~ , . 21· 19:s:,"" · 13.2..- ~-i3~12.8.~;..12.6~ -12.4S..:, tal ~). )/3 ,Figure 6: Lim its of the horizontal versus vertical beta func-tion maxima (left) and chromaticities (light).-13.8· 13.6The working point is chosen such as to avoid low or-der svstematic resonances. The tune space considered ispresented in Fig. 5. The red and blue lines represent sys-tematic and random resonan ces. The solid lines are norma!resonances and the dashed lines are skew. From the left plotwhere resonances up to 3rd order are traced , it is inferredthat the best integer tunes are to; Qy) = (13, 8) and , asusual , resonance free space is found close to the diagonal,A zoom around this area is shown on the right plot, whereresonances up to 4th order are traced. The actual workingpoin t sits very close to the 4th order (2,-2) systematic res-onance. The horizontal 3rd integer reso nance is also sys-tem atic and thi s may affect the resonant slow extraction ef-ficiency. Although DA with sextupoles seems not to be af-fected [13], a working point below the diagonal and abovethe half integers (13 .5,8 .5) may be advantageous. How-ever, it is difficu lt to be reached with present beta functionand space cons traints and without retuning the LSS optics.The purple curve represents the area enclosing around 50matched working poin ts tuned by changing the phase ad-vance of the NMC mod ule. In thi s way, the transition en-ergy is varied from 25 to 10Gi. For the lower tran sitionenergy (higher horizontal tune), the maximum verti ca l betafunction exceeds 70m in a single location of the disper-sion suppressor (Fig.6 left). The horizontal chromatici ty isslightly increased by 10% (Fig.6 right).................... ................... ........ ....... ..................For redu cing the effect of systematic resonances , a 3-periodic ring may be considered. The optics for this op-tion are presented in Fig.7. The optics for the straightsection is sim ilar to the racetrack design but shortened byremoving 2 FODO cells by either side. The NMC mod-ule is identical with slightl y sm aller drift space between

Linear optics design of negative momentum compaction lattices for PS2

CERN Document Server

LINEAR OPTICS DESIGN OF NEGATIVE MOMENTUM COMPACTIONLATTICES FOR PS2Y. Papaphilippou, J. Barranco, W. Bartmann, M. Benedikt, C. Carli, B. GoddardCERN, Geneva, Switzerland,R. de Maria, S. Peggs, D. Trbojevic, Brookhaven National Laboratory, Upton, NY, USAAbstractIn view of the CERN Proton Synchrotron proposed re-placement with a new ring (PS2), a detailed optics designhas been undertaken following the evaluation of several lat-tice options. The basic arc module consists of cells provid-ing negative momentum compaction. The straight sectionis formed with a combination of FODO and quadrupoletriplet cells, to accommodate the injection and extractionsystems, in particular the H- injection elements. The arc ismatched to the straight section with a dispersion suppres-sor and matching module. Different lattices are comparedwith respect to their linear optics functions, tuning ﬂexibil-ity and geometrical acceptance properties.LATTICE DESIGN CONSTRAINTSThe upgrade strategy of the Large Hadron Collider(LHC) [1] is heavily based on the renewal of the CERNinjector complex including the replacement of the old Pro-ton Synchrotron (PS) with a new ring (PS2). Some basicparameters guiding the optics design of PS2 are displayedin Table 1. A proton beam with kinetic energy of 4 GeVwill be injected directly from the Super-Conducting ProtonLinac (SPL) [2], accelerated up to 50 GeV and extractedtowards either the SPS or a dedicated experimental area.The LHC ion beams should be injected from the existingion complex. The injection and extraction energy choice isdriven from space charge tune-shift and instability consid-erations in the PS2 and SPS [3]. The ring circumferenceis ﬁxed to 3000π/7m, i.e. 15/77 of the SPS circumfer-ence (2200πm) for better ﬁlling and synchronization pur-poses [4]. After a detailed analysis and review process [5],a lattice that avoids transition was considered as the pre-ferred option. Several versions of Negative MomentumCompaction (NMC) lattices aiming to low imaginary tran-sition values were ﬁrst considered [7, 8]. This was basedon the need for fast synchrotron motion to achieve the highenergy longitudinal manipulations necessary for the LHCbunch pattern production, with an RF system similar tothe one of the actual PS. Considering a pre-choped beamfrom the SPL with the required bunch spacing and accel-erated with a 40MHz RF in the PS, relaxes the constrainton the transition energy which has still to be ﬂexible allow-ing fast RF gymnastics. This option imposes a challenginglarge tunability RF system for accelerating the ion beams aswell, with RF frequency range from below 20MHz up to 40MHz. The NMC arcs are ﬁlled with conventional dipolesnot exceeding 1.7 T and quadrupoles with pole tip ﬁeldsTable 1: Lattice constraints for the PS2.Parameter [unit] valueInjection kinetic energy [GeV] 4Extraction kinetic energy [GeV] 50Circumference [m] 1346.4Transition energy [GeV] ImaginaryMaximum bending ﬁeld [T] 1.7Maximum quadrupole gradient [T/m] 17Maximum beta functions [m] 60Maximum dispersion function [m] 6Minimum drift space for dipoles [m] 0.6Minimum drift space for quads [m] 1.2below 1.2 T, at a radius of 70 mm. The beta functions anddispersion should be kept below 60 and 6 m respectivelyto assure conformable geometrical aperture. Finally, andtaking into account space constraints for correctors and in-strumentation, coil ends and vacuum equipments, the driftspace around quadrupoles is set to at least 1.2m and dipolesare separated with a drift of 0.6m.Figure 1: Horizontal (blue), vertical (red) beta functionsand dispersion (green) of the NMC modulePS2 RING OPTICSThe backbone of the NMC ring is the arc module whoseoptics is shown in Fig. 1. The total length is 80m and ﬁveof these cells form the arc. The design is based on a com-bination of FODO cells and quadrupole doublets (in total4 quadrupole families). The bending strength is varied be-tween these two structures, in order to create a dispersionoscillation. Imposing a negative dispersion at the entranceof the module, drives the momentum compaction factor αcProceedings of PAC09, Vancouver, BC, Canada TH6PFP044Beam Dynamics and Electromagnetic FieldsD01 - Beam Optics - Lattices, Correction Schemes, Transport 3805to negative values. Three dipoles in the FODO part of theNMC is the best choice for reducing dipole lengths to be-low 4m, but the module gets slightly longer. The inclusionof central bends in the middle of the cell has the advantageof increasing the dipole ﬁlling factor but also has an impacton the ﬁnal αc value, as this point is associated with highpositive dispersion. The best compromise was found by in-cluding a single central bend. This module alone providesa relatively low transition energy of γt = 26i. In orderto achieve low imaginary transition energies, focusing hasto be increased resulting to high phase advances especiallyin the horizontal plane with μx = 267◦ and μy = 158◦.The maximum betas and dispersion are kept below 55mand 3m, respectively.MTEBK ExtK ES MS1 MS2 InjK InjS MTEBKT Fast Inj H- Inj Extraction Extraction Figure 2: Layout of the injection-extraction LSS (top) andoptics (bottom) for accommodating an injection schemewith foil (left) or laser stripping (right).The two long straight sections (LSS) will be ﬁlled withRF cavities, injection and extraction elements. The generallayout of the future LHC injector chain implies to installall PS2 injection and ejection elements in the same longstraight section. In consequence, the LSS has to be longenough to accommodate all beam transfer systems and pro-tection equipment like dumps or even collimators [9]. Onthe top of Fig. 2, the layout of the injection and extrac-tion LSS is displayed. The different systems enable the H−charge exchange injection, fast ion injection and fast, slowand multiturn ejections. The LSS is around 145m long,formed by a series of FODO cells with 90◦ phase advanceand a central quadrupole triplet with a long drift to housethe H− injection system. Two options are considered forstripping (foil and laser) [10] with two optics alternatives,presented in the bottom part of Fig. 2. In the usual strip-ping by foil option, the width of the foil can be minimizedby a mismatch of the optics functions between the injectedand circulating beam, imposing βx,y ≈ 22.5m at the injec-tion point (Fig. 2 bottom, left) and injected beam beta ofβx,y > 10 m. Considering laser stripping, the laser peakpower is proportional to the vertical beam size if the beam-laser interaction is horizontal and an optics with minimumvertical beam size at the waist is necessary (Fig. 2 bottom,right). This latest optics is limited by the available aper-ture at the last two triplet quadrupoles which in any casehave to be enlarged to accommodate the injected/extractedtrajectories.Figure 3: Horizontal (blue), vertical (red) beta functionsand dispersion (green) of the dispersion suppressor.Figure 4: Horizontal (blue), vertical (red) beta functionsand dispersion (green) of one quarter of the PS2 ring.The link between the LSS and the NMC arcs is providedby dispersion suppressor and matching cells. The disper-sion suppression can be also achieved by ﬁxing the totalphase advance of the arc to a multiple of 2π [11] as in theJ-PARC main ring [12]. Although this option provides acompact arc and partial resonant compensation, the addi-tional dispersion oscillations induced lead to larger max-imum dispersion and aperture requirements. This optionalso would necessitate wide tuning ﬂexibility of the LSSwhich is limited in the PS2 case by the beam transfer con-straints. The optics of the tuned cell can be found in Fig. 3.The ﬁrst half of the suppressor is similar to a NMC halfmodule for zeroing dispersion and matching to the straightsection optics. Six independent quadrupole families areneeded to achieve the matching constraints, while keepingthe maximum beta functions to reasonable levels.The optics functions for a quarter of the ring are plot-ted in Fig. 4 for tunes of Qx = 13.25, Qy = 8.21 andtransition energy of γt = 36i. The ring is composed of166, 3.78m-long dipoles with 1.7T ﬁeld at top energy, 132quadrupoles in 17 families of 6 types with maximum gra-dients of 18T/m. The natural chromaticities are not high(ξx, ξy) = (−22,−13) considering that dispersion is largeenough allowing efﬁcient correction with low chromaticTH6PFP044 Proceedings of PAC09, Vancouver, BC, Canada3806Beam Dynamics and Electromagnetic FieldsD01 - Beam Optics - Lattices, Correction Schemes, Transportsextupole strengths.Figure 5: Tune space area with resonance lines up to 3rdorder (left) and zoom around the chosen working point withresonances up to 4th order (right).The working point is chosen such as to avoid low or-der systematic resonances. The tune space considered ispresented in Fig. 5. The red and blue lines represent sys-tematic and random resonances. The solid lines are normalresonances and the dashed lines are skew. From the left plotwhere resonances up to 3rd order are traced, it is inferredthat the best integer tunes are (Qx, Qy) = (13, 8) and, asusual, resonance free space is found close to the diagonal.A zoom around this area is shown on the right plot, whereresonances up to 4th order are traced. The actual workingpoint sits very close to the 4th order (2,-2) systematic res-onance. The horizontal 3rd integer resonance is also sys-tematic and this may affect the resonant slow extraction ef-ﬁciency. Although DA with sextupoles seems not to be af-fected [13], a working point below the diagonal and abovethe half integers (13.5,8.5) may be advantageous. How-ever, it is difﬁcult to be reached with present beta functionand space constraints and without retuning the LSS optics.The purple curve represents the area enclosing around 50matched working points tuned by changing the phase ad-vance of the NMC module. In this way, the transition en-ergy is varied from 25 to 100i. For the lower transitionenergy (higher horizontal tune), the maximum vertical betafunction exceeds 70m in a single location of the disper-sion suppressor (Fig.6 left). The horizontal chromaticity isslightly increased by 10% (Fig.6 right).50 55 60 65 70 75 59 60 61 62 63 64 65 y x -14 -13.8 -13.6 -13.4 -13.2 -13 -12.8 -12.6 -12.4 -12.2 -24 -23 -22 -21 -20 -19 y x Figure 6: Limits of the horizontal versus vertical beta func-tion maxima (left) and chromaticities (right).For reducing the effect of systematic resonances, a 3-periodic ring may be considered. The optics for this op-tion are presented in Fig.7. The optics for the straightsection is similar to the racetrack design but shortened byremoving 2 FODO cells by either side. The NMC mod-ule is identical with slightly smaller drift space between0.0 10. 20. 30. 40. 50. 60. 70. 80.                               s (m)0.010.20.30.40.50.60.x(m),y(m)-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.0Dx(m) x  y Dx0.0 20. 40. 60. 80. 100.                               s (m)5.10.15.20.25.30.35.40.45.50.55.x(m),y(m)0.00.10.20.30.40.50.60.70.80.91.0Dx(m) x  yDx0.0 10. 20. 30. 40. 50. 60.                               s (m)0.06.513.019.526.032.539.045.552.058.565.0x(m),y(m)-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.03.5Dx(m) x  y Dx0.0 400. 800. 1200. 1600.                               s (m)0.06.513.019.526.032.539.045.552.058.565.0x(m),y(m)-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.03.5Dx(m) x  y DxFigure 7: Optics of a 3-periodic PS2 module (top left), LSS(top right), suppressor (bottom left) and ring (bottom right).dipole and quad (1.1 instead of 1.2m), thus reducing thetotal length to 77.5m. Each arc is formed by 3 such mod-ules. This module achieves γt = 25i for similar phaseadvances (270◦ and 163.7◦). The minimum dispersion isslightly larger and the beta functions the same as for thenominal ring. The dispersion suppressor is shortened byreducing drift space between magnets. The ﬁnal ring hasa γt = 38i and is tuned for (Qx, Qy) = (14.25, 9.3)The number of dipoles is slightly increased (172) and theirlength reduced. The number of quads remains identical.The slightly bigger chromaticities of -22.8 and -15.7 shouldnot affect the design. The ﬁnal choice of this alternativewill be driven by non-linear dynamics simulations includ-ing space-charge and space constraints for the beam trans-fer elements.REFERENCES[1] R. Garoby, EPAC08, Genova, Italy, p.3734, 2008.[2] F. Gerigk (ed.), Conceptual design of the SPL II, CERN-2006-006.[3] M. Benedikt, these proceedings; E. Shaposhnikova,HB2008, Nashville, TN, US, 2008.[4] R. Garoby, CERN internal AB-Note-2007-020.[5] http://indico.cern.ch/conferenceDisplay.py?confId=31855[6] W. Bartmann, et al. PAC 2007, Albuquerque, NM, USA,p.739, 2007.[7] Y. Papaphilippou et al, HB2008, Nashville, TN, USA, 2008.[8] D. Trbojevic et al, EPAC 2008, Genova, Italy, p. 370, 2008.[9] J. Barranco, these proceedings.[10] J. Uythoven et al. these proceedings.[11] Yu. Senichev, A. Chechenin, J. Exp. Th. Phys., 105-5, 988,2007.[12] Accelerator Technical Design Report for JPARC, KEK Re-port 2002-13.[13] Y. Papaphilippou et al., these proceedings.Proceedings of PAC09, Vancouver, BC, Canada TH6PFP044Beam Dynamics and Electromagnetic FieldsD01 - Beam Optics - Lattices, Correction Schemes, Transport 3807

Linear optics design of negative momentum compaction lattices for PS2

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