Towards the ultimate storage Beijing Advanced ring： The lattice design for Photon Source

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Chinese Physics C Vo1.37，No.5(2013)057003ring：The lattice design forBeijing Advanced Photon Sourcexu Gang(徐刚) JIAO Yi(焦毅)Institute of High Energy Physics，Chinese Academy of Sciences，Beijing 100049，ChinaAbstract： A storage ring-based light source，Beijing Advanced Photon Source(BAPS)，is proposed to store a5 GeV low-emittance electron beam and to provide high-briliance coherent radiation.In this paper，we report oureforts of pushing down the emittance of BAPS to approach the so-caled ultimate storage ring，while fixing thecircumference to about 1200 m.Tb helD deal with the challenge of beam dynamics associated with the intrinsic，verystrong nonlinearities in an ultralow emittance ring.a combination of severa1 progressive technologies is used in thelinear optics design and nonlinear optimization，such as a modifed theoretical minimum emittance cell with small-aperture magnets.quasi-3rd-order achromat，theoretical ana1)rzer based on Lie Algebra and Hamiltonian analysis，multi.objective genetic algorithm and frequency map analysis.These technologies enable us to obtain satisfactorybeam dynamics in one lattice design with natural emittance of 75 pm。

K ey words： ultimate storage ring，emittancej linear optics design，nonlinear optimizationPACS：29.20.db，41.85.-P，29.27.-a DOI：10.1088/1674-1137/37/5/0570031 Introduct ionA storage ring-based light source，Beijing AdvancedPhoton Source(BAPS)，is planned to be built in Beijingto satisfy the increasing requirements of high-briliancecoherent radiation from the user community.A baselineof BAPS was designed to provide a 5 GeV electron beamwith natural emittance of 1.6 nanometers(nm)by using48 double bend achromat fDBA1 cells，and of 0.5 nm byusing additional damping wigglers[1，within a circum-ference of 1209 m. Recently we continuously pusheddown the emittance of BPAS in order to approach theso-called ultimate storage ring.i.e..a storage ring withemittance in both transverse planes at the difractionlimit for the range of X-ray wavelengths of interest forscientific community.For BAPS with an electron beamof 5 GeV.it requires a reduction of the emittance toseveral tens of picometers(pm)。

As a result of the equilibrium between the radia-tion damping and quantum fluctuation.the natural emit-tance of the electron beam in a storage ring o is givenby[2] 2 (1)where CTq3.83×10- 。m； L is the Lorenz factor； LB/0 is the bending radius of a single dipole，with LBbeing the dipole length and 0 the bending angle；3 isthe horizontal damping partition number；and(H)dip。1。

is the average over the storage ring dipoles of thetion日 D：2 D D：flxDfunc-where z 1 8 and 1 are the horizontal Courant-Snyderparameters，D∞and D are the horizontal dispersion andits derivative。

In a storage ring with uniform dipoles， ≈1，the the-oretical minimum emittance fTME)is derived by mini-mizing(H>aip。l with symmetric dispersion in the dipole3，-Cq7O3。 。-12vqg4with the optical functions at the dipole center(expressedwith subscript of zero)satisfying.。 LB， ：0：。， 。 24p D ：00In Eqs.(3)and(4)，an asterisked quantity means thequantity is evaluated when the exact TME condition isfulfilled。

In this paper，a TME cell refers to a unit cell satisfy-ing the conditions in Eq.(4)and having horizontal phaseadvance of half-cel 142 degrees.It consists of onedipole and symmetric quadrupole structure outside，withfocusing quadrupole(QF)closer to the dipole(or withfocusing gradient combined into the dipole1. A TME-like cel refers to a unit cell with similar layout and withReceived 2 July 2012Supported by Special fund of Chinese Academy of Sciences(H92931 IOTA)###2013 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Acad emy of Sciences and the Institute ofModern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd057003-1Chinese Physics C Vo1.37，No.5(2013)057003optical functions at the dipole center close to.but not ex-actly on，the conditions in Eq.(4).In Ref.4]，one of theauthors(JIA0 Y)highlights one kind of TME-like cellsnamed the modified-TME cel1.in which the defocusingquadrupole f QD 1 is closer to the dipole or the defocns-ing gradient is directly combined into the dipole.Fig.1presents the configurations of a modified-TME cell anda conventiona1 TME-like cell for a TME cell1. Studyshows that such kind of TM E-like cell allows minima1emittance of the order of about 3 times of the theoreticalminimum，phase advance per half cell below 7/2 relaxedoptical functions，moderate natural chromaticities，andmost importantly，a compact layout[4]。

中 - -·.[ i中L0 LI. ·Fig.1. Two TME-like cell configurations，with QDcloser to the dipole in a modified-TME cell(theupper plot)，and with QF closer to the dipole in aconventional TME-like cell or in a TME cel(thelower plot)。

Because of the ability of approaching emittance of thetheoretical minimum，a TME or TME-like cell has beenadopted as the basic unit cell in many ultralow emittancedesigns[5-9].From Eq.(3)，the TME is proportional to. To reduce the natural emittance of an electron stor-age ring composed of TME or TME-like cels and withfixed beam energy and circumference，the most efectivewav is to reduce the bending angle of the dipole 0.whichmeans increasing the number of the dipoles Nd as wellas the number of the cells Nd in a ring associatedwith decreasing cel length.To reach a small cel length。

and at the same time.to achieve relatively 1ow natu-ral chromaticities，we use the modifed-TM E cells withcombined.function dipoles and smal1.aperture magnetsin our design。

According to a rough scaling that S。( /0 with Sthe integral strength of the chromaticity.correction sex。

tupoles I4l，decreasing the emittance in an electron stor-age ring unavoidably leads to an increase of the requiredsextupole strength.In an ultralow emittance design，therequired sextupole strength can be so strong that thenonlinear dynamics associated with the very large non-linear geometric and chromatic aberrations from the sex-tupoles becomes a great chalenge to the performance ofthe storage ring.To preserve large enough injection ac-ceptances as pushing down the emittance to several tensof pm region，more advanced optimization technologiesor methods compared to those used for the present 3rdgeneration rings，are essentially required.In our design，we construct a quasi.3rd.order achromat within everyeight superperiods through fine tuning of the phase ad-vance，so as to approximately cancel most of the 3rd-and4th-order resonances9-l11；we use multi-families of sex-tupoles and octupoles to minimize the residual 3rd-and4th-order resonance driving terms，and to control othernonlinear terms to an acceptable leve1.Using a theoreti-cal analyzer based on Lie Algebra and Hamiltonian anal-ysis，we obtain expressions of the nonlinear terms withrespect to the sextupole and octupole strengths；we thenset three objectives characterizing the chromaticity，de-tune and resonance terms and search for a set of optimalsolutions with non-dominated sorting genetic algorithmI (NSGA-I 121)；after numerica1 tracking with ATcode[13andequency map analysis[14]，we can findthe best solution that promises a large enough dynamicaperture(DA)。

In the folloice design forBAPS reaching natural emittance of 75 pm.Linear op-tics is shown in Section 2.where we foCUS on the designof the modifed.TME unit cell and the quasi.3rd.orderachromat.Nonlinear dynamics is discussed in Section 3。

with an emphasis on the extraction of the chromaticity，detune and resonance terms with the theoretical ana-lyzer and subsequent optimization with NSGA-II.Con-clusions are given in Section 4。

2 Linear optics designAs mentioned above，as we pursue an ultralow emit。

tance ring composed of TME-like cells；the number ofdipoles Nd and hence the number of cells M increaseas the emittance decreases. Limited by the construc-tion budget，the BAPS circumference is expected to bearound 1200 m. As a balance of the requirements oflarger N，for lower emittance and of more straight sec-tions for insertion devices fID1，we adopt 32 supercellsconsisting ofseven-bend.achromat f7BA1.Each supercellincludes five modifed-TME unit cells and two dispersionmatching cells at two ends to produce a dispersion-freestraight section，as shown in Fig.2.The bending angleof the outer dipole in a supercel is set to 1/2 fnot farfrom the exact value 3-1/015])of the bending angle ofthe inner dipoles to obtain a minimum emittance.Over-all there are equivalently 1 92 dipoles in the ring.FromEq.(3)，the theoretical minimum emittance 27.7 pmwith 1。

As a simple but reasonablethat the matching cell lengthmodifed-TME unit cell lengthW ith an additional assumption057003-2assumption，we presume(Ld )is 1.5 times the( t)，i.e.Ld 1.5 L t。

that the straight section.日-小Y 十上 Chinese Physics C Vo1.37，No.5(2013)057003121O86420- 2O 0 5 1 0 1 5 2 0 2 5 3.0 3 5longitudinal position/mFig.5. Optical functions in a modified-TME unitcell with( ， )(137c/32，7/64)。

The linear optics design on one hand should alowlow beta functions in the straight section for high bril-1iance(optimal beta LID/2 9，with LID the IDlength)，and on the other hand，should promise largeenough acceptance for of-axis injection.Like that atNSLS-II1 71，we merge two 7BA supercells into onesuperperiod. Each superperiod has a high-beta 10 mstraight section for injection and a low-beta 6 m straightsection for IDs.The phase advance of each superperiod ischosen to be 12卅 /4 u xTr/8 in horizontal plane and47cn/4Sv x Tr/8 in vertical plane，where 5v and uare the expected decimal portions of the working pointof the ring.Fig.6 shows the optical functions in a super-period.Note that the third term for the phase advanceis much smaller than the sum of the first two. Thus，every eight superperiods makes an approximate identitytransformation and forms a quasi-3rd-order achromat。

Such a design helps to approximately cancel the 3rd-andmost of the 4th-order resonances and hence to facilitatethe subsequent nonlinear optimization(see Section 3 formore discussions)。

- - - - I -， 、、I--- l- -1r ，r F 1 l。-q I- Ir - H ll0 10 20 30 40 50 60 70 80longitudinal position/mFig.6. Optical functions in a superperiod consist-ing of two 7BA supercels。

Sixteen identical superperiods compose a 1263.4 mring with natural emittance of 75 pm.The main param-eters of the ring are summarized in Table 1。

Table 1. M ain parameters for the BAPS 5 GeVstorage ring。

parameters fluxenergy/GeVcircumference/mhorizontal damping partition number Jnatural emittance/pmworking point(H/V)natural chromaticities(H/V)number of 7BA achromatsnumber of high-beta 10 m straight sectionsbeta functions in high-beta straightsection(H/V)/mnumber of low-beta 6 m straight sectionsbeta functions in low-beta straightsection (H/V)/mdamping times(x/y/z)/msenergy spreadmomentum compaction01263.41.407598.4/34.3- 189/-113321641/4.7164.5/1.720/2s/17.48x10-43.86x10-53 N onliear optim izationThe aim of the nonlinear optimization is to obtain1arge enough DA and momentum acceptance for of-axisin ection and a good performance of the ring.As men。

tioned above，the sextupole strength increases as theemittance decreases. In spite of the adoption of themodifiedTME cell with relatively low phase advanceand relatively long combined-function dipoles.the re-quired sextupole strengths to compensate the naturalchromaticity are still very large，i.e.， f290 m-3 andKsd：-274 m。，provided only two families of sextupolesfSF and SD1 are used for chromaticity correction.Strongnonlinearities induced by the sextupoles may limit DAto a few millimeters(mm)or even lower.Compared tothe nonlinear optimization for the 3rd generation rings，more advanced tools or methods are required to analyzeand control the nonlinear terms to obtain a satisfactorybeam dynamicsAs a start of the optimization，and also for demon-stration of the efrect of the quasi-3rd-order achromat。

we first perform numerical tracking(without magneticerrors)and FMA for the case with only two familiesof chromaticity-correction sextupoles. The results areshown in Fig.7. The on-momentum horizontal DA is7.5 mm，larger than the requirement(/>4 mm)of of-axisinjection with pulsed sextupoles f18I.The resonanceshave small driving terms fverified by the folowing anal-ysis)due to approximate cancellation，and hence do not057003-4Ⅲ/-口当0p) 8 ( 窝p) l(pIlo∞-∞ 如 m 0Ⅲ，-日 op- 00 ( I是p) I10∞-Chinese Physics C Vo1.37，No.5(2013)0570033 53 O2 52 O1 5l O0 5O- 8 -6 -4 -2 0 2 4 6x/mmtune footprint with structure resonance(up to 5th order)纛 - t、 。i i：； ：-；豢974 97 6 97 8 98 O 98 2 984VFig.7. (color online)The dynamic aperture and frequency map obtained after tracking of 1024 turns with AT codefor BAPS ring lattice with only two families of chromaticity-correction sextupoles.The colors from blue to red，represent the stabilities of the particle motion，from stable to unstable。

cause significant distortions inIn the absence of magneticthe frequency map(FM)。

errors，the particles canpass through the integer resonances at x4.5 mm and"：1.5 mm without loss.However，the rule of thumb isthat the integer resonances are always dangerous in a re-alistic machine and cannot be passed.One can foreseethat when the integer resonances are more excited dueto magnetic field errors and misalignments.al the or-bits beyond the integer resonances will become unstable，leading to a signifcant shrinkage in DA。

Further study shows that the large detune terms areresponsible for particles quickly reaching the integer res-onances. In order to minimize the detune terms，andat the same time.to control the other nonlinear termsto an acceptable level，we use an additional four fam-ilies of chromaticity-correction sextupoles and six fam-ilies of harmonic sextupoles and octupoles for nonlin-ear optimization.To understand the combined eifects ofmulti-families of sextupoles and octupoles，one of the au-thors fXU G1 develops a theoretical analyzer based onLie Algebra and Hamiltonian dynamics(see Appendixfor a short introduction)，from which one can obtain an-alytical expressions of the detune fup to the 2nd order)，chromaticity(up to the 4th order)，and the resonancedriving terms(up to the 6th order)with respect to thesextupole and octupole strengths.It is worth mention-ing here that the resonance driving terms obtained bythe analyzer are somewhat diferent from those obtainedwith normal form method19l which，however，does notafect the efectiveness of the analyzer in measuring theresonance strengths.We then make multi-objective ge-netic optimization with NSGA.II by setting three objec。

tive functions fl，f2and3 to characterize the detunechromaticity and resonance terms，respectively.Fig.8presents the Pareto-optimal solutions obtained after 500generations with NSGA-II.Among the obtained opti-mal solutions，we select those providing good balance ofthree objectives and veriry them with numerical track-ing and FMA.Finaly we obtain one optimal set of thesextupole and octupole strengths.The nonlinear termsfor this solution are listed in Table 2 and compared withthose for the case with only two families of chromaticity-correction sextupoles.One can see that after optimiza-tion the detune terms are smaler，while the chromaticityand resonance terms do not increase a lot.The corre-sponding on-momentum DA and FM are shown in Fig.9。

In this case，particles reach the integer resonance at rel-atively large amplitudes，i.e.，x6 mm and y2.6 mm，which promises of-axis injection in the high-beta i0 mstraight section。

Table 2.Nonlinear terms with only two families of sextupole(TFS)or multi-families of sextupoles and octupoles(MFSO)for BAPS ring。

竺 !!!：!first order detune(dQz/dJx，dQ∞/dJu，dQ /dJy)second order detune(dQ2z/d ，dQ2y/d )horizontal chromaticities( ， ， ， )vertical chromaticities( ，晶，器， )sum of resonance driving terms(1st，2nd，3rd，4th，5th，6th)TFS M FSO(9.6，3.8，7.2)×10。

f1.1，1.3)x10(0，-1271，-7874，1.5x10 )(0，-442，-2.8x10 ，1.1×10 )f3×104，1×1010，3.8×105，1×1012，7×1016， 6×10231(5.8，0.6，0.6)×10f3.2，3.81×i0f0，-769，1.2×100，4×10 )(0，-755，-2.2×100，9×1O0)f3x104，7×10lo，3.9×105，2.5×1012，1×1016， 1×10261057003-51驺、 ， 2 3 4 5 6 ； 7 8 9 Chinese Physics C Vo1.37，No.5(2013)057003Further study shows that this optimal solution al-lows momentum acceptance of-3% which ensures longenough Touschek lifetime.up to 5 h. The emittancecan be further reduced to 16 pm by instaling in total60 m superconducting damping wigglers(3.3 T，3.1 cmperiod 1 in the straight sections.Another option is to in。

crease the circumference to 1 500 m.Studies show thatthe natural emittance can be reduced to 30 40 pm.Inthis case，the required total length of the damping wig-glers to achieve emittance of 10 pm level is expected tobe smaler.One interesting and important factor relatedto an ultralow design is to achieve a round beam.i.e。

with equivalent transverse emittances. 0ne of the au-thors(XU G1 has proposed a novel method of producinga locally-round beam by using solenoid and anti-solenoid，which however，wil be discussed in detai1 elsewhere I 2 I。

W ith this method，we can achieve an electron beam withemittance岛 8 pm in the presence of the dampingwigglers without introducing great perturbations to theglobal beam dynamics of the ring。

3 53 O2 52 O1.51 O0 5O - - ，· - f f ～ - 。- - ， d ∞ - - ㈣ n - . 1. -童 。

。 . 儡 嚣 i ～ 搿兽孽辱瓣瓣 - 著 .j未 墨 毒 兽 毒 ， l-。

∞ - - '- 日- ～ - ～ i i 二 二 - 1 00.80.6O 40.201Fig.8. Pareto-optima1 solutions obtained after 500generations with NSGA-lI.Three ob jectives areused with fl ，and h characterizing the de-tune，chromaticity and resonance terms，respec-tively.The star denotes the best solution found，which provides large dynamic aperture。

tune footprint with structure resonance(up to 5th order)：- l /·- - 10 --8 --5 --4 --2 0 2 4 6 8 lO 98 0 98 1 98 2 98 3 98 4 98 5x/mmFig.9. (color online)The dynamic aperture and frequency map obtained after tracking of 1024 turns with AT codefor BAPS ring lattice by using 12 families of sextupoles and 6 families of octupoles with strengths obtained byNSGA-II.The colors from blue to red，represent the stabilities of the particle motion，from stable to unstable。

4 ConlusionsDecreasing the natural emittance of an electron stor-age ring to several tens of picometers will lead to a greatchallenge to the linear optics design and nonlinear op-timization.In this paper，we present one lattice designfor the 5 GeV BAPS storage ring with natural emit。

tance of 75 pm.We adopt the modified-TME cel withsmal-aperture magnets as the unit cel，so as to realizeOcompact layout and control the required field strengthsof the chromaticity-correction sextupoles.We make ev-ery eight superperiods a quasi.3rd.order achromat so asto approximately cancel most of the 3rd-and 4th.orderresonances.Using a theoretical analyzer we obtain theexpressions of the nonlinear terms with respect to thesextupole and octupole strengths.The obtained depen-dency enables us to search for a set of solutions by opti-mization with NSGA-II.After numerical tracking andfrequency map analysis.we find one optima1 set of the057003-62 3 4 5 6 7 8 9 K卯 ∞ 如 加 m ∞4 寸： 寸： 3 3 3 3 3 3 3 3 3 3 32 3 4 5 6 7 8 9 m- - - - - - - - -I- 、，tChinese Physics C Vo1.37，No.5(2013)057003sextupole and octupole strengths which promises of-axisin 1 . ..iection and ong enough lifetime To summarize thetechniques used in the lattice design for BAPS help usdeal with the great challenge associated with the intrin-sic，strong nonlinearities，which would also be beneficialfor other ultralow emittance designs.In the end，we haveAppendix Ato state that except the sextupoles and octupoles men-tioned in this paper，there are other nonlinearity sourcesin a ring，such as magnetic field error and misalignment，sinusoidal field of the damping wigglers and undulators，whose efects on the beam dynamics should and will beconsidered in our next work。

Computation of analytical Hamiltonian in a storage with the jth final coordinate being analytic in X0ringA theoretical analyzer was developed on the M athematicaplatform to compute the one-turn Hamiltonian as a functionof more than 23 strength variables of sextupole，octupole，de-capole and dodecapoles.From the one-turn Hamiltonian.onecan get the analytical formula of the high order chromatici-ties，detune and resonance terms。

In the Lie Algebra framework[21]，the map for an accel-erator element with index i can be represented as exponentialoperator， ∑oX e-：LiHi：XIX ：Hi： )(A1) Owhere ( ，Pz，Y，Py) ，X1 and X0 refer to the initial andfinal canonical coordinates，Li is the length of the element，and Hi is the Hamiltonian describing the particle motion inthe element，which is in the form of22]Hihz 2x2/2-(1九 )、/ 二 二 -5/B#，(A2)with 等( 。 )hlxyk2下X3-3xy2 T3x2y-y3危3-xa--6x 2y2q-y4九34x3y2-44xya. 4-x5--110x3 y2- 5xy4九4-5xa-y- lO x2-y3y5 5-x6--1-5x-ay2 15x-2y-4-5xy4 5-6x-Sy--2 0x 3y3--6xy5，where hl/p， v/c)ki and hi are the regular and skewstrength of the multipoles.5 is momentum deviation relativeto the reference momentum and is taken as a parameter here。

From Eq.(A1)，we obtain the corresponding Taylor mapof the element ftruncated to 5th power of the canonical vari。

ables)through straightforward derivations，X1 ( )， (A3)X1(J)∑RjkXo(k)Rjk，Xo(k)Xo(1)kl∑ Rjkl Xo(k)Xo(OXoOn)Xo(n)Xo(o)lm n0In this way，the Taylor maps for al the elements can beobtained and they can be composed to the one-turn Taylormap in a similar form with Eq.(A3).During the calculation，the map is always truncated to the 5th power of the variables。

Note that because of the truncation.the Taylor map is notsymplectic。

Considering the one-turn Hamiltonian has the followingfclrm日efF∑hjkXo(j)Xo(k)·J∑ hjkz 。Xo(j)X。( )Jklm noxXo(1)Xo(m)Xo(n)Xo(o) (A4)From the correspondence of the Lie map and Taylor map，we can find a set of linear equations between the coeficientsof the one-turn Hamiltonian and the one-turn Taylor map。

Solving the linear equations order by order，one can obtainhjk，hjkl， ，hjktmn0。

We make canonical transformation from the coordinate-momentum variables(z，Pz，Y，P )to action-angle variables( ，西， ， )，using the generating function ofn(z， ，， )-等 -百y2Y tan(a) tan( ). (A5) The new Hamiltonian is in the formHeftlne H0( ， ， )H1( ， ， ， )G( ， ， ， ，5) (A6)From H0 one can derive the analytical expressions of thechromaticity and detune terms by taking the derivative ofH0 with respect to 6 and respectively；and taking thederivative of H1 one can obtain the expressions of the reso-nance coemcients。

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