第30卷第12期2006年12月
高能物理与核物理
HIGH ENERGY PHYSICS AND NUCLEAR PHYSICS
Vol.30,No.12
Dec.,2006
An Alternative Lattice Design for ILC Damping Ring*
理所当然是什么意思
SUN Yi-Peng1,2;1)GAO Jie2GUO Zhi-Yu1
1(Key Laboratory of Heavy Ion Physics,Ministry of Education&School of Physics,
Peking University,Beijing100871,China)
2(Institute of High Energy Physics,CAS,Beijing100049,China)
Abstract The ILC damping rings should provide beams with very low natural emittances for the linear c
ollider to reach the required luminosity,and at the same time,the damping rings also need to have a large acceptance to ensure good injection efficiency for high emittance,high energy spread beam from the positron source.Meeting the above requirements is a real challenge for the ILC damping ring lattice design.In order to reduce the cost for ILC damping rings,an alternative lattice different from the baline configuration design has been done with modified FODO cells，and the total quadrupole number has been reduced by half.The new lattice has been optimid to have good dynamic apertures.
Key words ILC,damping ring,FODO cell,lattice,dynamic aperture
1Introduction
The International Linear Collider(ILC)which is bad on superconducting RF acceleration technology requires the damping rings to provide beams with extremely small equilibrium emittances,and large acceptance[1].Another main design criteria for the damping ring comes from the requirement of provid-ing a long beam pul of1ms containing2820bunches (normal parameter t[2]),corresponding to an ap-proximately300km long bunch train.To keep the damping ring’s circumference reasonable,the bunch train has to be stored in a compresd mode with much smaller 雷锋摘抄
bunch spacing than in the linacs.Con-quently,each bunch has to be individually injected and ejected.The ring circumference is then deter-mined both by pul width of the injection and ex-traction kicker system and electron-cloud(ion)ef-fects.Bad on the studies of the various configu-ration options,the ILC damping ring Baline Con-figuration Design(BCD)decides that the positron damping ring should consist of two(roughly circular) rings of approximately6km circumference in a single tunnel and the electron damping ring should consist of a single6km ring,assuming that thefill pattern allows a sufficient gap for clearing ions[2,3],and the basic requirements of the positron damping ring de-sign which is shown in Table1.
Table1.Beam parameters for positron ring. injected emittanceγεx(y)0.01m·rad extracted emittanceγεx/γεy8×10−6/2×10−8m·rad damping time28ms
pul length1ms
number of bunches2820
particles per bunch2×1010
In the ILC damping ring baline design,a lattice of circumference of∼6km using TME arc cells has b
een ud[4].In this paper,with the aim to reduce the cost of the damping rings,a new lattice with mod-ified FODO arc cells is designed to be an alternative of the baline design.
In the following ctions,the linear lattice design, the chromaticity correction,and the optimisation of
Received21June2006
*Supported by National Natural Science Foundation of China(10525525) 1)E-mail:ypsun@mail.ihep.ac
改变自己英文1190—1195
第12期孙一鹏等：国际未来直线对撞机(ILC)阻尼环Lattice设计1191 the dynamic aperture will be prented.
2The linear lattice design
2.1The arc cell
There are120arc cells in all for a6km damping
ring,therefore each arc cell provides a bending angle
ofπ/60for the beam.The FODO arc cell length is
lected as38.9m and the pha advance per arc cell
is90◦/90◦for the horizontal and vertical betatron
motion,respectively.According to Eq.(1),the maxi-
mum and the minimum value of the beta functions of
the FODO arc cell is66.2m and11.4m,respectively.
From Eq.(2),the maximum and the minimum hori-
zontal dispersion function is1.37m and1.02m,respec-
tively which are too large to get a reasonable bunch
length of6mm.
β±=L P
1±sin
µ
2
sinµ
,(1)
D±=L Pφ
1±
1
2曹操几个儿子
sin
µ
2
4sin2
µ
2
,(2)
whereβ+andβ−are the maximum and minimum value of the beta functions,D+and D−are the max-imum and minimum value of the horizontal dispersion function,L P is the length of the cell,µis the pha advance of the cell,andφis the bending angle in one arc cell.To have smaller rms dispersion value,the length of the drifts that are between the quadrupoles is adjusted and the ultimate value of the two long drifts are13.7m and1.55m with the maximum and the rms horizontal dispersion function being1.15m and0.77m,
respectively.
Fig.1.Lattice functions in an arc cell.
The lattice functions in an arc cell are shown in Fig.1and the main parameters of the arc cell are shown in Table2.
Table2.Main parameters of the arc cell.
cell length38.9m
max/min beta functions66.1m/11.4m max/RMS dispersion 1.15m/0.77m pha advance x/y0.25/0.2
5
2.2The RF and wiggler cell
The vertical damping time can be written as fol-lows:
τy=
3C
r e cγ3I2
=
3C
r e cγ3(I2a+I2w)
,(3) where C is the circumference,r e is the classical elec-tron radius,c is the speed of light,γis the normalized beam energy,and I2is the cond synchrotron inte-gral over dipoles and
wigglers.
Fig.2.Lattice functions in RF(a)and wiggler cell(b).
The equilibrium horizontal emittance can be writ-ten as:
γεx=
C qγ3I5
J x I2
,(4) where C q≈3.84×10−13m,J x≈1for parate func-tion magnet,and I5is determined by the arc ctions.
1192高能物理与核物理(HEP&NP)第30卷
The wiggler ction is designed to provide enough damping.The average beta functions in this ction are about10m,and totally40such wiggler ctions have been ud for positron damping ring.The lat-tice functions in a RF cell and in a wiggler cell are shown in Fig.2.
2.3The long straight ctions
There are two long straight ctions which are designed to be dispersion free.One is designed for th
e injection and the extraction of the beam.The incoming electrons(or positrons)are injected into the empty RF buckets in the ring,rather than be-ing added to the contents of bunches already orbiting the ring.In addition,the emittance of an incoming positron bunch is very large.As a result,the injection and extraction optics must be designed to accommo-date the beam characteristics while still achieving high injection and extraction efficiencies.
The injection and extraction optics are designed to accommodate either of two different types of kick-ers that are studied at Fermilab:a puld kicker system with6ns ri time(and longer fall time)and a Fourier kicker,which is shown in Fig.
3.
Fig.3.The injection and extraction optics.
As for the other long straight ction,it should be stresd that it can be changed to give a proper pha advance which plays a dominant role in the optimization of the dynamic aperture.
2.4The dispersion suppressors
The inrtions match the dispersion function be-tween arc ctions and straight ctions.There are two kinds of dispersion suppressor inrtions.One kind matches dispersion without affecting the alpha and beta functions,and the other suppress the dis-persion function by only modifying the focal length of two quadrupoles,which modifies both the alpha and beta functions.
Here we do not u the usual method to make a dispersion suppressor,but inrt three quadrupoles into the arc cell to get a dispersion suppressor,as shown in Fig.
4.
Fig.4.Lattice functions in a dispersion suppressor.
2.5The whole ring
The layout of the ring has been designed with two-fold symmetry.There are totally ten long straight ctions.The RF cavity and the wigglers have been installed in eight of them.The other two long straight ctions are designed to accommodate various injec-tion and extraction schemes,also ud to adjust the pha advance.
The lattice functions for the whole ring are shown in Fig.5and the principal lattice parameters are list退票时间限制
ed in Table
3.
Fig.5.Lattice functions for the whole ring.
第12期孙一鹏等：国际未来直线对撞机(ILC)阻尼环Lattice设计1193
Table3.The principle lattice parameters. circumference/m6614.057 energy/GeV5
arc cell FODO
tune53.6368/48.7552 natural chromaticity−65.1/−65.3 momentum compaction(10−4) 4.11 transver damping time/ms25/25 Norm.natural emittance/(mm·mrad) 4.2
RF voltage/MV46.6 synchrotron tune0.093 synchrotron pha/(◦)169.2
RF frequency/MHz650
RF acceptance/(%) 2.68
natural bunch length/mm 5.96
natural energy spread/(10−3) 1.28最大挖掘机
3Chromaticity correction
3.1Chromaticity correction and DA
Two family xtupoles in the arc cell are ud to correct thefirst order chromaticity to zero.The pha advance per arc cell is90◦/90◦for the horizontal and vertical betatron motion,respectively.There are to-tally ten arc ctions and twelve arc cells in each arc ction.Therefore,the pha advance per ar
c c-tion is3πfor both the horizontal and vertical motion. This kind of cond order achromat helps to cancel all driving terms of the third order resonances generated by the xtupoles within each arc ction.
The pha advance between the arc ctions,that is to say in the straight ctions,also plays a dominant role in determining the dynamic aperture.Roughly, it is thought that the best pha advance in the straight ction is nearly a multiple of360◦,which makes the whole straight ction an identity trans-formation and therefore maximizes the symmetry of the ring and minimizes the number of excited high order resonances[5].
In our study it is found that not only the pha ad-vance in the straight ction but also between the last xtupole in one arc ction and thefirst xtupole in the next arc ction are very important.The straight ctions are optimized to give a reasonable horizontal pha advance which is near2π,and a vertical pha advance which is nearπ.
The position of the two family xtupoles in the arc cell has been adjusted to give a good chromaticity property and a larger dynamic aperture.The varia-tion of tune with momentum spread±1%is shown in Fig.6(a).After tracking for512turns,the dynamic aperture of both on-momentum and off-momentum particles is obtained,which is shown in Fig.6(b)[6]
国外节日
.
Fig. 6.The tune variation with momentum
spread±1%(a);the dynamic aperture with
momentum spread up to±2.5%(b).
It can be calculated from the injected beam’s emittance and the beta functions at the injection point of the ring that the horizontal and vertical bunch sizes of the injected beam are approximately 6.6mm and4.2mm,respectively.It can be en from Fig.5(b)that the dynamic aperture is ten times the vertical injected beam size and eight times the hori-zontal injected beam size for on momentum particles, and the lattice also gives a good dynamic aperture for offmomentum particles.
As the nonlinear wiggler effect is not included in the tracking process,we u analytical method to cal-culate the dynamic aperture of the damping ring with ideal nonlinear wigglers[7].One has the dynamic aper-ture limited by one wiggler as follows(vertical and horizontal respectively):
A N
w,y
(s)=
3βy(s)
β2
y,m
ρw
k y
√
L w
,(5)
1194高能物理与核物理(HEP&NP)第30卷
A N
w,x (s)=
βy(s)
βx(s)
(A2
N w,y
(s)−y2),(6)
whereβy(s)andβx(s)are the unperturbed beta func-tions,βy,m is the vertical beta function at the middle of the wiggler,ρw is the radius of curvature of the wiggler peak magneticfield B0,L w=N wλw is the wiggler length,k y=2π/λw,λw is the period length of the wiggler,and N w is the cell number of a wiggler.
Assuming that the dynamic aperture of the ring without the wiggler’s effects is A y and that there are M wigglers to be inrted inside the ring at different places,one has the total dynamic aperture expresd as:
A total,y(s)=
1
1
A y(s)2
+
M
j=1
1
A j,w,y(s)2
,(7)
where A j,w,y denotes the dynamic aperture limited by the j-th wiggler.
Using Eqs.(5),(6)and(7),the dynamic aper-ture of both on-momentum and off-momentum parti-cles which includes nonlinear wiggler effects is shown in Table4,where off-momentum particles’dynamic aperture is also calculated.A y0and A x0are the verti-cal and horizontal dynamic apertures with linear wig-gler model.A y and A x are the corresponding vertical and horizontal dynamic apertures including nonlinear wiggler effects.
Table4.DA using nonlinear wiggler.
∆P/P A y/mm A x/mm A y0/mm A x0/mm 02539.53850
−1%22.8363448
±2.5%2133.182836
It can be en from Table4that the dynamic aper-ture of the designed damping ring,where nonlinear wiggler effect has been taken into account,is greater than six times the injected beam size for on momen-tum particles andfive times the injected beam size for offmomentum particles.
3.2FMA analysis
Frequency map analysis(FMA)is introduced for the demonstration and understanding of the chaotic behavior of a dynamical system.The application to particle accelerator dynamics is done in the ca that the motion of a single particle in a storage ring is described in a surface of ction of the beam by a symplectic map of dimension4or6[8].
Here FMA is ud to optimize the working points and the dynamic aperture.The optimized result is shown in Fig.7where2500particles distributed at the range of ven times the injected bunch size are tracked for1024turns to do the frequency map anal-ysis
on the lattice.
Fig.7.FMA analysis for damping ring DA.
4Conclusions
To reduce the ILC damping ring cost,an alterna-tive lattice with modified FODO cells has been de-signed.The number of quadrupoles in the whole ring has been decread by half compared with the origi-nal ILC damping ring BCD design.The equilibrium emittances,bunch length,acceptance,dynamic aper-tures,and the damping time can fulfill the require-ments for the ILC
damping ring.The dynamic aper-ture is large enough including the nonlinear wiggler effects and it can be optimized further by adjusting the pha advance of the long straight ction.Fre-quency map analysis has been ud on this lattice during the optimization of the dynamic apertures by choosing the working point far away from the dan-gerous resonance line,and the result shows that the lattice can be further optimized.
The author would like to thank XU Gang,QIN Qing,JIAO Yi and other colleagues in the physics group for their helpful discussions.
第12期孙一鹏等：国际未来直线对撞机(ILC)阻尼环Lattice设计1195
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国际未来直线对撞机(ILC)阻尼环Lattice设计*
孙一鹏1,2;1)高杰2郭之虞1
1(北京大学重离子物理研究所北京100871)
2(中国科学院高能物理研究所北京100049)
摘要为了使国际未来直线对撞机达到高的亮度，对撞机的阻尼环必须要有很小的自然发射度.同时，阻尼环要有很大的接收度来接受来自正电子注入系统的大能散、大发射度的正电子束团.对于磁聚焦结构的设计来说，同时达到以上要求是一件很有挑战性的工作.为了降低ILC阻尼环的造价采用修正的FODO结构设计了可以满足ILC要求的阻尼环备用Lattice，并使四极磁铁的数量比起ILC阻尼环BCD设计减少了一半.
关键词国际未来直线对撞机(ILC)阻尼环FODO结构磁聚焦结构动力学孔径
2006–06–21收稿
腊肉炒什么最好吃
*国家自然科学基金(10525525)资助
1)E-mail:ypsun@mail.ihep.ac
