lattice design c.c. kuo 郭錦城 july 27 ~ august 6, 2014 安徽省黃山市休寧 ocpa accelerator...

Lattice Design

C.C. Kuo郭錦城

July 27 ~ August 6, 2014安徽省黃山市休寧

OCPA Accelerator School, 2014, CCKuo- 1

outline

• Introduction• Linear beam dynamics• Linear lattice • Nonlinear beam dynamics• Example• Imperfection, correction and lifetime


Accelerator Lattice• (Magnet) Lattice: The arrangement of the accelerator elements

(usually magnets) along beam path for guiding or focusing charged particles is called “Magnet Lattice” or “Lattice”.

• Regularity: The arrangement can be irregular array or repetitive regular array of magnets. A transfer line from one accelerator to another is usually irregular.

• Periodicity: The repetitive regular array is called periodic lattice. Usually the circular accelerator lattice is in a periodic form.

• Symmetry: In a circular accelerator, the periodic lattice can be symmetric. Usually, the lattice is constructed from cells and then super-periods. A number of super-periods then complete a ring.

• Design goal: The goal of lattice design is to obtain simple, reliable, flexible, and high performance accelerators that meet users’ request.


BEPCII －双环高亮度正负电子对撞机

IR 超导磁铁

超导高频腔

馬力第五届 OCPA 加速器学校

Collider: 1.89GeVSR: 2.5GeV

e+e-

e+e-


A Lattice with FODO Arc and Doublet straight

王生 OCPA 2010散裂中子源 CSNSRCS


Lattice consists of FODO arc (with missing gap) and doublet dispersion free straight section

Arcs: 3.5 FODO cells, 315 degrees phase advance

Straights: doublet, 6.5×2+9.3 m long drifts at each straight

Gap of Dipole : 175mm Max. Aper. of Quadrupoles: 308mm

A Lattice with Triplet CellsLattice consists of 16 triplet cells, with a gap in the middle of arc and dispersion free straight section .

Arcs: Triplet cells as achromatic insertion Straights: Triplet, 3.85×2+11 m long drifts at

each straight Gap of Dipole : 160mm Max. Aper. of Quadrupoles: 265mm

王生 OCPA 2010散裂中子源 CSNSRCS


SSRF150 MeV Linac3.5 GeV Booster3.5 GeV Storage Ring

趙振堂等OCPA Accelerator School, 2014, CCKuo- 7

20 DBA cells

Hefei Storage RingHLS

-5

0

5

10

15

20

250 2 4 6 8 10 12 14 16

SD

Theoretical Curve of x

Measured Value of x

Theoretical Curve of y

Measured Value of yy

x

SDQ6

Q5

Length(m)

Q8

Q7B3

SFSF

B2

Q4

Q3B1Q2

Q1

x,y(

m)

劉祖平 . OCPA2004_satellite meeting

800 MeV66.13m


The Hefei Light Source Upgrade

Q4Q3Q2Q1 B

S1 S3S2 S4

HLS Upgrading

e<40 nm·rad I>300 mA

E=800MeV


(2012 OCPA School 張闖 )

NSRRC TLS

偏轉磁鐵 Bending Magnet

四極磁鐵 Quadrupole Magnet

注射脈衝磁鐵 Pulsed Injection Magnet

六極磁鐵 Sextupole Magnet

高頻系統 RF Cavity

儲存環 Storage Ring(1.51 GeV)

增能環 Booster Ring(1.51 GeV)

光束線 Beamline

線型加速器LINAC

傳輸線 Transport Line

插件磁鐵 Insertion Device

6 TBA

FODO cell in TLS boosterOCPA Accelerator School, 2014, CCKuo- 10

TPSEnergy : 3 GeV

Beam current: 500 mAEmittance: 1.6 nm-rad.

Straight Section: 7 m (18); 12 m (6)Lattice structure: Double-Bend

Circumference: 518.4 mRF: 500 MHz

Linac energy : 150 MeV Booster energy: 3 GeV

Booster circumference: 496.8 mBooster emittance: 10 nm-rad

Lattice structure: MBARepetition rate: 3 Hz


AccumulatorCooler

SynchrotronFast extractionSlow extraction

External target Internal target

12 Tm12GeV—C6+, 120GeV—U72+

CSRm Layout

夏佳文OCPA-School-08

CSRm CSRe

Circumference (m) 161.0014 128.8011

Average radius (m) 8RSSC=34RSFC=25.62416 4/5RCSRm=20.499328

Geometry Race-track Race-track

Max. energy (MeV/u) 2800 (p)

1100 (C6+)

500 (U72+)

2000 (p)

750 (C6+)

500 (U92+)

B (Tm) 0.81/12.05 0.50/9.40

B(T) 0.10/1.60 0.08/1.60

Ramping rate (T/s) 0.05 ~ 0.4 -0.1 ~ -0.2

Repeating circle (s) ~17 (~10s for Accumulation )

Acceptance Fast-extraction mode Normal mode

A h ( mm-mrad) 200 (p/p = 0.3 %) 150 (p/p =0.5%)

A v ( mm-mrad) 40 75

p/p (%) 1.4 (h= 50 mm-mrad)

2.6 (h= 10 mm-mrad)

OCPA2012H1-Hadron Accelerator Complex


Lattice Design Procedure-(1)

1) For a modern accelerator, lattice design work usually takes some years to finalize the design parameters. It is an iterative process, involving users, funding, accelerator physics, accelerator subsystems, civil engineering, etc.

2) It starts from major parameters such as energy, size, etc. 3) Then linear lattice is constructed based on the building

blocks. Linear lattice should fulfill accelerator physics criteria and provide global quantities such as circumference, emittance, betatron tunes, magnet strengths, and some other machine parameters.



4) Design codes such as MAD, OPA, BETA, Tracy, Elegant, AT, BeamOptics,…. are used for the matching of lattice functions and parameters calculations.

5) Usually, a design with periodic cells is needed in a circular machine. The cell can be FODO, Double Bend Achromat (DBA), Triple Bend Achromat (TBA), Quadruple Bend Achromat (QBA), Multi-Bend Achromat (MBA or nBA) or some combination types.

① Combined function or separated function magnets are selected.② Maximum magnetic field strengths are constrained. (room-

temperature or superconducting magnets, bore radius or chamber profile, etc.)

③ Using matching subroutines to get desired machine functions and parameters.



6) To get stable solution of the off-momentum particle, we need to put sextupole magnets and RF cavities in the lattice beam line. Such nonlinear elements induce nonlinear beam dynamics and the dynamic acceptances in the transverse and longitudinal planes need to be carefully studied in order to get sufficient acceptances. (for long beam current lifetime and high injection efficiency)

7) For the modern high performance machines, strong sextupole fields to correct high chromaticity will have large impact on the nonlinear beam dynamics and it is the most challenging and laborious work at this stage.



8) In the real machines, there are always imperfections in the accelerator elements. Therefore, one needs to consider engineering/alignment limitations or errors, vibrations, etc. Correction schemes such as orbit correction, coupling correction, etc., need to be developed. (dipole correctors, skew quadrupoles, beam position monitors, etc)

9) Make sure long enough beam current lifetime, e.g., Touschek lifetime, in the real machine including insertion devices, etc.

10) To achieve a successful accelerator, we need to consider not only lattice design but many issues, which might be covered in this school.


Lattice Design Process for a Light Source

1) User requirements2) Lattice type (FODO, DBA, TBA, QBA, MBA)3) Linear lattice 4) Nonlinear dynamic aperture tracking5) Longitudinal dynamics6) Error tolerance analysis7) Satisfying user requirements (if NO, go back to

step 2)8) ID effect, lifetime, COD, coupling, orbit stability,

instabilities, injection, etc.


Circular machine need dipole magnets

Lorentz force = centrifugal force

2

0vmFevBF centrL

For relativistic particle

• Total required dipole magnet length = 2pr• Synchrotron radiation loss per turn for electron ~ 8.85x10-5β3E4/r

[GeV] and for proton ~ 7.78x10-18β3E4/ r [GeV] • Usually for the high energy particle with constrained ring

circumference, maximum energy is limited by synchrotron radiation power compensation for electron/positron, while proton /heavy ion are limited by dipole magnetic field strength.

]/[

][3.0]/1[

1

uGeVE

TBm

B

e

p

e

vm 0

GeVin ,/)(10

)/(][]/1[

19

2

EceVE

mVsBe

p

TBem

TLS: 1.5 GeV, B=1.43T, r=3.49m, Circumference=120m; LHC: 7000GeV, B=8.3T, r=2.53km, Circumference=27km

]/[

][3.0]/1[

1ion for :

uGeVE

TB

A

ZmNote


• Need quadrupoles to get restoring force for particle deviating from the ideal (design) orbit.

• Quadrupole field can be combined into a dipole magnet and this is called combined-function magnet.

• Need sextupoles to provide chromaticity corrections.• Sextupole also can be combined into dipole and/or quadrupole

magnet.• Dipole correctors and skew quadrupoles are used for orbit and

coupling control. These elements can be separated or combined into other magnets.

• Octupoles can be used to get tune spread as a function of betatron oscillation amplitude and help reduce beam instabilities.

• Higher-order magnetic multi-pole fields are usually unwanted and inevitable in real magnets. These fields need to be minimized in design and manufacturing to avoid the shrinkage of the “dynamic aperture”—the maximum allowed particle motion in transverse planes. It is usually around 100 ppm in the good field region.

Basic magnet elements


Magnetic field expressionFor transverse fields in Frenet-Serret (curvilinear ) coordinate system:

0:focusing define nk

)( ,2sext skew :

2 ),(sext :

, quad skew :

, quad :

dipole vert.:

dipole :

2220202

2022

202

10101

10101

000

000

zxaBBxzaBBa

xzbBBzxbBBb

xaBBzaBBa

zbBBxbBBb

aBBa

bBBb

xz

xz

xz

xz

x

z

ower)negative(l(upper), charge positive:

|!

1 ,|

!

1

,))((1

)(1

00

00

00

zxnx

n

nzxnz

n

n

n

nnnxz

x

B

nBa

x

B

nBb

jzxjabjBBB

nz

n

nn

n

nzz x

B

Bkx

n

kBxB

000

1,

!|)(


Betatron equation of motion (negative charge particle) :

Frenet-Serret (curvilinear ) coordinate system, negative charge case:

0

20

22)(

ˆˆ

/

B

Bvz

B

Bvxx

zBxBB

BvedtpdF

xs

zs

zx

20

0

20

02

)1(

)1(

x

p

p

B

Bz

x

p

p

B

Bxx

x

z

2

22 , ,

ds

xdxxvxvts


Hill’s equation for on-momentum particle :

Frenet-Serret (curvilinear ) coordinate system, negative charge :

)(

)11

(

000

000

B

BxkBBB

B

Bx

x

B

BBB

zxz

zzz

,1

)( ,11

)(

)(

)(

002

0

0

z

B

BsK

x

B

BsK

B

BzsKz

B

BxsKx

xz

zx

xz

zx

20

0

20

02

)1(

)1(

x

p

p

B

Bz

x

p

p

B

Bxx

x

z


Frenet-Serret (curvilinear ) coordinate system, negative charge :

)(

)11

(

000

000

B

BxkBBB

B

Bx

x

B

BBB

zxz

zzz

Hill’s equation linear motion for off-momentum particle (assuming Bz,x=0) :

20

0

20

02

)1(

)1(

x

p

p

B

Bz

x

p

p

B

Bxx

x

z

002

,1

)(,)1(

)1

)(

)1(

1(

p

p

x

B

Bskx

skx z

xx

0))()((then xsKsKx xx )(1

))()((

oDsKsKD xx

)()](2

[),(1

)( where 222

oskKsksK xxxx

Neglecting Chromatic perturbation term )(sK x

)(

1)()()(

0)()()(

ssDsKsD

sxsKsx

x

x

)()(Let sDsxx


Main magnets

h

NIB 0

2

01

11

2

,

R

NIB

xBBzBB zx

3

02

22

22

/6

)(2

1,

RNIB

zxBBxzBB zx

dipole

quadrupole

sextupole

B field limits (example): Dipole < 1.5TQuad poletip < 0.7TSext poletip < 0.4T


Matrix formalism in linear beam dynamics:

)(

)(),(

)(

)(

0

00 sy

syssM

sy

sy

(1) Focusing quadrupole

||

1lim)length focal(,0 ,

1/1

01

,0 ,cossin

sin1

cos),(

0

00

Kf

f

ssKKKK

KK

KssM

(2) Defocusing quadrupole

0for , 1/1

01

,0 ,

||cosh||sinh||

||sinh||

1||cosh

),( 00

f

ssK

KKK

KK

KssM

(3) Drift space K=0

00 ,10

1),( ssssM


(4) pure sector dipole:

,cossin1

sincos),( 0

ssM x

10

1),( 0

ssM z

/ smallfor 10

1),( 0

ssM x

non-deflecting plane (ignoring fringe field):

/


(5) pure rectangular dipole due to wedge in both ends:

1/1

01cossin

1sincos

1/1

01),( 0

xxx ff

ssM

)2

tan(11

where,1/1

01

10

1

1/1

01),( 0

zzz

z fffssM

In deflecting plane:

In non-deflecting plane (neglecting fringe field):

)2

tan(11

where10

sin1),( 0

xx f

ssM

if cossin1

sincos

12

1

),(

2

0

zzz

zz

fff

fssM


Beam dynamics in transport line

• In open transport lines the phase space can be transferred using transfer matrix piecewise. We need initial condition.

• In terms of Courant-Snyder parameters, there are relations between initial and final points along the beam path.

)sin(coscossin1

sin)sin(cos

),(

2

2

1

21

21

21

21

211

1

2

12

ssM

1

12231 '),(),()...,(

's

nn

snx

xssMssMssM

x

x

1

12

2'

),('

ssx

xssM

x

x


Courant –Snyder Parameters in periodic cells

2221

1211MMM

MM

sincos

sincossin

sinsincos M JI

22 1,0)Trace( ,

IJJJ

sincos seigenvalue ie i

]2/)[(cos 22111 MM sin/12M sin2/)( 2211 MM

where transformation is for one period or one turn.

Using similarity transformation for any beam line:1

121122 ),(M)(M),(M)(M ssssss

1

1

1

2222221

221

2212211222112111

2122211

211

2

2

2

2

2

MMMM

MMMMMMMM

MMMM

kJkIJIM kk sincos)sincos(


Floquet transformation

)()( ,0)( sKLsKysKy Hill’s equation:

)()()( siesawsy

)()( swLsw

01

)(3

wwsKw 2

1

w

)()( sLs

Transformation matrix in one period:

sincossin

sinsincossincossin

)(1sinsincos

2

2

2

M

ww

w

wwwww

)(

)(1)( ,

2

(s)-(s) ),()(

22

s

sssws


Ls

s

i

s

dsdsseasy

0

0 )(,

)()(,)(

0

)sin(coscossin1

sin)sin(cos

),(M

22

1

21

21

21

21

2111

2

12

ss

1

1

1

1

22

2

2

12

01

cossin

sincos10

),(M

ss


Stability of FODO Cell and Optimum Phase Advance Per Cell

FODO cell: ,/1 ,2

1

2

1 kfQFLQDLQF length quad half

)/)(/1(2/1 , with 21/1

)1(2212*

2

2*

2

2

fLfLffff

f

Lf

f

LL

f

L

M dfFODO

LfLfLxx

1 / with

1

)1(QF) of middle(

2

1

)1(QD) of middle(

2

Lxx

Maximum betatron function can be minimized with optimum phase advance per FODO cell:

345.760 0

d

d x


L1:DRIFT,L=1L2:DRIFT,L=1QFH :QUADRUPOLE,L=.5/2, K1=0.5QDH :QUADRUPOLE,L=.5/2, K1=-0.5BD :SBEND,L=3,ANGLE=TWOPI/96HSUP :LINE=(QFH,L1,BD,L2,QDH)FSUP :LINE=(HSUP,-HSUP)FODO :LINE=(2*FSUP)RING :LINE=(48*FSUP,)

86.865075.5),5.*25.0/(1

0,sin

)sin1(2,sin ,21cos

sincossin

sinsincos

)/)(/1(2/1 , with 21/1

)1(221

1/1

01

10

1

1/2

01

10

1

1/1

01

length quad half,/1 ,2

1

2

1 :cell half FODO

22

2

2*

2

2*

2

2

Lf

Lf

L

f

L

fLfLffff

f

Lf

f

LL

f

L

M

f

L

f

L

fM

kfQFLQDLQF

df

Minimizing beta in one planephase advance of 76.3° each FODO cell

Minimizing beta in both planes phase advance of 90° each FODO cell


)(2

1 TuneBetatron

,, s

ds

zxzx

)cos()( :solution General 0 asy

Normalized coordinate:

s dw

)(

1,

y

)(cos)( 0 vaw

022

2

wd

wd

A simple harmonic oscillation in normalized coordinate.

Courant-Snyder Invariant and Emittance:

222

22

a2emittance''2)',(

))'((1

)',(

JyyyyyyC

yyyyyC

2

2

))('()()(' :divergence Beam

))(()()(:size Beam

sDss

sDss


/xw

)'(11

xxd

dw

a

M. Sands SLAC-121


Off-momentum orbit

1

)(

DsKD x

periodic are )(),(),(')(' ),()( ssKsDLsDsDLsD

1

)(

)(

1

)(

)(

10

),(

1

)(

)(

0

0

0

0

0

sD

sD

MsD

sDdssM

sD

sD

,0 if sin

1

)cos1(1

x

x

x

xx K

sKK

sKK

d

0 if

||sinh||

1

)||cosh1(||

1

x

x

x

xx K

sKK

sKK

d

x

B

Bskx

skx z

xx

1

)(,)1(

)1

)(

)1(

1(

2 )()(Let sDsxx

0))((then xKsKx xx

)(1

))((

oDKsKD xx )()](2

[),(1

)( where 222

oskKsksK xxxx

To the lowest order in


Off-momentum orbit

matrix 33

1

)(

)(

1

)(

)(

0

0

sD

sD

MsD

sD

100

sincossin)/1(

)cos1(sincos

xM

100

10

2/1

0

xM

1002

tan210

)cos1(sin1

xM

pure sector dipole:

pure rectangular dipole:

100

10

2/1

0

xM

100

011

001

/f-M x

100

011

001

/fM z thin lens quadrupole:


2

1)( where

sK x

Dispersion for periodic latticeFor periodic lattice, the closed-orbit condition:

11001232221

131211

D

D

MMM

MMM

D

D

2211

23112113

2211

23122213

2

)1(

2

)1(

MM

MMMMD

MM

MMMMD

2313, Solving MM

100

)sincos1( sinsincossin

sin)sincos1(sinsincos

M DD

DD


Dispersion in a FODO CellFODO cell—thin lens approximation:

100

)2/1)(/(/1/

)2/(/1

100

01/1

001

100

/10

)2/(1

100

01/1

0012

22

2/1 fLLfLfL

LLfL

fL

LL

fM FODO

1

0

1

0 2/1

- D

M

D

FODO 0D

)2

1(

)2

1(

2

2

f

LfD

f

LfD


length cell halfL length, quad half,/1 kf

Summary in a FODO cell


)sin1(sin

),sin1(sin

sin2

),sin1(sin

2 ),sin1(

sin

2

length) quad fullfor :Note(

221

22D22

1

22F

222

QDQF

LD

LD

f

LLL

fff

xx

xDxF

21

2

21

21

2

21

1

2

2

2

1

2

2

2

2QD1QF

21cos

21cos

)2(sin

1 ),2(

sin

1

)2(sin

1 ),2(

sin

1

strength) quad fullfor :note( ,

ff

L

f

L

f

L

ff

L

f

L

f

L

f

LL

f

LL

f

LL

f

LL

ffff

y

x

yyD

xyF

yxD

xxF

Integral Representation of Dispersion FunctionSame as dipole field error representation, we can substitute

000

)( ,1

p

psDx

p

p

B

Bco

action dispersion])'([1

2

1)',(

2

1

|))()(|cos()(

sin2

)()(

22

DDDDDHJ

dtstts

sD

xxx

d

xxx

Cs

s

x

x

x

Jd is constant in the region without dipole.


Achromatic Lattice

• In principle, dispersion can be suppressed by one focusing quadrupole and one bending magnet, if no strength and space constraints.

• With one focusing quad in the middle between two dipoles, one can get achromat condition. Due to mirror symmetry of the lattice w.r.t. to the middle quad, one can find, by simple matrix manipulation, D’ at quad center should be zero to get achromat solution.

• Beam line in between two bend is called arc section. Outside arc section, we can match dispersion to zero. This is so called double bend achromat (DBA) structure.

• We need quads outside arc section to match the betatron functions, tunes, etc.

• Similarly, one can design triple bend achromat (TBA), quadruple bend achromat (QBA), and multi-bend achromat (MBA or nBA) structure.

• For FODO cells structure, dispersion suppression section at both ends of the standard cells.


DBA

1

0

0

100

10

2/1

100

010

01

100

01)2/(1

001

1

01

LLL

f

Dc

)2

1( ),

2

1(

2

111 LLDLLf c

Where f is the focal length of half quad, θ and L are the bending angle and dipole length, L1 is the distance between end of dipole to center of quad.

Consider a simple DBA cell with a single quadrupole in the middle. In thin-lens approximation, the dispersion matching condition

It shows the quad strength becomes smaller with longer distance and the dispersion at quad center becomes higher with longer distance and larger bend angle.

By splitting quad into two pieces and being moved away from the center symmetrically, we can reduce the dispersion function and also quad strength.


DBA-1

NSLS-Xray ring2.5GeV, 8-fold

NSLS-VUV ring0.7GeV, 4-fold

• R. Chasman and K. Green proposed in 1975.

• One quad between two bends in the arc.

• Also called Chasman-Green lattice.• Doublet or triplet quads in straights for

beta function control.• Less flexibility.• Usually emittance is high. • Chromaticity is not so high, sextupole

scheme is simple.• Note: both rings with combined-function

dipoles.


DBA-2• Expanded C-G lattice with 4 quads in

between 2 bends as shown for ESRF lattice. (6GeV 32 cells, 6.7 nm-rad)

• Chromaticity is high (-131, -31), need more families of sextrupoles for nonlinear beam dynamics corrections

• Quadrupole triplets in straights for beta function control.

• Emittance reduction by breaking achromat.

• Upgrade program for lower emittance and increase of straight length for more IDs.

• APS also use same scheme and upgrade program is in progress.

ESRF

APS 7 GeV, 40-fold ESRF upgrade


DBA-3

• Expanded C-G lattice in Elettra (2GeV, 260m, 7 nm-rad)

• Three quads in between 2 bends• Can optimize emittance to close to

theoretical values.• But the drift space in the arc is too

long. ID length is limited.• Chromaticity (-41, -13). Use

harmonic sextupoles.


DBA-4

SSRF 4-fold 2.86 nm-rad at 3.5 GeV• Components compacted C-G

lattice is the trend in the modern light sources.

• Breaking achromat (dispersion in the ID straights) is usually adopted. Emittance is extremely low.

• Need harmonic sextupole families to optimize beam dynamics effects.

• ID lengths are varied for different experimental purpose. Long straight can accommodate more IDs in one straight.

TPSHalf super-period


DBA-5

SOLEIL 2.75 GeV 4-fold

ALBA 3 GeV 4-fold

• Increase useful straights in between 2 bends in the arcs.

• Straights can be three types.


TBA

SLS 4.5 nmm 2.4 GeV

TLS

ALS

• Triple-bend achromat (TBA) structure with better flexibility compared with the C-G lattice.

• ALS, TLS, PLS-I, SLS, HLS… adopted this type.

• In TLS, ALS, PLS-I,… no harmonic sextupoles needed.

• Lower the emittance in these lattices need add harmonic sextupoles for nonlinear optimization.

• SLS has 8°—14°—8° bend angle in the TBA structure and lower the emittance.

• SLS has three types of straights.• SLS uses harmonic sextupoles.


QBA

B1=8.823°B2=6.176°

B1B2 B2B1

TPS QBA518.4mTune x:26.27Tune y: 13.30Emittance:3.2 nmNat. chromaticity x:-65.8Y: -27.3


FODO Lattice(missing dipole cases)

TLS booster 119 nm-rad at 1.3 GeV

OCPA Accelerator School, 2014, CCKuo- 51f

Focusing and defocusing strengths are different

Dispersion suppression in FODO type structure• For FODO cells structure, dispersion suppression section at

both ends of the standard cells.• Usually, the solution can be with one bend (same or smaller

angle) and proper phase advance. It can be with two bends with smaller angle. The figure shown below is the case with two bends and same quadrupole strengths and relations are as :

Bend angle and phase advance in normal half FODO cell

: and

21

21

22

)sin4

1(

)sin4

11(


10084

122

142

412

212

21

100

012

1001

100

102

1

100

011

001

100

102

1

100

012

1001

2

2

2

2

3

2

2

2

2

f

L

f

L

f

L

f

L

f

L

f

LL

f

LL

f

L

M

f

LL

f

LL

fM

1

0

1

0

0

21

D

MM

L

f

LLD ,

2 )2/sin( ,

)2/(sin

))2/sin(21

1(

2

2, , )

sin4

1(

)sin4

11(

21

21

22


Dispersion suppression in FODO type structure

Dispersion suppression in FODO cell

f

L

2sin,2, ,

)sin4

1(

)sin4

11(

21

21

22

θ1 θ1 θ2 θθ2 θ


Dispersion suppression

Same dipole Separated functions dipole


MBA:PEP-X (SLAC)

PEP-X, 6GeV, 11pm-rad 2.2 km


MBA: MAX-IV (Sweden)

MAX-IV, 3GeV, 0.3 nm-rad20-cell, 528m


MBA: Sirius (Brazil)

Sirius (Brazil) 5BA with superbend

3 GeV, 518 m, 0.28 nm

BAPS 20*13BA


徐刚

5GeV, 1272m, 36pm-rad (bare) 10/10 pm-rad with damping wiggler/full coupling

60

ESRF existing (DBA) cell• Ex = 4 nmrad, 7GeV

• tunes (36.44,13.39)• nat. chromaticity (-130, -58)

Proposed HMB cell• Ex = 160 pmrad, 7GeV

• tunes (75.60, 27.60)• nat. chromaticity (-97, -79)

Andrea Franchi ESRF

ESRF(France) Hybrid Multi-Bend (HMB) lattice

• multi-bend for lower emittance• dispersion bump for efficient

chromaticity correction => “weak” sextupoles (<0.6kT/m)

• no need of “large” dispersion on the three inner dipoles => small Hx and Ex

Momentum compaction

i

iic DC

dsD

CC

Cds

DC

111

,

• For small bend angle and large circumference DBA lattice, the 1st-order momentum compactor can be around 10-4.

• Higher-order momentum compaction will play important role in the longitudinal motion if the 1st-order is small.

• One can design very low alpha or negative alpha lattice by properly adjusting dispersion in the dipole so that the integration is small.

• Inverting dipole bend also can get low alpha.

22

2

,

1

)2/(sin2

)( :cell FODO

x

DFFODOc L

DD

ring of radius average:6

negative6

1

2

length dipole :L

moduleDBA half:

)26

1(

:cellDBA

2

,

00

003

RR

D

L

D

L

D

L

D

L

DBAc

m

mc


Low alpha lattice

651

2321

10~10

11

dsD

CC

Cc

Low alpha lattice can have shorter bunch length (a few ps) for time-resolved experiments or THz source

Need sextupole/octupole to control a2 and a3 to near zero

NewSUBARU invert bend

TPS High emittance

Low emittance


C.C. Kuo ipac2013

Chromaticity Effects

For off-momentum particle, gradient errors:

4

1-

4

1

4

1-

4

1

)(

,)(

dskdsK

dskdsK

skK

skK

zzzzz

xxxxx

zz

xx

Chromaticity:

Natural chromaticity: kdsnat 4

1-

FODO cell (N cells):

)/fβN(βFODOnat minmax4

1-

)0/(

)0/(

)0/(

0

0

0

ppf

ppf

ppf


)()](2

[

),(1

)(

0))()((

22

2

oskK

sksK

xsKsKx

xx

xx

xx

Sextupoles

Need focusing strength proportional to displacement

Sextupole field 0/ 0 pp

0/ 0 pp

x

squadrupole

sextupole

sextupole

D>0

Focal length

xxk )(

022

20

2

0

22

0

2

0

|/ , ),(2

zxz

xz xBBxzB

B

B

Bzx

B

B

B

B


Chromaticity correction

For a large (negative) natural chromaticity, the tune shift is large with typical energy offset and can cause beam storage lifetime reduction induced by transverse resonances. We need sextupole magnets installed in the storage ring to increase the focusing strength for larger energy beam. Two types of sextupoles, i.e., focusing and defocusing sextupoles, are properly situated. To avoid head-tail instability, a slightly positive chromaticity is preferred.

0

2

0

where][ andB

BSzSD

B

Bx

SDΔKSDΔK zx ,

][

4

1-

][ 4

1-

dsDSDSk

dsDSDSk

DDFFzzz

DDFFxxx

022

20

2

0

22

0

2

0

|/ , ),(2

zxz

xz xBBxzB

B

B

Bzx

B

B

B

B

xSDB

BDxx z ][ oforder first the to,Let

0


,1

)( ,11

)(

)(

)(

002

0

0

z

B

BsK

x

B

BsK

B

BzsKz

B

BxsKx

xz

zx

xz

zx

FODO cell

FF

D

F

DF

DF

DfS

DfS

ff

ffDS

ffDS

LD

LD

f

LLL

2D

1F

21

221

22D

221

22F

221

2222

1

22

222

1 ,

1

If

sin1

sin

2

11

,sin1

sin

2

11

)sin1(sin

),sin1(sin

sin2

),sin1(sin

2 ),sin1(

sin

2


Design Rules for sextupole scheme• To minimize chromatic sextupoles strengths, it should be located

near quadrupoles at least for FODO cells, where βxDx and βzDx are maxima.

• A large ratio of βx/βz for the focusing and βz/βx for the defousing sextupoles are needed.

• The families of sextupoles should be arranged to minimize resonance strengths.

• For strong focusing (low emittance) lattice, strong chromatic sextupole fields are needed to correct chromaticity and such strong nonlinear fields can induce strong nonlinear chromatic effects as well as geometric aberrations. Phase cancellations can help reduce nonlinear chromatic aberrations. Harmonic sextupoles are installed at proper positions to further reduce resonance strengths of geometric aberrations.


Nonlinear Effects of Chromatic Sextupoles

, ,)()(

))(()( 22

DxxxzsSzsKz

zxsSxsKx

z

x

xx

DDw

xwwww ~

, , Normalized coordinate:

,~

2

1~/

~~ 22/520

220

2/120

20 DSDkDD

22/520

220

220

20 2

1 SwSDwkwww

22/520

220

2/320

20 2

1/ Swkwww


• The first two terms cancel the chromatic aberrations to first order locally. However, in real machine, local cancellation might be not perfect and beta-beat and higher harmonics of chromatic terms in the circular machine still exist .

• The third term is betatron amplitude dependent perturbation and called geometric aberrations. -I transformation scheme is to put sextupoles in (2n+1) π phase advance apart in a periodic lattice and this scheme can compensate the effects.

22/520

220

220

20 2

1 SwSDwkwww


Chromatic Aberrations

If chromaticity is corrected locally, there is no induced beta-beat and no half integer resonances. However, in reality, we might need several families of sextupoles to get small beta-beat

Colliders need to have local chromatic aberration correction with sextuples at interaction point to ensure small beta-beat at IP and keep beam size as small as possible (high luminosity).

)()( SDksK Chromatic gradient errors with sextupoles

Beta-beat:


dttstDtStkt |])()(|(2cos[))()()()((sin2

(s)0

0

-I transformation

1000

0100

0010

0001

IM xySyyxSx ),(2

1 22

000

0

020

20

0

0

0

0

0

0

00

0

0

0

0

1

)(2

1

100

0100

02

11

2

10001

1000

0100

0010

0001

)(

yyxS

y

xyxS

x

y

y

x

x

xS

ySxS

y

y

x

x

MI

y

y

x

x

s

Cancellation of geometric aberration after a complete transformation matrix through one unit.


Adding second sextupole:

advance phase )12(

0

0

0

0

0

0

0

0

n

y

y

x

x

y

y

x

x

Real machine

For periodic lattice, compensation can be across one or several cells. Thick-lens sextupoles together with errors in real machine such as COD, quad gradient errors, and working point selection always lead some limitation in its effectiveness.

Selection of working tune is different from above.Diamond Light Sourcenominal lattice working tune (27.23, 12.36)

Diamond phase cancellation scheme. Horizontal tune ~ 29.12 due to long straight perturbation


Nonlinear Hamiltonian

s

yyyyyy

s

xxxxxx

dsss

dsss

0

0

)( ,)(

)( ,)(

),( ),,( yyxx JJ are pairs of conjugate phase-space


]cos33)[cos(12

2

)]2cos()2cos(cos2)[(4

2

))(cos(2

)3(2

)( ),(

2

1

),,(

)3(2

)()(

2

1),,,(

2/32/3

2/12/13

233

22220

30

232222

xxxx

yxyxxyxyx

xxxx

yyxx

yyxxyx

sSJ

sSJJV

sJx

xyxsS

VyKpxKpH

syxVHH

xyxsS

yKpxKpypxpH

Nonlinear Hamiltonian

Periodic in s, Fourier expansion:

dsesSeG xx sjx

j ])3()(3[2/3,0,3 )(

24

2


...

)2cos(

)2cos(

)3cos(),,,(

,2,12/1

,2,1

,2,12/1

,2,1

,0,32/3

,0,3

yxyx

yxyx

xxyyxxyyxx

JJG

JJG

JGJJJJH

Nonlinear Effects of Chromatic Sextupoles

Concatenation of sextupoles perturbation to the betatron motion can induce nonlinear betatron detuning.

yx νν 2Sum resonance:

Difference resonance:

Parametric resonance:

Other higher-order resonance:

yx νν 2

xx νν 3 ,

...22 ,4 yxx ννν

yyxyxx ,,

yyyxxyyy

yxyxxxxx

JJνν

JJνν

0

0

Detuning coefficient:


20020112

22000111

20011110

2110019

28

27

26

2102005

2100204

2300003

2101102

2210001

|))(2(||))(2(|

|)(||)(|

|/||/||/|

|)2(||)2(|

|)3(||)(||)(|

yx

yx

yyyxxx

yxyx

xxx

hphp

hphp

dJdpdJdpdJdp

hphp

hphphpf

Sextupole Hamiltonian

0])()[(22

1

])()[(222

])([

)(

pmlkji

ml

yn

kj

xn

Nquad

nn

mlkjipn

ml

yn

kj

xn

Nsxt

nnjklmp

ynxn

ynxn

eLb

eDLbh

In the single resonance approach, first 9 terms of first-order driving sources: 4 Chromatic terms, 5 Geometry terms 9 families can get first-order optimal solutions (ref: J. Bengtsson, The sextupole scheme for the SLS, SLS-Note 9/97)

For the first 12 terms: 4 Chromatic terms(P9~P12), 5 Geometry terms(P1~P5),3 Amplitude dependent terms (P6~P8) (second-order terms and critical) need to be minimized (by Powell’s method in OPA).


Usually, we can get good solution for the first 12 terms optimization. However, we can further reduce 2nd-order terms.

13 terms in 2nd -order of sextupole strength3 tune shift with amplitude8 octupole-like driving terms 4νx , 2 ν x±2 ν y , 4νy , 2 ν x , 2 ν y

2 terms generating second-order chromaticity

This nonlinear optimization is very important for the low emittance, high chromaticity lattice. Iterations between linear and nonlinear schemes are proceeded to get acceptable solution.

Objective Function Including Higher-order Driving terms


TLS Storage Ring

Lattice type TBAOperational energy 1.5 GeVCircumference 120 mNatural emittance 25.6 nm-rad (achromat)Natural energy spread 0.075%Momentum compaction factor 0.00678Damping time Horizontal 6.959 ms Vertical 9.372 ms Longitudinal 5.668 msBetatron tunes horizontal/vertical 7.18/4.13Natural chromaticities Horizontal -15.292

Vertical -7.868Radiation loss per turn (dipole) 128 keV

For TLS, do we need more than two sextupole families?

Only two family of sextupoles for chromaticity correction.SD=-5.3(m-2), SF=7.62(m-2)


TLS Storage Ring

If only put sextupoles in one section (locally) to control chromaticity, SD=-23.7 (1/m2) and SF=26.6 (1/m2), dynamic aperture reduce significantly due to lack of phase cancellation.

-30 -20 -10 0 10 20 30X [mm]

0

2

4

6

8

10

Y [mm]

-30 -20 -10 0 10 20 30X [mm]

0

2

4

6

8

10

Y [mm]

Dynamic aperture for distributed sextupoles

Local sextupoles only


TLS Storage RingWhat is the RF energy acceptance?

ID No ,128 :TLS

sin

1

)/1(cos1(sin

)(

0

0

0

1

0

02

0

keVU

U

eVq

qqcph

eV

p

p

s

c

sacc

Dynamic aperturePhase space

Beta and tune change vs energy

-30 -20 -10 0 10 20 30X [mm]

0

2

4

6

8

10

Y [mm]

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

RF Voltage (MV)

RF E

nerg

y ac

-ce

ptan

ce (%

)


Betatron Tune and Nonlinear Resonance

integers are ,,

6

pmn

pmn yx integers are ,, pmn

pmn yx

Systematic resonances Random resonances

Tune selection in the lattice very important. Tune should be away from integer, half integer and third-order resonances. Introduction of sextupoles and nonlinear field errors in the magnets can drive higher-order nonlinear resonances. Particle tracking study and fequency (tune)-map analyses can further provide optimization information in tune selection.


Linear Lattice Matching• Computer design codes are usually used in the matching. (for example

MAD)• Matching method such as simplex command is to minimize the penalty

function by simplex method. Make sure enough varying parameters for the selected constraints.

• Starting from unit cell and impose constraints on optical functions such as D, D’ at both ends of dipoles, local and global betatron functions, phase advance per cell, etc. Weighting factors are also given.

• Construct super-period structure and do the same matching process with different constraints.

• Maximum strengths of quads are limited.• Ring tunes are matched.• Not always able to find stable solutions and need change initial conditions

for matching.• Examine the global parameters and fitted parameters. If satisfactory, go to

nonlinear optimization.


Linear matching

strength quad :

),..,...,( ,... 2111

i

mjiinT

k

kkkkfffffF Objective function

mn k

k

k

K

f

f

f

F

.

. and

.

.2

1

2

1

m

nnn

m

m

k

f

k

f

k

f

k

f

k

f

k

fk

f

k

f

k

f

A

....

..........

....

....

21

2

2

2

1

2

1

2

1

1

1

m=n, with equal weighting factor and only linear functions considered,

)( 01

0 FFAKK ideal

Iteration needed due to nonlinearity.

)( 00 KKAFFideal

With different weightings and constraints of objective functions, or m<n case, least square or nonlinear optimization methods can be used. Design codes usually provide useful tools to get solutions but maybe not what one wants. In MAD code, minimization by gradient or simplex method can be employed. Local minimum maybe the unwanted solutions. Initial values of quad strength should be changed.Some tools using genetic algorithm to get desired linear optics.


TPS Lattice

Nonlinear optimizationQuality factor: Tune shift with amplitude, Phase space plots,Tune shift with energy, Dynamic aperture (on and off momentum, 4D, 6D), Frequency Map AnalysisCodes: OPA, BETA, Tracy-2, MAD, Patricia, AT, elegant, etc.8 families of sextupoles are used.Chromaticities are corrected to slightly positive.Weighting factors, sextupole families, positions are varied. Effects on the dynamic aperture in the presence of ID, field errors, chamber limitation, alignment errors, etc. are studied.


S1 S2 SD SD S4 S3 S5 S6 SD SD S6 S5 SF SF

TPS Sextupole scheme

OPA


Only Chromatic Sextupoles


8 families of sextupoles for nonlinear optimization.Chromaticity of +5 in both planes are still with acceptable dynamic aperture and energy acceptance.

OPA

Nonlinear Optimization with Sextupoles


nx,y vs. X

nx,y vs. Y

Phase SpaceBetatron Function vs Energy

Tune Shifts vs. Amplitude and EnergyDynamic Aperture

nx,y vs. dp/p


RF Energy Aperture 3.5 MV RF

(no chamber limitation)

pp

dsL

dsL

/2

2

1

1

101.2,104.2

factors compaction Momentum

21

12

0

02

0

01

32

41

TPS

Tracy-2 analytical

ID No ,85.0 :TPS

sin

1

)/1(cos1(sin

)(

0

0

0

1

0

02

0

MeVU

U

eVq

qqcph

eV

p

p

s

c

sacc


Dynamic Aperture and Frequency Map Analysis

Tune diffusion rate:

D= log10((Dnx)2+ (Dny)2)1/2

for tune difference between

the first 512 turns and

second 512 turns

4uy=53

ux+3u

y=66

2ux +2u

y =79

3uy=40ux+3uy=66

4uy=53

3uy=402ux+2uy=79

6ux=157


mmRB

B25at 10 4

With multipole errors

Liouville theorem foundation of accelerator physics

“Beam phase space is a constant”

• Liouville’s theorem :the density of points representing particles in (6-D) (x, p) phase space is conserved if any forces conservative and differentiable . i.e., the forces must be divergence free in momentum space (p-divergence =0).

• radiation and dissipation do not satisfy the p-divergence requirement, but magnetic forces and (Newtonian) gravitational forces do.

• No or very slow time dependence in the Hamiltonian system.• Note: acceleration keeps (x,p) phase space constant, but reduces

(x, x’) phase space , no violation of Liouville theoremi.e. normalized emittance

Joseph Liouville, 1809 – 1882

0 pp

20 1/1,/, cvN

emittance geometric0


Non-conservation of emittance

• Coupling (horizontal-vertical, chromaticity,..)• Scattering (gas, intra-beam, beam-beam,

through foil)• Damping (synchrotron radiation, electron

cooling, stochastic cooling, laser cooling)• Filamentation (nonlinearities, phase space

conserved, but emittance growth)• Instabilities (wake-fields,…)• Space charge effects


Transverse beam Emittance• Transverse emittance in a linac is determined by the beam source (gun)

emittance. The normalized emittance is a constant, but geometric emittance is reduced during acceleration due to adiabatic damping.

• In a booster synchrotron, the geometric beam emittance is damped first due to adiabatic damping during energy ramping and then reach equilibrium natural emittance.

• The natural emittance in the storage ring is determined by the equilibrium between radiation damping and quantum excitation of synchrotron radiation.

• In proton or heavy-ion rings, e-cooling or other damping schemes are needed.

• The natural emittance in a ring is determined by the lattice design.• The natural emittance can be reduced by adding damping wigglers,

Robinson wigglers, or change the bend radius, using longitudinal gradient dipole, etc.

• The diffraction limited emittance is

E (keV) = 1.24 / λ (nm)

4


Synchrotron radiation and damping• Charged particle under acceleration (through EM field, etc.) will radiate

EM wave.• The radiation in synchrotron accelerator is called synchrotron radiation. Properties of synchrotron radiation: High Intensity, Continuous Spectrum,

Excellent Collimation, Low Emittance, Pulsed-time Structure, Polarization.

Energy loss due to synchrotron radiation in circulating particle is replenished by rf cavities with longitudinal electric field.

Higher energy particle loss more energy and in average the energy is damped to the equilibrium energy. (longitudinal damping)

In transverse planes, radiation in a cone with some angle compensated by the longitudinal electric field and also particle is damped in transverse phase space. (transverse damping)

Radiation is a quantum process and cause diffusion and excitation. energy spread in equilibrium

Longitudinal and transverse motions are coupled through dispersion function natural emittance in equilibrium


Damping of Synchrotron Oscillations

Relativistic electron loss energy due to radiation

2

4

2 E

Cc

P

40

2

40

0 2

ECdsECPdtU )/(10846.8 35 GeVmC

Plus gain energy from rf:

0| ,)( 0 EEdE

dUWEWUEU

)(1)(

0

EWVeTdt

Ed

02 22

2

sE dt

d

dt

d

ET

Ve

T

W csE

0

2

0

,2

)cos()( 0 tAet stE )2(

22 0

0

0

DET

U

T

WE

dssKsD

dsdssK

sD

dipole2

1

22

))(21

)((2

1

))(21

()(

D

E

P

ET

UE

0

0Isomagnetic separated function ring

4

0

0

2

EC

R

c

T

UP

E

ET

dt

d

T

EUeV

dt

Edc

0

0

and ,)()()(

Isomagnetic


:parameter D

Damping of Vertical Betatron Oscillation

aAzzAA

zAz ,)( ,sin ,cos 222

z

s

p pp

After emitting radiation and rf acceleration:

E

uz

p

pzz

0

0

0 2

11

ET

U

A

A

Tdt

dA

A

Average one turn

E

P

ET

Uz 22 0

0

E

UzzzAA 022 )()

E

UAAA 0

2

22/)( 22 Az


Damping of Horizontal Betatron Oscillations

After emitting radiation :

Average one turn:

E

P

ET

Ux 2

)1(2

)1(0

0

DD

ee xxE

EDxxxx

E

EDxx

,

E

uDxx

E

uDxx ee ,

Change in betatron amplitude and average in betatron phase:

E

uxDDxxxxxAA )( 22

dscE

Pxx

dx

dB

BDxAA

)

21(

E

UdsdsK

D

E

U

A

A

2

112

20

1

220 D

Including rf acceleration:

E

U

A

A

21( 0)D


Robinson theorem

4 Ezx JJJ

Robinson damping partition theorem

Radiation damping coefficients

1 ,2

)2(2

2

)1(2

0

00

00

0

00

0

00

0

z

Es

zz

xx

JET

U

JET

U

JET

U

JET

U

D

D


21

:conditionstability ring,electron function combinedFor

etc. ggler,Robison wi usingor orbit changingby change need

,concerned) (emittance machinelepton for usedBut

samll) very /Emachine(hardron for worksstill

4,1,12

:ring focusing strongfunction combinedFor

2,11

:ring focusing strongfunction separatedFor

0

D

D

D

D

U

JJJ

JJJ

Ex

Ex

z

z

,

,

dssKsD

dsdssK

sD

dipole2

1

22

))(21

)((2

1

))(21

()(

D

Energy spread

tAE scosSynchrotron oscillation without fluctuation and damping:

With radiation and damping:

222

2 uA

dt

Ad

E

N

emissionphoton of rate :N

In equilibrium: EuA 22

2

1N

Define EE uA 2

22

4

1

2N

23

32

/1324

55

2

3

P

cu Nwhere

PJ

E

EE

2

For isomagnetic ring:

Eq

E

JC

E

22)(

32

22 /1

/1)(

Eq

E

JC

Em

mcCq

131083.3332

55


Horizontal Emittance

mCJ

C qx

xq

x

xx

x 132

32

2

1083.3,/1

||/

H

Natural emittance

Radiation emission results in change of betatron coordinates:

E

uDx

E

uDx ,

Change of Courant –Snyder invariant after average:

22 )(E

ua H

2

2

2

1x

xxxx

xx DDD

H

Including damping termand average one turn: 2

222

2E

ua

dt

a s

x

HN

Equilibrium:

x

xsx

x

E

ua

2

2

22

2

1

HN

23

32

/1324

55

2

3

P

cu N


Vertical emittanceFor an ideal flat accelerator without errors, due to the finite emission angle of synchrotron, we can get natural vertical emittance as:

radm101/ρJ

ρ/βCε 13

2y

3y

qy

||

However, due to spurious vertical dispersion by errors,

2 ||

yyyyy2yyy

132

y

3y2

qy DβDD2αDγrad,m101/ρJ

ρ/γCε

HH

Betatron coupling from skew quads, etc., also generate vertical emittance coupled from horizontal emittance.


))((1

where

,||

)21

(

||

1

1

2'2

35

24

33

22

1

xxxxxx

x

x

dsI

dsKI

dsI

dsI

dsI

H

H

242424

35320

24

0

42322

13

4252

1

/ ,/2 ,/1 (5)

)/(1085.8)/()3/4(

2/per turn loss(4)Energy

)2/()/( spreadEnergy (3)

1083.3/)332/55(

)/( Emittance (2)

2/ comp. Mom. (1)

IIIIJIIJ

GeVmmcrC

IECU

IIICE

mmcC

IIIC

RI

Ex

qE

q

qx

c

D

Radiation integral and electron beam properties

Radiation integral


FODO cell

2sin

2 ),

2sin1(

sin

2 ),

2sin1(

sin

2

f

LLLDF

)2

sin2

11(

)2/(sinD ),

2sin

2

11(

)2/(sinD

2D2F

LL

23

2 )2

sin2

11(

)2/sin(1)(2/(sin

)2/cos(

LFH

23

2 )2

sin2

11(

)2/sin(1)(2/(sin

)2/cos(

LDH

)

2sin1(

)2

sin21

1(

)2

sin1(

)2

sin21

1(

)2/(sin

)2/cos(22

32

H

323

232 1

)2/cos()2/(sin

)2/(sin)4/3(1 qx

qFODOx CJ

CF


FODO Lattice

• FODO lattices are commonly used in colliders, booster synchrotrons in which emittance is not pushed to extremely small values.

• In high energy ring for particle experiments, dipole magnets with FODO cells occupy most of the ring path.

• Light source design usually try to have as large percentage of straights for IDs, FODO type lattice usually not adopted for dedicated light sources.

32 qME FC F~1.2 for phase advance ~ 138° per cellF~ 2.5 for 90°

323

232 1

)2/cos()2/(sin

)2/(sin)4/3(1 qx

qFODOx CJ

CF

1Let xJ

1xJ


Minimum Emittance in DBADBA: small bend angle At bend entrance,D0=D0’=0 Bend length L

Optimum emittance:

S.Y. Lee, “Emittance optimization in three- and multiple-bend achromats”, Phy. Rev. E, 52, 1940, 1996 and “Accelerator Physics”, World Scientific, 2nd edition, 2004

Each MEDBA module with horizontal phase advance across each dipole of 156.70, dispersion matching 1220, and about 1.2 unit of horizontal tune per DBA cell.

0

00

2000

)(

)(

2)(

s

ss

sss

sinsin')('

)cos1()cos1(')(

0

00

DsD

sDDsD

0

22

0

))(')(')()(2)(')((1

)(1

dssDsDsDssDsdss HH

)2043

( 0003

H )2043

( 000

32

x

qx J

C

000

xx 15,)15/6( min,0min,0

x

q

x

qMEDBA J

C

J

C 3232

0645.0154


Minimum Emittance in DBA

For non-achromat lattice, minimization of I5 lead to

0)2/(,0)2/( achromat)-(non TME sDs

2

1,

6

1,15,

15

80

2000 DD

x

qx J

C 32

1512

1

only one type of dipole. beta and dispersion functions are symmetric at dipole center


Design Emittance• In practice, real machine will not be able to reach

theoretical minimum emittance because of the constraints in betatron function limitation, tune range, nonlinear sextupole scheme, and engineering limitation, etc.

• Usually, a few factor larger in real machine is feasible. • For 24-cell 3GeV, theoretical minimum emittance is

1.92 nm-rad for the achromatic DBA. In real design, it can reach 4.9 nm-rad.

• For non-achromat configuration, we can reach 1.6 nm-rad (as compared with the TME of 0.64 nm-rad)


Minimum Emittance in TBA

Matching optical functions (with quad) between MEDBA module (outer dipoles L1) and ME (central dipole L2) results in: (ref. S.Y. Lee, Accelerator Physics, World Scientific)

21

31

22

32 3

ρ

L

ρ

L 127.76 and phase advance

x

qTME

x

qMETBA J

C

J

C 31

231

2

1512

1,

154

1

2M cell, ain angle bend Total:

23)2(1512

1

1512

13

3

231

2

MJ

C

J

C

x

q

x

qTME

isomagnetic ring: 13/1

2123/11 )32(2 ,3

1


Emittance comparison• One can compare the minimum emittance among DBA,

TBA and QBA if same number of dipoles are used in the ring.

• From the formula in the previous slide, we can get

MEDBAMEDBA

3

3/1MEQBA

MEDBAMEDBA

3

3/1METBA

55.031

2

66.032

3


Effective emittance

xx

xExeff

D

2)(1

section IDin 2

''

2''2

22,

2222,

xxxxxxxID

xEIDxeffx

effx

DDDDH

xxxx

H

With dispersion in the long straights, we can reduce the natural emittance by a factor of 3, but the effective emittance is

At the symmetry point of the long straights,

TPS emittance

0

1

2

3

4

5

0 0.05 0.1 0.15 0.2

dispersion (m)

emitta

nce

(nm

-rad

)

natural emittance

effective emittance


Closed Orbit Distortion

In reality, dipole field errors distributed around the ring:

|))'()(|cos(sin2

)'()()',( where

')'(

)',()(

ssss

ssG

dsB

sBssGsy

cs

sco

N

iiiico sss

ssy

1

|))()(|cos()(sin2

)()(

In dipole angular kick form:

ii

i dsB

sB

)(

rmsrmsco N

sy

|sin|22,

due to quadrupole misalignment f

yy

B

xBz

)/(

The coefficient in curly brackets is called sensitivity or amplification factor.

,

av

av,

|sin|22rmsqrmsco xN

fx


N

iijii

j

jco ssss

sy1

|))()(|cos()(sin2

)()(

|))()(|cos()(sin2

)(iji

jji sss

sA

mcon yA ,

1

The distorted orbit can be minimized at least at the BPM to the desired value so that

COD correction, SVD method:


ID effect

)cosh()cos( ykskBB wwwy 0xB )sinh()sin( ykskBB wwws

The equations of motion:

Insertion devices cause some effects on beam dynamics like betatron tune shifts, optical functions perturbation, emittance variation, multipole field effects, etc. The field of a wiggler can be modeled as:

),sin()sinh(2

)2sinh()(sin

),cos()cosh(1

2

2

2

2

2

2

skykp

k

yksk

ds

yd

skykds

xd

www

x

w

w

w

w

www

average focusing strength:22

2

2

1)(sin

ww

wsk

vertical tune shift:

28 w

wyy

L

Beta beat in vertical plan can be evaluated

Vertical beta perturabtion due to Wiggler 20 in TLS (Kuo, et al PAC95)

w20Tune-shift with gap for w20 at TLS


Correction Algorithm1. By the following linear

relations, construct the response matrix A of bare lattice.

2. Use SVD method to restore the perturbed optics back to the optics of the bare lattice as much as possible by minimizing (AΔk +b).

3. Where Δk is the tuning strength of the quadrupoles, and b is the perturbed optics.

4. One can choose arbitrary positions to restore optics with different weighting.

nq

y

x

y

x

n k

k

k

A

Q

Q

w

w

wy

y

x

x

2

1

1

12

1

1

1

1

1

SW60Current (A) 0.4

λ (mm) 60Nperiod 8By (T) 3.5Bx (T)L (m) 0.45

Gap (mm) 17

Total power (kW) 13.39


Beta Beating and phase beating correction

(a) Horizontal beta-beating with SW60

(c) Horizontal phase-beating with SW60

(b) Vertical beta-beating with SW60

(d) Vertical phase-beating with SW60


Emittance with IDs

undulator helical ),2/(

undulatorplanar ),4/(

/

undulator helical ,/

undulatorplanar ,)3/(4

||

1

2 ,

,21

ID with Emittance

ID without Emittance

24

24

04

0

3

3

35

22

20

350

424020

5502

4020

5020

ww

ww

w

wwID

wwID

wIDw

xxxxxxxID

dipoleh

IDhIDh

ww

wqx

qx

LEC

LECU

ECU

L

LdsI

f

fdsfI

IIII

IIC

II

IC

H

HH

HH

H

HH

undulators helical ,)1(

)1(

undulatorsplanar ,)1(

)38

1(

undulator helical ,

undulatorplanar ,3

8

3

8

0

00

0

00

0

000

0

000

0

50

5

w

w

w

w

h

w

w

w

w

w

h

w

x

x

w

h

ww

wh

w

h

ww

whw

UU

UU

BfB

UU

UU

BfB

U

U

Bf

B

U

U

f

U

U

Bf

B

U

U

f

I

I

If Bw > (3πfh/8)B0 then emittance will increase for planar undulator Installing IDs in the non-dispersive straights, where fh is very large,

emittance always decreases. For a TME lattice, fh=0.25, and for a tyical well-designed distributed

dispersion lattice, fh=0.5~0.8 OCPA Accelerator School, 2014, CCKuo- 117

Energy spread with IDs

undulators helical ,)1/()1(

undulatorsplanar ,)1/()3

81(

)()(

/1

/1)(

)()(

undulator helical ,

undulatorplanar ,3

8

3

8

undulator helical ,/

undulatorplanar ,)3/(4 ,

2

000

0002

0

2

202

3032

0

42

322

000

0

000

0

30

3

3

3

32

0

30

w w

www

w w

www

EE

w

wEq

E

www

w

www

ww

ww

ww

w

U

U

U

U

B

B

U

U

U

U

B

B

EE

II

II

EII

IC

E

U

U

B

B

U

U

f

U

U

B

B

U

U

I

I

L

LII

If Bw > (3π/8)B0 then energy spread will increase for planar undulator.

undulator helical ),2/(

undulatorplanar ),4/(

/

24

24

0

4

0

ww

ww

wLEC

LECU

ECU


Touschek scattering

L

accxzx

x

acc

l

e

T

dsssss

ss

C

L

cNr

02'

2

'

3

2

)()()()(

)()(

1.

8

1

21

• Transverse momentum can be transferred to longitudinal plane by Coulomb scattering in the same bunch and energy loss/gain is boosted by a factor of p=px = p x /x.

• Particle can be lost due to energy acceptance (rf or transverse) limitation or aperture(dynamic or physical) limitation.

• The effects were first recognized by B.Touschek in 1963 in Frascati e+-e- storage ring ADA.

• Touschek lifetime can be expressed as:

where V=83/2 xyL, =[ acc/x’ ]2, N is the number of electrons per bunch.

duu

edue

u

ueC

uu )2ln3(5.0

ln

25.1)(

Bruck formula


Touschek lifetime vs RF gap voltage

summary Lattice design of accelerators is an iterative process. Different options might be presented for comparison and evaluation. FODO lattice usually adopted in rings for proton and heavy ions. DBA, TBA, or MBA are adopted for light sources which require extremely small

emittance Linear lattice design is to match requirements of betatron, dispersion functions,

betatron tunes, and other global parameters. Nonlinear optimization with sextupole scheme is more complicated in modern

light sources. Correction schemes for orbit, optics, coupling should be studied. Make sure enough aperture (energy, dynamic and physical ) for injection and

lifetime with the existence of errors (magnetic field errors, alignment errors, etc.) Effects in the presence of insertion devices need to be well investigated. Instability issues are also very important and make sure that impedances of

vacuum components are well controlled. Feedback systems for fast orbit correction, instabilities, etc., are required.


Some of the materials, figures, etc., are from references:1. S.Y. Lee, Accelerator Physics, World Scientific, 2nd edition,

2004.2. Helmut Wiedemann, Particle Accelerator Physics, Springer,

3rd edition, 2007.3. Klause Wille, The Physics of Particle Accelerators, Oxford

University Press, 20004. A. W. Chao and Maury Tigner, Handbook of Accelerator

Physics and Engineering, World Scientific, 3rd edition, 1999.5. And other resources, public web pages, etc.These lecture notes are only for this School.


Thank you for your attention


lattice design c.c. kuo 郭錦城 july 27 ~ august 6, 2014 安徽省黃山市休寧 ocpa accelerator...

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