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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Transcript of 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
OCPA Accelerator School, 2014, CCKuo- 2
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.
OCPA Accelerator School, 2014, CCKuo- 3
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-
OCPA Accelerator School, 2014, CCKuo- 4
A Lattice with FODO Arc and Doublet straight
王生 OCPA 2010散裂中子源 CSNSRCS
OCPA Accelerator School, 2014, CCKuo- 5
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
OCPA Accelerator School, 2014, CCKuo- 6
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
OCPA Accelerator School, 2014, CCKuo- 8
The Hefei Light Source Upgrade
Q4Q3Q2Q1 B
S1 S3S2 S4
HLS Upgrading
e<40 nm·rad I>300 mA
E=800MeV
OCPA Accelerator School, 2014, CCKuo- 9
(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
OCPA Accelerator School, 2014, CCKuo- 11
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
OCPA Accelerator School, 2014, CCKuo- 12
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.
OCPA Accelerator School, 2014, CCKuo- 13
Lattice Design Procedure-(2)
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.
OCPA Accelerator School, 2014, CCKuo- 14
Lattice Design Procedure-(3)
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.
OCPA Accelerator School, 2014, CCKuo- 15
Lattice Design Procedure-(4)
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.
OCPA Accelerator School, 2014, CCKuo- 16
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.
OCPA Accelerator School, 2014, CCKuo- 17
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
OCPA Accelerator School, 2014, CCKuo- 18
• 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
OCPA Accelerator School, 2014, CCKuo- 19
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,
!|)(
OCPA Accelerator School, 2014, CCKuo- 20
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
OCPA Accelerator School, 2014, CCKuo- 21
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
OCPA Accelerator School, 2014, CCKuo- 22
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
OCPA Accelerator School, 2014, CCKuo- 23
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
OCPA Accelerator School, 2014, CCKuo- 24
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
OCPA Accelerator School, 2014, CCKuo- 25
(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):
/
OCPA Accelerator School, 2014, CCKuo- 26
(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
OCPA Accelerator School, 2014, CCKuo- 27
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
OCPA Accelerator School, 2014, CCKuo- 28
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(
OCPA Accelerator School, 2014, CCKuo- 29
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
OCPA Accelerator School, 2014, CCKuo- 30
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
OCPA Accelerator School, 2014, CCKuo- 31
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
OCPA Accelerator School, 2014, CCKuo- 32
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
OCPA Accelerator School, 2014, CCKuo- 33
)(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
OCPA Accelerator School, 2014, CCKuo- 34
/xw
)'(11
xxd
dw
a
M. Sands SLAC-121
OCPA Accelerator School, 2014, CCKuo- 35
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
OCPA Accelerator School, 2014, CCKuo- 36
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:
OCPA Accelerator School, 2014, CCKuo- 37
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
OCPA Accelerator School, 2014, CCKuo- 38
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
OCPA Accelerator School, 2014, CCKuo- 39
length cell halfL length, quad half,/1 kf
Summary in a FODO cell
OCPA Accelerator School, 2014, CCKuo- 40
)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.
OCPA Accelerator School, 2014, CCKuo- 41
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.
OCPA Accelerator School, 2014, CCKuo- 42
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.
OCPA Accelerator School, 2014, CCKuo- 43
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.
OCPA Accelerator School, 2014, CCKuo- 44
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
OCPA Accelerator School, 2014, CCKuo- 45
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.
OCPA Accelerator School, 2014, CCKuo- 46
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
OCPA Accelerator School, 2014, CCKuo- 47
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.
OCPA Accelerator School, 2014, CCKuo- 48
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.
OCPA Accelerator School, 2014, CCKuo- 49
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
OCPA Accelerator School, 2014, CCKuo- 50
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(
OCPA Accelerator School, 2014, CCKuo- 52
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
OCPA Accelerator School, 2014, CCKuo- 53
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
OCPA Accelerator School, 2014, CCKuo- 54
Dispersion suppression
Same dipole Separated functions dipole
OCPA Accelerator School, 2014, CCKuo- 55
MBA:PEP-X (SLAC)
PEP-X, 6GeV, 11pm-rad 2.2 km
OCPA Accelerator School, 2014, CCKuo- 56
MBA: MAX-IV (Sweden)
MAX-IV, 3GeV, 0.3 nm-rad20-cell, 528m
OCPA Accelerator School, 2014, CCKuo- 57
MBA: Sirius (Brazil)
Sirius (Brazil) 5BA with superbend
3 GeV, 518 m, 0.28 nm
BAPS 20*13BA
OCPA Accelerator School, 2014, CCKuo- 59
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
OCPA Accelerator School, 2014, CCKuo- 61
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
OCPA Accelerator School, 2014, CCKuo- 62
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
OCPA Accelerator School, 2014, CCKuo- 63
)()](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
OCPA Accelerator School, 2014, CCKuo- 64
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
OCPA Accelerator School, 2014, CCKuo- 65
,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
OCPA Accelerator School, 2014, CCKuo- 66
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.
OCPA Accelerator School, 2014, CCKuo- 67
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
OCPA Accelerator School, 2014, CCKuo- 68
• 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
OCPA Accelerator School, 2014, CCKuo- 69
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:
OCPA Accelerator School, 2014, CCKuo- 70
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.
OCPA Accelerator School, 2014, CCKuo- 71
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
OCPA Accelerator School, 2014, CCKuo- 72
Nonlinear Hamiltonian
s
yyyyyy
s
xxxxxx
dsss
dsss
0
0
)( ,)(
)( ,)(
),( ),,( yyxx JJ are pairs of conjugate phase-space
OCPA Accelerator School, 2014, CCKuo- 73
]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
OCPA Accelerator School, 2014, CCKuo- 74
...
)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:
OCPA Accelerator School, 2014, CCKuo- 75
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).
OCPA Accelerator School, 2014, CCKuo- 76
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
OCPA Accelerator School, 2014, CCKuo- 77
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)
OCPA Accelerator School, 2014, CCKuo- 78
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
OCPA Accelerator School, 2014, CCKuo- 79
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 (%
)
OCPA Accelerator School, 2014, CCKuo- 80
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.
OCPA Accelerator School, 2014, CCKuo- 81
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.
OCPA Accelerator School, 2014, CCKuo- 82
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.
OCPA Accelerator School, 2014, CCKuo- 83
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.
OCPA Accelerator School, 2014, CCKuo- 85
S1 S2 SD SD S4 S3 S5 S6 SD SD S6 S5 SF SF
TPS Sextupole scheme
OPA
OCPA Accelerator School, 2014, CCKuo- 86
Only Chromatic Sextupoles
OCPA Accelerator School, 2014, CCKuo- 87
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
OCPA Accelerator School, 2014, CCKuo- 88
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
OCPA Accelerator School, 2014, CCKuo- 89
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
OCPA Accelerator School, 2014, CCKuo- 90
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
OCPA Accelerator School, 2014, CCKuo- 91
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
OCPA Accelerator School, 2014, CCKuo- 92
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
OCPA Accelerator School, 2014, CCKuo- 93
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
OCPA Accelerator School, 2014, CCKuo- 94
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
OCPA Accelerator School, 2014, CCKuo- 95
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
OCPA Accelerator School, 2014, CCKuo- 96
: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
OCPA Accelerator School, 2014, CCKuo- 97
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
OCPA Accelerator School, 2014, CCKuo- 98
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
OCPA Accelerator School, 2014, CCKuo- 99
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
OCPA Accelerator School, 2014, CCKuo- 100
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
OCPA Accelerator School, 2014, CCKuo- 101
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.
OCPA Accelerator School, 2014, CCKuo- 102
))((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
OCPA Accelerator School, 2014, CCKuo- 103
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
OCPA Accelerator School, 2014, CCKuo- 104
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
OCPA Accelerator School, 2014, CCKuo- 105
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
OCPA Accelerator School, 2014, CCKuo- 106
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
OCPA Accelerator School, 2014, CCKuo- 107
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)
OCPA Accelerator School, 2014, CCKuo- 108
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
OCPA Accelerator School, 2014, CCKuo- 109
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
OCPA Accelerator School, 2014, CCKuo- 110
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
OCPA Accelerator School, 2014, CCKuo- 111
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
OCPA Accelerator School, 2014, CCKuo- 112
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:
OCPA Accelerator School, 2014, CCKuo- 113
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
OCPA Accelerator School, 2014, CCKuo- 114
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
OCPA Accelerator School, 2014, CCKuo- 115
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
OCPA Accelerator School, 2014, CCKuo- 116
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
OCPA Accelerator School, 2014, CCKuo- 118
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
OCPA Accelerator School, 2014, CCKuo- 119
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.
OCPA Accelerator School, 2014, CCKuo- 121
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.
OCPA Accelerator School, 2014, CCKuo- 122
Thank you for your attention
OCPA Accelerator School, 2014, CCKuo- 123