### (DBA) cell

```Lattice Design
C.C. Kuo

July 27 ~ August 6, 2014

OCPA Accelerator School, 2014, CCKuo- 1
outline
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Introduction
Linear beam dynamics
Linear lattice
Nonlinear beam dynamics
Example
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
Collider: 1.89GeV 铁
SR: 2.5GeV
e-
e+
e+
e馬力 第五届OCPA加速器学校
OCPA Accelerator School, 2014, CCKuo- 4

RCS
A Lattice with FODO Arc and Doublet straight
Lattice consists of FODO
arc (with missing gap)
and doublet dispersion free
straight section
Arcs: 3.5 FODO cells, 315 degrees
Straights: doublet, 6.5×2+9.3 m
long drifts at each straight
Gap of Dipole : 175mm
OCPA Accelerator School, 2014, CCKuo- 5

RCS
A Lattice with Triplet Cells
Lattice 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
OCPA Accelerator School, 2014, CCKuo- 6
SSRF
150 MeV Linac
3.5 GeV Booster
3.5 GeV Storage Ring
20 DBA cells

OCPA Accelerator School, 2014, CCKuo- 7
HLS
Hefei Storage Ring
Length(m)
25
0
2
4
x
20
y
x,y(m)
15
6
8
10
12
14
16
Theoretical Curve of x
Measured Value of x
Theoretical Curve of y
Measured Value of y
10
5
0
Q1
-5
Q4 SF
Q2
B1 SD Q3
SF Q5
B2
800 MeV
66.13m

OCPA Accelerator School, 2014, CCKuo- 8
Q6 SD
Q8
B3
Q7
(2012 OCPA School 張闖)
E=800MeV
S1
S2
Q1
Q2
S3
B
Q3
S4
Q4
I>300 mA
OCPA Accelerator School, 2014, CCKuo- 9
NSRRC TLS
6 TBA

(1.51 GeV)

LINAC

(1.51 GeV)
FODO cell in TLS booster
OCPA Accelerator School, 2014, CCKuo- 10
TPS
Energy : 3 GeV
Beam current: 500 mA
Straight Section: 7 m (18); 12 m (6)
Lattice structure: Double-Bend
Circumference: 518.4 m
RF: 500 MHz
Linac energy : 150 MeV
Booster energy: 3 GeV
Booster circumference: 496.8 m
Lattice structure: MBA
Repetition rate: 3 Hz
OCPA Accelerator School, 2014, CCKuo- 11
Layout
Accumulator
Cooler
Synchrotron
Fast extraction
Slow extraction
CSRm
CSRe
Circumference (m)
161.0014
128.8011
4/5RCSRm =20.499328
Geometry
Race-track
Race-track
2800 (p)
1100 (C6+)
500 (U72+)
2000 (p)
750 (C6+)
500 (U92+)
0.81/12.05
0.50/9.40
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
Max. energy (MeV/u)
External target
Internal target
B (Tm)
B(T)
A
h
A
v
p/p (%)
200
(p/p = 0.3 %)
Normal mode
150 (p/p =0.5%)
40
75
1.4
2.6
12 Tm
12GeV—C6+, 120GeV—U72+
OCPA2012

OCPA Accelerator School, 2014, CCKuo- 12
OCPA-School-08
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.
 Combined function or separated function magnets are selected.
 Maximum magnetic field strengths are constrained. (roomtemperature or superconducting magnets, bore radius or
chamber profile, etc.)
 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)
2)
3)
4)
5)
6)
7)
User requirements
Lattice type (FODO, DBA, TBA, QBA, MBA)
Linear lattice
Nonlinear dynamic aperture tracking
Longitudinal dynamics
Error tolerance analysis
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
m0v 2
FL  evB  Fcentr 


m0 v
e
p
  B
e
For relativistic particle
B[T ]
B(Vs / m2 )
[1 / m]  e
e
, E in GeV
9

p
E 10 (eV ) / c
1
Note : for ion
1

[1 / m]  0.3
1

[1 / m]  0.3
B[T ]
E[GeV / u ]
Z
B[T ]
A E[GeV / u ]
TLS: 1.5 GeV, B=1.43T, =3.49m, Circumference=120m;
LHC: 7000GeV, B=8.3T, =2.53km, Circumference=27km
• Total required dipole magnet length = 2
• Synchrotron radiation loss per turn for electron ~ 8.85x10-5β3E4/
[GeV] and for proton ~ 7.78x10-18β3E4/ [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.
OCPA Accelerator School, 2014, CCKuo- 18
Basic magnet elements
• 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.
OCPA Accelerator School, 2014, CCKuo- 19
Magnetic field expression
For transverse fields in Frenet-Serret
(curvilinear ) coordinate system:
1
1 
( Bz  jBx )    (bn  jan )(x  jz ) n ,
B0 
 n 0
1  n Bz
1  n Bx
bn 
|
, an 
| x  z 0
n x  z 0
n
B0 n! x
B0 n! x
 : posit ivecharge(upper),negat ive(lower)
b0 : dipole Bz  B0b0
a0 : vert.dipole
Bz  B0b1 x, Bx  B0b1 z
b2 : sext
Bx  B0 a0
Bz   B0 a1 z , Bx  B0 a1 x
Bz  B0b2 ( x 2  z 2 ), Bx  2 B0b2 xz
a2 : skew sext
Bz  2 B0 a2 xz, Bx  B0 a2 ( x 2  z 2 )
kn n
1  n Bz
Bz ( x) |z 0  B0   x , kn 
B0  x n
n n!
OCPA Accelerator School, 2014, CCKuo- 20
define focusing: kn  0
Frenet-Serret (curvilinear ) coordinate system, negative charge case:


 
F  dp / dt  ev  B

B  Bx xˆ  Bz zˆ
2

v
2
s
   Bz


x

(


x
)


B0 


2
v
B

z   s x

B0 

Betatron equation of motion (negative charge particle) :
d 2x
s  vt, x  v x, x  2
ds
2
x
Bz p0
x 2



x



(
1

)
2


B0  p



 z   Bx p0 (1  x ) 2

B0  p

OCPA Accelerator School, 2014, CCKuo- 21
Frenet-Serret (curvilinear ) coordinate system, negative charge :
x
Bz p0
x 2



x



(
1

)
2


B0  p



 z    Bx p0 (1  x ) 2

B0  p

Hill’s equation for
on-momentum particle :
Bz  B0  (
1


1 Bz
B
x z )
B0  x
B0 
Bz  B0  B0  (k x x 
Bz
)
B0 
B z



x

K
(
s
)
x


x

B0 


 z   K ( s ) z   Bx
z

B0 
1
1 Bz
1 Bx
K x (s)  2 
, K z ( s)  
,

B0  x
B0  z
OCPA Accelerator School, 2014, CCKuo- 22
Frenet-Serret (curvilinear ) coordinate system, negative charge :
x
Bz p0
x 2



x



(
1

)
2


B0  p



 z    Bx p0 (1  x ) 2

B0  p

Bz  B0  (
1


1 Bz
B
x z )
B0  x
B0 
Bz  B0  B0  (k x x 
Bz
)
B0 
Hill’s equation linear motion for off-momentum particle (assuming Bz,x=0) :
x  (
k x ( s)
1 

1 Bz
p

)
x

,
k
(
s
)

,


x
 2 (1   ) 1  
 (1   )
B0  x
p0
then x  ( K x (s)  K x (s))x  0
where K x ( s) 
1
2
D  ( K x ( s)  K x ( s))D 
 k x ( s), K x  [
Neglecting Chromatic
perturbation term K x (s)
Let x  x (s)  D(s)
2
2
 k x ( s)]  o( 2 )
 x ( s)  K x ( s) x ( s)  0

D( s)  K ( s) D( s)  1
x

 (s)

OCPA Accelerator School, 2014, CCKuo- 23
1

 o( )
Main magnets
Bx   B1 z , Bz   B1 x
B1 
dipole B  
2 0 NI
R2
0 NI
h
B field limits (example):
Dipole < 1.5T
Sext poletip < 0.4T
Bx  B2 xz , Bz 
1
B2 ( x 2  z 2 )
2
sextupole
B2  60 NI / R 3
OCPA Accelerator School, 2014, CCKuo- 24
Matrix formalism in linear beam dynamics:
 y ( s0 ) 
 y(s) 


  M ( s, s0 )


 y (s) 
 y ( s0 ) 
(3) Drift space K=0
1


sin K  
 cos K 
M ( s , s0 )  
K
, K  0,   s  s0
  K sin K 
cos K  

0
 1
1
,   0, f (focallengt h)  lim
 
 0 | K | 
 1/ f 1 
1


sinh | K | 
 cosh | K |
M ( s , s0 )  
|K|
, K  0,   s  s0
 | K | sinh | K |
cosh | K | 

0
 1
 , for   0
 
1
/
f
1


1 
M ( s, s0 )  
,   s  s0
 0 1
OCPA Accelerator School, 2014, CCKuo- 25
(4) pure sector dipole:
 sin  
 cos


M x ( s , s0 )   1
,


sin

cos

 



  / 
1 
M x ( s, s0 )  
 for small    / 
 0 1
non-deflecting plane (ignoring fringe field):
1 
M z ( s, s0 )  

 0 1
OCPA Accelerator School, 2014, CCKuo- 26
(5) pure rectangular dipole due to wedge in both ends:
In deflecting plane:
 1
M x ( s, s0 )  
1 / f x
1
M x ( s, s0 )  
0
0  cos
 1
1   sin 
 
 sin  
 1
cos 1 / f x

0

1 
 sin  
1
1
1

 where  tan( )
fx 
2

In non-deflecting plane (neglecting fringe field):
 1
M z ( s, s0 )  
 1/ f z
0  1   1


1  0 1   1 / f z


 1
fz
M z ( s, s0 )  

 2


2
 f
 z fz
0
1 1

, where  tan( )
1
fz 
2

  cos
 1
    sin 
1    
fz 

 sin  

if   
cos 

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.
 x
 x
   M (sn , sn1 )...M (s3 , s2 )M (s2 , s1 ) 
 x'  sn
 x'  s1
• In terms of Courant-Snyder parameters, there are
relations between initial and final points along the
beam path.
 x
 x
   M (s2 , s1 ) 
 x'  s 2
 x'  s1

2

(cos  1 sin )


1
M ( s2 , s1 )  
1  1 2
  2
sin  1
cos
 




1 2
1 2

OCPA Accelerator School, 2014, CCKuo- 28




1
(cos   2 sin ) 
2

1 2 sin
Courant –Snyder Parameters in periodic cells
 M 11
 M 21
M  

J  

M 12 

M 22 
M
 sin 
 cos   sin 

  I cos  J sin 
 
cos   sin  
   sin 
 
, T race(J )  0, J 2   I    1   2
 
eigenvalues   ei  cos  i sin 
M k  ( I cos  J sin )k  I cosk  J sin k
  cos1[(M11  M 22 ) / 2]
  M12 / sin 
  (M11  M 22 ) / 2 sin 
where transformation is for one period or one turn.
Using similarity transformation for any beam line:
M(s2 )  M(s2 , s1 )M(s1 )M(s2 , s1 )1
2
  2   M 11
  
   2     M 11M 21
   M 2
21
 2 
 2M 11M 22
M 11M 22  M 12 M 21
 2M 21M 22
 1 
 
 M 12 M 22  1 
  
M 222
 1 
OCPA Accelerator School, 2014, CCKuo- 29
M 122
Floquet transformation
y  K (s) y  0, K (s  L)  K (s)
Hill’s equation:
y(s)  aw(s)e
i ( s )
w( s  L)  w( s)
1
w  K ( s ) w  3  0
w

1
w2
 (s  L)   (s)  
Transformation matrix in one period:
 cos  ww sin 
  cos   sin 
w2 sin 
 sin 


2

M    1  ( ww) sin  cos  ww sin    
  sin 
cos   sin  
2

w


 (s)
2
1


( s)
 (s)  w2 (s),  (s)  ,  ( s) 
2
 ( s)
OCPA Accelerator School, 2014, CCKuo- 30
y( s)  a  e
 i
d
ds
, ( s  s0 )  
,  
 ( )
 (s)
s0
0
s
L

2

(cos  1 sin )
1

M( s2 , s1 )  
1  1 2
 2

sin  1
cos

1 2
1 2

 2

M( s2 , s1 )    2


2

0 
 cos
1 
 sin
 2 




1
(cos   2 sin ) 

2



sin 

cos 


1 2 sin
1
1
1
1
OCPA Accelerator School, 2014, CCKuo- 31

0 


1 


Stability of FODO Cell and Optimum Phase Advance Per Cell
FODO cell:
M FODO

L2
1  2 2
f


*
 1/ f

1
1
QF  L  QD  L  QF , f  1 / k,   half quad length
2
2
L 
)
f 
*
2
2  with f f   f d  f , 1 / f  2(1  L / f )(L / f )
L
1 2 2 
f 
2 L(1 

  x (middle of QF)   x  L
 (  1)
 x (middle of QD)   x   L
 1
 (  1)
2
with   f / L    1  f  L
 2 1
Maximum betatron function can be minimized with
optimum phase advance per FODO cell:

d x
 0  0  76.345
d
OCPA Accelerator School, 2014, CCKuo- 32
L1:DRIFT,L=1
L2:DRIFT,L=1
BD
:SBEND,L=3,ANGLE=TWOPI/96
HSUP :LINE=(QFH,L1,BD,L2,QDH)
FSUP :LINE=(HSUP,-HSUP)
FODO :LINE=(2*FSUP)
RING :LINE=(48*FSUP,)
1
1
QF  L  QD  L  QF , f  1 / k,   half quad lengt h
2
2
0  1 L  1
0  1 L  1
0





1  0 1  2 / f 1  0 1   1 / f 1 
FODO half cell :
 1
M  
 1/ f

L2
L 
1  2 2 2 L(1  ) 
f
f 
M 
wit h f f   f d  f , 1 / f *  2(1  L / f )(L / f 2 )
2

L 
*

1
/
f
1

2


f2 

 sin 
 cos   sin 


 


sin

cos



sin



(1  sin 2 )
L2
L 

 cos  1  2 , sin 2  ,   2 L
,  0
f
f
sin 
f  1 /(0.25* .5), L  5.5    86.86507
OCPA Accelerator School, 2014, CCKuo- 33
Minimizing beta in
one planephase
each FODO cell
Minimizing beta in
both planes 
90° each FODO cell
BetatronT une  x, z 
1
2
ds
  x , z ( s)
General solution: y(s)  a  cos(  0 )
Normalized coordinate: w 
w( )  a cosv(  0 )
y

, 
d


   ( ) 
1
s
d 2w 2
 w  0  A simple harmonic oscillation in normalized coordinate.
2
d
Courant-Snyder Invariant and Emittance:
C ( y, y ' ) 
1

( y 2  (y  y ' ) 2 )
C ( y, y ' )  y 2  2yy' y '2  e  emittance 2 J  a 2
Beam size :  (s)  e (s)  ( D(s)  ) 2
Beam divergence:  ' (s)  e (s)  ( D' (s)  ) 2
OCPA Accelerator School, 2014, CCKuo- 34
1 dw
1

(x  x ' )
 d

a

w x/ 
M. Sands SLAC-121
OCPA Accelerator School, 2014, CCKuo- 35
Off-momentum orbit
x  (
k x ( s)
1 

1 Bz

)
x

,
k
(
s
)

x
 2 (1   ) 1  
 (1   )
B x
Let x  x (s)  D(s)
then x  (K x (s)  K x ) x  0
D  ( K x ( s)  K x ) D 
1

 o( )
where K x ( s ) 
To the lowest order in   D  K x ( s) D 
1
2
 k x ( s ), K x  [
2
2
 k x ( s )]  o( 2 )
1

D(s  L)  D(s), D' (s  L)  D' (s), K (s),  (s) are periodic
 D( s) 
 D ( s0 ) 
 D ( s0 ) 

  M ( s, s0 ) d 



 D( s0 )   M  D( s0 ) 
 D( s )   
0
1 
 1  

 1 


 1 


 1

(1  cos K x s) 

K x
 if K  0,
d 
x
1


sin
K
s
x
  K

x


 1

(1  cosh | K x |s) 

 | Kx |
 if K  0
d 
x
1


  | K | sinh | K x |s 
x


OCPA Accelerator School, 2014, CCKuo- 36
Off-momentum orbit
 D( s) 
 D ( s0 ) 






D
(
s
)

M
D
(
s
)



0 
 1 
 1 




3  3 matrix
pure sector dipole:
where K x ( s) 
cos


M x    (1 /  ) sin 

0

 sin 
cos
0
1
2
 (1  cos ) 

sin 


1

 1   / 2 


 0 1
 
0 0
1 

 0
Mx
pure rectangular dipole:
Mx
1

 0

0

 sin 
1
 (1  cos ) 
 
0
1
2 t an
2
 1

thin lens quadrupole: M x   -1/f
 0

 1   / 2 


 0 1
 
0 0
1 

 0
Mx



0 0

1 0
0 1 
 1

M z  1/f
 0

OCPA Accelerator School, 2014, CCKuo- 37
0 0

1 0
0 1 
Dispersion for periodic lattice
For periodic lattice, the closed-orbit condition:
 D   M 11
  
 D    M 21
1  0
  
M 12
M 22
0
M 13  D 
 
M 23  D 
1  1 
D
M 13 (1  M 22 )  M 12 M 23
2  M 11  M 22
D 
M 13 M 21  (1  M 11 ) M 23
2  M 11  M 22
Solving M13 , M 23
 sin 
(1  cos   sin  ) D   sin  D 
 cos   sin 


M     sin 
cos   sin   sin  D  (1  cos   sin  ) D 


0
0
1


OCPA Accelerator School, 2014, CCKuo- 38
Dispersion in a FODO Cell
FODO cell—thin lens approximation: f  1 / k,   half quad length,L  half cell length
M 1/ 2 FODO
 1

 1 / f
 0

0 0   1 L L2 /(2  )   1


1 0  0 1
L /    1/ f
0 1   0 0
1   0
 D- 
 D 




 0   M 1/ 2 FODO  0 
 1 
 1 




0 0

1 0 
0 1 
1  L / f
L

  L / f 2 1 L / f
 0
0

D  0
  f2
L
D

(
1

)


2f

2
f
L

D 
(1 
)


2f
OCPA Accelerator School, 2014, CCKuo- 39
L2 /(2  )


( L /  )(1  L / 2 f ) 

1

Summary in a FODO cell
f QF   f QD  f ( Note: for full quad length)
2L
2L
L
(1  sin 2 ),  xD 
(1  sin 2 ),
 sin 2
sin 
sin 
2f
L
L
DxF  2  (1  12 sin 2 ), DxD  2  (1  12 sin 2 )
sin 2
sin 2
 xF 
f QF  f1 , f QD   f 2 (not e: for full quad st rengt h)
 xF
1
L2
1
L2

(2 L  ),  xD 
(2 L  )
sin  x
f2
sin  y
f1
 yF
1
L2
1
L2

(2 L  ),  yD 
(2 L  )
sin  x
f2
sin  y
f1
L
L
L2
cos x  1 


f1
f 2 2 f1 f 2
L
L
L2
cos y  1 


f1 f 2 2 f1 f 2
OCPA Accelerator School, 2014, CCKuo- 40
Integral Representation of Dispersion Function
Same as dipole field error representation, we can substitute
B
p 1
p

, xco  D( s )
B0 
p0 
p0
 x ( s)
D( s) 
2 sin  x
Jd 
s C

s
 x (t )
cos( x  | x (t )   x ( s ) |) dt

1
1 1
H ( D, D ' ) 
[ D 2  (  x D' x D) 2 ]  dispersion action
2
2 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
Consider a simple DBA cell with a single quadrupole in the middle.
In thin-lens approximation, the dispersion matching condition
1
0 0  1 L1 0  1 L L / 2  0 
 Dc  
  


 
0


1
/(
2
f
)
1
0
0
1
0
0
1

  


 0 
1 
0
0 1  0 0 1  0 0
1  1 
  
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.
1
1
1
f  ( L1  L), Dc  ( L1  L)
2
2
2
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
OCPA Accelerator School, 2014, CCKuo- 43
DBA-1
• 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.
NSLS-Xray ring
2.5GeV, 8-fold
NSLS-VUV ring
0.7GeV, 4-fold
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.
APS
7 GeV,
40-fold
OCPA Accelerator School, 2014, CCKuo- 45
ESRF
DBA-3
• Expanded C-G lattice in Elettra
• 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-4SSRF 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
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.
TPS
Half super-period
OCPA Accelerator School, 2014, CCKuo- 47
DBA-5
• Increase useful straights
in between 2 bends in
the arcs.
• Straights can be three
types.
SOLEIL 2.75 GeV 4-fold
ALBA 3 GeV 4-fold
OCPA Accelerator School, 2014, CCKuo- 48
TBA
• Triple-bend achromat (TBA)
structure with better flexibility
compared with the C-G lattice.
• ALS, TLS, PLS-I, SLS, HLS…
• In TLS, ALS, PLS-I,… no harmonic
sextupoles needed.
• Lower the emittance in these lattices
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.
TLS
ALS
SLS 4.5 nmm 2.4 GeV
OCPA Accelerator School, 2014, CCKuo- 49
QBA
B2 B1
B1 B2
B1=8.823°
B2=6.176°
TPS QBA
518.4m
Tune x:26.27
Tune y: 13.30
Emittance:3.2 nm
Nat. chromaticity
x:-65.8
Y: -27.3
OCPA Accelerator School, 2014, CCKuo- 50
FODO Lattice
(missing dipole cases)
Focusing and defocusing strengths are different
TLS booster 119 nm-rad at 1.3 GeV
OCPA Accelerator School, 2014, CCKuo- 51f
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 :
1
)
4 sin 2 
1
1   (
)
2
4 sin 
  1   2
 2   (1 
 and  :
Bend angle and phase
FODO cell
OCPA Accelerator School, 2014, CCKuo- 52
Dispersion suppression in FODO type structure
 1

1
M  
 2f
 0


0 0  1 L

1 0  0 1


0 1  0 0


L2
 1
2f 2

 L
L2
M  

2
4f3
 2f

0


L 
 1
2  1
 
f
1  0


L
2 L1 
 2f
L2
1
2f 2
0
0
 D
 
 




0

M

M

 
1
2  0 
1
1
 
 



L 
 1
2  1
  
 2f
1  0


L  
 
2 L 1 
4
f

 

L
L2  
2 1 
 2  
4
f
8 f 


1



0 0  1 L

1 0  0 1


0 1  0 0

0 0 
1 0

0 1 
1
L (1  sin(  / 2))
L
2
D
,
sin(

/
2
)

, L  
2
sin ( / 2)
2f
1




(
1

)
 2
4 sin 2 

,   1   2 ,   2
1
 1   (
)

4 sin 2 
OCPA Accelerator School, 2014, CCKuo- 53
Dispersion suppression in FODO cell
θ1 θ1 θ2 θ2 θ θ
1




(
1

)
2

 L
4 sin 2 

,   1   2 ,   2 , sin 
1
2 f
 1   (
)
2

4 sin 
OCPA Accelerator School, 2014, CCKuo- 54
Dispersion suppression
Same dipole
Separated functions dipole
OCPA Accelerator School, 2014, CCKuo- 55
MBA:PEP-X (SLAC)
2.2 km
OCPA Accelerator School, 2014, CCKuo- 56
MBA: MAX-IV (Sweden)
20-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
ESRF(France) Hybrid Multi-Bend (HMB) lattice
ESRF existing (DBA) cell
• Ex = 4 nmrad, 7GeV
• tunes (36.44,13.39)
• nat. chromaticity (-130, -58)
• 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
Proposed HMB cell
• Ex = 160 pmrad, 7GeV
• tunes (75.60, 27.60)
• nat. chromaticity (-97, -79)
60
Andrea Franchi ESRF
Momentum compaction
C 1 1 D
1
C   ds  ,  c 
  ds    Di i

C  C 
C i
( DF  DD )
2
1
FODO cell :  c, FODO 

 2
2
2L
sin ( / 2)  x
D
DBA cell :
D0 D0
c 
( 
 )
Lm 6 L 2
 3 1
Lm : half DBA module
L : dipole lengt h
D0 D0
1

   negative
L 2
6
 c , DBA 
 2
6R
• For small bend angle and large
circumference DBA lattice, the 1storder momentum compactor can be
around 10-4.
• Higher-order momentum
compaction will play important role
in the longitudinal motion if the 1storder 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.
OCPA Accelerator School, 2014, CCKuo- 61
Low alpha lattice
c 
C 1
1

C  C
D
  ds
  1   2   3 2
1  105 ~ 106
TPS High emittance
C.C. Kuo ipac2013
Low emittance
Low alpha lattice can have
shorter bunch length (a few ps)
for time-resolved experiments or
THz source
Need sextupole/octupole to
control 2 and 3 to near zero
NewSUBARU invert bend
OCPA Accelerator School, 2014, CCKuo- 62
Chromaticity Effects
K x   k x ( s ) ,
x  ( K x ( s)  K x ( s))x  0
K z   k z ( s )
K x ( s) 
1
-1
 x 


K
ds

 x k x ds 
x
x


4
4
1
-1
 z 


K
ds

 z k z ds 
z
z
4 
4 
K x  [
Chromaticity:

Natural chromaticity:
1
2
2

2
 k x ( s),
 k x ( s)]  o( 2 )


 nat
-1

 kds

4
FODO cell (N cells):
 FODO 
nat
-1
N(βmax  βmin )/f  
4
OCPA Accelerator School, 2014, CCKuo- 63
f (p / p0  0)
f (p / p0  0)
f (p / p0  0)
Sextupoles
x
D>0
p / p0  0
sextupole
s
sextupole
p / p0  0
Focal length
Need focusing strength
proportional to displacement
k ( x)  x
Sextupole field
Bx
Bz
B2
B

( x 2  z 2 ),
 2 xz, B2   2 Bz / x 2 | x  z 0
B0  2 B0 
B0 
B0 
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.
B
Bz
B
B
 2 ( x 2  z 2 ), x  2 xz, B2   2 Bz / x 2 |x  z 0
B0  2B0 
B0  B0 
Let x  x  D , to thefirst order of  
and
Bz
 [ SD ]x
B0 
Bx
B
 [ SD ]z where S  2
B0 
B0 
ΔK x  SD , ΔK z  SD
-1



 x 4   x [k x  S F DF  S D DD ]ds

-1
 z 
 z [k z  S F DF  S D DD ]ds

4

OCPA Accelerator School, 2014, CCKuo- 65
B z

 x  K x ( s ) x   B 

0

 z   K ( s ) z   Bx
z

B0 
1
1 Bz
1 Bx
K x (s)  2 
, K z ( s)  
,

B0  x
B0  z
FODO cell
2L
2L
2L
(1  sin 2 ),  D 
(1  sin 2 ),
 sin 2
sin 
sin 
f
L
L
1

1

DF 
(
1

sin
),
D

(
1

sin
D
2
2
2
2)
2 
2 
sin 2
sin 2
F 
sin 2
1
1
S F 

,
2
1

fDF 2 f  1  2 sin 2
sin 2
1
1
SD  
 2
fDD
2 f  1  12 sin 2
If f1   f 2
S F 
1
1
, SD 
f1 DF
f 2 DF
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
x  K x ( s) x   S ( s)(x 2  z 2 )
z  K z ( s) z  S ( s) xz, x  x  D ,
Normalized coordinate:
w w  w , w 
x
x
D
~
, w  D 

1 2 5/ 2 2
  w     /    kw    0  Sw
w
2
1 2 5/ 2 ~ 2
~
2 ~
2 1/ 2
2 2 ~
D  0 D   0  /   0  kD   0  SD  ,
2
2
0
2
0
3/ 2
2
0
2
1
w  02 w    02  2 kw    02  2 SDw     02  5 / 2 Sw2
2
OCPA Accelerator School, 2014, CCKuo- 68
x
1 2 5/ 2 2
  w     kw     SDw     0  Sw
w
2
2
0
2
0
2
2
0
2
• 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.
OCPA Accelerator School, 2014, CCKuo- 69
Chromatic Aberrations
K (s)  (k  SD)
Beta-beat:
 (s)



2 sin  0
  (t )(k (t )  S (t ) D(t )) cos[2
0
(  |  ( s)   (t ) |]dt
If chromaticity is corrected locally, there is no induced betabeat and no half integer resonances. However, in reality, we
might need several families of sextupoles to get small betabeat
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).
OCPA Accelerator School, 2014, CCKuo- 70
-I transformation
 1 0 0 0 


0

1
0
0


M  I  
0 0 1 0 


 0 0 0  1


1
x   S( x 2  y 2 ), y  Sxy
2
 x0    1 0 0 0  1
x
  
 
 1
 x0   0  1 0 0   Sx0
 x 

(

I
)
M

 2
s
 y
y0   0 0  1 0  0
  
 

 y 
 y   0 0 0  1
 1
 0
 0 
 x0    x0 
  



 x0    x0 
 y     y   (2n  1) phase advance
 0  0
 y    y 
 0  0
0
0
1
Sy0
2
0
1
0 Sx0
1
 x0

0  x0  

   1
2
2
0  x0   S( x0  y0 )  x0 

    2
0  y0  
 y0



1  y0    Sx0 y0  y0 
Cancellation of geometric
aberration after a
complete transformation
matrix through one unit.
OCPA Accelerator School, 2014, CCKuo- 71
Real machine
For periodic lattice, compensation can be across one or several
cells. Thick-lens sextupoles together with errors in real machine
always lead some limitation in its effectiveness.
Diamond phase
cancellation scheme.
Horizontal tune ~ 29.12 due
to long straight perturbation
Selection of working tune
is different from above.
Diamond Light Source
nominal lattice working
tune (27.23, 12.36)
OCPA Accelerator School, 2014, CCKuo- 72
Nonlinear Hamiltonian
S ( s) 3
1
2
2
2
2


( x  3xy 2 )
H ( p x , x, p y , y )  ( p x  K x x  p y  K y y ) 
2
2
H  H 0  V3 ( x, y, s)
S ( s) 3
1
2
2
2
2


( x  3xy 2 )
H 0  ( p x  K x x  p y  K y y ), V3 
2
2
x  2 J x  x cos( x   ( s)  x )
2 1/ 2
J x J y  x1/ 2  y S ( s )[2 cos x  cos( x  2 y )  cos( x  2 y )]
4
2 3/ 2 3/ 2
J x  x S ( s)[cos3 x  3 cos x ]

12
V3  
s
ds
0
x
 x   x   x ( s)  x ,  x ( s)  
s
ds
0
y
 y   y   y ( s)  y ,  y ( s)  
( J x ,x ), ( J y , y )
are pairs of conjugate phase-space
OCPA Accelerator School, 2014, CCKuo- 73
Nonlinear Hamiltonian
Periodic in s, Fourier expansion:
H ( J x ,  x , J y ,  y )   x J x  y J y   G3, 0, J x3 / 2 cos(3 x     3,0, )

  G1, 2, J 1x / 2 J y cos( x  2 y     1, 2, )

  G1, 2, J 1x / 2 J y cos( x  2 y     1, 2, )

 ...
G3,0, e
j
2
j [ 3  x ( s ) ( 3 x  ) ]
3/ 2


S
(
s
)
e
ds
x

24
OCPA Accelerator School, 2014, CCKuo- 74
Nonlinear Effects of Chromatic Sextupoles
Sum resonance:
νx  2νy  
Difference resonance:
Parametric resonance:
νx  2νy  
νx  , 3νx  
Other higher-order resonance:
4νx  , 2νx  2νy  ...
Concatenation of sextupoles perturbation to
the betatron motion can induce nonlinear
betatron detuning.
ν x  ν x 0   xx J x   xy J y
ν y  ν y 0   xy J x   yy J y
Detuning coefficient:
 xx , xy , yy
OCPA Accelerator School, 2014, CCKuo- 75
Sextupole Hamiltonian
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)
Nsxt
h jklmp   (b2 L) n 
jk
2
xn

l m
2
yn
e
l m
2
yn
Dnp e
i[( j  k ) xn  ( l  m ) yn ]
n
[
 (b1L) n 
jk
2
xn

i[( j  k ) xn  ( l  m ) yn ]
] p 0
n
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).
f  p1 | h21000 ( x ) |2  p2 | h10110 ( x ) |2  p3 | h30000 (3 x ) |2
 p4 | h10020 ( x  2 y ) |2  p5 | h10200 ( x  2 y ) |2
 p6 | d x / dJ x |2  p7 | d x / dJ y |2  p8 | d y / dJ y |2
 p9 | h11001 ( x ) |2  p10 | h00111 ( y ) |2
 p11 | h20001 ( 2 x ( )) |2  p12 | h00201 ( 2 y ( )) |2
OCPA Accelerator School, 2014, CCKuo- 76
Objective Function Including Higher-order Driving
terms
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 strength
3 tune shift with amplitude
8 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.
OCPA Accelerator School, 2014, CCKuo- 77
TLS Storage Ring
For TLS, do we need more than two
sextupole families?
Lattice type
TBA
Operational energy
1.5 GeV
Circumference
120 m
Natural emittance
0.075%
Momentum compaction factor 0.00678
Damping time
Horizontal
6.959 ms
Vertical
9.372 ms
Longitudinal
5.668 ms
Betatron tunes horizontal/vertical 7.18/4.13
Natural chromaticities
Horizontal
-15.292
Vertical
-7.868
Radiation loss per turn (dipole) 128 keV
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
10
10
8
8
Y [mm]
Y [mm]
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.
6
4
6
4
2
2
0
-30
-20
-10
0
X [mm]
10
20
30
Dynamic aperture for distributed
sextupoles
0
-30
-20
-10
0
X [mm]
10
20
Local sextupoles only
OCPA Accelerator School, 2014, CCKuo- 79
30
TLS Storage Ring
What is the RF energy
acceptance?
(
eV sin s
p 2
) acc  0
( q  1  cos1 (1 / q)
p0
h c cp0
q
eV0
1

U 0 sin s
T LS:U 0  128keV , No ID
RF Energy
acceptance (%)
3
2
Beta and tune change vs energy
1
0
0
0.5
10
Y [mm]
8
6
4
2
0
-30
-20
-10
0
X [mm]
10
20
30
Phase space
Dynamic aperture
OCPA Accelerator School, 2014, CCKuo- 80
RF1Voltage (MV)
1.5
2
2.5
Betatron Tune and Nonlinear Resonance
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 higherorder nonlinear resonances. Particle tracking study and fequency
(tune)-map analyses can further provide optimization information in
tune selection.
Systematic resonances
Random resonances
n x  m y  6 p
n x  m y  p
n, m, p are integers
n, m, p are integers
OCPA Accelerator School, 2014, CCKuo- 81
Linear Lattice Matching
• Computer design codes are usually used in the matching. (for example
• 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
Objective function
F T   f1
f1 ... f n , f i  f i (k1 , k2 ,...k j ,..km )



F 




f1 
 k1 

 
f2 
 k2 
.  and K   . 

 
. 
 . 

k 
fn 
 m
 f1

 k1
 f 2
A   k1

 ..
 f n
 k
 1
f1
k 2
f 2
k 2
..
f n
k 2
f1 

k m 
f 2 
.. ..
k m 

.. .. .. 
f n 
.. ..
k m 
.. ..
Fideal  F0  A( K  K0 )
m=n, with equal weighting factor and only linear functions considered,
K  K0  A1 ( Fideal  F0 )
Iteration needed due to nonlinearity.
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
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 optimization
Quality factor: Tune shift with amplitude, Phase space
plots,Tune shift with energy, Dynamic aperture (on and off
momentum, 4D, 6D), Frequency Map Analysis
Codes: 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
TPS Sextupole scheme
OPA
S1 S2
SD SD
SF
S4 S3
S5 S6 SD SD
OCPA Accelerator School, 2014, CCKuoSF86
S6 S5
Only Chromatic Sextupoles
OCPA Accelerator School, 2014, CCKuo- 87
Nonlinear Optimization with Sextupoles
OPA
8 families of sextupoles for nonlinear optimization.
Chromaticity of +5 in both planes are still with
acceptable dynamic aperture and energy acceptance.
OCPA Accelerator School, 2014, CCKuo- 88
Phase Space
Betatron Function vs Energy
Tune Shifts vs. Amplitude and Energy
Dynamic Aperture
x,y vs. X
x,y vs. Y
OCPA Accelerator School, 2014, CCKuo- 89
x,y vs. dp/p
RF Energy Aperture
3.5 MV RF
(no chamber limitation)
TPS
Momentumcompactionfactors
1  2.4 10 4 ,  2  2.1103
1 
1   0 ds
L0 
2
1  0 1 
2   
 ds
L0  2

  1  2 2 p / p
Tracy-2
(
p 2
eV sin s
)  0
( q  1  cos 1 (1 / q)
p0
h c cp 0
q
acc
eV0
1

U 0 sin s
TPS : U 0  0.85MeV , No ID
OCPA Accelerator School, 2014, CCKuo- 90
analytical
Dynamic Aperture and Frequency Map Analysis
4uy=53
ux+3uy=66
3uy=40
2ux+2uy=79
3uy=40
4uy=53
6ux=157
With multipole errors
B
 104 at R  25mm
B
Tune diffusion rate:
D= log10((x)2+ (y)2)1/2
for tune difference between
the first 512 turns and
second 512 turns
OCPA Accelerator School, 2014, CCKuo- 91
Liouville theorem
foundation of accelerator physics
“Beam phase space is a constant”
Joseph Liouville,
1809 – 1882
• 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).  p p  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 theorem
i.e. normalized emittance
e N  e 0 ,   v / c,   1 / 1   2
e 0  geometricemittance
OCPA Accelerator School, 2014, CCKuo- 92
Non-conservation of emittance
• Coupling (horizontal-vertical, chromaticity,..)
• Scattering (gas, intra-beam, beam-beam,
through foil)
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
• 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

• The diffraction limited emittance is e 
E (keV) = 1.24 / λ (nm)
4
OCPA Accelerator School, 2014, CCKuo- 94
• Charged particle under acceleration (through EM field, etc.) will radiate
EM wave.

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.

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
c
E4
P 
C 2
2

U 0   Pdt 
U0
c
E4
 P  

C
T0 2R

d (E ) 1 
 (eV  WE )
dt
T0
 (t )  Ae
2
ds

2

C E0

4
C  8.846105 m /(GeV )3
d 2
d

2

e
V
W
2

2





0
c
E
s
E 
, s  
dt 2
dt
2T0
T0 E
E 
cos(st  0 )
D parameter :
4
dU
U ( E )  U 0  WE , W 
| E  E0
dE
d (E ) eV ( )  U ( E )
d
E

, and
  cT0
dt
T0
dt
E
Plus gain energy from rf:
 E t
C E0
U
W
 0 ( 2  D)
2T0 2T0 E
Isomagnetic
 D( s) 1
  ds 
D  
( 2  2 K ( s ))ds   2 
  
  
1
1

D
(
s
)(
 2 K ( s ))dipole ds
2

2

1
Isomagnetic
separated function
ring
OCPA Accelerator School, 2014, CCKuo- 96
U 0  P 
E 

ET0
E
Damping of Vertical Betatron Oscillation
z  A cos , z  
A

sin  , A2  z 2  ( z) 2 , A  a 
After emitting radiation and rf acceleration:
p
u
p
E
U
AA    2 z z )   ( z ) 2  0
E
z   z
p
z
p
p 
  z
Average one turn
s
(z)   A / 2
2
2
A2 U 0
AA  
2 E
U0
1 dA 1 A


A dt T0 A
2 ET0
 P 
U0
z 

2ET0 2E
OCPA Accelerator School, 2014, CCKuo- 97
Damping of Horizontal Betatron Oscillations
E
E
 x  xe , x  x  D
 x  xe
E
E
u
u




x



x


D
,

x



x


D
e

e
E
E
Change in betatron amplitude and average in betatron phase:
u
AA  x x   2 x x  ( Dx    2 Dx )
E
x P
2 dB
AA  x D(1 
x  )
ds
B dx
 cE
x  x  D
Average one turn:
1

U
A U 0  D 
1   1
  2 K  2  ds   2 ds   D 0

A
2 E   
   
2E

Including rf acceleration:
U
A
 (1  D ) 0
A
2E
 x  (1  D)
P 
U0
 (1  D) 
2ET0
2E
OCPA Accelerator School, 2014, CCKuo- 98
Robinson theorem
U0
x 
(1  D)  J x 0
2T0 E
U0
z 
 J z 0
2T0 E
U0
s 
( 2  D )  J E 0
2T0 E
U0
0 
, Jz 1
2 ET0
For separatedfunctionstrongfocusing ring :
D  1, J x  J z  1, J E  2
For combinedfunctionstrongfocusing ring :
D  2, J x  1, J z  1, J E  4
still works for hardronmachine(U 0 /E verysamll)
Robinson damping partition theorem
Jx  Jz  JE  4
 D( s) 1
  ds 
D  
( 2  2 K ( s ))ds   2 
  
  
1
1

D
(
s
)(
 2 K ( s ))dipole ds
2

2

1
But used for leptonmachine(emittanceconcerned),
need changeD by changingorbit or using Robison wiggler, e
For combinedfunctionelectronring, stabilitycondition:
1  D  2
OCPA Accelerator School, 2014, CCKuo- 99
Synchrotron oscillation without fluctuation and damping:
E  A cosst
d  A2 
 A2 
 2
 Nu 2
dt
E
In equilibrium:
Define
N : rateof photonemission
1
 A2   Nu 2 E
2
2

A
 1
 E2 
 Nu 2 E
2
4
3 55
 3  P 
c 3
where N u  
2 24 3  1 /  2 
2
(
E
E
) 2  Cq
2
J E 1 /  2 
1 /  3  Cq 
For isomagnetic ring:
E
E 
2E
J E  P 
55 
 3.831013 m
32 3 m c
2
( )  Cq
E
JE
2
OCPA Accelerator School, 2014, CCKuo- 100
Horizontal Emittance
Radiation emission results in change of betatron coordinates:
x   D
u
u
, x   D
E
E
Change of Courant –Snyder invariant after average:
2
 x  
1  2 
H x   Dx    x Dx  Dx  
 x 
2

 
2
 a 2  N u 2 H  s
Including damping term   a 
 2

and average one turn:
dt
x
E2
3 55
 3  P 
2
N u  
c 3
Equilibrium:
2
2
2
2


1
/


24
3


N

u

H

1
x x
2
s
u 2
2
 a   H ( )
E
a    x
2
E
2

x
Natural emittance
3
 x2

H
/
|

|

2
13
x
ex 
 Cq 
,
C

3
.
83

10
m
q
2
x
J x 1 /  
x
OCPA Accelerator School, 2014, CCKuo- 101
Vertical emittance
For an ideal flat accelerator without errors, due to the finite
emission angle of synchrotron, we can get natural vertical
emittance as:
 β y / | ρ |3 
13
ε y  Cq

10
2
J y 1/ρ 
However, due to spurious vertical dispersion by errors,
ε y  Cq γ 2
H y / | ρ |3 
J y 1/ρ 2 
 1013m  rad, H y  γ y Dy2  2α y Dy Dy  β y Dy2
Betatron coupling from skew quads, etc., also generate vertical
emittance coupled from horizontal emittance.
OCPA Accelerator School, 2014, CCKuo- 102
Radiation integral and electron beam properties
I1  
I2  
I3  
x
ds

1

2
(2) Emittancee x  Cq 2 I 5 /( I 2  I 4 )
Cq  (55 / 32 3 ) / m c  3.831013 m
ds
1
||
3
(3) Energyspread ( E / E ) 2  Cq 2 I 3 /(2 I 2  I 4 )
ds
x 1
(
 2 K )ds
 2
H
I5  
ds,
|  |3
I4  
whereH 
(1) Mom.comp. c  I1 / 2R
1
x
( x2  ( x x   x x' ) 2 )
(4)Energyloss per turnU 0  C E 4 I 2 / 2
C  (4 / 3)r0 /(m c2 ) 3  8.85105 m /(GeV )3
(5) J x  1  I 4 / I 2 , J E  2  I 4 / I 2 , D  I 4 / I 2
OCPA Accelerator School, 2014, CCKuo- 103
FODO cell
F 
2L

2L
 2L

(1  sin ),  D 
(1  sin ),
 sin
sin 
2
sin 
2
f
2
DF 
L
1

L
1

(
1

sin
),
D

(
1

sin
)
D
sin 2 ( / 2)
2
2
sin 2 ( / 2)
2
2
cos( / 2)
1
 2
(
1

sin
)
3
sin ( / 2)(1  sin( / 2)
2
2
cos( / 2)
1
 2
2
H D  L
(1  sin )
3
sin ( / 2)(1  sin( / 2)
2
2
H F  L 2
1
 2
1
 2

(
1

sin
)
(
1

sin
) 
 cos( / 2) 
2
2 
2
2
H  

3
 
2 sin ( / 2)  (1  sin  )
(1  sin ) 
2
2 

2
1

(
3
/
4
)
sin
( / 2) 1
2 3
e x  FFODO Cq   3
Cq 2 3
sin ( / 2) cos( / 2) J x
OCPA Accelerator School, 2014, CCKuo- 104
FODO Lattice
2
1

(
3
/
4
)
sin
( / 2) 1
2 3
e x  FFODO Cq   3
Cq 2 3
sin ( / 2) cos( / 2) J x
Let J x  1
e ME  FCq 
2
3
F~1.2 for phase advance ~ 138° per cell
F~ 2.5 for 90°
• 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.
OCPA Accelerator School, 2014, CCKuo- 105
Jx  1
Minimum Emittance in DBA
DBA: small bend angle
At bend entrance,D0=D0’=0 Bend length L
 ( s)   0  2 0 s   0 s 2
 (s)   0   0 s
 (s)   0

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
D(s)  D0  D0 ' s   (1  cos )   (1  cos )
D' (s)  D0 ' sin   sin 

1
1
H   H ( s)ds   (  ( s) D'2 ( s)  2 ( s) D( s) D' ( s)  D'2 ( s))ds
0
0
2 3
 0  0
3 0
C

  0  0  0
q
H   (  
)
ex 
(  
)
3 4 20
J
3 4 20
x
e x e x

0
Optimum emittance:
 0  0
e MEDBA
Cq 2 3
0,min  (6 / 15), 0,min  15
Cq 2 3

 0.0645
Jx
4 15J x
Each MEDBA module with
each dipole of 156.70, dispersion
matching 1220, and about 1.2 unit
of horizontal tune per DBA cell.
OCPA Accelerator School, 2014, CCKuo- 106
Minimum Emittance in DBA
For non-achromat lattice, minimization of I5 lead to
1
ex 
12 15
Cq 2 3
Jx
T ME(non- achromat)  (s   / 2)  0, D(s   / 2)  0
8
1 2
1

0 
,  0  15, D0   , D0   
6
2
15
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
• For non-achromat configuration, we can reach 1.6
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)
L32
L13
3 2
2
ρ2
ρ1
1 Cq  
1 Cq  

, e TME 
4 15 J x
12 15 J x
3
1
2
e METBA
isomagnetic ring:
1 
2
3
1
1
1/ 3

,


2




(
2

3
)1
2
1
2
1/ 3
3
1 Cq  
1 Cq 


12 15 J x
12 15 J x
2
e TME
  127.76
3
1
2





 ( M  2)3 3  2 


 : T otalbend angle in a cell, M  2
OCPA Accelerator School, 2014, CCKuo- 109
3
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
3
e METBA
 3 

e MEDBA  0.66e MEDBA
1/ 3 
 23 
3
e MEQBA
 2 

e MEDBA  0.55e MEDBA
1/ 3 
1 3 
OCPA Accelerator School, 2014, CCKuo- 110
Effective emittance
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
e
 x x'  xx'
the long straights,
2
2
2
2
x ,eff
e x ,eff  e x2  H ID E2 e x
e eff  e x 1
H ID  x Dx2  2 x Dx Dx'   x Dx'2 in ID section
TPS emittance
5
natural emittance
4
effective emittance
3
2
1
0
0
0.05
0.1
0.15
0.2
dispersion (m)
OCPA Accelerator School, 2014, CCKuo- 111
( E Dx ) 2
e xx
Closed Orbit Distortion
In reality, dipole field errors distributed around the ring:
yco ( s )  
s c
s
B( s ' )
G ( s, s ' )
ds '
B
 (s) (s' )
where G ( s, s ' ) 
cos(  |  ( s )   ( s ' ) |)
2 sin 
In dipole angular kick form:
 (s) N
yco ( s ) 
 i  ( si ) cos(  |  ( s )  ( si ) |)

2 sin  i 1
 s 
y co,rms 
N  rms
2 2 | sin  |
xco,rms


 av


N q xrms ,


 2 2 f av | sin  |

i 
B(si )
dsi
B
(Bz / x )
y
y 
B
f
The coefficient in curly brackets is
called sensitivity or amplification
factor.
OCPA Accelerator School, 2014, CCKuo- 112
COD correction, SVD method:
y co ( s j ) 
A ji 
 (s j )
N


2 sin 
i 1
 (s j )
i
 ( si ) cos(  |  ( s j )   ( si ) |)
 ( si ) cos(  |  ( s j )   ( si ) |)
2 sin
The distorted orbit can be minimized at least at the BPM to the
desired value so that


 n   A yco,m
1
OCPA Accelerator School, 2014, CCKuo- 113
ID effect
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:
Bx  0
By  Bw cos(kws) cosh(kw y)
Bs  Bw sin(k w s) sinh(k w y)
The equations of motion:

d 2x
1

cosh(k w y) cos(k w s),

2
w
ds

 2
2
 d y  sin (k w s) sinh(2k w y)  p x sinh(k w y) sin(k w s),
2

2k w
w
 w2
 ds
sin 2 (k w s )
average focusing strength:
vertical tune shift:
 y 
 y Lw
 w2

1
2  w2
8 w2
Beta beat in vertical plan can be evaluated
w20
Tune-shift with gap
for w20 at TLS
Vertical beta perturabtion due to
Wiggler 20 in TLS (Kuo, et al
PAC95)
OCPA Accelerator School, 2014, CCKuo- 114
Correction Algorithm
1.
2.
3.
4.
By the following linear
relations, construct the
response matrix A of bare
lattice.
Use SVD method to restore
the perturbed optics back to
the optics of the bare lattice
as much as possible by
minimizing (AΔk +b).
Where Δk is the tuning
and b is the perturbed
optics.
One can choose arbitrary
positions to restore optics
with different weighting.
Current (A)
λ (mm)
Nperiod
By (T)
Bx (T)
L (m)
Gap (mm)
Total power (kW)
 w1






w2

  x1 
 x1 
  
  y1 
  y1 
 k1 
  





k




2 
x1 

A
  
  







wn   y1 
 k nq 
  


 Qx 


 Q y 
SW60
0.4
60
8
3.5
0.45
17
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
8Bw U w
 8 0 U w

, planar undulator

3

f

U
3

f
B
U
I 5w  h w 0
h 0
0
I 50  I 5 w

2

Emittance with ID e x  Cq
B U
I 50   0 U w
I 20  I 40  I 2 w  I 4 w
 w w , helical undulator
 f h  w U 0 f h B0 U 0
1
2
I 50  f hH ID  3 ds  f hH ID 2 ,

0
8Bw U w

(
1

w 3f B U )
H dipole

2
2
fh 
, H   x x  2 x x x   x x
h 0
0
, planar undulators

H ID
U
 (1   w )
3

4
H
L
/(
3

)
,
planar
undulator
1
 ID w
w
w U0
ex 
I 5 w  H ID 
ds  
3
3
w|  |

H ID Lw /  w , helicalundulator
Bw U w
e
x0

4
(
1

)

U 0  C E /  0

w f h B0 U 0
, helical undulators
C E 4 Lw /(4w2 ), planarundulator

U
Uw  
(1   w )

4
2
C E Lw /(2w ), helicalundulator
w U0

Emittancewithout ID e x 0  Cq 2
I 50
I 20  I 40
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
4 Lw /( 3 w3 ), planar undulator
I 30  2 , I 3 w  
3
0
 Lw /  w , helical undulator
8 Bw U w
 8 0 U w

, planar undulator

3

U
3

B
U
I 3w

w
0
0
0

I 30
  0 U w  Bw U w , helical undulator

B0 U 0
 f w U 0

I3

1  I 3 w / I 30
( E ) 2  C q 2
 ( E ) 02
E
(I2  I4 )
E
1  I 2 w / I 20
2
8 Bw U w
Uw

(
1

)
/(
1

), planar undulators


3

B
U
U
w
w
E 2
E 2 

0
0
0
(
) (
)0  
E
E
(1   Bw U w ) /(1   U w ), helical undulators

w B0 U 0
w U0

U 0  C E 4 /  0
4
2
C
E
L
/(
4

), planar undulator


w
w

Uw  
4
2
C
E
L
/(
2

), helical undulator


w
w

If Bw > (3π/8)B0 then energy
undulator.
OCPA Accelerator School, 2014, CCKuo- 118
Touschek scattering
• 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
• Touschek lifetime can be expressed as: Bruck formula
 e (s) 2 
C  acc'  
   x (s)  
1  r e2 cN  1 L
.
 
ds
3  
'
2
 T 1  8   l  L 0  x (s) z (s) x (s) e acc (s)
2
where V=83/2 xyL, =[ eacc/x’ ]2, N is the number of electrons per
bunch.
C ( )  1.5e


  ln u
2 

e u
e du  0.5(3   ln   2) 
du
u
u

u
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, 2000
4. 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