S. Guiducci, INFN-LNF
Seventh International Accelerator School for Linear Colliders
Hosted by Raja Ramanna Centre for Advanced Technology
4 December 2012
Outline
 A3.1 - DR Basics: Introduction to Damping Rings
 Role of the damping rings in the ILC accelerator complex
 Review parameters and constraints of CLIC and ILC damping rings
 Identify key challenges
 A3.2 - DR Basics: General Linear Beam Dynamics
 Review the basic physics of storage rings including the linear beam
dynamics
 A3.3 - LER Design: Radiation Damping and Equilibrium Emittance
 A3.4 - LER Design: Damping Ring Lattices
 A3.5 – DR Technical systems
 A3.6 – Beam Dynamics
 A3.7 – R&D Challenges and Test Facilities
 A3.8 – Circular Colliders
These slides have been presented at the 2010 LC school by Mark Palmer
2
Storage Ring Basics
Now we will begin our review of storage ring basics. In particular,
we will cover:
–
–
–
–
–
–
–
Ring Equations of Motion
Betatron Motion
Emittance
Transverse Coupling
Dispersion and Chromaticity
Momentum Compaction Factor
Radiation Damping and Equilibrium Beam Properties
October 31, 2010
A3 Lectures: Damping Rings - Part 1
3
Equations of Motion
Particle motion in electromagnetic fields is governed by the
Lorentz force: dp
dt

e EvB

with the corresponding Hamiltonian:


2 1/2
H  c  m2c 2  P  eA 


H
H
x
, Px  
,...
Px
x
 e
For circular machines, it is convenient to convert to a curvilinear
coordinate system and change the independent variable from time
to the location, s-position, around the ring.
r
y
s
In order to do this we transform
x
r
to the Frenet-Serret
coordinate system.
Reference Orbit
The local radius of
r  r0  xxˆ  yyˆ
curvature is denoted by r.
0
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A3 Lectures: Damping Rings - Part 1
4
Equations of Motion
With a suitable canonical transformation, we can re-write the
Hamiltonian as:

2

x    H - e 
2
2 2
H = - 1   

m
c   px  eAx    p y  eAy  
2
c

 r  

1/2
2
Using the relations
E  H  e ,
 eAs
E2
2 2
p
m c
2
c
and expanding to 2nd order in px and py yields:
2

x  1 x r 
2
  eA
H  - p 1   
p

eA

p

eA




x
x
y
y
s




2p
 r
which is now periodic in s.
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A3 Lectures: Damping Rings - Part 1
5
Equations of Motion
Thus, in the absence of synchrotron motion, we can generate the equations of
motion with:
H
H
H
H
x 
, px  
, y 
, py  
px
x
p y
y
which yields:
2
By p0 
rx
x

x  2 
1   , top / bottom sign for + / - charges
r
Br p  r 
and
y 
Bx p0 
x
1  
Br p  r 
2
Note: 1/Br is the beam rigidity
and is taken to be positive
Specific field configurations are applied in an accelerator to achieve the desired
manipulation of the particle beams. Thus, before going further, it is useful to
look at the types of fields of interest via the multipole expansion of the
transverse field components.
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6
Magnetic Field Multipole Expansion
Magnetic elements with 2-dimensional fields of the form
B  Bx  x, y  xˆ  By  x, y  yˆ
can be expanded in a complex multipole expansion:

By ( x, y )  iBx ( x, y )  B0   bn  ian  x  iy 
n
n 0
n

By
1
with bn 
n ! B0 x n
 x , y   0,0 
1  n Bx
and an 
n ! B0 x n
 x , y   0,0 
In this form, we can normalize to the main guide field strength,
-Bŷ, by setting b0=1 to yield:
1
e
By  iBx    By  iBx  

Br
p0
October 31, 2010
1

b

r
n 0
n
 ian  x  iy  for  q
A3 Lectures: Damping Rings - Part 1
n
7
Multipole Moments
Upright Fields
Skew Fields
Dipole (q  90°):
Dipole:
e
By   x
p0
e
Bx  0
p0
e
Bx  ky
p0
e
By  kx
p0
e
Bx  kskew x
p0
e
By  kskew y
p0
Sextupole (q  30°):
Sextupole:
e
1
By  m  x 2  y 2 
p0
2
e
1
Bx   mskew  x 2  y 2 
p0
2
e
By  mskew xy
p0
Octupole (q  22.5°):
Octupole:
e
1
Bx   rskew  x3  3xy 2 
p0
6
e
1
Bx  r  3x 2 y  y 3 
p0
6
e
1
By  r  x 3  3xy 2 
p0
6
October 31, 2010
e
By  0
p0
Quadrupole (q  45°):
Quadrupole:
e
Bx  mxy
p0
e
Bx   y
p0
A3 Lectures: Damping Rings - Part 1
e
1
By  rskew  3x 2 y  y 3 
p0
6
8
Equations of Motion (Hill’s Equation)
We next want to consider the equations of motion for a ring with
only guide (dipole) and focusing (quadrupole) elements:
p0
By  B0  kx  B0  r kx 1 and
e
p0
Bx 
ky  B0 r kx
e
Taking p=p0 and expanding the equations of motion to first order in
x/r and y/r gives:
1
x  K x  s  x  0,
Kx  s 
y  K y  s  y  0,
K y  s   k  s 
r s
2
k s
also commonly
denoted as k1
where the upper/low signs are for a positively/negatively charged
particle.
The focusing functions are periodic in s:
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A3 Lectures: Damping Rings - Part 1
K x, y  s  L   K x, y  s 
9
Solutions to Hill’s Equation
Some introductory comments about the solutions to Hill’s
equations:
– The solutions to Hill’s equation describe the particle motion around a
reference orbit, the closed orbit. This motion is known as betatron
motion. We are generally interested in small amplitude motions around
the closed orbit (as has already been assumed in the derivation of the
preceding pages).
– Accelerators are generally designed with discrete components which
have locally uniform magnetic fields. In other words, the focusing
functions, K(s), can typically be represented in a piecewise constant
manner. This allows us to locally solve for the characteristics of the
motion and implement the solution in terms of a transfer matrix. For
each segment for which we have a solution, we can then take a particle’s
initial conditions at the entrance to the segment and transform it to the
final conditions at the exit.
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A3 Lectures: Damping Rings - Part 1
10
Solutions to Hill’s Equation
Let’s begin by considering constant K=k:
x  kx  0
where x now represents either x or y. The 3 solutions are:
x( s )  a sin
 k s   b cos  k s  ,
x( s )  as  b,
x( s )  a sinh


k s  b cosh


ks ,
k 0
Focusing Quadrupole
k 0
Drift Region
k 0
Defocusing Quadrupole
For each of these cases, we can solve for initial conditions and
recast in 2×2 matrix form:
 x   m11 m12   x0 
  
    m
 x   21 m22  x0 
x  M  s s0  x0
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A3 Lectures: Damping Rings - Part 1
11
Transfer Matrices
We can now re-write the solutions of the preceding page in
transfer matrix form: 
1

 cos k
sin k 

Focusing

k

Quadrupole


cos k
  k sin k


 1 
Drift Region
M  s s0   

 0 1 

1

 cosh k
sinh k 
Defocusing
k

 Quadrupole


 k sinh k
cosh k



where  s  s .
 
 

 
 







0
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A3 Lectures: Damping Rings - Part 1
12
Transfer Matrices
Examples:
 0,
– Thin lens approximation:
M focusing
 1

 1 f
0

1
1
f = lim
0 K
M defocusing
 1

1 f
0

1
– Sector dipole (entrance and exit faces ┴ to closed orbit):
c.o.
M sector
October 31, 2010
 cos q
 1
  sin q
 r

r sin q   1

cos q    2
  r


1 

A3 Lectures: Damping Rings - Part 1
where q 
r
13
Transfer Matrices
Transport through an interval s0 s2 can be written as the product
of 2 transport matrices for the intervals s0 s1 and s1 s2:
M  s2 s0   M  s2 s1  M  s1 s0 
and the determinant of each transfer matrix is:
Mi  1
Many rings are composed of repeated sets of identical magnetic
elements. In this case it is particularly straightforward to write the
one-turn matrix for P superperiods, each of length L, as:
M ring  M  s  L s  
with the boundary condition that:
P
M s  L s  M s
The multi-turn matrix for m revolutions is then:
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A3 Lectures: Damping Rings - Part 1
M  s  
mP
14
Twiss Parameters
The generalized one turn matrix can be written as:
b sin 
 cos   a sin 

M 
  I cos   J sin 
cos   a sin  
 g sin 
Identity matrix
This is the most general form of the matrix. a, b, and g are known
as either the Courant-Snyder or Twiss parameters (note: they
have nothing to do with the familiar relativistic parameters) and 
is the betatron phase advance. The matrix J has the properties:
b 
a
2
2
J 
,
J


I

bg

1

a


g

a


The n-turn matrix can be expressed as: Mn  I cos  n   J sin  n 
which leads to the stability requirement for betatron motion:
Trace  M   2 cos   2
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A3 Lectures: Damping Rings - Part 1
15
The Envelope Equations
We will look for 2 independent solutions to Hill’s Equation of the
form:
x  s   aw  s  eiy  s  and x  s   aw  s  eiy  s 
Then w and y satisfy:
1
w  Kw  3  0
w
1
y 2
w
Betatron envelope
and
phase equations
Since any solution can be written as a superposition of the above
solutions, we can write [with wi=w(si)]:
w2

cosy  w2 w1 siny

w1

M  s2 s1  
 1  w1w1w2 w2 
 w1 w2 
siny     cosy
 
w1w2
 w2 w1 

October 31, 2010
A3 Lectures: Damping Rings - Part 1




w1
cosy  w1w2 siny 
w2

w1w2 siny
16
The Envelope Equations
Application of the previous transfer matrix to a full turn and direct
comparison with the Courant-Snyder form yields:
w2  b
a   ww  
b
2
the betatron envelope equation becomes
1
1
b   K b 
2
b
 b 2 
1  4   0


and the transfer matrix in terms of the Twiss parameters can
immediately be written as:

b2
 cos y  a1 sin y 

b1

M  s2 s1   
  1  a1a 2 sin y  a1  a 2 cos y

b1b 2
b1b 2

October 31, 2010

b1b 2 sin y



b1
 cos y  a 2 sin y  
b2

A3 Lectures: Damping Rings - Part 1
17
General Solution to Hill’s Equation
The general solution to Hill’s equation can now be written as:
x  s   A b x  s  cos y x  s   0  where y x  s   
s
0
ds
bx  s 
We can now define the betatron tune for a ring as:
turn
1
Qx   x 

2
2

s C
s
ds
where C  ring circumference
bx  s
If we make the coordinate transformation:
x
and   s  
1
s
ds
bx  s
 x 0
bx
we see that particles in the beam satisfy the equation for simple
harmonic motion:
d 2z
2


xz 0
2
d
z
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18
The Courant-Snyder Invariant
With K real, Hill’s equation is conservative. We can now take
x  s   A b x  s  cos y x  s   0  and
x  s   
A
bx  s
a  s  cos y
x
 s   0   sin y x  s   0 
After some manipulation, we can combine these two equations to
give:
2
Conserved
2
 a s

quantity
x
A2   
 x
x  b x  s  x 
bx  s  bx  s


Recalling that bg  1a2 yields:
A2    g  s  x2  s   2a  s  x  s  x  s   b  s  x2  s 
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A3 Lectures: Damping Rings - Part 1
19
Emittance
The equation
g  s  x2  s   2a  s  x  s  x  s   b  s  x2  s   
describes an ellipse with area .
For an ensemble of particles, each
following its own ellipse, we can
define the moments of the beam as:
x   x r  x, x dxdx
  x  x
2
x
 xx2     x 
a

b
g

b
Area = 

g
a
b
x

g
x   xr  x, x dxdx
    x 
 r  x, x dxdx
x  x  x r  x, x  dxdx  r 
2
x
2
x
x
x
 r  x, x dxdx
2
x
A2
The rms emittance of the beam is then
 rms   x2 x2   xx2  
2
which is the area enclosed by the ellipse of an rms particle.
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A3 Lectures: Damping Rings - Part 1
20
Coupling
Up to this point, the equations of motion that we have considered
have been independent in x and y. An important issue for all
accelerators, and particularly for damping rings which attempt to
achieve a very small vertical emittance, is coupling between the
two planes. For the damping ring, we are primarily interested in
the coupling that arises due to small rotations of the quadrupoles.
This introduces a skew quadrupole component to the equations of
motion.
x  K x  s  x  0  x  K x  s  x  k skew y  0
y  K y  s  y  0  y  K y  s  y  k skew x  0
Another skew quadrupole term arises from “feed-down” when the
closed orbit is displaced vertically in a sextupole magnet. In this
case the effective skew quadrupole moment is given by the
product of the sextupole strength and the closed orbit offset
kskew  myco
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A3 Lectures: Damping Rings - Part 1
21
Coupling
For uncoupled motion, we can convert the 2D (x,x′) and (y,y′)
transfer matrices to 4D form for the vector (x,x′,y,y′):
 Mfocusing
M 4D  s s0   
 0
0 
  MF


Mdefocusing   0 M D 
0
where we have arbitrarily chosen this case to be focusing in x.
The matrix is block diagonal and there is no coupling between the
two planes. If the quadrupole is rotated by angle q, the transfer
matrix becomes:
M skew
 M F cos 2 q  M D sin 2 q sin q cos q  M D  M F  
 

2
2
 sin q cos q  M D  M F  M D cos q  M F sin q 
and motion in the two planes is coupled.
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A3 Lectures: Damping Rings - Part 1
22
Coupling and Emittance
Later in this lecture series we will look in greater detail at the
sources of vertical emittance for the damping rings.
In the absence of coupling and ring errors, the vertical emittance of
a ring is determined by the the radiation of photons and the fact
that emitted photons are randomly radiated into a characteristic
cone with half-angle q1/2~1/g. This quantum limit to the vertical
emittance is generally quite small and can be ignored for presently
operating storage rings.
Thus the presence of betatron coupling becomes one of the
primary sources of vertical emittance in a storage ring.
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23
Dispersion
In our initial derivation of Hill’s equation, we assumed that the
particles being guided had the design momentum, p0, thus ignoring
longitudinal contributions to the motion. We now want to address
off-energy particles. Thus we take the equation of motion:
2
By p0 
rx
x
x  2  
1  
r
Br p  r 
and expand to lowest order in
and
which yields:
x
p
d
r
p0
d
x  K  s  x 
r solution, xb(s). If we
We have already obtained a homogenous
denote the particular solution as D(sd, the general solution is:
x  xb  s   D  s  d
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24
Dispersion Function and Momentum Compaction
The dispersion function satisfies:
with the boundary conditions:
D  K (s) D  1 r
D  s  L   D  s ; D  s  L   D  s 
The solution can be written as the sum of the solution to the
homogenous equation and a particular solution:
 D  s2  
 D  s1    d 

  M  s2 s1  
 
 D  s2  
 D  s1    d  
which can be expressed in a 3×3 matrix form as:
 D  s2  
 D  s1  

  M  s2 s1  d  



  D  s1   ,
 D  s2    
0
1 
 1  



 1 
October 31, 2010
d
where d   
 d
A3 Lectures: Damping Rings - Part 1
25
Momentum Compaction
We can now consider the difference in path length experienced by
such an off-momentum particle as it traverses the ring. The path
xc.o.
length of an on-momentum particle is given by:
C
ds
r
For the off-momentum case, we then have: C  d  D  s ds  I d
1
 r
I1 is the first radiation integral.
The momentum compaction factor, ac, is defined as:
ac 
October 31, 2010
C C
d
I1

C
A3 Lectures: Damping Rings - Part 1
26
The Synchrotron Radiation Integrals
I1 is the first of 5 “radiation integrals” that we will study in this
lecture. These 5 integrals describe the key properties of a storage
ring lattice including:
–
–
–
–
Momentum compaction
Average power radiated by a particle on each revolution
The radiation excitation and average energy spread of the beam
The damping partition numbers describing how radiation damping is
distributed among longitudinal and transverse modes of oscillation
– The natural emittance of the lattice
In later sections of this lecture we will work through the key
aspects of radiation damping in a storage ring
October 31, 2010
A3 Lectures: Damping Rings - Part 1
27
Chromaticity
An off-momentum particle passing through a quadrupole will be
under/over-focused for positive/negative momentum deviation.
This is chromatic aberration. Hill’s equation becomes:
x   K 0  s 1  d   x  0
We will evaluate the chromaticity by first looking at the impact of
local gradient errors on the particle beam dynamics.
October 31, 2010
A3 Lectures: Damping Rings - Part 1
28
Effect of a Gradient Error
We consider a local perturbation of the focusing strength
K = K0+K. The effect of K can be represented by including a
thin lens transfer matrix in the one-turn matrix. Thus we have
M K
and
 1

 K
0

1
b sin 
 cos   a sin 

M1turn  


g
sin

cos


a
sin



b sin  0
0
 cos  0  a sin  0
 1




g
sin

cos


a
sin


K
1

0
0
0 

With 0, we can take the trace of the one-turn matrix to
give:
1
cos   0     cos  0  bK sin  0
2
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A3 Lectures: Damping Rings - Part 1
29
Effect of a Gradient Error
Using the relation:
cos  0     cos  cos 0  sin  sin 0
we can identify:
1
  b K
2
Thus we can write: Q 
1
bK
4
and we see that the result of gradient errors is a shift in the
betatron tune. For a distributed set of errors, we then have:
1
Q 
4
 bKds
which is the result we need for evaluating chromatic aberrations.
Note that the tune shift will be positive/negative for a
focusing/defocusing quadrupole.
October 31, 2010
A3 Lectures: Damping Rings - Part 1
30
Chromaticity
We can now write the betatron tune shift due to chromatic
aberration as:
1
d
Q 
bKds  
b Kds


4
4
The chromaticity is defined as the change in tune with respect to
the momentum deviation:
C
Q
d
Because the focusing is weaker for a higher momentum particle,
the natural chromaticity due to quadrupoles is always negative.
This can be a source of instabilities in an accelerator. However,
the fact that a momentum deviation results in a change in
trajectory (the dispersion) as well as the change in focusing
strength, provides a route to mitigate this difficulty.
October 31, 2010
A3 Lectures: Damping Rings - Part 1
31
Sextupoles
Recall that the magnetic field in a sextupole can be written as:
e
1
By  m  x 2  y 2 
p0
2
e
Bx  mxy
p0
Using the orbit of an off-momentum particle
e
we obtain
B  mD s d y s  mx s y
x
and
 
b
 
b
x  xb  s   D  s  d
 
p0
e
1
1
By  mD  s  d xb  s   mD 2  s  d 2  m  xb2  s   yb2  s  
p0
2
2
where the first terms in each expression are a quadrupole feeddown term for the off-momentum particle. Thus the sextupoles can
be used to compensate the chromatic error. The change in tune
due to the sextupole is
Q 
October 31, 2010
d
4
 mD  s b  s  ds
A3 Lectures: Damping Rings - Part 1
32
Summary
During the last portion of today’s lecture, we have begun our walk
through the basics of storage/damping ring physics.
We will pick up this discussion tomorrow with the effect known as
radiation damping which is central to the operation of all lepton
collider, storage and damping rings.
Once we have completed that discussion we will look in greater
detail at the lattice choices that have been made for the damping
rings and how these lattices are presently being forced to evolve.
In the first part of today’s lecture we had an overview of the key
design issues impacting the damping ring lattice. The homework
problems will provide an opportunity to become more familiar with
some of these issues.
October 31, 2010
A3 Lectures: Damping Rings - Part 1
33
Bibliography
1.
2.
3.
4.
5.
6.
7.
8.
9.
The ILC Collaboration, International Linear Collider Reference Design Report
2007, ILC-REPORT-2007-001,
http://ilcdoc.linearcollider.org/record/6321/files/ILC_RDR-August2007.pdf.
S. Y. Lee, Accelerator Physics, 2nd Ed., (World Scientific, 2004).
J. R. Rees, Symplecticity in Beam Dynamics: An Introduction, SLAC-PUB-9939,
2003.
K. Wille, The Physics of Particle Accelerators – an introduction, translated by J.
McFall, (Oxford University Press, 2000).
S. Guiducci & A. Wolski, Lectures from 1st International Accelerator School for
Linear Colliders, Sokendai, Hayama, Japan, May 2006.
A. Wolski, Lectures from 2nd International Accelerator School for Linear Colliders,
Erice, Sicily, October 2007.
A. Wolski, Lectures from 4th International Accelerator School for Linear Colliders,
Beijing, China, October 2009.
A. Wolski, J. Gao, S. Guiducci, ed., Configuration Studies and Recommendations
for the ILC Damping Rings, LBNL-59449 (2006). Available online at:
https://wiki.lepp.cornell.edu/ilc/pub/Public/DampingRings/ConfigStudy/DRConfigRe
commend.pdf
Various recent meetings of the ILC and CLIC damping ring design teams.
October 31, 2010
A3 Lectures: Damping Rings - Part 1
34