Overview of Damping Rings in Linear Colliders
S. Guiducci, INFN-LNF
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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
2
These slides have been presented at the 2010 LC school by Mark Palmer
October 31, 2010
A3 Lectures: Damping Rings - Part 1
3
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
4
Equations of Motion
Particle motion in electromagnetic fields is governed by the
Lorentz force:
with the corresponding Hamiltonian:
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.
In order to do this we transform
to the
Frenet-Serret
coordinate system.
The local radius of
curvature is denoted by
October 31, 2010
A3 Lectures: Damping Rings - Part 1
5
Equations of Motion
With a suitable canonical transformation, we can re-write the
Hamiltonian as:
Using the relations
and expanding to 2
nd
order in p
x
and p
y
yields:
which is now periodic in s.
October 31, 2010
A3 Lectures: Damping Rings - Part 1
6
Equations of Motion
Thus, in the absence of synchrotron motion, we can generate the equations of
motion with:
which yields:
and
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.
Note:
1/
B
is the
beam rigidity
and is taken to be positive
October 31, 2010
A3 Lectures: Damping Rings - Part 1
7
Magnetic Field Multipole Expansion
Magnetic elements with 2-dimensional fields of the form
can be expanded in a complex multipole expansion:
In this form, we can normalize to the main guide field strength,
-B
ŷ
, by setting
b
0
=1
to yield:
October 31, 2010
A3 Lectures: Damping Rings - Part 1
8
Multipole Moments
Upright Fields
Dipole:
Quadrupole:
Sextupole:
Octupole:
Skew Fields
Dipole (
°)
Quadrupole (
°)
Sextupole (
°)
Octupole (
°)
October 31, 2010
A3 Lectures: Damping Rings - Part 1
9
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:
Taking
p=p
0
and expanding the equations of motion to first order in
x
and
y
gives
where the upper/low signs are for a positively/negatively charged
particle.
The focusing functions are periodic in s:
also commonly
denoted as
k
1
October 31, 2010
A3 Lectures: Damping Rings - Part 1
10
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.
October 31, 2010
A3 Lectures: Damping Rings - Part 1
11
Solutions to Hill’s Equation
Let’s begin by considering constant
K=k
:
where
x
now represents either
x
or
y.
The
3 solutions are:
For each of these cases, we can solve for initial conditions and
recast in 2
×2
matrix form:
Focusing Quadrupole
Defocusing Quadrupole
Drift Region
October 31, 2010
A3 Lectures: Damping Rings - Part 1
12
Transfer Matrices
We can now re-write the solutions of the preceding page in
transfer matrix form:
where
Focusing
Quadrupole
Defocusing
Quadrupole
Drift Region
October 31, 2010
A3 Lectures: Damping Rings - Part 1
13
Transfer Matrices
Examples:
–
Thin lens approximation:
–
Sector dipole (entrance and exit faces
┴
to closed orbit)
:
October 31, 2010
A3 Lectures: Damping Rings - Part 1
14
Transfer Matrices
Transport through an interval
s
0
s
2
can be written as the product
of 2 transport matrices for the intervals
s
0
s
1
and
s
1
s
2
:
and the determinant of each transfer matrix is:
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:
with the boundary condition that:
The multi-turn matrix for m revolutions is then:
October 31, 2010
A3 Lectures: Damping Rings - Part 1
15
Twiss Parameters
The generalized one turn matrix can be written as:
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:
The n-turn matrix can be expressed as:
which leads to the stability requirement for betatron motion:
Identity matrix
October 31, 2010
A3 Lectures: Damping Rings - Part 1
16
The Envelope Equations
We will look for 2 independent solutions to Hill’s Equation of the
form:
Then
w
and
satisfy:
Since any solution can be written as a superposition of the above
solutions, we can write [with
w
i
=w
(
s
i
)
]:
Betatron envelope
and
phase equations
October 31, 2010
A3 Lectures: Damping Rings - Part 1
17
The Envelope Equations
Application of the previous transfer matrix to a full turn and direct
comparison with the Courant-Snyder form yields:
the betatron envelope equation becomes
and the transfer matrix in terms of the Twiss parameters can
immediately be written as:
October 31, 2010
A3 Lectures: Damping Rings - Part 1
18
General Solution to Hill’s Equation
The general solution to Hill’s equation can now be written as:
We can now define the
betatron tune
for a ring as:
If we make the coordinate transformation:
we see that particles in the beam satisfy the equation for simple
harmonic motion:
October 31, 2010
A3 Lectures: Damping Rings - Part 1
19
The Courant-Snyder Invariant
With K real, Hill’s equation is conservative. We can now take
After some manipulation, we can combine these two equations to
give:
Recalling that
yields:
Conserved
quantity
October 31, 2010
A3 Lectures: Damping Rings - Part 1
20
Emittance
The equation
describes an ellipse with area
For an ensemble of particles, each
following its own ellipse, we can
define the moments of the beam as:
The rms emittance of the beam is then
which is the area enclosed by the ellipse of an
rms particle
.
October 31, 2010
A3 Lectures: Damping Rings - Part 1
21
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.
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
October 31, 2010
A3 Lectures: Damping Rings - Part 1
22
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
′
):
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
, the transfer
matrix becomes:
and motion in the two planes is coupled.
October 31, 2010
A3 Lectures: Damping Rings - Part 1
23
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
1/2
~1/
. 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.
October 31, 2010
A3 Lectures: Damping Rings - Part 1
24
Dispersion
In our initial derivation of Hill’s equation, we assumed that the
particles being guided had the design momentum,
p
0
, thus ignoring
longitudinal contributions to the motion. We now want to address
off-energy particles. Thus we take the equation of motion:
and expand to lowest order in and which yields:
We have already obtained a homogenous solution,
x
(s)
. If we
denote the particular solution as
D(s
, the general solution is
October 31, 2010
A3 Lectures: Damping Rings - Part 1
25
Dispersion Function and Momentum Compaction
The dispersion function satisfies:
with the boundary conditions:
The solution can be written as the sum of the solution to the
homogenous equation and a particular solution:
which can be expressed in a 3
×3 matrix form as:
October 31, 2010
A3 Lectures: Damping Rings - Part 1
26
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
length of an on-momentum particle is given by:
For the off-momentum case, we then have:
I
1
is the first
radiation integral
.
The momentum compaction factor,
c
, is defined as:
October 31, 2010
A3 Lectures: Damping Rings - Part 1
27
The Synchrotron Radiation Integrals
I
1
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
28
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:
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
29
Effect of a Gradient Error
We consider a local perturbation of the focusing strength
K = K
0
+
. The effect of
can be represented by including a
thin lens transfer matrix in the one-turn matrix. Thus we have
and
With
we can take the trace of the one-turn matrix to
give:
October 31, 2010
A3 Lectures: Damping Rings - Part 1
30
Effect of a Gradient Error
Using the relation:
we can identify:
Thus we can write:
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:
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
31
Chromaticity
We can now write the betatron tune shift due to chromatic
aberration as:
The chromaticity is defined as the change in tune with respect to
the momentum deviation:
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
32
Sextupoles
Recall that the magnetic field in a sextupole can be written as:
Using the orbit of an off-momentum particle
we obtain
and
where the first terms in each expression are a quadrupole feed-
down 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
October 31, 2010
A3 Lectures: Damping Rings - Part 1
33
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
34
Bibliography
1.
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
.
2.
S. Y. Lee,
Accelerator Physics, 2
nd
Ed., (World Scientific, 2004)
.
3.
J. R. Rees,
Symplecticity in Beam Dynamics: An Introduction
, SLAC-PUB-9939,
2003.
4.
K. Wille,
The Physics of Particle Accelerators – an introduction,
translated by J.
McFall, (Oxford University Press, 2000).
5.
S. Guiducci & A. Wolski, Lectures from 1
st
International Accelerator School for
Linear Colliders, Sokendai, Hayama, Japan, May 2006.
6.
A. Wolski, Lectures from 2
nd
International Accelerator School for Linear Colliders,
Erice, Sicily, October 2007.
7.
A. Wolski, Lectures from 4
th
International Accelerator School for Linear Colliders,
Beijing, China, October 2009.
8.
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
9.
Various recent meetings of the ILC and CLIC damping ring design teams.
This content provides insights into the basics of damping rings in linear colliders, covering topics such as ring equations of motion, betatron motion, emittance, transverse coupling, dispersion, and momentum compaction factor. It delves into the equations of motion governing particle behavior in electromagnetic fields and the Hamiltonian transformations involved. The discussion also highlights the importance of specific field configurations in manipulating particle beams within accelerators.
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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: ( e E v B dt dp ) = + 1/2 ( ) 2 H = + P eA + 2 2 with the corresponding Hamiltonian: c m c H e H = = , ,... x P x P x x 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. In order to do this we transform to the Frenet-Serret coordinate system. The local radius of curvature is denoted by r y s x 0r Reference Orbit = + + r r xx yy 0 October 31, 2010 A3 Lectures: Damping Rings - Part 1 4
Equations of Motion With a suitable canonical transformation, we can re-write the Hamiltonian as: ( 2 2 2 1 m c c 1/2 ) 2 H -e x ( ) 2 ( ) 2 H = - + p eA p eA eA x x y y s 2 E c Using the relations H = = 2 2 , E e p m c 2 and expanding to 2nd order in px and py yields: + 1 x x p ( ) 2 ( ) 2 H - + + 1 p p eA p eA eA x x y y s 2 which is now periodic in s. October 31, 2010 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 y = = , , , x p y p x p x p y x y which yields: 2 B B + p p x x y = + top/bottom sign for +/- charges 1 , 0 x 2 and 2 x B B p x Note: 1/B is the beam rigidity and is taken to be positive = + 01 p y 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. October 31, 2010 A3 Lectures: Damping Rings - Part 1 6
Magnetic Field Multipole Expansion Magnetic elements with 2-dimensional fields of the form ( , x B B x y x = ) ( ) + , B x y y y can be expanded in a complex multipole expansion: ( )( ) n + = + x iy + ( , ) B x y ( , ) iB x y B b ia 0 y x n n = 0 n n B x n 1 ! 1 ! B x y = = with and x b a n n n n n B n B ( ) ( = ) 0 0 ( ) ( = ) , 0,0 x y , 0,0 x y In this form, we can normalize to the main guide field strength, -B , by setting b0=1 to yield: 1 1 e p ( ) ( ) ( )( ) n + = + = + x iy + for B iB B iB b ia q y x y x n n B = 0 n 0 October 31, 2010 A3 Lectures: Damping Rings - Part 1 7
Multipole Moments Upright Fields Skew Fields Dipole ( = ) eB p Dipole: eB p eB p eB p = = = 0 = 0 y x x y y x 0 0 0 0 Quadrupole ( = ) eB k p Quadrupole: eB p eB p eB p = = = = x ky kx k y x skew x y y skew 0 0 0 0 Sextupole ( = ) 1 2 p Sextupole: eB p 1 2 eB p eB ( ) ( ) = = = 2 2 2 2 m x y mxy m x y y x x skew 0 0 0 eB p = m xy y skew 0 Octupole ( = ) 1 6 p Octupole: eB ( ) 1 6 eB p ( ) = 3 2 3 skew r x xy = 2 3 3 r x y y x x 0 0 1 6 1 6 eB p eB p ( ) ( ) = = 3 2 2 3 3 3 r x xy skew r x y y y y 0 0 October 31, 2010 A3 Lectures: Damping Rings - Part 1 8
Equations of Motion (Hills Equation) We next want to consider the equations of motion for a ring with only guide (dipole) and focusing (quadrupole) elements: p e p e ( ) = + = = = and 1 0 0 B B kx B kx B ky B kx 0 0 0 y x Taking p=p0 and expanding the equations of motion to first order in x andy gives 1 ( ) s x ( ) s ( ) + = = 0, x K K k s ( ) ( ) k s x x 2 s ( ) s y ( ) s + = = 0, y K K also commonly denoted as k1 y y where the upper/low signs are for a positively/negatively charged particle. The focusing functions are periodic in s: ( ) ( ) s + = K s L K , , x y x y October 31, 2010 A3 Lectures: Damping Rings - Part 1 9
Solutions to Hills 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. October 31, 2010 A3 Lectures: Damping Rings - Part 1 10
Solutions to Hills Equation Let s begin by considering constant K=k: + = 0 x kx where x now represents either xor y.The 3 solutions are: ( ( ) , x s as b = + ) ( ) = + ( ) x s sin cos , 0 a ks b ks k Focusing Quadrupole = 0 k Drift Region ( ) ( ) = + ( ) x s sinh cosh , 0 a k s b k s k Defocusing Quadrupole For each of these cases, we can solve for initial conditions and recast in 2 2 matrix form: = ( x = M x x m m m m ) 0 x x 0 11 12 0 21 22 s s x 0 October 31, 2010 A3 Lectures: Damping Rings - Part 1 11
Transfer Matrices We can now re-write the solutions of the preceding page in transfer matrix form: cos = s s = ( ) ( ) 1 sin k k Focusing Quadrupole k ( ) ( ) sin cos k k k 1 0 1 ( ) Drift Region M s s 0 ( ) ( ) 1 cosh sinh k k k Defocusing Quadrupole ( ) ( ) sinh cosh k k k where 0. October 31, 2010 A3 Lectures: Damping Rings - Part 1 12
Transfer Matrices Examples: Thin lens approximation: 1 = 0, lim f K 0 1 1 0 1 1 0 1 = = M M focusing defocusing 1 f f Sector dipole (entrance and exit faces to closed orbit): c.o. cos 1sin sin 1 = = where M sector cos 1 2 October 31, 2010 A3 Lectures: Damping Rings - Part 1 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: ( 2 0 s s = M M ) ( ) ( ) M s s s s 2 1 1 0 i= M 1 and the determinant of each transfer matrix is: 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: ( ) P L s = + M M s L s ( s ring ) ( ) s with the boundary condition that: + = M M ( ) s mP The multi-turn matrix for m revolutions is then: M October 31, 2010 A3 Lectures: Damping Rings - Part 1 14
Twiss Parameters The generalized one turn matrix can be written as: + cos sin sin = = + M I J cos Identity matrix sin sin cos sin This is the most general form of the matrix. and 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 Jhas the properties: = = I = + 2 2 J J , 1 The n-turn matrix can be expressed as: which leads to the stability requirement for betatron motion: ( ) ( ) = + n M I J cos sin n n ( ) = M Trace 2cos 2 October 31, 2010 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 e = and ( ) s ( ) ( ) s ( ) s i i = x aw s e 1 w Then w and satisfy: + = Betatron envelope and 0 w Kw 3 1 = phase equations 2 w Since any solution can be written as a superposition of the above solutions, we can write [with wi=w(si)]: w w 1 cos sin sin w w ww 2 2 1 2 ( ) 1 = M s s ( ) 1 2 2 1 + 1 2 1 www w ww w w w w w w 2 + 1 2 sin cos cos sin ww 1 1 1 2 2 1 2 October 31, 2010 A3 Lectures: Damping Rings - Part 1 16
The Envelope Equations Application of the previous transfer matrix to a full turn and direct comparison with the Courant-Snyder form yields: w = 2 = = ww 2 the betatron envelope equation becomes 2 1 2 1 + + = 1 0 K 4 and the transfer matrix in terms of the Twiss parameters can immediately be written as: ( ) + cos sin sin 2 1 1 2 ( ) 1 = M s s 2 1 + 1 ( ) + sin cos cos sin 1 2 1 2 1 2 2 1 2 1 2 October 31, 2010 A3 Lectures: Damping Rings - Part 1 17
General Solution to Hills Equation The general solution to Hill s equation can now be written as: ds s ( ) x s ( ) s ( ) s ( ) s = + = where cos A ( ) s 0 x x x 0 x We can now define the betatron tune for a ring as: 1 ds s C + = = = = where ring circumference turn Q C ( ) s x x 2 2 s x If we make the coordinate transformation: 1 x ds s ( ) s = = and z ( ) s 0 x x x we see that particles in the beam satisfy the equation for simple harmonic motion: 2 d z d + = 2 x 0 z 2 October 31, 2010 A3 Lectures: Damping Rings - Part 1 18
The Courant-Snyder Invariant With K real, Hill s equation is conservative. We can now take ( ) ( ) ( ) cos x s = + and cos x s A s s 0 x x A ( ) ( ) s ( ) s ( ) s x s = + + + sin ( ) 0 0 x x After some manipulation, we can combine these two equations to give: = = + 2 Conserved quantity ( ) ( ) x s 2 x ( ) s x + x 2 A x ( ) s x s x Recalling that = + yields: ( ) s x ( ) s ( ) ( ) ( ) s x s x s ( ) s x ( ) s = = + + 2 2 2 2 A October 31, 2010 A3 Lectures: Damping Rings - Part 1 19
Emittance x The equation ( ) s x ( ) s ( ) ( ) ( ) s x s x s ( ) s x ( ) s + + = 2 2 2 describes an ellipse with area For an ensemble of particles, each following its own ellipse, we can define the moments of the beam as: ( ) ( ( ) ( xx x x x x = x Area = ( ) ) x x dxdx x x dxdx = = , , x x x x ) ( 2 2 ( ) x x dxdx x x dxdx = = 2 x 2 x , , x x x x )( ) ( ) x x dxdx = 2 , r x x 2 A The rms emittance of the beam is then which is the area enclosed by the ellipse of an rms particle. = = 2 x 2 x 2 xx rms 2 October 31, 2010 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. ( ) ( ) 0 y y K s y y + = ( ) ( ) s y + = + + + + = = 0 0 x K s x x K s x k y x x skew k 0 K x y skew 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 skew k = my co October 31, 2010 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 ): M 0 M 0 ( ) focusing 0 F = = M s s 4D 0 M M 0 defocusing D 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 , the transfer matrix becomes: + = ( + ) 2 2 M M M M M M cos sin M sin cos cos M F D D F M ( ) skew 2 2 sin cos sin D F D F and motion in the two planes is coupled. October 31, 2010 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 1/2~1/ . 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. October 31, 2010 A3 Lectures: Damping Rings - Part 1 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 B B + p p p p x x y = + 1 0 x 2 and expand to lowest order in and which yields: x = 0 ( ) + = x K s x We have already obtained a homogenous solution, x (s). If we denote the particular solution as D(s) , the general solution is ( ) s ( ) = + x x D s October 31, 2010 A3 Lectures: Damping Rings - Part 1 24
Dispersion Function and Momentum Compaction The dispersion function satisfies: + = ( ) ( ) 1 ( D ) K s D ( ) ( ) D s D s with the boundary conditions: + = + = ; D s L D s L The solution can be written as the sum of the solution to the homogenous equation and a particular solution: ( ) ( ) ( ) ( ) + D s D s D s D s d d ( ) 2 1 = M s s 2 1 2 1 which can be expressed in a 3 3 matrix form as: ( ) ( ) 1 ( ) ( ) 1 D s D s D s D s ( ) 2 1 M s s d d d 2 1 = = , where d 2 1 0 1 October 31, 2010 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 length of an on-momentum particle is given by: . . c o x ( ) = C ds D s For the off-momentum case, we then have: I1 is the first radiation integral. = = C d s I 1 The momentum compaction factor, c, is defined as: I C C C = = 1 c October 31, 2010 A3 Lectures: Damping Rings - Part 1 26
The Synchrotron Radiation Integrals I1is 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: ( )( s ) + = 1 0 x K 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+ . The effect of can be represented by including a thin lens transfer matrix in the one-turn matrix. Thus we have = 1 0 1 M K K and + cos sin sin = M 1 turn + sin cos sin cos sin sin 1 0 1 0 0 0 = sin cos sin K 0 0 0 With = + we can take the trace of the one-turn matrix to give: ( ) 0 cos cos + = 1 2 sin K 0 0 October 31, 2010 A3 Lectures: Damping Rings - Part 1 29
Effect of a Gradient Error ( ) 0 cos + = cos cos sin sin Using the relation: 0 0 1 2 1 4 we can identify: K Thus we can write: = Q K 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 = Kds Q 4 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 4 = Kds Kds Q 4 The chromaticity is defined as the change in tune with respect to the momentum deviation: Q = C 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: eB mxy p 1 2 eB p ( ) = = 2 2 m x y x y 0 0 ( ) s ( ) Using the orbit of an off-momentum particle we obtain x mD s p eB p = + x x D s eB ( ) ( ) s ( ) s = + y mx y 0 1 2 1 2 ( ) s ( ) s ( ) s ( ) s ( ) s and m x = + + 2 2 2 2 mD x mD y y 0 where the first terms in each expression are a quadrupole feed- down 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 ( ) ( ) mD s = Q s ds 4 October 31, 2010 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. 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. 2. 3. 4. 5. 6. 7. 8. 9. October 31, 2010 A3 Lectures: Damping Rings - Part 1 34
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