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Damping Ring Design
Andy Wolski
University of Liverpool/Cockcroft Institute
International Accelerator School for Linear Colliders
Sokendai, Hayama, Japan
21 May, 2006
Outline and Learning Objectives
1. Introduction: Basic Principles of Operation
2. Lattice Design and Parameter Optimization
You should be able to explain the issues involved in choosing the principal parameters for
the damping rings, including the circumference, beam energy, lattice style, and RF
frequency.
3. Beam Dynamics
You should be able to explain the physics behind important beam dynamics phenomena,
including coupling, dynamic aperture, space charge effects, microwave instability,
resistive-wall instability, fast ion instability and electron cloud. You should be able to
describe the impact of these effects on damping ring design. For some effects (space
charge, microwave, resistive-wall and fast ion instability), you should be able to estimate
the impact on damping ring performance, using simple linear approximations.
4. Technical Subsystems
You should be able to describe the principles of operation behind important technical
subsystems in the damping rings, including the injection/extraction kickers, fast feedback
systems and the damping wiggler. For the damping wiggler, you should be able to
explain the issues involved in choosing between the various technology options.
2
Prerequisites
These lectures assume:
•
undergraduate level physics knowledge:
– electromagnetism;
– some classical mechanics;
– special relativity.
•
knowledge of accelerator physics in electron storage rings:
–
–
–
–
–
–
–
–
transverse focusing and betatron motion;
effect of RF cavities, momentum compaction and synchrotron motion;
definition of beta functions and dispersion;
definition of betatron and synchrotron tunes;
chromaticity;
description of dynamics using phase-space plots;
emittance (geometric and normalized) and its relationship to beam size;
synchrotron radiation effects, including radiation damping, quantum excitation,
equilibrium emittance, energy spread and bunch length;
– definition of synchrotron radiation integrals.
3
Part 1
Principles of Operation
Introduction: Basic Principles of Operation - Performance Specs
The performance parameters are determined by the sources, the
luminosity goal, interaction region effects and the main linac
technology.
ILC parameters determining damping ring requirements
Particles per bunch
Average current in main linac
1×1010 - 2×1010
Upper limit set by disruption at IP.
< 9.5 mA
Upper limit set by RF technology.
Set by cryogenic cooling capacity.
Partially determines required damping time.
Machine repetition rate
5 Hz
Linac RF pulse length
< 1.2 ms
Upper limit set by RF technology.
Particles per pulse
> 5.6×1013
Lower limit set by luminosity goal.
Injected normalized emittance
0.01 m-rad
Set by positron source.
Partially determines required damping time.
Extracted normalized emittances
8 m horizontally
Set by luminosity goal.
20 nm vertically
Extracted bunch length
< 6 mm
Upper limit set by bunch compressors.
Extracted energy spread
< 0.15%
Upper limit set by bunch compressors.
5
Introduction: Basic Principles of Operation - Need for Compression
Synchrotron radiation damping times are of the order of 10 - 100 ms.
Linac RF pulse length is of the order of 1 ms.
Therefore, damping rings must store (and damp) an entire bunch train
in the (~ 200 ms) interval between machine pulses.
Particles per bunch
1×1010
Particles per pulse
5.6×1013
Number of bunches
5600
Average current in main linac
9.5 mA
Bunch separation in main linac
168 ns
Train length in main linac
0.94 ms = 283 km
We must compress the bunch train to fit into a damping ring.
This is achieved by injecting and extracting bunches one at a time.
6
Introduction: Basic Principles of Operation - Injection/Extraction
Most storage rings use off-axis injection, in which synchrotron
radiation damping is used to merge an off-axis injected bunch, with a
stored bunch. The acceptance of the ring must be much larger than
the injected bunch size, and the injection process necessarily takes
several damping times.
In the damping rings, acceptance and damping time are at a premium,
because of the large emittance of the injected positron bunches.
Therefore, we use on-axis injection, in which full-charge bunches are
injected on-axis into empty RF buckets. Fast kickers are used to
deflect the trajectory of incoming (or outgoing) bunches. The kickers
must turn on and off quickly enough so that stored bunches are not
deflected. The kicker rise/fall times must be a few ns: this is
technically challenging.
7
Introduction: Basic Principles of Operation - Injection/Extraction
trajectory of
incoming beam
following
bunch
empty
RF bucket
injection
kicker
preceding
bunch
trajectory of
stored beam
1. Kicker is OFF.
“Preceding” bunch exits
kicker electrodes.
Kicker starts to turn ON.
2. Kicker is ON.
“Incoming” bunch is
deflected by kicker.
Kicker starts to turn OFF.
3. Kicker is OFF by the
time the following
bunch reaches the
kicker.
8
Introduction: Basic Principles of Operation - Train (De)compression
Consider a damping ring with h stored bunches, with bunch
separation t.
If we fire the extraction kicker to extract every nth bunch, where n is
not a factor of h, then we extract a continuous train of h bunches, with
bunch spacing n×t.
4
1
5
3
2
6
5
4
3
2
1
An added complication is that we want to have regular gaps in the fill
in the damping ring, for ion clearing (see later in lecture).
9
Introduction: Basic Principles of Operation - ILC Baseline Configuration
Single damping ring for electrons.
Two (stacked) damping rings for positrons.
Circumference 6695 m.
5 GeV beam energy.
650 MHz RF.
Bunch
Charge
Number of
Bunches
(1010)
Bunch Spacing Bunch Spacing
in Damping Ring
in Linac
Average Current
in Linac
Beam Pulse
Length
(ns)
(ns)
(mA)
(ms)
0.97
5782
3.08
189
8.2
1.09
0.99
5658
3.08
182
8.7
1.03
1.29
4346
3.08
272
7.6
1.18
1.54
3646
4.62
312
7.9
1.14
2.02
2767
6.15
363
8.9
1.00
10
Introduction: Basic Principles of Operation - Summary
The damping rings parameter regime is set by constraints on other systems:
the sources (injected beam parameters);
bunch compressors (extracted bunch length and energy spread);
main linac (bunch charge and bunch spacing; pulse length; rep rate);
luminosity goals (total charge per pulse; extracted emittances);
IP (bunch charge).
The bunch train in the linac is of order 300 km long, and must be compressed to be
stored in the damping rings. This is achieved by injecting/extracting individual
bunches.
Injection in the damping rings must be on-axis.
Single-bunch, on-axis injection is achieved by the use of fast kickers, which turn on
and off in the space between two bunches.
Kickers with rise/fall times of a few ns are technically challenging, and a key
component of the damping rings.
11
Part 2
Lattice Design and Parameter Optimization
You should be able to explain the issues involved in choosing the principal
parameters for the damping rings, including the circumference, beam energy,
lattice style, and RF frequency.
Lattice Design and Parameter Optimization: Circumference
Lower limit ~ 3 km: the smaller the damping ring, the shorter the
distance between bunches. This makes the ring more difficult:
Injection/extraction kickers need shorter rise and fall times.
Electron cloud build-up is sensitive to bunch spacing, and it becomes
increasingly difficult to avoid electron cloud instabilities as the ring gets smaller.
In smaller rings, it becomes difficult to provide sufficient gaps in the fill for ion
clearing, so the beam becomes susceptible to ion instabilities.
Upper limit ~ 17 km: space-charge, acceptance and cost.
Space-charge tune shifts (in a linear model) are proportional to the
circumference. Large tune shifts can lead to emittance growth and particle loss.
The cost of very large (~17 km) rings may be reduced by using a “dogbone”
layout, in which long straight sections share the tunnel with the main linac…
…but these long straights generate chromaticity, which breaks any symmetry for
off-energy particles and limits the acceptance.
13
Lattice Design and Parameter Optimization: Circumference
Lower limit on circumference from injection/extraction kickers:
consider the bunch spacing in the damping rings with 1×1010 particles
per bunch (“low-Q” parameter set, desirable to ease IP limitations).
To achieve the desired luminosity with 1×1010 particles per bunch, we
need ~ 6000 bunches.
In a 3 km ring, without any ion-clearing gaps, the bunch separation
with 6000 bunches is 1.67 ns. The (challenging) goal for present
kicker R&D is to achieve rise/fall times of 3 ns.
To achieve the “low-Q” parameter set, and allow kicker rise/fall times
of 3 ns, the damping ring circumference should be at least 6 km.
14
Lattice Design and Parameter Optimization: Circumference
Lower limit on circumference from electron cloud:
Electron-cloud effects will be discussed in more detail later. Briefly,
electrons are generated in a storage ring by ionisation of the residual
gas, or by photoemission prompted by synchrotron radiation. Under
some circumstances, the number of electrons (generally in a proton or
positron ring) can increase rapidly to roughly the neutralization level.
The electrons can interact with the high-energy beam, and lead to
beam instability.
Build-up of electron cloud can be suppressed by solenoids (in fieldfree regions) or by appropriate treatment of the surface of the vacuum
chamber, but becomes difficult as the bunch spacing gets shorter.
Electron cloud build-up and instabilities must generally be studied
using simulation codes.
15
Lattice Design and Parameter Optimization: Circumference
Simulated build-up of electron cloud in a dipole of a 6 km damping ring.
(SEY = Peak Secondary Electron Yield)
16
Lattice Design and Parameter Optimization: Circumference
Growth in projected vertical beam size as a function of the number of
turns in a 6 km damping ring, for electron cloud densities between
1.2×1011 m-3 and 1.8×1011 m-3
17
Lattice Design and Parameter Optimization: Circumference
Comparison between electron cloud instability thresholds and cloud densities, in
various damping rings under various conditions.
18
Lattice Design and Parameter Optimization: Circumference
The beam ionizes residual gas in the vacuum chamber, and the ions
drive transverse bunch oscillations. There must be frequent gaps in
the bunch train so that the ion densities stay low. In the damping
rings, we always expect to see some ion instability, but with sufficient
gaps, this can be controlled using a feedback system.
19
Lattice Design and Parameter Optimization: Circumference
The space-charge tune shifts are proportional to the circumference.
Using a linear approximation for the space-charge forces, the
(incoherent) vertical tune shift is given by:
 y
1 2re
 y  
ds
3 
4  0  y  x   y 
C
where  is the line density of charge in the bunch. Generally, we
want to keep the tune-shifts below approximately 0.1 to avoid
emittance growth.
In reality, the space-charge force is not linear, and the above
expression may significantly over-estimate the impact of space-charge
effects. For a proper characterization, we need to do tracking.
20
Lattice Design and Parameter Optimization: Circumference
Tune-scan of emittance
growth from space-charge
in a 17 km DR lattice.
(Flat beam in long straights.)
Tune-scan of emittance
growth from space-charge
in a 6 km DR lattice.
21
Lattice Design and Parameter Optimization: Circumference
Tune-scan of emittance
growth from space-charge
in a 17 km DR lattice.
Flat beam in long straights.
Tune-scan of emittance
growth from space-charge
in a 17 km DR lattice.
Coupled (round) beam in
long straights.
22
Lattice Design and Parameter Optimization: Circumference
Acceptance is an important issue. The 17 km (dogbone) lattices have
poor symmetry, which makes it very difficult to achieve the necessary
dynamic aperture.
3inj
3inj
Dynamic aperture with magnet errors, and energy deviation.
Left: 17 km dogbone lattice. Right: 6 km circular lattice.
23
Lattice Design and Parameter Optimization: Circumference
Summary of circumference issues:
The damping ring circumference is a compromise between effects that
favor a smaller circumference (space-charge, acceptance, cost) and
effects that favor a larger circumference (electron cloud, fast ion
instability, kicker performance).
After considering a wide range of issues in some detail, the decision
was taken in the ILC to adopt a baseline specification of a single 6.6
km damping ring for the electrons, and two 6.6 km damping rings for
the positrons. Two rings for the positrons are needed to increase the
bunch spacing, in order to mitigate electron cloud effects.
24
Lattice Design and Parameter Optimization: Damping Time
The beam emittances evolve as:
2t 
2t 



 t    t 0 exp      t  1  exp   
  

  
where t=0 is the injected normalized emittance, t= is the
equilibrium emittance, and  is the damping time.
To damp from an injected normalized vertical emittance of ~ 0.01 m
to an extracted normalized vertical emittance of ~ 20 nm (6 orders of
magnitude), we need to store the beam for ~ 7 damping times.
Given the store time of 200 ms in the ILC, the damping time needs to
be <30 ms.
25
Lattice Design and Parameter Optimization: Beam Energy
Like the circumference, the beam energy is a compromise between
competing effects.
Favoring a higher energy:
Damping times (shorter at higher energy; less wiggler is needed)
Collective effects (instability thresholds are higher at higher energy; spacecharge, intrabeam scattering, etc. are weaker effects at higher energy).
Favoring a lower energy:
Emittance (easier to achieve lower transverse and longitudinal emittances at
lower energy)
Cost (magnets are weaker, RF voltage is lower).
26
An aside: the damping wiggler
The damping time in a storage ring depends on the rate of energy loss
of the particles through synchrotron radiation. In the damping rings,
the rate of energy loss can be enhanced by insertion of a long wiggler,
consisting of short (~ 10 cm) sections of dipole field with alternating
polarity.
y
z
x
The magnetic field in the
wiggler can be approximated by:
By = Bw sin(kzz)
27
Lattice Design and Parameter Optimization: Beam Energy
The (transverse) damping time in a storage ring is given by:
E0
  2 T0
U0
U0 
C
2
4
0 2
E I
I2  
1

2
ds
where E0 is the beam energy; U0 is the energy loss per turn; T0 is the
revolution period;  is the local bending radius of the magnets; and
C = 8.846×10-5 m/GeV3 is a physical constant.
If the energy loss U0 is dominated by a wiggler of length Lw and peak
field Bw, then the damping time  scales as:
T0 1

E0 Lw Bw2
28
Lattice Design and Parameter Optimization: Beam Energy
The natural energy spread in a storage ring is given by:
 2  12 Cq 2
I3
I2
I3  
1

3
ds
where  is the relativistic factor,
and Cq = 3.832×10-13 m is a physical constant.
Performing the integrals for a wiggler, we find:
Lw Bw2
I 2   2 ds 
2

2 B 
1
I3  
4 Lw Bw3
ds 
3
3

3  B 
1
I3
8 Bw

I 2 3 B
If the energy loss is dominated by a wiggler with peak field Bw (so we count only
the wiggler contribution to the energy spread) then:
 2  12 Cq 2 
8 Bw
3 B
Note the scaling with energy and wiggler field:
   E0 Bw
29
Lattice Design and Parameter Optimization: Beam Energy
Finding the correct energy is a complicated multi-parameter
optimization, and depends on many assumptions. However, if we
consider just the damping time and energy spread, and assume
reasonable wiggler parameters, we can find a realistic range for the
energy.
   E0 Bw
 < 0.13%
Lw = 200 m
Bw = 1.6 T
T0 = 6.6 km/c
 < 27 ms
5 GeV < E0 < 5.5 GeV
T0 1

E0 Lw Bw2
30
Lattice Design and Parameter Optimization: Energy and Polarization
Considering just the damping time and the energy spread sets the energy scale at a
few GeV. A more thorough optimization will include collective effects (spacecharge, intrabeam scattering, instability thresholds) which generally get worse at
lower energy, and costs, which generally increase with energy.
Once an appropriate energy range is found, the exact energy must be chosen so as to
avoid spin depolarization resonances (which are a function of energy).
The spins of particles in the beam precess in the field of the dipoles (and wiggler).
The number of complete rotations of the spin is the spin tune = G, where
G = 0.00115965 is the anomalous magnetic moment of the electron. Resonances
can occur which may depolarize the beam rapidly. To avoid these resonances, the
spin tune is usually chosen to be a half integer, i.e. (for integer n):
G  n  12
31
Lattice Design and Parameter Optimization: Lattice Styles
Various configurations are possible for the arc cells, e.g.:
FODO
DBA (Double Bend Achromat)
TME (Theoretical Minimum Emittance)
The style of arc cell influences the natural emittance (and also the momentum
compaction, and other parameters).
In general, the natural emittance of an electron storage ring is given by:
 x  Cq 2
where
J x  1
I4
I2
I5  
H
3
ds
I5
J x I2
I2  
1
2
ds
H   2  2    2
If the dipoles have zero quadrupole component, then the damping partition number
Jx  1.
32
Lattice Design and Parameter Optimization: Lattice Styles
The natural emittance in any style of lattice depends on the lattice
functions (beta function and dispersion) in the dipoles and wigglers.
The minimum emittance that can be achieved depends on the style of
lattice, and can be written (in the absence of any wiggler, and
assuming no quadrupole component in the dipole):
 x ,min
3
F
2

Cq 
Jx
12 15
where F is a factor depending on the lattice style, and  is the bending
angle of a single dipole.
Note that most lattice designs do not achieve the minimum possible
emittance, because of a variety of constraints (momentum
compaction, dynamic aperture, engineering limitations…)
33
Lattice Design and Parameter Optimization: Lattice Styles
FODO Lattice: F  100
34
Lattice Design and Parameter Optimization: Lattice Styles
Double Bend Achromat (DBA) Lattice: F = 3
35
Lattice Design and Parameter Optimization: Lattice Styles
Theoretical Minimum Emittance (TME) Lattice: F = 1
36
Lattice Design and Parameter Optimization: Lattice Styles
The TME lattice is often preferred for the damping rings, because:
- a very low equilibrium emittance is achieved with relatively few arc cells,
making the design economic;
- the number of dispersion-free straights is relatively small, so there is no need
to match the dispersion to zero outside every arc cell (as in a DBA).
The minimum emittance in a TME lattice is achieved with the lattice
functions taking specific values at the center of each dipole:
L
x 
2 15
L
x  
24
where L is the length of the dipole.
37
Lattice Design and Parameter Optimization: Lattice Styles
If the energy loss in the ring is completely dominated by the wiggler, then the
natural emittance is given by:
 wig
3
3
B
8
e 

Cq 
 x 2
15  mc 
kw
where x is the mean beta function in the wiggler. Note that the specification is
usually in terms of the normalized emittance , and that in a wiggler-dominated
lattice, this is independent of the beam energy.
Where both arcs and wigglers contribute to the energy loss, the equilibrium
emittance can be written:
 tot   arc
J x ,arc
J x ,arc  Fw
  wig
Fw
J x ,arc  Fw
where arc, Jx,arc are the natural emittance and damping partition number in the
absence of the wiggler, and Fw = I2,wig/I2,arc is the ratio of the energy loss in the
wiggler to the energy loss in the arcs.
38
Lattice Design and Parameter Optimization: Lattice Styles
Putting it together (an exercise for the student!):
1.
Given the ring circumference and the beam energy, the field in the arc dipoles
determines the damping time (in the absence of the wiggler). Hence, we can
calculate the additional energy loss needed from the wiggler to give the
specified damping time.
2.
Given the ratio of energy loss in the wiggler to energy loss in the arcs, and
some reasonable wiggler parameters (peak field and period), we can calculate
the maximum tolerable emittance in the arcs (absent wiggler) to achieve the
specified equilibrium emittance.
3.
Given the emittance in the arcs (in the absence of the wiggler), we can decide
the lattice style and number of arc cells appropriate for our lattice design.
There are many other issues that need to be considered when designing the lattice:
- momentum compaction
- chromaticity
- dynamic aperture…
39
Lattice Design and Parameter Optimization: RF Frequency
As with most other parameters, there is no clear “correct” choice for the RF
frequency.
Favoring a higher frequency:
Easier to achieve a shorter bunch for a lower total RF voltage.
Higher harmonic number for a given circumference (potentially) allows greater flexibility
in fill patterns - in practice, this is a complicated issue.
Favoring a lower frequency:
Power sources (klystrons) get more difficult at higher frequency.
In addition, it is desirable to have an RF frequency in the damping rings that is a
simple subharmonic of the main linac RF frequency. This simplifies phase-locking
between the damping ring and the main linac.
Presently, the baseline for the ILC is an RF frequency of 650 MHz (half of the main
linac RF frequency). This is a non-standard RF frequency. The other choice
considered was 500 MHz, which is widely used in synchrotron light sources.
40
Lattice Design and Parameter Optimization: RF Frequency
The bunch length in a storage ring is given by:
p
 z  c 
s
where c is the speed of light, p is the momentum compaction, s is
the synchrotron frequency, and  is the energy spread.
The synchrotron frequency is given by:
s2  
eVRF  RF
 p coss 
E0 T0
sin s  
U0
eVRF
where VRF is the RF voltage, E0 is the beam energy, U0 is the energy
loss per turn, s is the synchronous phase, and T0 is the revolution
period.
41
Lattice Design and Parameter Optimization: Summary
Given a set of performance specifications, a number of parameters
can be chosen to minimize technical risk and cost.
The parameters that need to be chosen include:
circumference
beam energy
lattice style
RF frequency
Choice of values for the various parameters is frequently a
compromise between competing effects.
42
Lattice Design and Parameter Optimization: Circumference
Lower limit ~ 3 km: the smaller the damping ring, the shorter the
distance between bunches. This makes the ring more difficult:
Injection/extraction kickers need shorter rise and fall times.
Electron cloud build-up is sensitive to bunch spacing, and it becomes
increasingly difficult to avoid electron cloud instabilities as the ring gets smaller.
In smaller rings, it becomes difficult to provide sufficient gaps in the fill for ion
clearing, so the beam becomes susceptible to ion instabilities.
Upper limit ~ 17 km: space-charge, acceptance and cost.
Space-charge tune shifts (in a linear model) are proportional to the
circumference. Large tune shifts can lead to emittance growth and particle loss.
The cost of very large (~17 km) rings may be reduced by using a “dogbone”
layout, in which long straight sections share the tunnel with the main linac…
…but these long straights generate chromaticity, which breaks any symmetry for
off-energy particles and limits the acceptance.
43
Lattice Design and Parameter Optimization: Beam Energy
Finding the correct energy is a complicated multi-parameter
optimization, and depends on many assumptions. However, if we
consider just the damping time and energy spread, and assume
reasonable wiggler parameters, we can find a realistic range for the
energy.
   E0 Bw
 < 0.13%
Lw = 200 m
Bw = 1.6 T
T0 = 6.6 km/c
 < 27 ms
5 GeV < E0 < 5.5 GeV
T0 1

E0 Lw Bw2
44
Lattice Design and Parameter Optimization: Lattice Styles
Equilibrium emittance is a key issue in the choice of lattice style.
The minimum emittance from the arcs (in the absence of a wiggler is):
 arc,min
3
F
2

Cq 
Jx
12 15
where F ~ 100 (FODO), F = 3 (DBA), F = 1 (TME).
The wiggler contributes an emittance:
 wig
3
3
  wig
Fw
J x ,arc  Fw
B
8
e 

Cq 

 x 2
15  mc 
kw
and the total emittance is:
 tot   arc
J x ,arc
J x ,arc  Fw
45
Lattice Design and Parameter Optimization: RF Frequency
As with most other parameters, there is no clear “correct” choice for the RF
frequency.
Favoring a higher frequency:
Easier to achieve a shorter bunch for a lower total RF voltage.
Higher harmonic number for a given circumference (potentially) allows greater flexibility
in fill patterns - in practice, this is a complicated issue.
Favoring a lower frequency:
Power sources (klystrons) get more difficult at higher frequency.
In addition, it is desirable to have an RF frequency in the damping rings that is a
simple subharmonic of the main linac RF frequency. This simplifies phase-locking
between the damping ring and the main linac.
Presently, the baseline for the ILC is an RF frequency of 650 MHz (half of the main
linac RF frequency). This is a non-standard RF frequency. The other choice
considered was 500 MHz, which is widely used in synchrotron light sources.
46
Part 3
Beam Dynamics
You should be able to explain the physics behind important beam
dynamics phenomena, including coupling, dynamic aperture, space
charge effects, microwave instability, resistive-wall instability, fast ion
instability and electron cloud. You should be able to describe the impact
of these effects on damping ring design. For some effects (space charge,
microwave, resistive-wall and fast ion instability), you should be able to
estimate the impact on damping ring performance, using simple linear
approximations.
Beam Dynamics: Vertical Emittance
Betatron oscillations of a particle are excited when the particle emits a photon at a
point of non-zero dispersion.
The energy of the particle changes
If the particle was following a closed orbit, then (because of the dispersion) it will no
longer be doing so.
emitted photon
on-energy
closed orbit
off-energy (dispersive)
closed orbit
particle trajectory
The equilibrium emittance is determined by the balance between radiation damping
and quantum excitation.
48
Beam Dynamics: Vertical Emittance and the Radiation Limit
In a perfectly aligned lattice lying in a horizontal plane and containing only normal (i.e. nonskew) elements, there is no vertical dispersion and no coupling of the betatron oscillations.
Vertical oscillations are excited only by the “recoil” from photons emitted with some angle to
the horizontal plane, so…
…the vertical opening angle of the synchrotron radiation places a fundamental lower limit on
the vertical emittance.
 y ,min
13 Cq

55 J y
3


ds
y

2
1

 ds
In this formula, y is the vertical beta function, and  is the local (horizontal) bending radius.
Note that the fundamental limit on the geometric (not normalized) vertical emittance is
independent of the beam energy. This is because the increased photon energy at higher
electron energy cancels the increased beam rigidity, and the decrease in the vertical opening
angle of the radiation (~1/).
Generally, for ILC damping ring lattices, we find y,min < 0.1 pm; other effects generating
vertical emittance are much more significant.
49
Beam Dynamics: Vertical Emittance Sources
The dominant sources of vertical emittance in storage rings are:
vertical dispersion generated from vertical steering
- caused by dipole tilts or vertical quadrupole misalignments
vertical dispersion generated from the coupling of horizontal dispersion into the
vertical plane
- caused by quadrupole tilts or vertical sextupole misalignments
direct coupling of horizontal motion into the vertical plane
- caused by quadrupole tilts or vertical sextupole misalignments*
The ILC damping rings require a vertical emittance that is of order 0.5% of the
horizontal emittance. Generally, alignment errors in an “uncorrected” storage ring
result in a vertical emittance that is of similar order of magnitude to the horizontal
emittance. A long process of beam-based alignment and error correction is needed
to bring the emittance ratio to the level of 1% or less.
*An exercise for the student!
50
Beam Dynamics: Vertical Emittance and Vertical Dispersion
Vertical dispersion is directly analogous to horizontal dispersion.
3
H

ds
  y
 y  Cq
J y  1  2 ds
2
Vertical dispersion is generated by a multitude of steering and coupling errors,
rather than by the main dipoles (as is the case for horizontal dispersion). Assuming
that the errors are uncorrelated, we can write:
 y2 2
 y  2J

y
Note that the vertical emittance is proportional to the mean square of the vertical
dispersion.
Correcting the vertical dispersion in a tuning ring is an important step in tuning and
correction for achieving low vertical emittance.
51
Beam Dynamics: Vertical Emittance and Vertical Dispersion
Correcting the vertical dispersion can generally be achieved by steering.
Various techniques can be applied. At the KEK-ATF, some success has been achieved by:
1. Use beam-based alignment to determine the beam offsets in the quadrupoles.
2. Steer the beam to the centers of the quadrupoles.
3. Find the vertical dispersion by measuring the change in vertical closed orbit with respect to RF
frequency.
4. Make small steering changes to minimize the vertical dispersion.
Correction of vertical dispersion in the
KEK-ATF. Over a period of time, the
RMS vertical dispersion is reduced from
~ 4 mm to ~ 2 mm. A factor of 2
reduction in the RMS vertical dispersion
implies a factor of 4 reduction in the
vertical emittance contributed by vertical
dispersion.
Plot courtesy of Mark Woodley, SLAC.
52
Beam Dynamics: Vertical Emittance and Coupling
In a storage ring, coupling is characterized by non-zero elements outside the
principal block diagonals in the single-turn transfer matrix.
If the ring is tuned away from coupling resonances, there are three distinct tunes
(frequencies of oscillation of particles in the lattice), corresponding to three degrees
of freedom. Motion associated with just one tune is referred to as a normal mode.
If only one normal mode is excited for a given particle, then only one frequency is
observed in a Fourier analysis of the motion of that particle.
The tunes are found from the eigenvalues of the single-turn matrix.
The normal modes are found from the eigenvectors of the single-turn matrix.
The three beam invariant emittances (I, II, III) describe the amplitudes of each of
the normal modes (modes I, II and III) averaged over all the particles in the beam.
In an uncoupled lattice, the normal modes correspond to horizontal, vertical and
longitudinal motion. In the presence of coupling, motion associated with a normal
mode does not lie entirely in any one plane (horizontal, vertical or longitudinal).
Quantum excitation in dipoles and wigglers usually generates horizontal betatron
motion. In the presence of coupling, horizontal motion is a mixture of three normal
modes: in this case, quantum excitation in the dipoles and wigglers generates
emittance (larger than the radiation limit) in all three modes.
53
Beam Dynamics: Vertical Emittance and Coupling
Without
coupling



0

0
 0 0

 0 0
0  

0  
With
coupling






  

  
  

  
• Single-turn matrix is block-diagonal.
• Normal modes correspond to the coordinate axes, x and y.
• Quantum excitation occurs in the
horizontal (x) plane.
• The vertical (or mode II) equilibrium
emittance is limited only by the vertical
opening angle of the radiation.
• Single-turn matrix contains non-zero
elements off the block-diagonal.
• Normal modes are rotated with respect to
the co-ordinate axes, x and y.
• Quantum excitation occurs in the
horizontal (x) plane, which is a mixture of
the mode I and mode II normal modes.
• The mode II equilibrium emittance is larger
than the radiation limit.
y
x
mode II
mode I
54
Beam Dynamics: Vertical Emittance and Coupling
The dominant sources of coupling in a storage ring are:
quadrupole rotations
sextupole vertical misalignments
Correcting the vertical dispersion is relatively straightforward, requiring only
measurement of the vertical dispersion and appropriate steering corrections.
Correcting the coupling is more complex. In practice, coupling is often
characterized in terms of the change in vertical closed orbit with respect to a change
in horizontal steering.
Orbit Response Matrix (ORM) Analysis uses the following procedure:
1. A complete orbit response matrix is measured, consisting of the change in horizontal
and vertical orbit at each BPM, with respect to small changes in each of the horizontal
and vertical steering magnets.
2. A lattice model (including BPM and corrector gains and tilts, and skew errors) is
fitted to the orbit response matrix.
3. Information on the skew errors from the fitted model is used to determine appropriate
corrections, so as to minimize the coupling.
55
Beam Dynamics: Vertical Emittance and Ring Design
The sensitivities of the closed orbit and equilibrium vertical emittance to various
magnet misalignments (quadrupole tilts and sextupole vertical misalignments)
depend on the quadrupole and sextupole strengths, lattice functions and tunes.
Closed orbit amplification:
yco2
2
yquad
 y2

8 sin 2  y
2


k
L

 y 1
Quadrupole tilts:
y
J x 1  cos 2 x cos 2 y 
J z 2
2


  k1L  
2 x  x y
2
 quad
4 sin 2  y
4 J y cos 2 x  cos 2 y  quads
2
2



k
L

 y x 1
quads
Sextupole vertical misalignments:
y
y
2
sext
J x 1  cos 2 x cos 2 y 
J z 2


  k 2 L  
2 x  x y
4 sin 2  y
4 J y cos 2 x  cos 2 y  quads
contribution from coupling
2
 
y
2
x
k2 L2
quads
contribution from dispersion
56
Beam Dynamics: Vertical Emittance and Ring Design
For many sets of magnet misalignments (all sets with the same RMS), there will be
a wide range of equilibrium vertical emittances:
10,000 sets of sextupole misalignments in the PPA lattice
57
Beam Dynamics: Vertical Emittance and Ring Design
The approximate expressions for the alignment sensitivities describe the average
behavior fairly well.
58
Beam Dynamics: Vertical Emittance and Ring Design
Example: sextupole alignment senstivity. This is defined as the RMS sextupole
vertical misalignment that is expected to generate the specified equilibrium vertical
emittance in an otherwise perfect lattice. (Larger is better). Note that this takes no
account of beam-based alignment and tuning procedures.
There is a wide variation in sextupole (and quadrupole) alignment sensitivities,
depending on the lattice design.
59
Beam Dynamics: Dynamic Aperture
A lattice constructed from only dipoles and quadrupoles has large chromaticity.
Quadrupoles focus higher-energy particles less strongly than lower-energy particles.
Without chromatic correction, the tunes rapidly get smaller as the energy increases.
The natural chromaticity of a linear lattice is always negative.
Negative chromaticity is a problem.
The tunes of off-energy particles can cross linear resonances, and their trajectories
become unstable.
Various collective instabilities need to be suppressed by positive chromaticity.
Chromaticity can be corrected using sextupoles.
Focusing strength is a function of horizontal position.
If the sextupoles are located where there is some dispersion, off-energy particles follow
trajectories that are off-center in the sextupoles.
Located appropriately, sextupoles can be used to provide additional (reduced) focusing
for higher-(lower-) energy particles.
Sextupoles are a problem.
Large-amplitude trajectories are subject to nonlinear forces, and become unstable.
60
Beam Dynamics: Dynamic Aperture
The dynamic aperture is the range of amplitudes (betatron and synchrotron) over
which particle trajectories are stable.
Dynamic aperture is important for light sources, because it is often a limitation on
the beam (Touschek) lifetime. For linear collider damping rings, a large dynamic
aperture is necessary to ensure good injection efficiency.
For ILC, average injected beam power into the damping rings is 225 kW.
Losing even a small portion of the injected beam can quickly cause damage.
The dynamic aperture can be complicated to characterize.
Boundary is not necessarily smooth or well-defined.
There may be “holes” within the dynamic aperture.
The stability of a given trajectory may be very sensitive to tuning errors or magnet
multipole errors.
Lattice design and optimization is an important but difficult task.
Some general rules can be applied, e.g. keep the natural chromaticity as small as possible;
design the lattice so that sextupole strengths are as small as possible.
Some tools are available for detailed characterization, which can be useful for guiding
design changes to improve the dynamic aperture.
Ultimately, we rely on tracking, tracking, tracking…
61
Beam Dynamics: Dynamic Aperture - FODO Example
A phase space portrait is produced by:
taking a set of particles with regular spaced over a range of betatron amplitudes;
tracking the particles over some number of turns;
plotting the phase space coordinates of every particle on every turn.
Phase space portraits are useful for giving a “rough and ready” picture of nonlinear
effects (tune shifts and resonances).
Horizontal phase space portrait (tune = 0.28)
62
Beam Dynamics: Dynamic Aperture - FODO Example
Dynamic aperture depends strongly on the tune of the lattice.
tune = 0.25
tune = 0.31
tune = 0.33
tune = 0.36
63
Beam Dynamics: Dynamic Aperture and Sextupoles
To achieve a good dynamic aperture, we need to keep the sextupole strengths low.
This means designing a lattice with a low natural chromaticity, and finding good
locations for the sextupoles.
The chromaticity of a lattice is given by:
d x
1
1


k
ds

 x x k 2 ds
x 1


d
4
4
d y
1
1
y 


k
ds

 y x k 2 ds
y 1
d
4 
4 
x 
We see that to correct the horizontal chromaticity, we need xk2 > 0,
and to correct the vertical chromaticity, we need xk2 < 0.
We resolve the conflict by locating sextupoles with xk2 > 0 where x >> y,
and sextupoles with xk2 < 0 where y >> x.
To keep the sextupole strengths as small as possible, we need locations with large
dispersion, and well-separated beta functions…
64
Beam Dynamics: Dynamic Aperture and Sextupoles, TME Lattice
SD SF
SF SD
x
SF: k2 > 0
SD: k2 < 0
x
x
65
Beam Dynamics: Dynamic Aperture
Dynamic aperture plots often show the maximum initial values of stable trajectories
in x-y coordinate space at a particular point in the lattice, for a range of energy
errors.
The beam size (injected or equilibrium) can be shown on the same plot.
Generally, the goal is to allow some significant margin in the design - the measured
dynamic aperture is often significantly smaller than the predicted dynamic aperture.
This is often useful for comparison, but is not a complete characterization of the
dynamic aperture: a more thorough analysis is needed for full optimization.
5inj
5inj
OCS: Circular TME
TESLA: Dogbone TME
66
Beam Dynamics: Frequency Map Analysis
A more complete characterization of the dynamics can be carried out using
Frequency Map Analysis.
Track a particle for several hundred turns through the lattice.
Use a numerical algorithm (e.g. NAFF; or interpolated Fourier-Hanning) to determine the
betatron tunes with high precision.
Continue tracking for several hundred more turns.
Find the tunes for the second set of tracking data.
Plot the tunes on a resonance diagram; use a color scale to represent the change in tunes
between the first and second sets of tracking data (the “diffusion rate”).
67
Beam Dynamics: Acceptance
The injection efficiency depends on:
the total acceptance of the ring (including dynamical and physical apertures);
the 6D distribution of the injected beam.
Optical and mechanical designs of the damping ring must allow some acceptance
margin over the anticipated injected distribution, to allow for errors. The goal is for
100% injection efficiency.
Estimates of injection efficiency for a given design can be made by tracking a
simulated distribution, including physical apertures, magnet field errors etc.
68
Beam Dynamics: Acceptance
Ax

  x x 2  2 x xpx   x p x2
Estimate of required physical aperture
in the damping wiggler in seven
representative lattice designs, for a
given injection (e+) distribution.
69
Beam Dynamics: Collective Effects
So far, the beam dynamics effects we have looked at are vertical emittance, and
acceptance. We have treated these in a way that does not consider interactions
between the particles: the results we obtain are independent of bunch charge.
In the real world, there are many effects that depend directly on the bunch charge.
These can be very complicated effects. Important ones for the damping rings, that
we shall consider briefly, include:
space charge;
intrabeam scattering (see Susanna Guiducci’s lectures);
microwave instability;
coupled-bunch instabilities;
fast-ion instability;
electron-cloud.
The observed phenomena associated with each effect can vary widely, depending on
the exact conditions in the machine. Not all these effects can be modeled with
sufficient accuracy or completeness, to allow completely confident predictions to be
made.
70
Beam Dynamics: Space Charge
Each particle in the bunch sees electric and magnetic fields from all the other
particles in the bunch.
FE
FM
For a bunch moving a close to the speed of light, the magnetic force almost cancels
the electric force. Viewed in the rest frame of the bunch, there is no magnetic force
(neglecting the relative motion of the particles within the bunch); but the expansion
driven by the Coulomb forces is slowed by time dilation when viewed in the lab
frame.
To calculate the effects of the space-charge forces, we should use the fields of a
Gaussian bunch. The expressions are complicated (look them up!) so we use a
linear expansion…
71
Beam Dynamics: Space Charge in the Linear Approximation
An expression for the vertical space-charge force (normalized to the reference
momentum) expanded to first order in y is:
Fy  2
z
y
 3  y  x   y 
re
where re is the classical radius of the electron;  is the beam energy; z is the
longitudinal density of particles in the bunch; x, y are the rms bunch sizes.
The vertical force (integrated around the lattice) results in a vertical tune shift:
 y  
Fy
1

ds
y

4
y
Since the density depends on the longitudinal position in the bunch, and the force
Fy is really nonlinear, every particle experiences a different tune shift; therefore, the
tune shift is really a tune spread, or an “incoherent” tune shift.
72
Beam Dynamics: Space Charge in the Linear Approximation
The space charge incoherent tune shift can be written:
 y z
 y  
ds
2 3   y  x   y 
re
Note the factor 1/3; for high-energy electron storage rings, this generally suppresses
the space charge forces so that the effects are negligible. However, the tune shift
becomes appreciable (~ 0.1 or larger) when:
the longitudinal charge density is high;
the vertical beam size is very small;
the circumference of the ring is large.
The damping rings will operate at reasonably high bunch charges and very small
vertical emittances. Therefore, we need to consider space charge effects,
particularly in configuration options with a large circumference (e.g. the dogbone
rings, with circumference ~ 17 km).
73
Beam Dynamics: Space Charge Effects in the Damping Rings
To estimate the impact of space charge forces on damping ring performance, we
need to go beyond the linear approximation. For example, we can perform tracking
simulations, where we include the full nonlinear form of the space charge forces.
In the damping rings, we typically observe some emittance growth.
74
Emittance growth from space charge calculated by tracking in SAD (K. Oide)
Beam Dynamics: Space Charge Effects in the Damping Rings
The emittance growth observed depends on the tunes of the lattice.
Tune scan of emittance growth from space charge in a
17 km lattice calculated by tracking in SAD (K. Oide)
75
Beam Dynamics: Space Charge and Coupling Bumps
Space charge forces can be reduced by increasing the vertical beam size. In an
uncoupled lattice, this can be done (for a given emittance) by increasing the beta
function; but this makes the beam more sensitive to disruptive effects such as stray
magnetic fields.
An alternative is to use a “coupling transformation” that makes the horizontal
emittance contribute to the vertical as well as the horizontal beam size. Even if the
vertical emittance is orders of magnitude smaller than the horizontal, the beam can
then be made to have a circular cross-section, without increasing the beta functions.
In the dogbone damping rings, an appropriate transformation can be used at the
entrance to the long straight, and a corresponding transformation can be used at the
exit of the long straight, to remove the coupling and make the beam flat again.
Since there is no radiation emitted from the beam in the straight, the emittances are
preserved.
76
Beam Dynamics: Space Charge and Coupling Bumps
skew quadrupoles
x 2  11I  I  11II II
xy  13I  I  13II II
y 2   33I  I   33II  II
Lattice functions at the entrance to a long straight with a
coupling transformation. The value of  33I gives the
contribution of the “horizontal” emittance to the vertical
beam size.
77
Beam Dynamics: Space Charge and Coupling Bumps
Coupling bumps do not necessarily solve the problem: although they mitigate space
charge effects, they can drive resonances that themselves lead to emittance growth.
Tune scan of emittance growth in a
17 km lattice, with space charge,
without coupling bumps.
Tune scan of emittance growth in a
17 km lattice, with space charge,
and with coupling bumps.
78
Beam Dynamics: Microwave Instability
Particles can interact directly (space charge; intrabeam scattering).
Particles in a bunch can also interact indirectly, via the vacuum chamber.
The electromagnetic fields around a bunch must satisfy Maxwell’s equations.
The presence of a vacuum chamber imposes boundary conditions that modify the fields.
Fields generated by the head of a bunch can act back on particles at the tail, modifying
their dynamics and (potentially) driving instabilities.
Wake fields following a point
charge in a cylindrical beam
pipe with resistive walls.
(Courtesy, K. Bane)
79
Beam Dynamics: Wake Functions
Finding analytical solutions for the field equations is possible in some simple cases.
Generally, one uses an electromagnetic modeling code to solve numerically for a
given bunch shape in a specified geometry.
It is useful to determine the “wake function” W//(z), W(z) for a given component,
which gives the field behind a point unit charge integrated over the length of the
component. For a bunch distribution (z):
 ( z )  
p y ( z )  
re

re


   zW  z  zdz
//
z´
z
y

 y( z)  zW

z
 z  zdz
s
z
z=0
where (z) is the energy deviation of a particle at position z in the bunch, and py(z) is
the normalized transverse momentum of a particle at position z in the bunch.
Generally, the wake functions are found numerically, by solving Maxwell’s equations.
80
Beam Dynamics: Wake Function and Impedance
Consider the longitudinal wake averaged over an entire storage ring. Suppose that
the storage ring is filled with an unbunched beam so that the particle density is:
  z   0   exp  i
z 

 c 
The energy change of a particle in one turn is:
  z   
re


   zW  z  zdz
//
z

z
i
re 
    0   e c
 z

re c

i


W//  z  z dz 

z
e c  Z //  
where we have defined the impedance:
Z //   
and we assume that Z//(0) = 0.
1
  i z W  z dz
exp

 //
c
c 

81
Beam Dynamics: Wake Function and Impedance
The change in energy deviation per turn is:
  z  
re c

i
z
e c  Z //  
which can be written:
E  z 
 I ; z Z //  
e
or, in other words, V = IZ, just as one would expect from an impedance.
Now we need to find the effect of the impedance on the beam…
82
Beam Dynamics: Impedance and Beam Evolution
The evolution of the beam distribution (,;t) obeys the Vlasov equation:
    


0
t


where  is the azimuthal coordinate in the accelerator (i.e. distance around the ring,
in radians). This equation is just a continuity equation in phase space. We suppose
that the distribution is uniform, plus some perturbation of defined frequency:
 ,  ; t   0 ( )  n   exp in  nt 
We can also write:
  0 1   
  Z // n 
I 0 0
E / e 2
   d e
i  n  n t 
n
Our goal is to find the mode frequency n: this gives the time evolution of the
perturbation. If n has a positive imaginary part, then the beam distribution is
unstable and the perturbation will grow exponentially with time.
83
Beam Dynamics: Impedance and Beam Evolution
Making the appropriate substitutions into the Vlasov equation and expanding to first
order in the perturbation , we find the equation:
n0  n    iZ // n 
I 0 0 0
n  d

E / e 2 
Integrating both sides over , we find the dispersion relation:
1  iZ // n 
I 0 0
E / e 2
0 / 
 n0  n d
The dispersion relation is an integral equation for the mode frequency n, given an
impedance Z//(). This is not easy to solve; even if we have a solution for n and
we find that the beam is unstable, we cannot really say anything about the long-time
evolution of the distribution, because we have assumed that the perturbation is
small. We have also ignored the fact that the beam is bunched, and particles
perform synchrotron oscillations. A better approach is to use a numerical code to
solve the Vlasov equation directly, and watch the evolution of quantities like the
energy spread.
84
Beam Dynamics: Microwave Instability and Keill-Schnell Criterion
Using the dispersion relation, and making some crude assumptions about the form
of the impedance, we find that the beam goes unstable when:
2
Z
  p   z
 Z0
n
2 N 0 re
This is the Keill-Schnell criterion. It gives the threshold of an instability which
appears as a density modulation in the beam, where the wavelength of the
modulation is C/n (for ring circumference C). The impedance is crudely
characterized as Z(n0)/n = constant; this is not really a satisfactory approximation.
Note that p is the momentum compaction, and  is the energy spread. If either of
these quantities is zero, then the beam is unstable. Having non-zero values for these
quantities stabilizes the beam through Landau damping. As the density
modulation develops, it tends to be smeared out because particles with different
energies () move around the ring at different rates (p), which tends to “smear
out” the modulation.
85
Beam Dynamics: Microwave Instability and Damping Ring Design
The microwave instability is often observed as an increase in energy spread in the
beam. This needs to be avoided in the damping rings, because any increase in
longitudinal emittance will make operation of the bunch compressors difficult. An
instability can also appear in a “bursting” mode, where there is a dramatic increase
in energy spread which damps down, before growing again. This type of instability
in the SLC damping rings caused significant problems.
In the ILC damping rings, the energy spread, bunch length, beam energy and
number of particles per bunch are all specified (or limited) from other
considerations. To avoid the microwave instability, the options are:
– Design a lattice with high momentum compaction. This leads to a very large RF
voltage (which is expensive and has its own risks) and a high synchrotron tune (which
can lead to a limited energy acceptance).
– Design and build a chamber with a very low impedance. This is technically
challenging.
In practice, we may need to have both a large momentum compaction and a
chamber with very low impedance. It’s a challenge to get the balance right.
86
Beam Dynamics: Coupled-Bunch Instabilities
As well as the short-range wakefields acting over the length of a single bunch, there
are also long-range wakefields that act over the distance between bunches. The
principal sources of long-range wakefields are:
- resistive-wall wakefield, resulting from the modifications to the fields in the vacuum
chamber that arise when the walls of the chamber are not perfectly conducting.
- higher-order modes (HOMs) in the RF cavities (and other chamber cavities).
Oscillations of the electromagnetic fields in cavities are excited by a bunch passage;
modes with high Q damp slowly, and can persist from one bunch to the next.
Resistive-wall wakefields depend on the vacuum chamber geometry (larger
chambers have lower wakefields) and material (better conducting materials have
lower wakefields). Cavity HOMs depend principally on the geometry, and vary
significantly from one design to another. Various techniques are used in cavity
design to damp the HOMs to acceptable levels.
The effects of long-range wakefields include the growth of coherent oscillations of
the individual bunches, with growth rates depending on the fill pattern and beam
current. In high-current rings, feedback systems are often needed to suppress the
coherent motion of the bunches, thereby keeping the beam stable.
87
Beam Dynamics: Coupled-Bunch Instabilities
s
2
py
1
y
s
We can describe the kick on the trailing particle (2) from the wakefield of the
leading particle (1) in terms of a wake function (N0 is the bunch charge):
p y , 2  
re

N 0W  s  y1
In a storage ring containing M bunches, we construct the equation of motion:
r c
yn t    yn t    e N 0 
 T0
k
2
betatron
oscillations
  kC  m  n C y  t  kT  m  n T 
W
 m


0
0
M
M




m 0
M 1
multiple multiple
turns
bunches
88
Beam Dynamics: Coupled-Bunch Instabilities
The equation of motion (from the previous slide) is:
r c
yn t    yn t    e N 0 
 T0
k
2
  kC  m  n C y  t  kT  m  n T 
W
 m


0
0
M
M




m 0
M 1
We try a solution of the form:
n
yn t   exp  2i  exp  i  t 
M

spatial (bunch number)
dependence
time dependence
Substituting this solution into the equation of motion, we find an equation that gives
us (in principle) the mode frequency  for a given mode number . As usual, the
imaginary part of  gives the instability growth (or damping) rate.
89
Beam Dynamics: Coupled-Bunch Instabilities
In a coupled-bunch instability, the bunches
perform coherent oscillations.
The mode number  gives the phase advance
from one bunch to the next at a given moment
in time.
The examples here show the modes ( = 0, 1, 2
and 3) in an accelerator with M = 4 bunches.
From A. Chao, “Physics of Collective Beam
Instabilities in Particle Accelerators,” Wiley (1993).
90
Beam Dynamics: Resistive-Wall Instability
Each mode can have a different growth (or damping) rate. For the ILC damping
rings, the resistive-wall wakefields are expected to lead to a resistive-wall
instability, with the fastest modes having growth times of the order of 10 turns. This
is much faster than the synchrotron radiation damping rate, and close to the limit of
the damping rates that can be provided by fast feedback systems.
The transverse resistive-wall wakefield for a chamber with length L and circular
cross-section of radius b is given (for z<0) by:
W  z  
2

c L
 c b3
1
z
Implications for the ILC damping rings are:
- beam pipe radius must be as large as possible to keep the wakefields small - note that
the wakefield (and hence the growth rates) vary as 1/b3;
- beam pipe must be constructed from a material with good electrical conductivity (e.g.
aluminum) to keep the wakefields small - note that the wakefields vary as 1/c
91
Beam Dynamics: Resistive-Wall Instability
For the resistive-wall instability, the growth (damping) rate for the fastest mode is
found to be:
1
 ( )
sgn   
MN 0 re c 2
 3
b  T0 2c0

where M is the total number of bunches, N0 is the number of particles per bunch, re
is the classical radius of the electron, b is the beam-pipe radius,  is the relativistic
factor at the beam energy,  is the betatron frequency, T0 is the revolution period,
c is the conductivity of the vacuum chamber material, 0 is the revolution
frequency. Also, if  is the betatron tune, and N is the integer closest to , then
we define:
   N  
 12    
1
2
Note that if  is positive (tune below the half-integer), then the fastest mode is
damped; if  is negative (tune above the half-integer), then the fastest mode is
antidamped. It therefore helps if the lattice has betatron tunes that are below the
half-integer.
92
Beam Dynamics: Resistive-Wall Instability
Resistive-wall growth rates in a 6 km ILC damping ring lattice:
1000
1500
100
1
Growth Rate s
Growth Rate s
1
2000
1000
500
0
10
1
500
0.1
1000
0.01
0
500
1000
1500
2000
Mode Number
2500
3000
1750
Linear scale:
All modes.
Note:
2000
2250
2500
2750
Mode Number
3000
3250
Log scale:
Unstable modes only.
Revolution frequency  50 kHz.
Synchrotron radiation damping time  25 ms.
93
Beam Dynamics: Fast-Ion Instability
Residual gas molecules in the
vacuum chamber are ionized by the
passage of bunches of electrons.
During the passage of a train of
closely-spaced bunches, the ion
density can reach levels such that
the dynamics of the bunches towards
the rear of the bunch train are significantly affected.
This is a complicated effect to analyze, but the growth rates may be estimated from:
1


1 2
c
 yky
3 3 i i
where the ion frequency spread i/i  0.3 (generally); and the ion focusing is:
ky 
i re
 y  x   y 
i   i
p
N 0 nb
kT
where x, y are the beam sizes; i is the ion line density; i is the ionization cross section; p
is the residual gas pressure; N0 the number of particles per bunch; nb is the number of
bunches.
94
Beam Dynamics: Fast-Ion Instability
When calculating the growth rates, we need to take into account the fact that the beam sizes
change with position in the lattice, and with time during the damping process. We also need
to take into account the fact that ions with low mass may not be “trapped” by the bunch train.
The trapping condition is:
N 0 rp sb
A
2 y  x   y 
where sb is the bunch spacing; rp is the classical radius of the proton. At injection, all ions
are trapped because the beam sizes are relatively large. Different ions become “released” at
different times in different sections of the lattice, depending on the lattice functions.
Ion trapping, growth
rates and tune shift in
a 6 km ILC damping
ring lattice, during the
damping cycle.
95
Beam Dynamics: Fast-Ion Instability
Ion effects have been observed at a number of storage rings (ALS, PLS, Tristan,
PEP-II, KEK-B), but quantitative studies are difficult because few existing rings are
capable of reaching the parameter regime where the effect is significant. The main
problem is in achieving the very small vertical beam size where the ion focusing
becomes large. Therefore, ion effects are still being studied.
Using the present models, growth rates from fast-ion instability in the ILC damping
rings are expected to be fast (of order 10 s). The implications for the design are:
- The vacuum system must be capable of achieving very low pressures (<1 ntorr), to
reduce the number of ions produced;
- There must be regular gaps in the fill in the damping rings, to clear the ions and
prevent large densities being accumulated. Typically, gaps of ~ 40 ns are required
every ~ 40 bunches.
96
Beam Dynamics: Electron Cloud Effects
Electron cloud effects in positron rings are analogous to ion effects in electron rings.
During the passage of a bunch train, electrons are generated by a variety of
processes (photoemission, gas ionization, secondary emission). Under certain
circumstances, the density of electrons in the vacuum chamber can reach levels that
are high enough to affect significantly the dynamics of the positrons. When this
happens, an instability can be observed.
In positron damping rings, the build-up of electron cloud is usually dominated by
secondary emission, in which primary electrons are accelerated in the beam
potential, and hit the walls of the vacuum chamber with sufficient energy to release
a number of secondaries.
The critical parameters for the build-up of the electron cloud are:
- charge of the electron bunches;
- the separation between the electron bunches;
- the properties of the vacuum chamber (particularly, the number of secondary electrons
emitted per incident primary electron = the Secondary Emission Yield or SEY);
- the presence of a magnetic or electric field (e-cloud can be worse in dipoles and
wigglers);
- the beam size (which affects the energy with which electrons strike the walls).
97
Beam Dynamics: Electron Cloud Effects
Secondary emission yield (SEY) is critical for build-up of electron cloud.
Measurements of SEY
of TiZrV (NEG)
coating, F. le Pimpec,
M. Pivi, R. Kirby.
98
Beam Dynamics: Electron Cloud Effects
Simulations of electron-cloud build-up need to include all relevant effects (chamber
surface, beam pattern, magnetic and electric fields etc.)
Depending on the SEY, peak cloud density can vary by orders of magnitude.
Simulation of e-cloud build-up in an ILC damping ring, by Mauro Pivi, using Posinst.
99
Beam Dynamics: Electron Cloud Effects
Interaction between the beam and the electron-cloud is a complicated phenomenon.
In the ILC damping rings, the dominant instability mode is expected to be a “headtail” instability, which may appear as a blow-up of vertical emittance.
The effects are best studied by simulation. Various effects need to be taken into
account, including the density enhancement (by an order of magnitude) that can
occur in the vicinity of the beam during a bunch passage.
Simulation of vertical emittance
growth in a 6 km ILC damping
ring in the presence of electron
cloud of different densities.
K. Ohmi.
100
Beam Dynamics: Electron Cloud Effects
To avoid instabilities associated with electron cloud, we expect to need to keep the
average electron cloud density below ~ 1011 m-3.
This will require keeping the peak SEY of the chamber surface below ~ 1.1, which
will be a challenging task. Presently, three main approaches are being investigated:
- Coating the aluminum vacuum chamber (peak SEY ~ 2) with a low SEY material, for
example TiN or TiZrV.
- Cutting grooves in the vacuum chamber surface to “trap” and re-absorb low-energy
secondary electrons before they can be accelerated by the beam.
- Using clearing electrodes.
Currently, an active research program is under way to find the most effective
technique.
The present ILC baseline specifies two 6 km positron damping rings, precisely to
allow sufficient bunch separation in each ring so that the electron cloud does not
build up to dangerous levels. If an effective suppression technique can be found,
only one ring may be needed.
101
Beam Dynamics: Suppressing E-Cloud with Low-SEY Coatings
Achieving a peak SEY below 1.2 seems possible with sufficient conditioning.
Reliability/reproducibility and durability are concerns.
102
Beam Dynamics: Suppressing E-Cloud with a Grooved Chamber
Electrons entering the grooves release secondaries which are reabsorbed at low
energy (and hence without releasing further secondaries) before they can be
accelerated in the vicinity of the beam.
103
Beam Dynamics: Suppressing E-Cloud with a Grooved Chamber
Measurements suggest that grooves can be very effective at suppressing secondary
emission, and will be tested experimentally in PEP-II later this year. Wakefields are
a concern, but if the grooves are cut longitudinally, should be ok.
M. Pivi and G. Stupakov 104
Beam Dynamics: Suppressing E-Cloud with Clearing Electrodes
Low-energy secondary electrons emitted from the electrode surface are prevented
from reaching the beam by the electric field at the surface of the electrode. This
also appears to be an effective technique for suppressing build-up of electron cloud.
105
Part 4
Technical Subsystems
You should be able to describe the principles of operation behind
important technical subsystems in the damping rings, including the
injection/extraction kickers, fast feedback systems and the damping
wiggler. For the damping wiggler, you should be able to explain the
issues involved in choosing between the various technology options.
Technical Subsystems
Damping rings, like any storage ring, can be broken down into a number of
technical subsystems which are closely interrelated:
Vacuum system
Main magnets (dipoles, quadrupoles, sextupoles)
Wiggler (insertion devices)
RF system
Diagnostics and instrumentation
Fast feedback system
Orbit and coupling control (steering magnets, skew quadrupoles…)
Injection and extraction system
Control system
Cryogenics (for superconducting RF or magnets)
Conventional facilities (tunnel, water…)
Alignment and supports
Personnel protection system
…
107
Technical Subsystems
All of these are important.
I will cover in a very superficial way, just three of them:
Vacuum system
Main magnets (dipoles, quadrupoles, sextupoles)
Wiggler (insertion devices)
RF system
Diagnostics and instrumentation
Fast feedback system
Orbit and coupling control (steering magnets, skew quadrupoles…)
Injection and extraction system
Control system
Cryogenics (for superconducting RF or magnets)
Conventional facilities (tunnel, water…)
Alignment and supports
Personnel protection system
…
108
Technical Subsystems: Injection/Extraction Principles
trajectory of
incoming beam
following
bunch
empty
RF bucket
injection
kicker
preceding
bunch
trajectory of
stored beam
1. Kicker is OFF.
“Preceding” bunch exits
kicker electrodes.
Kicker starts to turn ON.
2. Kicker is ON.
“Incoming” bunch is
deflected by kicker.
Kicker starts to turn OFF.
3. Kicker is OFF by the
time the following
bunch reaches the
kicker.
109
Technical Subsystems: Injection/Extraction Kickers
Several different types of fast kicker are possible. For the ILC damping rings, the
injection/extraction kickers are composed of two parts:
- fast, high-power pulser;
- stripline electrodes.
Again, several technologies are possible for the fast, highpower pulser. We do not consider this part of the kicker,
except to note that the parameters for the ILC damping
rings are very challenging, and pulser development is ongoing.
The stripline electrodes are comparatively
straightforward: we will look at these in a little more
detail. Physically, they are fairly simple, consisting of
two plates, connected to a high-voltage line, between
which the beam travels. The design is fairly challenging,
because of the need to provide a large on-axis field while
maintaining field quality and physical aperture; and the
need to match the impedance to the power supply.
Kicker stripline designs.
S. de Santis.
110
Technical Subsystems: Injection/Extraction Kickers
Let us take a simplified model of the stripline
electrodes, consisting of two infinite parallel
plates. The beam travels in the z direction. We
apply an alternating voltage between the plates:
V  V0 e
y
z
it
x
From Maxwell’s equations, there are electric
and magnetic fields between the plates:
E x  E0 ei ( kz t )
By 
E0 i ( kz t )
e
c
A particle traveling in the +z direction with speed c will experience a force:
Fx  qEx  vz By   q1   E0ei 1 t
For an ultra-relativistic particle,   1, and the electric and magnetic forces almost
exactly cancel: the resultant force is small. But for a particle traveling in the
opposite direction to the electromagnetic wave,   –1, and the resultant force is
twice as large as would be expected from the electric force alone.
111
Technical Subsystems: Injection/Extraction Kickers
Let us calculate the deflection of a particle traveling between a pair of stripline
electrodes. Let us suppose that there is a voltage pulse of amplitude V and length
2L traveling along the electrodes, which consist of infinitely wide parallel plates of
length L separated by a distance d:
x
d
z
L
V
2L
The change in (normalized) horizontal momentum of the particle is:
p x 
Fx L
V L
2
p0 c
E ed
where E is the beam energy. In reality, we can account for the fact that the
electrodes are not infinite parallel plates by including a geometry factor, g.
112
Technical Subsystems: Injection/Extraction Kickers
The kickers must inject and extract individual bunches, without affecting preceding
or following bunches. This means that the rise and fall times of the kickers must be
less than the time between bunches.
The “effective” rise and fall times of the kickers include:
- the rise/fall time of the voltage pulse;
- the time taken for the pulse to “fill” the electrodes, and for the electrodes to “empty”
at the end of the pulse.
Even for an absolutely “hard-edged” voltage pulse (unphysical!) the effective
rise/fall time is 2L/c, where L is the length of the stripline electrodes. For example,
if the electrodes are 30 cm long, then the effective rise/fall time for a hard-edged
voltage pusle is 2 ns: this is an appreciable fraction of the minimum bunch
separation of 3.08 ns (= 2 RF buckets for RF frequency of 650 MHz).
Shorter stripline electrodes help to achieve a faster rise/fall time, but provide
proportionately less “kick”. The solution is to use a large number (20? 40?) of short
electrode pairs in series. This also helps reduce the overall pulse-to-pulse jitter (by
a factor 1/N, for N sets of electrodes).
113
Technical Subsystems: Injection/Extraction Kickers
voltage pulse
Stage 1: Leading bunch must exit
kicker before voltage pulse arrives.
target
bunch
kicker
Stage 2: Voltage pulse fills kicker as
target bunch arrives.
Stage 3: Voltage pulse fills kicker
while target bunch is between
striplines.
Stage 4: Voltage pulse exits kicker
before trailing bunch arrives.
114
Technical Subsystems: Injection/Extraction Kickers
How large a kick is needed? Consider the extraction optics:
septum
kicker
quadrupole
If the beta function at the kicker is x,k, the beta function at the septum is x,s, and
the betatron phase advance is x,k-s, then the transfer matrix element R21 from the
kicker to the septum is:
R12   x,k  x,s sin  x,k  s
In other words, assuming that the bunch is on-axis at the kicker, we have:
xs  R12px,k   x,s  x,k sin  x,k s
The optics should be designed with large beta function at the kicker and phase
advance of /2 from the kicker to the septum. A large beta function at the septum
does not really help, because it increases the beam size: aperture is the issue.
115
Technical Subsystems: Injection/Extraction Kickers
Suppose that we require a beam offset at the septum of 30 mm (engineering
constraint); that the beta functions at the kicker and septum are each 50 m; and that
we have the optimal phase advance from kicker to septum. Then the deflection
required from the kickers is simply:
px ,k 
xs
 0.6 mrad
 x ,k  x ,s sin  x ,k s
The deflection provided is:
p x ,k  2 g
V L
E ed
Let us take L = 20 cm, d = 2 cm, g = 0.6, E = 5 GeV. For a deflection of 0.6 mrad,
this implies that the voltage between the striplines needs to be ~ 250 kV. This far
exceeds the capability of any fast, high-power pulser. A more reasonable, but still
very challenging goal is 10 kV. The total required deflection is then achieved by
using 25 pairs of electrodes in series (total length 5 m).
116
Technical Subsystems: Fast Feedback Systems
Fast feedback systems are needed to damp coupled-bunch instabilities (e.g.
resistive-wall instability), that appear as coherent oscillations of bunches in the
beam. They allow a stable beam to be maintained at high currents.
Conceptually, they are reasonably straightforward:
single bunch
shown at
different times
y
pick-up
py
kicker
amplifier
In practice, these are very challenging technical systems, requiring high
performance from the pick-ups, from the fast, high-bandwidth power amplifiers and
from the kickers.
117
Technical Subsystems: Fast Feedback Systems
Fast feedback systems are needed to damp both longitudinal and transverse
coupled-bunch instabilities in the damping rings.
Let us consider operation of a feedback system from the point of view of the beam
dynamics. Our goal will be to find an expression for the damping rate in terms of
the gain, g, defined by:
p y , 2  g  y1
where y1 is the measured beam position at the pick-up, and py,2 is the kick
provided by the kicker.
The amplifier performance is the main limiting factor on the gain (and hence on the
damping rate) that can be achieved. We are also interested in residual noise that is
excited on the beam by the feedback system, because of noise in the pick-up signal
or the amplifier.
118
Technical Subsystems: Fast Feedback Systems
Consider a bunch that has initial betatron amplitude (invariant action) J1, and
betatron phase 1 at the pick-up. The transverse offset is:
y1  21 J1 cos 1
If the phase advance from the pick-up to the kicker is 12, then the normalized
transverse momentum at the kicker will be:
p y,2  
2 J1
sin 1  12    2 cos1  12 

2 J1
cos1    2 sin 1 
2
2
where we have assumed the optimal case 12 = /2. The kicker now provides a
deflection:
p y , 2  g  y1
Writing the action…
2 J1   1 y12  21 y1 p y ,1  1 p y2,1
2 J 2   2 y22  2 2 y2  p y , 2  p y , 2   1  p y , 2  p y , 2 
2
119
Technical Subsystems: Fast Feedback Systems
…we find (after some algebra):
J 2  J1 1  2 g 1 2 cos 2 1  g 2 1 2 cos 2 1 
Over many turns, we can average the phase angle:
cos 2 1  12
in which case we find:
J 2  J1 1  g 12  12 g 2 1 2   J1 exp  g 12 
where we have assumed in the last step that g12 << 1. We see that on average,
the action damps exponentially:
t 
t 


J t   J 0 exp   g 1 2   J 0 exp   2

T



0 
FB 
where the feedback damping time is:
 FB 
T0
2 g 1 2
120
Technical Subsystems: Fast Feedback Systems
Damping times of around 20 turns can be achieved using modern fast feedback
systems. However, one potential drawback of very fast damping is a larger
“excitation” of bunch jitter from noise on the pick-up signal. Let us calculate how
large we expect the jitter to be, for a given damping rate and noise level.
If we replace the pick-up signal:
y1  y1  y
where y is a noise term, then repeating the previous analysis, we find:
2
2
dJ  2 g y
2


J
dt
2T0
 FB
The equilibrium action is:
J equ 
 FB
4T0
 2 g 2 y 2 
g
8
2
y 2
1
121
Technical Subsystems: Fast Feedback Systems
Let us make an order-of-magnitude estimate of the pick-up noise limit, beyond
which the beam jitter is larger than 10% of the beam size. This specification on the
jitter can be written:
y
y

2 y J y
 y y
 0.1

J y  0.005 y
For a vertical emittance of 2 pm, the limit on the action Jy is therefore 10-14 m.
Suppose we require a damping time of 10 turns, and that 1  2  10 m. We have:
 FB
T0

1
2 g 1 2

g  0.005 m 1
Finally, we find:
y 2 
8
g
1
Jy
2

y 2  4 μm
The noise on the pick-up should be less than 4 m. This is a reasonable goal. If
necessary, the specification can be relaxed by increasing 1.
122
An aside: the damping wiggler
The damping time in a storage ring depends on the rate of energy loss
of the particles through synchrotron radiation. In the damping rings,
the rate of energy loss can be enhanced by insertion of a long wiggler,
consisting of short (~ 10 cm) sections of dipole field with alternating
polarity.
y
z
x
The magnetic field in the
wiggler can be approximated by:
By = Bw sin(kzz)
123
Technical Subsystems: Damping Wigglers
Principal issues for the wiggler are as follows:
- Field quality: wigglers have intrinsically nonlinear fields, which can limit the dynamic
aperture.
- Physical aperture: a large physical aperture is needed to ensure good injection efficiency;
the high field strengths (~1.6 T) required in the damping wigglers are easier to achieve at
large aperture with some technologies (superconducting) than with others (permanent or
electromagnetic).
- Power consumption: the running costs (electricity) for electromagnetic wigglers are high.
For the size of wigglers needed in the damping rings, the costs can run into $Ms.
Superconducting wigglers take relatively little power; permanent or hybrid wigglers take
zero power.
- Resistance to radiation damage is a concern for permanent magnet material (loses field
strength, or becomes activated) and superconducting magnets (energy deposition can lead
to quenching). Electromagnets are relatively robust in high radiation environments.
- Materials and construction costs: Permanent magnet material is very expensive.
Superconducting wigglers are fairly complex and involved. Electromagnets use relatively
cheap materials, and construction is relatively straightforward.
124
Technical Subsystems: Damping Wigglers in the KEK-ATF
125
Technical Subsystems: Damping Wigglers
The key wiggler parameters from point of view of beam dynamics are:
-
peak field
period
aperture
field quality
These are strongly connected with the wiggler technology, and should not really be
considered independently of the technology options. However, we can consider the
beam dynamics impact of different parameter choices, to arrive at an understanding
of what parameters we would ideally like.
We consider mainly the peak field and the period, and consider an idealized model
of a wiggler, i.e. a perfectly periodic device with infinite width. The field in such a
device is given by:
By = Bw sin(kzz)
where Bw is the peak field and kz = 2/z where z is the wiggler period.
126
Technical Subsystems: Damping Wigglers
Let us start by considering the trajectory of the beam through the wiggler. Let z be
the distance along the magnetic axis of the wiggler (a straight line). Then the
equation of motion for a particle following the reference trajectory of the beam is:
d 2 x By Bw


sin k z z 
2
dz
B B
where B = P0/e is the beam rigidity (P0 is the reference momentum). The solution
is simply:
x
Bw 1
sin k z z   aw sin k z z 
B k z2
The path length in one period is:
w
w
0
0
 ds  
2
dx
1    dz  w 1  12 aw2 k z2
 dz 
127
Technical Subsystems: Damping Wigglers
For a beam of energy 5 GeV in a wiggler with peak field 1.6 T and period 0.4 m, the
amplitude of the trajectory oscillations is roughly aw  389 m. The difference in
path length between the beam trajectory and the wiggler period is approximately 105 : we shall ignore this difference in the following calculations.
w
We shall estimate the wiggler contribution to:
- the synchrotron radiation damping rates;
- the equilibrium energy spread;
- the natural emittance.
Note: in the following we assume that the damping partition numbers, which describe the
way the damping is “distributed” between the three degrees of freedom in the beam, are
equal to 1 for the transverse planes, and 2 for the longitudinal plane. This is the usual
situation if there is no quadrupole component in the dipole magnet fields.
128
Technical Subsystems: Damping Wigglers
The synchrotron radiation damping rate in a storage ring is given by:
C E03

I2
  T0
1
Lw Bw2
I 2   2 ds 
2

2 B 
1
(wiggler only)
The equilibrium energy spread is given by:
I
1
   Cq 2 3
2
I2
2
I3  
4 Lw Bw3
ds 
3
3

3  B 
1
I3
8 Bw

I 2 3 B
(wiggler only)
To keep the energy spread under control, we need to limit the peak field. An upper
limit on the energy spread is placed by the beam dynamics in the bunch
compressors. To compensate the limit on the peak field in the damping rate, we can
increase the overall length of the wiggler.
Note that the damping rate and equilibrium energy spread are independent of the
wiggler period: only the peak field matters.
129
Technical Subsystems: Damping Wigglers
The natural emittance in a storage ring is given by:
I
 0  Cq 5
I2
2
I5  
4  x Lw Bw5
ds 
3
5

15k z2  B 
H
8  x Bw3
I5

I 2 15k z2  B 3
(wiggler only)
(wiggler only)
We see that the emittance contribution is proportional to the cube of the peak field;
and is proportional to the square of the period. It is also directly proportional to the
beta function, so smaller beta functions help to reduce the emittance.
To maintain a low emittance, we need to limit the peak field, and keep the period
short. The wiggler period is important because the larger the period, the larger the
dispersion generated by the bending in the wiggler, and the larger the quantum
excitation.
130
Technical Subsystems: ILC Damping Wigglers
The baseline configuration for the ILC damping rings specifies superconducting
wigglers, because of the large physical aperture that can be achieved. Radiation
effects are a concern; but electromagnetic wigglers are not attractive because of the
very high power consumption, and the relatively narrow physical aperture that is
needed to achieve the field strength.
Presently, the wiggler peak field is specified at 1.6 T, and the period at 0.4 m, and
the total length of wiggler is around 200 m. This allows the specifications for
damping time (< 25 ms), natural emittance (< 8 m normalized) and energy spread
(< 0.15%) to be achieved…
…but there is probably scope for optimization…
131