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:

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.

Prerequisites:

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.

Slide4:

Part 1
Principles of Operation

Introduction: Basic Principles of Operation - Performance Specs:

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

Introduction: Basic Principles of Operation - Need for Compression:

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.
We must compress the bunch train to fit into a damping ring.
This is achieved by injecting and extracting bunches one at a time.

Introduction: Basic Principles of Operation - Injection/Extraction:

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.

Introduction: Basic Principles of Operation - Injection/Extraction:

Introduction: Basic Principles of Operation - Injection/Extraction trajectory of stored beam trajectory of incoming beam preceding bunch following bunch empty RF bucket injection kicker 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.

Introduction: Basic Principles of Operation - Train (De)compression:

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.
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). 1 5 2 3 4 6 1 2 3 4 5

Introduction: Basic Principles of Operation - ILC Baseline Configuration:

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.

Introduction: Basic Principles of Operation - Summary:

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.

Slide12:

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:

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.

Lattice Design and Parameter Optimization: Circumference:

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.

Lattice Design and Parameter Optimization: Circumference:

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 field-free 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.

Lattice Design and Parameter Optimization: Circumference:

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)

Lattice Design and Parameter Optimization: Circumference:

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

Lattice Design and Parameter Optimization: Circumference:

Lattice Design and Parameter Optimization: Circumference Comparison between electron cloud instability thresholds and cloud densities, in various damping rings under various conditions.

Lattice Design and Parameter Optimization: Circumference:

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.

Lattice Design and Parameter Optimization: Circumference:

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:
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.

Lattice Design and Parameter Optimization: Circumference:

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.

Lattice Design and Parameter Optimization: Circumference:

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.

Lattice Design and Parameter Optimization: Circumference:

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. 3inj 3inj Dynamic aperture with magnet errors, and energy deviation. Left: 17 km dogbone lattice. Right: 6 km circular lattice.

Lattice Design and Parameter Optimization: Circumference:

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.

Lattice Design and Parameter Optimization: Damping Time:

Lattice Design and Parameter Optimization: Damping Time The beam emittances evolve as:
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.

Lattice Design and Parameter Optimization: Beam Energy:

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; space-charge, 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).

An aside: the damping wiggler:

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 x z By = Bw sin(kzz) The magnetic field in the
wiggler can be approximated by:

Lattice Design and Parameter Optimization: Beam Energy:

Lattice Design and Parameter Optimization: Beam Energy The (transverse) damping time in a storage ring is given by:
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:

Lattice Design and Parameter Optimization: Beam Energy:

Lattice Design and Parameter Optimization: Beam Energy The natural energy spread in a storage ring is given by:
where is the relativistic factor, and Cq = 3.832×10-13 m is a physical constant.
Performing the integrals for a wiggler, we find:
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:
Note the scaling with energy and wiggler field:

Lattice Design and Parameter Optimization: Beam Energy:

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. < 27 ms < 0.13% 5 GeV < E0 < 5.5 GeV Lw = 200 m
Bw = 1.6 T
T0 = 6.6 km/c

Lattice Design and Parameter Optimization: Energy and Polarization:

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 (space-charge, 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):

Lattice Design and Parameter Optimization: Lattice Styles:

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:
where
If the dipoles have zero quadrupole component, then the damping partition number Jx 1.

Lattice Design and Parameter Optimization: Lattice Styles:

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):
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…)

Lattice Design and Parameter Optimization: Lattice Styles:

Lattice Design and Parameter Optimization: Lattice Styles FODO Lattice: F 100

Lattice Design and Parameter Optimization: Lattice Styles:

Lattice Design and Parameter Optimization: Lattice Styles Double Bend Achromat (DBA) Lattice: F = 3

Lattice Design and Parameter Optimization: Lattice Styles:

Lattice Design and Parameter Optimization: Lattice Styles Theoretical Minimum Emittance (TME) Lattice: F = 1

Lattice Design and Parameter Optimization: Lattice Styles:

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:
where L is the length of the dipole.

Lattice Design and Parameter Optimization: Lattice Styles:

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:
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:
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.

Lattice Design and Parameter Optimization: Lattice Styles:

Lattice Design and Parameter Optimization: Lattice Styles Putting it together (an exercise for the student!):
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.
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.
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…

Lattice Design and Parameter Optimization: RF Frequency:

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.

Lattice Design and Parameter Optimization: RF Frequency:

Lattice Design and Parameter Optimization: RF Frequency The bunch length in a storage ring is given by:
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:
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.

Lattice Design and Parameter Optimization: Summary:

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.

Lattice Design and Parameter Optimization: Circumference:

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.

Lattice Design and Parameter Optimization: Beam Energy:

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. < 27 ms < 0.13% 5 GeV < E0 < 5.5 GeV Lw = 200 m
Bw = 1.6 T
T0 = 6.6 km/c

Lattice Design and Parameter Optimization: Lattice Styles:

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):
where F ~ 100 (FODO), F = 3 (DBA), F = 1 (TME).
The wiggler contributes an emittance:
and the total emittance is:

Lattice Design and Parameter Optimization: RF Frequency:

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.

Slide47:

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:

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.
The equilibrium emittance is determined by the balance between radiation damping and quantum excitation. particle trajectory off-energy (dispersive) closed orbit on-energy closed orbit emitted photon

Beam Dynamics: Vertical Emittance and the Radiation Limit:

Beam Dynamics: Vertical Emittance and the Radiation Limit In a perfectly aligned lattice lying in a horizontal plane and containing only normal (i.e. non-skew) 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.
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.

Beam Dynamics: Vertical Emittance Sources:

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!

Beam Dynamics: Vertical Emittance and Vertical Dispersion:

Beam Dynamics: Vertical Emittance and Vertical Dispersion Vertical dispersion is directly analogous to horizontal dispersion.
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:
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.

Beam Dynamics: Vertical Emittance and Vertical Dispersion:

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.

Beam Dynamics: Vertical Emittance and Coupling:

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.

Beam Dynamics: Vertical Emittance and Coupling:

Beam Dynamics: Vertical Emittance and Coupling x y mode II mode I Single-turn matrix is block-diagonal.
Normal modes correspond to the co-ordinate 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. Without
coupling With
coupling

Beam Dynamics: Vertical Emittance and Coupling:

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.

Beam Dynamics: Vertical Emittance and Ring Design:

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:
Quadrupole tilts:
Sextupole vertical misalignments: contribution from coupling contribution from dispersion

Beam Dynamics: Vertical Emittance and Ring Design:

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

Beam Dynamics: Vertical Emittance and Ring Design:

Beam Dynamics: Vertical Emittance and Ring Design The approximate expressions for the alignment sensitivities describe the average behavior fairly well.

Beam Dynamics: Vertical Emittance and Ring Design:

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.

Beam Dynamics: Dynamic Aperture:

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.

Beam Dynamics: Dynamic Aperture:

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…

Beam Dynamics: Dynamic Aperture - FODO Example:

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)

Beam Dynamics: Dynamic Aperture - FODO Example:

Beam Dynamics: Dynamic Aperture - FODO Example Dynamic aperture depends strongly on the tune of the lattice. tune = 0.25 tune = 0.31 tune = 0.36 tune = 0.33

Beam Dynamics: Dynamic Aperture and Sextupoles:

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:
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…

Beam Dynamics: Dynamic Aperture and Sextupoles, TME Lattice:

Beam Dynamics: Dynamic Aperture and Sextupoles, TME Lattice x x x SF SF SD SD SF: k2 > 0
SD: k2 < 0

Beam Dynamics: Dynamic Aperture:

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. 5inj 5inj OCS: Circular TME TESLA: Dogbone TME

Beam Dynamics: Frequency Map Analysis:

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”).

Beam Dynamics: Acceptance:

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.

Beam Dynamics: Acceptance:

Beam Dynamics: Acceptance Estimate of required physical aperture
in the damping wiggler in seven
representative lattice designs, for a
given injection (e+) distribution.

Beam Dynamics: Collective Effects:

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.

Beam Dynamics: Space Charge:

Beam Dynamics: Space Charge Each particle in the bunch sees electric and magnetic fields from all the other particles in the bunch.
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… FE FM

Beam Dynamics: Space Charge in the Linear Approximation:

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:
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:
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.

Beam Dynamics: Space Charge in the Linear Approximation:

Beam Dynamics: Space Charge in the Linear Approximation The space charge incoherent tune shift can be written:
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).

Beam Dynamics: Space Charge Effects in the Damping Rings:

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. Emittance growth from space charge calculated by tracking in SAD (K. Oide)

Beam Dynamics: Space Charge Effects in the Damping Rings:

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)

Beam Dynamics: Space Charge and Coupling Bumps:

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.

Beam Dynamics: Space Charge and Coupling Bumps:

Beam Dynamics: Space Charge and Coupling Bumps skew quadrupoles Lattice functions at the entrance to a long straight with a coupling transformation. The value of gives the contribution of the “horizontal” emittance to the vertical beam size.

Beam Dynamics: Space Charge and Coupling Bumps:

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.

Beam Dynamics: Microwave Instability:

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)

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):
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. Beam Dynamics: Wake Functions s z´ y z = 0 z

Beam Dynamics: Wake Function and Impedance:

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:
The energy change of a particle in one turn is:
where we have defined the impedance:
and we assume that Z//(0) = 0.

Beam Dynamics: Wake Function and Impedance:

Beam Dynamics: Wake Function and Impedance The change in energy deviation per turn is:
which can be written:
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…

Beam Dynamics: Impedance and Beam Evolution:

Beam Dynamics: Impedance and Beam Evolution The evolution of the beam distribution (,;t) obeys the Vlasov equation:
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:
We can also write:
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.

Beam Dynamics: Impedance and Beam Evolution:

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:
Integrating both sides over , we find the dispersion relation:
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.

Beam Dynamics: Microwave Instability and Keill-Schnell Criterion:

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:
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(n0)/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.

Beam Dynamics: Microwave Instability and Damping Ring Design:

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.

Beam Dynamics: Coupled-Bunch Instabilities:

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.

Beam Dynamics: Coupled-Bunch Instabilities:

Beam Dynamics: Coupled-Bunch Instabilities 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):
In a storage ring containing M bunches, we construct the equation of motion: s s y py betatron oscillations multiple turns multiple bunches

Beam Dynamics: Coupled-Bunch Instabilities:

Beam Dynamics: Coupled-Bunch Instabilities The equation of motion (from the previous slide) is:
We try a solution of the form:
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. spatial (bunch number) dependence time dependence

Beam Dynamics: Coupled-Bunch Instabilities:

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).

Beam Dynamics: Resistive-Wall Instability:

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:
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

Beam Dynamics: Resistive-Wall Instability:

Beam Dynamics: Resistive-Wall Instability For the resistive-wall instability, the growth (damping) rate for the fastest mode is found to be:
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:
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.

Beam Dynamics: Resistive-Wall Instability:

Beam Dynamics: Resistive-Wall Instability Resistive-wall growth rates in a 6 km ILC damping ring lattice: Linear scale:
All modes. Log scale:
Unstable modes only. Note: Revolution frequency 50 kHz. Synchrotron radiation damping time 25 ms.

Beam Dynamics: Fast-Ion Instability:

Beam Dynamics: Fast-Ion Instability This is a complicated effect to analyze, but the growth rates may be estimated from:
where the ion frequency spread i/i 0.3 (generally); and the ion focusing is:
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. 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.

Beam Dynamics: Fast-Ion Instability:

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:
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.

Beam Dynamics: Fast-Ion Instability:

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.

Beam Dynamics: Electron Cloud Effects:

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).

Beam Dynamics: Electron Cloud Effects:

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.

Beam Dynamics: Electron Cloud Effects:

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.

Beam Dynamics: Electron Cloud Effects:

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 “head-tail” 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.

Beam Dynamics: Electron Cloud Effects:

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.

Beam Dynamics: Suppressing E-Cloud with Low-SEY Coatings:

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.

Beam Dynamics: Suppressing E-Cloud with a Grooved Chamber:

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.

Beam Dynamics: Suppressing E-Cloud with a Grooved Chamber:

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

Beam Dynamics: Suppressing E-Cloud with Clearing Electrodes:

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.

Slide106:

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:

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
…

Technical Subsystems:

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
…

Technical Subsystems: Injection/Extraction Principles trajectory of stored beam trajectory of incoming beam preceding bunch following bunch empty RF bucket injection kicker 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.

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, high-power 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 on-going.
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.

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:
From Maxwell’s equations, there are electric and magnetic fields between the plates:
A particle traveling in the +z direction with speed c will experience a force:
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. x y z

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:
The change in (normalized) horizontal momentum of the particle is:
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. L d z x V 2L

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).

Technical Subsystems: Injection/Extraction Kickers Stage 1: Leading bunch must exit kicker before voltage pulse arrives. 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. voltage pulse kicker target bunch

Technical Subsystems: Injection/Extraction Kickers How large a kick is needed? Consider the extraction optics:
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:
In other words, assuming that the bunch is on-axis at the kicker, we have:
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. kicker quadrupole septum

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:
The deflection provided is:
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).

Technical Subsystems: Fast Feedback Systems:

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:
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. single bunch
shown at
different times pick-up amplifier kicker y py

Technical Subsystems: Fast Feedback Systems:

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:
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.

Technical Subsystems: Fast Feedback Systems:

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:
If the phase advance from the pick-up to the kicker is 12, then the normalized transverse momentum at the kicker will be:
where we have assumed the optimal case 12 = /2. The kicker now provides a deflection: Writing the action…

Technical Subsystems: Fast Feedback Systems:

Technical Subsystems: Fast Feedback Systems …we find (after some algebra):
Over many turns, we can average the phase angle:
in which case we find:
where we have assumed in the last step that g12 << 1. We see that on average, the action damps exponentially:
where the feedback damping time is:

Technical Subsystems: Fast Feedback Systems:

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:
where y is a noise term, then repeating the previous analysis, we find:
The equilibrium action is:

Technical Subsystems: Fast Feedback Systems:

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:
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:
Finally, we find:
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.

An aside: the damping wiggler:

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 x z By = Bw sin(kzz) The magnetic field in the
wiggler can be approximated by:

Technical Subsystems: Damping Wigglers:

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.

Technical Subsystems: Damping Wigglers in the KEK-ATF:

Technical Subsystems: Damping Wigglers in the KEK-ATF

Technical Subsystems: Damping Wigglers:

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:
where Bw is the peak field and kz = 2/z where z is the wiggler period. By = Bw sin(kzz)

Technical Subsystems: Damping Wigglers:

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:
where B = P0/e is the beam rigidity (P0 is the reference momentum). The solution is simply:
The path length in one period is:

Technical Subsystems: Damping Wigglers:

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 10-5w: we shall ignore this difference in the following calculations.
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.

Technical Subsystems: Damping Wigglers:

Technical Subsystems: Damping Wigglers The synchrotron radiation damping rate in a storage ring is given by:
The equilibrium energy spread is given by:
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. (wiggler only) (wiggler only)

Technical Subsystems: Damping Wigglers:

Technical Subsystems: Damping Wigglers The natural emittance in a storage ring is given by:
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. (wiggler only) (wiggler only)

Technical Subsystems: ILC Damping Wigglers:

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…

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