1. An Introduction to the ILC Damping Rings
Andy Wolski
University of Liverpool and the Cockcroft Institute
29 November, 2006

2. Principles of operation
Overview of this talk…
Introduction
Functions of damping rings in a linear collider.
Overview of present ILC damping rings configuration.
“Global” design constraints.
Principles of operation
Timing issues.
Radiation damping.
Equilibrium beam sizes.
Beam dynamics issues
Vertical emittance and coupling correction.
Dynamic aperture.
Some collective (intensity-dependent) effects.
Technical subsystems
Injection and extraction kickers.

3. Damping rings are critical to linear collider performance
Damping rings perform several important functions:
Reduce injected beam emittances by 6 orders of magnitude.
Remove jitter from the source, providing a highly stable beam for tuning of downstream systems.
Delay the beams from the source, allowing for feed-forwards.
ILC Damping Rings are large electron storage rings.
Circumference is 6.7 km.
Beam energy is 5 GeV.
Two rings (one for positrons, one for electrons) in a single tunnel.
Beam current is 400 mA.
Roughly 200 m of wiggler, to provide 25 ms damping time.
Introduction

4. Present ILC configuration has damping rings located “centrally” around the interaction region.
Introduction

5. e+ Introduction injection extraction Arc 2 (818 m) wiggler
short straight A (249 m)
short straight B (249 m)
shaft/large cavern A
wiggler
8 RF cavities
Arc 1 (818 m)
Arc 3 (818 m)
injection
long straight 1 (400 m)
long straight 2 (400 m) IP
extraction
small cavern 1
small cavern 2
Arc 6 (818 m)
Arc 4 (818 m)
10 RF cavities
wiggler
shaft/large cavern C
short straight D (249 m)
short straight C (249 m)
wiggler
Arc 5 (818 m)
Introduction

6. Damping ring designs are constrained by ILC parameters
An ILC “machine pulse” consists of one electron and one positron bunch train, extracted from the damping rings, accelerated in the main linacs and collided at the IP.
New bunch trains must simultaneously be produced by the sources, to replace the bunch trains extracted from the damping rings.
Machine pulse repetition rate is 5 Hz. Each bunch train consists of:
5782 bunches with 1010 particles per bunch.
Bunch separation of 189 ns in the main linacs.
Total bunch train length of 1.1 ms, or 328 km.
Positron and polarised electron sources produce bunches that are too large to generate much luminosity. We use damping rings to reduce the bunch emittances in the 200 ms between machine pulses.
The damping rings cannot be large enough to accommodate a full bunch train with bunches separated by 189 ns.
We “compress” the bunch trains in the damping rings by injecting and extracting individual bunches, to reduce the separation between bunches in the rings.
Introduction

7. Damping ring designs are constrained by ILC parameters
Bunch train length
Circumference
Injection/extraction kicker performance
Beam instabilities
Beam energy
Bunch charge
Injected bunch emittances
Damping times
Costs
Extracted bunch emittances
Lattice design
Technical systems performance
Machine pulse repetition rate
RF voltage
Availability and reliability
Main linac RF frequency
RF frequency
Introduction

8. Timing issues need to be carefully considered
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. 4 1 5 3 2 6 5 4 3 2 1
Principles of operation

9. Injection and extraction kickers must operate in ~ nanoseconds
trajectory of incoming beam
1. Kicker is OFF. “Preceding” bunch exits kicker electrodes. Kicker starts to turn ON.
injection kicker
following bunch
empty RF bucket
preceding bunch
trajectory of stored beam
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.
Principles of operation

10. Radiation damping
Charged particles emit photons (synchrotron radiation) in bending magnets.
The majority of photons are emitted within a cone of angle 1/ around the instantaneous direction of motion of the particle.
For ultra-relativistic particles,  is very large.
Particles lose longitudinal and transverse momentum in bending magnets.
Neglecting dispersion and chromaticity, the trajectory of the particle after emitting a photon is the same as it would be if no photon were emitted.
emitted photon
particle trajectory
closed orbit
bending magnet
Principles of operation

11. Radiation damping
In an RF cavity, the particle sees an accelerating electric field parallel to the closed orbit: the RF cavities in a storage ring restore the energy lost by synchrotron radiation.
The increase in momentum of a particle in an RF cavity is parallel to the closed orbit. This leads to a reduction in the amplitude of betatron oscillations of the particle.
particle trajectory
closed orbit
Principles of operation

12. Radiation damping
The combined effect of energy loss in the dipoles and energy gain in the RF cavities is to reduce the betatron amplitudes of particles over many turns around the ring.
The rate at which the betatron amplitudes are reduced is clearly related to the rate at which the particles lose energy in the dipoles.
A detailed analysis gives the result for the radiation damping time:
where E is the beam energy, U0 is the energy loss per turn, and T is the revolution period, and x is the decay constant for betatron oscillations:
Principles of operation

13. Radiation damping
Note that the emittance of a bunch of particles is given by the average of the square of the betatron amplitudes of particles in the bunch:
Therefore, the emittance decays with a time constant equal to half the radiation damping time:
Principles of operation

14. Damping time in the ILC damping rings
The damping time in the ILC damping rings is determined by the specifications on the injected and extracted emittances.
Injected positron emittance y(t = 0) = 0.01 m
Extracted positron vertical emittance y(t = 200 ms) = 20 nm
We find that the necessary damping time is:
To allow for effects such as quantum excitation (which slow the damping near equilibrium), we design for a damping time of around 25 ms.
Principles of operation

15. Radiation damping
To calculate the radiation damping time in a storage ring, we need to know the energy loss per turn, U0. This can be found from classical electromagnetic theory. The result is:
where E is the beam energy, C is a physical constant, with value:
 is the local radius of curvature of the trajectory of particles in the dipoles in the magnetic lattice, and the integral extends around the entire circumference.
Principles of operation

16. Radiation damping in the damping rings
Let us make a crude estimate of the damping time in a 6.7 km, 5 GeV electron storage ring.
Assuming that the dipoles make up a fraction D  10% of the circumference, the energy loss per turn is:
and the damping time is:
This is much too long: the ILC damping rings need a damping time of 25 ms.
We need to use wigglers to provide extra bending field without increasing the circumference.
Principles of operation

17. Damping wigglers reduce the damping time
A wiggler is simply a sequence of short, strong dipoles with adjacent dipoles bending the beam in opposite directions, so that the net deflection of the beam is zero.
The energy loss in a wiggler of length Lw and peak field Bw is:
To provide a damping time of 25 ms in a 6.7 km, 5 GeV storage ring with a wiggler of peak field 1.6 T, we need a total length of wiggler of 220 m.
By = Bw sin(kzz)
Principles of operation

18. Equilibrium beam sizes
The emission of radiation leads to excitation of the emittances as well as damping.
A particle following the closed orbit has zero betatron amplitude. If this particle radiates a photon, it loses energy. If this occurs at a location where the dispersion is non-zero, there is a new closed orbit for the particle.
The particle will start to make betatron oscillations around the new closed orbit: there has been an increase in emittance of the beam.
bending magnet
emitted photon
closed orbit
off-momentum closed orbit
particle
trajectory
Principles of operation

19. Equilibrium emittances
Excitation of the beam emittance by synchrotron radiation is a quantum effect, resulting from the emission of radiation in individual photons.
The equilibrium emittance of a beam is determined by the balance between radiation damping and quantum excitation.
The horizontal emittance is essentially determined by the dispersion generated by the main bending magnets.
Damping ring lattice designs generally use a style of arc lattice cell known as a “Theoretical Minimum Emittance” (or TME) cell, which achieves low dispersion in the bends.
With the large circumference of the ILC damping rings, it is possible (by using a large number of cells) to achieve low dispersion, and hence low emittance, using simple FODO arc lattice cells.
Principles of operation

20. Theoretical Minimum Emittance (TME) arc cell
Principles of operation

21. FODO arc cell
Principles of operation

22. Vertical emittance
The vertical emittance in a storage ring is generated by three effects:
The non-zero vertical opening angle of the synchrotron radiation (~ 1/) leads to a vertical “recoil” when a photon is emitted. This excites vertical betatron oscillations.
Typically, the vertical emittance generated by this effect is of the order of a few tenths of a picometer, around 10% of the emittance specification for the ILC damping rings.
Vertical orbit distortions lead to vertical dispersion, which generates vertical emittance in the same way as horizontal dispersion generates horizontal emittance.
The vertical orbit needs to be corrected at the level of a few tens of microns, to keep the vertical dispersion down to a few mm rms.
Skew quadrupole errors lead to the coupling of horizontal betatron motion into the vertical plane.
Vertical beam offset in the sextupoles (of the order of a few tens of microns) tends to dominate the sources of skew quadrupole field errors.
Beam dynamics issues

23. Equilibrium emittances
Circumference
Beam energy
Horizontal emittance
Vertical emittance
Bunch length
ILC Damping Rings
6,700 m
5 GeV
0.8 nm
2 pm
9 mm
KEK-ATF
139 m
1.28 GeV
1 nm
4.5 pm
4 mm
LBNL-ALS
197 m
1.9 GeV
5.6 nm
~ 10 pm
6 mm
PEP-II LER
2,200 m
3.1 GeV
50 nm
1,500 pm
11 mm
Note: this table gives the geometric emittances, . When quoting ILC parameters, it is usual to give the normalised emittances, , since these quantities are conserved during acceleration. For the ILC damping rings at 5 GeV,   10,000.
Beam dynamics issues

24. Ultra-low vertical emittance in the KEK-ATF
Y. Honda et al, “Achievement of Ultralow Emittance Beam in the Accelerator Test Facility Damping Ring,” Phys. Rev. Lett. 92, (2004).
Beam dynamics issues

25. Ultra-low vertical emittance in the KEK-ATF
Measuring beam sizes of a few microns presents a technical challenge. In the ATF, this is achieved using a laser wire.
A laser beam is focused to a waist of a few microns, and scanned across the electron beam. Gamma rays from Compton scattering are detected: the rate at which the gamma rays are produced gives the electron beam intensity as a function of transverse position.
Beam dynamics issues

26. Dynamic aperture
The emittance of the injected beam is orders of magnitude larger than the extracted (“damped”) beam.
The positron source produces a beam with normalised emittance 0.01 m.
For the ILC damping rings, this means a typical beam size is around 6 mm.
The vacuum chamber can be made with a physical aperture large enough to accommodate the injected beam without losses…
…but achieving a large dynamic aperture is more difficult.
The dynamic aperture in a storage ring is limited by higher-order field components (sextupoles, octupoles…)
The effect of higher-order field components is to introduce nonlinear terms into the equations of motion for a particle moving through the lattice.
At some amplitude, the nonlinear oscillations of a particle moving through the lattice become unstable: the amplitude of the oscillations increases rapidly, and the particle hits the vacuum chamber.
Beam dynamics issues

27. Dynamic aperture: example in a FODO lattice
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 issues

28. Dynamic aperture: example in a FODO lattice
The dynamic aperture depends strongly on the tune of the lattice.
tune = 0.25
tune = 0.31
tune = 0.33
tune = 0.36
Beam dynamics issues

29. Dynamic aperture is typically dominated by sextupole magnetsSF SD SD SF
Typically, the nonlinear dynamical effects in a storage ring are dominated by sextupoles.
The sextupoles are needed to correct the chromaticity (the change in tune with the energy of a particle).
To reduce the impact of the sextupoles on the dynamic aperture, the lattice has to be designed to keep the sextupole strengths as low as possible. x x x
Beam dynamics issues

30. Dynamic aperture is computed from tracking studies
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
Beam dynamics issues

31. Dynamic aperture: 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 issues

32. Collective effects
Particles are guided through an accelerator beamline using “external” fields from components such as magnets and RF cavities.
Particles in an accelerator will themselves generate fields that lead to interactions with:
the surrounding environment;
vacuum chamber
instrumentation and diagnostics
other particles in the beam.
As the number of particles in the accelerator increases, the interactions resulting from fields generated by the particles have greater impact, and can result in:
increases in emittance;
changes in betatron or synchrotron tune;
changes in the beam distribution;
coherent oscillations of whole bunches;
heating of components in the vacuum chamber.
Beam dynamics issues

33. Collective effects
Fundamentally, all “collective effects” are manifestations of electromagnetic interactions between particles in a beam, either directly, or mediated by vacuum chamber components, or by other particles in the beam pipe (ions or electrons).
Different types of collective effect are identified, depending on the circumstances in which the effect is observed, the dynamical mechanism that is involved, and its effect(s) on the beam.
Effects of concern for the damping rings include:
space charge
intrabeam scattering
Touschek effect
microwave instability
resistive-wall instability *
ion instabilities
electron cloud effects *
higher-order mode heating
Beam dynamics issues

34. Wakefields
The electromagnetic fields generated by a particle or a bunch of particles moving through a vacuum chamber are usually described as wakefields.
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 issues

35. Coupled-bunch instabilitiesy
py 2 1
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:
betatron oscillations
multiple turns
multiple bunches
Beam dynamics issues

36. Coupled-bunch instabilities
The equation of motion (from the previous slide) is:
We try a solution for the nth bunch of the form:
Substituting this solution into the equation of motion, we find an equation that gives us the mode frequency  for a given mode number .
The imaginary part of  gives the instability growth (or damping) rate.
time dependence
spatial (bunch number) dependence
Beam dynamics issues

37. 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.
Each mode can have a different growth (or damping) rate.
From A. Chao, “Physics of Collective Beam Instabilities in Particle Accelerators,” Wiley (1993).
Beam dynamics issues

38. Resistive-wall instability
For the ILC damping rings, the resistive-wall wakefields are expected to lead to an instability with the fastest modes having growth times ~ 20 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 issues

39. 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 issues

40. Electron cloud effects
In a positron (or proton) storage ring, electrons are generated by a variety of processes, and can be accelerated by the beam to hit the vacuum chamber with sufficient energy to generate multiple “secondary” electrons.
Under the right conditions, the density of electrons in the chamber can reach high levels, and can drive instabilities in the beam.
Important parameters determining the electron cloud density include:
the bunch charge and bunch spacing
the properties of the vacuum chamber surface (the “secondary electron yield”)
Beam dynamics issues

41. 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 issues

42. 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 issues

43. 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 to reducing the SEY 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.
Beam dynamics issues

44. Suppressing electron cloud with low-SEY coatings
Coatings being investigated include TiN and TiZrV.
Achieving a peak SEY below 1.2 seems possible after conditioning.
Reliability/reproducibility and durability are concerns.
Measurements of SEY of TiZrV (NEG) coating. (F. le Pimpec, M. Pivi, R. Kirby)
Beam dynamics issues

45. Suppressing electron 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 issues

46. Suppressing electron 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.
Beam dynamics issues

47. Injection and extraction kickers must operate in ~ nanoseconds
trajectory of incoming beam
1. Kicker is OFF. “Preceding” bunch exits kicker electrodes. Kicker starts to turn ON.
injection kicker
following bunch
empty RF bucket
preceding bunch
trajectory of stored beam
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

48. Injection and 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.
Several technologies are possible for the fast, high-power pulser. The parameters for the ILC damping rings are very challenging, and pulser development is on-going.
The stripline electrodes are comparatively straightforward: they consist of two plates, connected to a high-voltage line, between which the beam travels.
The stripline 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.
Technical subsystems

49. Injection and 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. y z x
Technical subsystems

50. Injection and 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 the (normalised) horizontal momentum of the particle is:
where E is the beam energy. In practice, 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

51. Injection and 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.
Technical subsystems

52. Injection and extraction kickers
How large a kick is needed? Consider the extraction optics:
Assuming a distance from the kicker to the septum of 50 m, and a required beam offset from the reference trajectory of 30 mm at the septum, the required kick is: If we assume E/e = 5 GV, L = 30 cm, d = 20 mm and g = 0.7, we find that the required voltage pulse is 143 kV. This is not realistic for a ~ ns pulser! The solution is to use multiple pairs of striplines, each producing a ~ 10 kV kick.
septum
kicker
quadrupole
Technical subsystems

53. Injection and extraction kickers
Tests of DSRD fast pulser.
(A. Krasnykh)
Output voltage 2.7 kV
Horizontal scale 1 ns/div
Technical subsystems

54. Injection and extraction kickers
Tests of FID fast pulser at KEK-ATF.
(T. Naito)
Technical subsystems

55. Final words
Basic principles for the damping rings are well understood, but design constraints (including cost considerations) lead to a number of very challenging issues:
Beam dynamics
Dynamic aperture must be large to accept the injected positron beam.
Extremely low vertical emittance of 2 pm must be achieved on a routine basis.
Many and varied collective effects threaten the beam quality and stability.
Technical subsystems
Damping wiggler must have high field quality.
Injection and extraction kickers require further R&D to demonstrate rise and fall times, pulse amplitude, repetition rate, stability and reliability.
Vacuum system must have low impedance, and have good performance from point of view of ion and electron cloud effects.
The ILC damping rings are the subject of a highly active R&D programme, involving nearly 150 people from more than 20 different institutions.