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## Damping Ring Design

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**Damping Ring Design**Andy Wolski University of Liverpool/Cockcroft Institute International Accelerator School for Linear Colliders Sokendai, Hayama, Japan 21 May, 2006**Outline and Learning Objectives**• 1. Introduction: Basic Principles of Operation • 2. Lattice Design and Parameter Optimization • You should be able to explain the issues involved in choosing the principal parameters for the damping rings, including the circumference, beam energy, lattice style, and RF frequency. • 3. Beam Dynamics • You should be able to explain the physics behind important beam dynamics phenomena, including coupling, dynamic aperture, space charge effects, microwave instability, resistive-wall instability, fast ion instability and electron cloud. You should be able to describe the impact of these effects on damping ring design. For some effects (space charge, microwave, resistive-wall and fast ion instability), you should be able to estimate the impact on damping ring performance, using simple linear approximations. • 4. Technical Subsystems • You should be able to describe the principles of operation behind important technical subsystems in the damping rings, including the injection/extraction kickers, fast feedback systems and the damping wiggler. For the damping wiggler, you should be able to explain the issues involved in choosing between the various technology options.**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.**Part 1**• Principles of Operation**Introduction: Basic Principles of Operation - Performance**Specs • The performance parameters are determined by the sources, the luminosity goal, interaction region effects and the main linac technology. ILC parameters determining damping ring requirements**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 • 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 trajectory of incoming beam 1. Kicker is OFF. “Preceding” bunch exits kicker electrodes.Kicker starts to turn ON. injection kicker following bunch emptyRF 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.**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). 4 1 5 3 2 5 4 3 2 1 6**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**• 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.**Part 2**• Lattice Design and Parameter Optimization • You should be able to explain the issues involved in choosing the principal parameters for the damping rings, including the circumference, beam energy, lattice style, and RF frequency.**Lattice Design and Parameter Optimization: Circumference**• Lower limit ~ 3 km: the smaller the damping ring, the shorter the distance between bunches. This makes the ring more difficult: • Injection/extraction kickers need shorter rise and fall times. • Electron cloud build-up is sensitive to bunch spacing, and it becomes increasingly difficult to avoid electron cloud instabilities as the ring gets smaller. • In smaller rings, it becomes difficult to provide sufficient gaps in the fill for ion clearing, so the beam becomes susceptible to ion instabilities. • Upper limit ~ 17 km: space-charge, acceptance and cost. • Space-charge tune shifts (in a linear model) are proportional to the circumference. Large tune shifts can lead to emittance growth and particle loss. • The cost of very large (~17 km) rings may be reduced by using a “dogbone” layout, in which long straight sections share the tunnel with the main linac… • …but these long straights generate chromaticity, which breaks any symmetry for off-energy particles and limits the acceptance.**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**• 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**• 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**• 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**• Comparison between electron cloud instability thresholds and cloud densities, in various damping rings under various conditions.**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**• 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**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**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**• 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**• 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**• 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**• 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**• The damping time in a storage ring depends on the rate of energy loss of the particles through synchrotron radiation. In the damping rings, the rate of energy loss can be enhanced by insertion of a long wiggler, consisting of short (~ 10 cm) sections of dipole field with alternating polarity. y z x The magnetic field in the wiggler can be approximated by: By = Bw sin(kzz)**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; andC = 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**• 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**• 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. < 0.13% Lw = 200 m Bw = 1.6 T T0 = 6.6 km/c < 27 ms 5 GeV < E0 < 5.5 GeV**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**• 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 numberJx 1.**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**• FODO Lattice: F 100**Lattice Design and Parameter Optimization: Lattice Styles**• Double Bend Achromat (DBA) Lattice: F = 3**Lattice Design and Parameter Optimization: Lattice Styles**• Theoretical Minimum Emittance (TME) Lattice: F = 1**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**• 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**• 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**• 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**• 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**• 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**• 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**• 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. < 0.13% Lw = 200 m Bw = 1.6 T T0 = 6.6 km/c < 27 ms 5 GeV < E0 < 5.5 GeV**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**• 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.**Part 3**• Beam Dynamics • You should be able to explain the physics behind important beam dynamics phenomena, including coupling, dynamic aperture, space charge effects, microwave instability, resistive-wall instability, fast ion instability and electron cloud. You should be able to describe the impact of these effects on damping ring design. For some effects (space charge, microwave, resistive-wall and fast ion instability), you should be able to estimate the impact on damping ring performance, using simple linear approximations.**Beam Dynamics: Vertical Emittance**• Betatron oscillations of a particle are excited when the particle emits a photon at a point of non-zero dispersion. • The energy of the particle changes • If the particle was following a closed orbit, then (because of the dispersion) it will no longer be doing so. • The equilibrium emittance is determined by the balance between radiation damping and quantum excitation. on-energy closed orbit emitted photon off-energy (dispersive) closed orbit particle trajectory**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**• 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!