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Linear Collider Damping Rings Andy Wolski Lawrence Berkeley National Laboratory USPAS Santa Barbara, June 2003 What do they look like? 30 m Wiggler Circumference 300 m Main Damping Ring Correction and 3 Trains of 192 bunches Extraction 1.4 ns bunch spacing Injection and RF 90 m Extraction Line 30 m Wiggler 110 m Spin Transfer Rotation Line 110 m Injection Line 231 m Predamping Ring NLC 2 Trains of 192 bunches Positron Rings 2 Operating Cycle in NLC/JLC MDRs • Each bunch train is stored for three machine cycles – 25 ms or 25,000 turns in NLC • Transverse damping time 4 ms • Horizontal emittance ×1/50, vertical ×1/7500 160 m Spin Extraction Rotation Line 30 m Wiggler 103 m Circumference Injection Correction and Line Extraction 300 m Main Damping Ring 3 Trains of 192 bunches 1.4 ns bunch spacing Injection and RF 30 m Wiggler 3 What do they look like? TESLA Damping Rings 4 Performance Specifications NLC MDR TESLA e+ Injected γε 150 µm rad 10 000 µm rad Extracted Horizontal γε 3 µm rad 8 µm rad Extracted Vertical γε 0.02 µm rad 0.02 µm rad Injected Energy Spread 1% full width 1% full width Extracted Energy Spread 0.1% rms 0.13% Extracted Bunch Length 4 mm 6 mm Bunch Spacing 1.4 ns 20 ns Bunches per Train 192 2820 Repetition Rate 120 Hz 5 Hz 5 Radiation Damping… • Longitudinal phase space – Particles perform synchrotron oscillations in RF focusing potential – Higher energy particles radiate energy more quickly in bends – At the equilibrium energy, the revolution period is an integer times the RF period (the synchrotron principle…) • Transverse phase space – Particles perform betatron oscillations around the closed orbit – Radiation is emitted in a narrow cone centered on the instantaneous direction of motion – Energy is restored by the RF cavities longitudinally – Combined effect of radiation and RF is a loss in transverse momentum • Damping time in all planes is given by: E0 J 2 T0 U0 6 …and Quantum Excitation • Radiation is emitted in discrete quanta • Number and energy distribution etc. of photons obey statistical laws • Radiation process can be modeled as a series of “kicks” that excite longitudinal and transverse oscillations 7 Synchrotron Oscillations Equilibrium orbit Dispersive orbit d p dt d eVRF 1 dU sin s RF U 0 dt E0T0 T0 dE d 2 2 d s2 0 dt 2 E dt 8 Longitudinal Damping ˆe t cos s t s E s ˆe t sin s t s E p eVRF RF s2 p cos s E0 T0 E0 J E E 2 T0 U0 I4 JE 2 I2 Problem 1 Show that: 1 1 1 I2 I4 2 2k1 ds 1 C0 ds 2 p C0 I1 ds 9 Quantum Excitation (Longitudinal) δ p ˆ 1 1 cos1 p ˆ cos s s 1 ˆ1 sin 1 ˆ sin u E0 τ 2 ˆ 2 2 u 2 u sin 1 ˆ ˆ 2 E0 E0 Including damping: ˆ d 2 1 ˆ 2 2 N u ds 2 2 dt E0 C0 E 10 Equilibrium Longitudinal Emittance • We have found that: ˆ d 2 1 ˆ 2 2 N u ds 2 2 dt E0 C0 E • From synchrotron radiation theory: 1 2 E02 I 3 1 N u ds 4Cq J E E I 2 I3 2 ds 3 C0 Problem 2 Find an expression for the equilibrium energy spread, and show that: 0et ,equ 1 et 11 Betatron Oscillations: Action-Angle Variables • It is often more convenient to describe betatron oscillations using action-angle variables: 2 J x 2 2xx x2 x tan x • The old variables are related to the new ones by: x 2 J cos x 2J sin cos • The equations of motion take the simple form: d 1 dJ 0 ds ds 12 Damping of Vertical Oscillations • Radiation is emitted in a narrow cone (angle ~1/γ) around instantaneous direction of motion, so vertical co-ordinate and momentum are not changed by photon emission • RF cavity changes longitudinal momentum, and hence the vertical direction of motion: py p y p p y1 y y1 1 y1 p p p p p • Averaging over all betatron phase angles gives (per turn): U0 J J Problem 3 E0 Show this! • Hence the equation of motion is: dJ U 0 J dt E0T0 13 Damping of Horizontal Oscillations • When a photon is emitted at a point where there is some dispersion, the co-ordinates with respect to the closed orbit change: u u x1 x x1 x E0 E0 • Taking the energy loss to first order and averaging around the ring, we find after some work: dJ I U U 1 4 0 J J x 0 J I ET dt 2 0 0 E0T0 14 Quantum Excitation of Betatron Motion • Let us now consider the second order effects. It is easy to show that the change in the action depends to second order on the photon energy as follows: 2 1 u J H H γη2 2αηη βη2 2 E0 • Averaging over the photon spectrum and around the ring, and including the radiation damping gives: dJ x 1 2 2 E02C0 N u 2 Hds J x dt x 2 I5 2 C q 2 Jx H J x x I 2 x I5 ds 3 15 Summary of Dynamics with Radiation d E0 dt inje 2t equ 1 e 2t J x x J y y J E E 2 U0 T0 I4 I4 J x 1 Jy 1 JE 2 I2 I2 I1 ds C U0 E04 I 2 1 2 I2 ds p 2 2 Cq 2 I3 1 JE I2 s I3 ds 3 I1 p 1 1 B y C0 I4 2 2k1 ds k1 B x eVRF RF s2 p cos s H E0 T0 I5 ds H 2 2 2 3 sin s U0 eVRF I5 0 C q 2 J x I2 16 The NLC TME Cell High field in dipole Sextupoles at high dispersion points, with separated betas Vertical focusing in the dipole Low dispersion and horizontal beta function in the dipole Cell length ≈ 5 m 17 H Function in the NLC TME Cell 18 The TESLA TME Cell Low field in dipole No vertical focusing in the dipole Sextupoles at high dispersion points Larger dispersion and horizontal beta function in the dipole Cell length ≈ 15 m 19 NLC and TESLA TME Cells Compared • NLC – Compact cell to keep circumference as short as possible – High dipole field for greater energy loss, reducing wiggler length – Short dipole requires very low values for dispersion and beta function – Gradient in dipole field to improve transverse dynamics • TESLA – Circumference fixed by bunch train and kicker rise/fall time – Long dipole for larger momentum compaction, longer bunch • Optimum lattice functions at center of dipole: L L 3 0 0 min Cq 2 2 15 24 J x12 15 – Obtained by minimizing I5 for a ring without a wiggler – It is not usually possible to control the dispersion and beta function independently 20 Two Simple Scaling Relationships Problem 4 Show that for an isomagnetic ring with the lattice functions tuned for minimum emittance: 2 8 2 Lcell Cq 3 N cellB0 C eme c 2 6 12 15 0 1 8 2 Lcell Cq 3 2 B0 C eme c 2 6 12 15 0 21 Scaling Relationships Applied to the NLC 1 bunch train γε0 = 3 μm Lcell = 6 m τ = Ntrain 1.6 ms 6 bunch trains 22 Scaling Relationships Applied to the NLC 2 bunch trains γε0 = 1 μm Lcell = 6 m τ = Ntrain 1.6 ms 6 bunch trains 23 Damping Wiggler • A wiggler reduces the damping time by increasing the energy loss per turn: C e 2 c 2C U0 E I 4 E0 B 2 ds 2 2 2 0 2 • Wiggler must be located where nominal dispersion is zero, otherwise there can be a large increase in the natural emittance • If horizontal beta function is reasonably small, wiggler can significantly reduce the natural emittance (through reduced damping time) • Drawbacks include possible detrimental effect on beam dynamics 24 Types of Wiggler • A wiggler is simply a periodic array of magnets, such that the field is approximately sinusoidal • Different technologies are possible: – Electromagnetic – Permanent magnet – Hybrid (permanent magnets driving flux through steel poles) • Choice of technology comes down to cost optimization for given requirements on field strength and quality • Both TESLA and NLC damping rings have opted for hybrid technology 25 Modeling the Dynamics in the Wiggler • Magnet design is produced using a standard modeling code • Field representation must be obtained in a form convenient for fast symplectic tracking sin mk x x sinh k y ,mn y cosnk z z mk x Bx cmn k y ,mn By cmn cosmk x x coshk y ,mn y cosnk z z cosmk x x sinh k y ,mn y sin nk z z nk z Bz cmn k y ,mn k y ,mn m 2 k x n 2 k z2 2 2 26 Fitting the Wiggler Field 27 Tracking Through the Field • Using an appropriate field representation (that satisfies Maxwell’s equations), one can construct a symplectic integrator: xnew m xold • M is an explicit function of the phase-space co-ordinates, and satisfies the symplectic condition (so the dynamics obey Hamilton’s Equations): M S M T S m M ij i x j 0 1 1 0 S 28 Dynamics in the NLC Wiggler Horizontal Kicks and Phase Space Vertical Kicks and Phase Space 29 Chromaticity • Chromaticity is the tune variation with energy • Quadrupole focusing strength gets smaller as particle energy increases • It can easily be shown that: x 1 4 x x k1ds 1 y y 4 y k1ds • Since beta functions peak at the focusing quadrupoles in the appropriate plane, the natural chromaticity is always negative • Chromaticity is connected to beam instabilities – particles with large energy deviation cross resonance lines – some collective effects (e.g. head-tail instability) are sensitive to the chromaticity 30 Correcting Chromaticity with Sextupoles sextupole k1= x k2 31 Dynamics with Sextupoles • Sextupoles can be used to correct chromatic aberrations… 1 x 4 x k1-ηx x k2ds 1 4 y y k1 x y k 2ds • …but introduce geometric aberrations and coupling: x 1 k 2l x 2 2 y k 2lY x • It is important to keep the required strengths to a minimum by designing the linear lattice functions for effective sextupole location 32 Dynamic Aperture • Geometric aberrations from sextupoles (and other sources) distort the transverse phase space, and limit the amplitude range of stable betatron oscillations Horizontal phase space of NLC TME cell Vertical phase space of NLC TME cell 33 Transverse and Longitudinal Aperture • Damping rings require a “large” dynamic aperture – Injected beam power ~ 50 kW average, and radiation load from any significant injection losses will destroy the ring – Nonlinear distortion of the phase space may lead to transient emittance growth from inability properly to match injected beam to the ring – For NLC Main Damping Rings, the target dynamic aperture is 15 times the injected rms beam size • We also need a large momentum acceptance – Injected beam has a large energy spread – Particles may be lost from insufficient physical aperture in dispersive regions, or through poor off-momentum dynamics – Particles within a bunch can scatter off each other, leading to a significant change in energy deviation (Touschek Effect) • It is important to perform tracking studies with full dynamic model and physical apertures 34 NLC Main Damping Ring Dynamic Aperture δ= -0.005 δ= +0.005 Dynamic Aperture On-Momentum 15× Injected Beam Size 35 Longitudinal Acceptance • The longitudinal acceptance has three major limitations: – Poor off-momentum dynamics – Physical aperture in dispersive regions – RF bucket height • Off-momentum dynamics can be difficult to quantify – see previous slides • Physical aperture can be a significant limitation – 1% momentum deviation in 1 m dispersion is a 1 cm orbit offset • RF bucket height comes from non-linearity of the longitudinal focusing – Previous study of longitudinal dynamics assumed a linear slope of RF voltage around the synchronous phase – Valid for small oscillations with synchronous phase close to zero- crossing 36 RF Bucket Height • The “proper” equations of longitudinal motion (without damping) are: d p dt d eVRF sin s RF sin s dt E0T0 • These may be derived from the Hamiltonian: H - 1 p 2 eVRF coss RF sin s RF E0T0 RF 2 d H dt d H dt 37 Longitudinal Phase Space • The Hamiltonian is a constant of the motion, which allows us to draw a phase-space portrait Stable fixed point Unstable fixed point S Separatrix RF VRF S coss s sin s 4eVRF RF 2 E0T0 RF p 2 38 Alignment Issues • The final luminosity of the collider is critically dependent on the vertical emittance extracted from the damping rings • In a perfectly flat lattice, the lower limit on the vertical emittance comes from the opening angle of the radiation – Gives about 10% of the specified values for NLC and TESLA • Magnet misalignments give the dominant contribution to the vertical emittance – Quadrupole vertical misalignments • Vertical dispersion • Vertical beam offset in sextupoles – Quadrupole rotations and sextupole vertical misalignments • Couple horizontal dispersion into the vertical plane • Couple horizontal betatron oscillations into the vertical plane 39 Betatron Coupling • In a damping ring, the dominant sources of betatron coupling are skew quadrupole fields – Normal quadrupoles have some “roll” about the beam axis – Sextupoles have some vertical offset with respect to the closed orbit • Particles with a horizontal offset get a vertical kick Particle on closed orbit Particle with horizontal amplitude Vertical kick depends on horizontal amplitude 40 Effects of Betatron Coupling • In action-angle variables, the “averaged Hamiltonian” for a coupled storage ring can be written: H x J x y J y n J x J y cos x y C0 ~ 2 • The equations of motion are: d x 2 ~ J x J y sin x y cos x y dJ x ~ Jy x ds ds C0 2 Jx d y 2 ~ J x J y sin x y cos x y dJ y ~ Jx y ds ds C0 2 Jy 41 Solutions to the Coupled Hamiltonian • The sum of the horizontal and vertical actions is conserved: J0 J x J y dJ 0 0 ds • There are fixed points at: 1 J x J 0 1 2 ~2 2 x y 1 J y J 0 1 2 ~2 2 • With radiation, the actions will damp to the fixed points 42 The Difference Coupling Resonance • The equilibrium emittance ratio is given by: Jy ~ 2 Jx ~ 42 2 • The measured tunes are given by: C0 d 1 ~2 2 1 2 x y 2 ds 2 43 What is the Coupling Strength? • We add up all the skew fields around the ring with an appropriate phase factor: C0 n s x y 2 x y n 1 s ~ n x y k s ei s ds n 2 0 C0 • ks is the skew quadrupole k-value. • For a rotated quadrupole or vertically misaligned sextupole, the equivalent skew fields are given by: k s k1 sin 2 k s k 2 y 44 Vertical Dispersion • In an electron storage ring, the vertical dispersion is typically dominated by betatron coupling – Emittance ratios of 1% are typical • For very low values of the vertical emittance, vertical dispersion starts to make a significant contribution • Vertical dispersion is generated by: – Vertical steering • vertically misaligned quadrupoles – Coupling of horizontal dispersion into the vertical plane • quadrupole rotations • vertical sextupole misalignments 45 Vertical Steering: Closed Orbit Distortion • A quadrupole misalignment can be represented by a kick that leads to a “cusp” in the closed orbit • We can write a condition for the closed orbit in the presence of the kick: y0 y 0 cos x x sin x x sin x y y M M 0 0 x sin x cos x x sin x • We can solve to find the distortion resulting from many kicks: y s1 y s ys1 2 sin y s cos y y s1 y s ds 46 Vertical Steering: Vertical Dispersion • The vertical dispersion obeys the same equation of motion as the vertical orbit, but with a modified driving term: y k1 y 1 differentiate wrt k1 y k1 y 1 1 y • We can immediately write down the vertical dispersion arising from a set of steering errors: y s1 y s k1 y cos y y s1 y s ds 1 y s1 2 sin y • Including the effect of dispersion coupling: y s1 y s1 2 sin y 1 y s k1 k 2 x y k s x cos y y s1 y s ds 47 Effects of Uncorrelated Alignment Errors • Closed orbit distortion from quadrupole misalignments: y2 Yq2 y k1l 2 y 8 sin 2 y • Vertical dispersion from quadrupole rotation and sextupole misalignment: y 2 q 2 y 2 Ys 2 y k1l x 2 y k 2 l x 2 y 2 sin 2 y y 8 sin 2 y • Vertical emittance generated by vertical dispersion: JE y 2 y 2 2 J y y 48 Examples of Alignment Sensitivities KEK- APS SLS ALS NLC MDR TESLA DR ATF Energy [GeV] 7 2.4 1.3 1.9 1.98 5 Circumference [m] 1000 288 140 200 300 17,000 γεx [µm] 34 23 2.8 24 3 8 γεy [nm] 140 70 28 20 19 14 Sextupole vertical [µm] 74 71 87 30 53 11 Quadrupole roll [µrad] 240 374 1475 200 511 38 Quadrupole jitter [nm] 280 230 320 230 264 76 • Note: Sensitivity values give the random misalignments that will generate a specified vertical emittance. In practice, coupling correction schemes mean that significantly larger misalignments can be tolerated. 49 Collective Effects • Issues of damping, acceptance, coupling are all single particle effects - they are independent of the beam current • Particles in a storage ring interact with each other (directly or via some intermediary e.g. the vacuum chamber) • A wide variety of collective effects limit the achievable beam quality, depending on the bunch charge or total current • The consequences of collective effects are – Phase space distortion and/or emittance growth – Particle loss • Damping rings have high bunch charges, moderate energies and small emittance – Vulnerable to a wide range of collective effects • Too wide a subject to enter into here! 50

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linear collider, International Linear Collider, Lattice Design, electron cloud, storage rings, dynamic aperture, the International, The Next, beam size, the wake

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posted: | 3/25/2011 |

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