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Physics of Incoherent Synchrotron Radiation Kent Wootton SLAC National Accelerator Laboratory

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Physics of Incoherent Synchrotron Radiation Kent Wootton SLAC National Accelerator Laboratory SLAC-PUB-17214 Physics of Incoherent Synchrotron Radiation Kent Wootton SLAC National Accelerator Laboratory US Particle Accelerator School Fundamentals of Accelerator Physics 23rd Jan 2018 Old Dominion University Norfolk, VA This work was supported by the Department of Energy contract DE-AC02-76SF00515. Circular accelerators – observation of SR in 1947 “A small, very bright, bluish- white spot appeared at the side of the chamber where the beam was approaching the observer. At lower energies, the spot changed color.” “The visible beam of “synchrotron radiation” was an immediate sensation. Charles E. Wilson, president of G.E. brought the whole Board of Directors to see it. During the next two years there were visits from six Nobel Prize winners.” J. P. Blewett, Nucl. Instrum. Methods Phys. Res., Sect. A, 266, 1 (1988). 2 Circular electron colliders – SR is a nuisance • “… the radiative energy loss accompanying the Source: Australian Synchrotron circular motion.” 4휋 푒2 훿퐸 = 훾4 3 푅 • Limits max energy • Damps beam size Source: CERN D. Iwanenko and I. Pomeranchuk, Phys. Rev., 65, 343 (1944). J. Schwinger, Phys. Rev., 70 (9-10), 798 (1946). 3 ‘Discovery machines’ and ‘Flavour factories’ • Livingston plot • Proton, electron colliders • Protons, colliding partons (quarks) • Linear colliders Linear colliders proposed Source: Australian Synchrotron 4 Future energy frontier colliders FCC-ee/ FCC/ SPPC d=32 km LEP/ Source: Fermilab 5 SR can be useful – accelerator light sources • Unique source of electromagnetic radiation • Wavelength-tunable • High intensity • Spatial coherence • Polarised • Pulsed Source: lightsources.org More in the next lecture – Storage Ring Light Sources 6 Outline – Synchrotron Radiation • Physics of synchrotron radiation • Characteristics of synchrotron radiation • Power, spectrum, flux • Universal spectrum of bending magnet SR • Critical photon energy • Accelerator physics implications • Damping rates • Equilibrium emittances 7 Electric field of an electric charge • Stationary charge • Field lines extend radially outward – • Acceleration • Finite speed of light, – causality • ‘Kinks’ in electric field distribution manifest as radiation 8 Electric field of an accelerated charge (휷 = ퟎ. ퟏ) http://www.shintakelab.com/en/enEducationalSoft.htm 9 Electric field of an accelerated charge (휷 = ퟎ. ퟗ) Detector http://www.shintakelab.com/en/enEducationalSoft.htm 10 Electric field, synchrotron radiation • Acceleration of a relativistic electron – – • Do virtual photons become real photons? • How much more power from transverse acceleration? 11 Power – linear acceleration 2 1 푒 푐 2 푃푅 = 2 2 푎 (charge rest frame) 6휋휖0 푚푐 1 푒2푐 d푝 2 1 d퐸 2 푃퐿 = 2 2 − 2 (lab frame) 6휋휖0 푚푐 d휏 푐 d휏 2 2 2 2 2 Using 퐸 = 푚0푐 + 푝 푐 , d푡 → 훾d휏 it can be shown that 1 푒2푐 d푝 2 1 푒2푐 d푝 2 푃퐿∥ = 2 2 = 2 2 (lab) 6휋휖0 푚푐 훾d휏 6휋휖0 푚푐 d푡 d푝 d퐸 Using = , d푡 d푠 1 푒2푐 d퐸 2 푃퐿∥ = 2 2 (lab) 6휋휖0 푚푐 d푠 J. Schwinger, Phys. Rev., 75, 1912 (1949). 12 Power – transverse acceleration 0 1 푒2푐 d푝 2 1 d퐸 2 푃퐿 = 2 2 − 2 (lab) 6휋휖0 푚푐 d휏 푐 d휏 1 푒2푐 d푝 2 푃퐿⊥ = 2 2 (lab) 6휋휖0 푚푐 d휏 Using d푡 → 훾d휏, 2 2 1 푒 푐 2 d푝 푃 = 훾2푃 푃퐿⊥ = 2 2 훾 (lab) 퐿⊥ 퐿∥ 6휋휖0 푚푐 d푡 d푝 2 퐸2 Using = 훽4 , d푡 휌2 2 4 4 푒 푐 4 훾 훾 푃퐿⊥ = 훽 2 (lab) 푃 ∝ 6휋휖0 휌 휌2 J. Schwinger, Phys. Rev., 75, 1912 (1949). J. Schwinger, ‘On Radiation by Electrons in a Betatron’, (2000), LBNL-39088. 13 Power – summary 2 4 푒 푐 4 훾 푃퐿⊥ = 훽 2 6휋휖0 휌 • Transverse a factor of 훾2 larger than longitudinal • 푃 ∝ 훾4 • Factor of 1013 for 푒−/푝+ (109 for 푒−/휇−) • Proton (or muon) circular colliders for high energy 1 • 푃 ∝ , large bending radius beneficial 휌2 Source: Australian Synchrotron 14 Energy loss per turn • Energy loss per turn 푈0 = ׯ 푃퐿⊥ d푡 2휋휌 • Isomagnetic lattice (only one bending radius 휌), d푡 = 훽푐 2 4 푒 3 훾 푈0 = 훽 3휖0 휌 퐸 GeV 4 푈 GeV = 퐶 , 0 훾 휌 m −5 −3 퐶훾 = 8.85 × 10 m GeV 15 Spectrum – Universal function of synchrotron radiation • Critical photon energy 3 ℏ푐훾3 휀 = ≡ ℏ휔 푐 2 휌 푐 • For electrons 퐸 GeV 3 휀 keV = 2.218 푐 휌 m = 0.665퐸 GeV 2퐵[T] • Synchrotron radiation spectrum 휔푐 - half the power above, half below ∞ d푃훾 푃훾 9 3 휔 푃훾 휔 = න 퐾 푥 d푥 = 푆 d휔 휔 8휋 휔 5 휔 휔 푐 푐 푥0 3 푐 푐 • Universal function 푆 H. Wiedemann, (2007) Particle Accelerator Physics, Springer-Verlag, p. 823 16 Angular distribution Spatial and spectral flux distribution of photon flux 2 2 d 푁ph 2 Δ휔 휔 훾 2 = 퐶Ω퐸 퐼 2 2 5/2 퐾2/3 휉 퐹 휉, 휃 d휃d휓 휔 휔푐 1+훾 휃 3훼 22 photons 퐶Ω = 2 2 = 1.3255 × 10 2 2 4휋 푒 푚푒푐 s rad GeV A 퐾 is the modified Bessel’s function, 2 2 퐾2 휉 2 2 2 훾 휃 1/3 퐹 휉, 휃 = 1 + 훾 휃 1 + 2 2 2 1 + 훾 휃 퐾2/3 휉 H. Wiedemann, Particle Accelerator Physics, Springer-Verlag, (2007). 17 Angular distribution – relativistic acceleration • Narrow cone of radiation 1 훾 • Tangential to the direction of acceleration D. H. Tomboulian and P. L. Hartman, Phys. Rev., 102, 1423 (1956). 18 Polarisation 퐵 (휋) (휎) • Synchrotron radiation highly polarised • Linear polarisations denoted with respect to magnetic field • Rings usually have a vertical main dipole field • 휎 – ‘senkrecht’, German for perpendicular (horizontal) • 휋 – parallel (vertical) 7 1 • Power 푃 = 푃, 푃 = 푃 휎 8 휋 8 19 Polarisation – angular distribution 휎 (horizontal) 휋 (vertical) • Typical electron, 퐸 = 3 GeV, 퐵 = 1.2 T, 휀푐 = 7.2 keV • Horizontal polarisation in orbit plane • Vertical polarisation above and below the orbit plane H. Wiedemann, (2007) Particle Accelerator Physics, Springer-Verlag, p. 814 20 Accelerator physics implications Protons are like elephants… … electrons (in rings) are like goldfish Source: http://www.mrwallpaper.com Source: http://www.theresponsiblepet.com 21 Liouville’s theorem • Liouville’s theorem • Emittance conserved under acceleration • SR introduces damping 푥′ 푥 22 Accelerator physics implications • Implications mostly for electron rings • So far, described synchrotron radiation using statistical distributions • Average energy loss per turn from SR results in damping • SR photons are both waves and quanta • Random quantised SR emission results in random excitation of electrons • Balance leads to equilibrium L. Rivkin, ‘Electron Dynamics with Synchrotron Radiation’, CERN Accelerator School: Introduction to Accelerator Physics, 5th Sep 2014, Prague, Czech Republic (2014). 23 Damping rates B • Synchrotron radiation damping means that the amplitude of single particle oscillations (betatron, synchrotron oscillations) are damped • Equilibrium determined by damping rates and lattice • At the instantaneous rate, the time for an electron to lose all its energy through synchrotron radiation • Damping time typically ~ms 24 SR damping – vertical • Electron performing vertical betatron oscillations • SR photons emitted in a very narrow cone, essentially parallel to electron trajectory with momentum 푝0 • Photon with momentum 푝훾 carries away transverse momentum of electrons • RF system replenishes only longitudinal momentum • Transverse momentum asymptotically damps 25 Damping rate – vertical • Consider betatron oscillation 퐴 푦 = 퐴 cos 휙 , 푦′ = sin 휙 훽(푠) • Oscillation amplitude 퐴2 = 푦2 + 훽 푠 푦′ 2 • Electron loses momentum to SR, gains only 훿퐸 energy at cavity 훿푦′ = −푦′ 퐸 • Change of amplitude 푑퐴 2퐴 = 훽2 푠 2푦′ 푑푦′ 훿퐸 퐴훿퐴 = −훽2 푠 푦′2 퐸 • Averaging over all betatron phases, 퐴2 2휋 퐴2 = 훽2 푠 푦′2 = ׬ sin2 휙d휙 2휋 0 2 M. Sands, ‘The Physics of Electron Storage Rings: An Introduction’, Stanford Linear Accelerator Center, Menlo Park, CA, SLAC-R-121 (1970). 26 Damping rate – vertical 퐴2 훿퐸 • Change in action 퐴훿퐴 = − 2 퐸 훿퐴 훿퐸 • Therefore = − 퐴 2퐸 • Summing all the energy losses around the ring must equal the total energy loss 푈0, therefore for one turn 푇0 Δ퐴 푈 Δ퐴 1 푑퐴 푈 = − 0 → = = − 0 퐴 2퐸 퐴푇0 퐴 푑푡 2퐸푇0 • Exponential damping 푒−휁푦푡 with damping coefficient 1 푈0 휁푦 = = 휏y 2퐸푇0 M. Sands, ‘The Physics of Electron Storage Rings: An Introduction’, Stanford Linear Accelerator Center, Menlo Park, CA, SLAC-R-121 (1970). 27 SR damping – longitudinal • SR photon emission carries away longitudinal momentum • Electron rings need significant RF voltage to maintain beam • SR emission is • Quantised – some electrons emit more, some less • Deterministic – high energy electrons emit more, low energy emit less • Design for phase stability B 28 Damping rate – longitudinal • An off-energy particle traverses a dispersive orbit Δ푥 퐷 Δ퐸 d푠′ = 1 + 푑푠 = 1 + 푥 푑푠 휌 휌 휌 퐸 d푠 Δ푥 • This particle radiates energy 푈′ over one turn d푠′ 1 1 퐷 Δ퐸 푈′ = ׯ 푃d푡 = ׯ 푃 d푠′ = ׯ 푃 1 + 푥 d푠 푐 푐 휌 퐸 • Differentiating with respect to energy d푈 1 d푃 퐷 d푃 Δ퐸 푃 ׯ + 푥 + d푠 = d퐸 푐 d퐸 휌 d퐸 퐸 퐸 • The average energy offset Δ퐸 should be zero, therefore 퐸 d푈 1 d푃 퐷 푃 ׯ + 푥 d푠 = d퐸 푐 d퐸 휌 퐸 K. Wille, The Physics of Particle Accelerators: An Introduction, Oxford University Press, Oxford, UK (2000). M. Sands, ‘The Physics of Electron Storage Rings: An Introduction’, Stanford Linear Accelerator Center, Menlo Park, CA, SLAC-R-121 (1970). Y. Papaphilippou, ‘Fundamentals of Storage Ring Design’, USPAS, Santa Cruz, CA, USA (2008). 29 Damping rate – longitudinal d퐵 d퐵 d푥 d퐵 퐷 퐷 • Using 푃 = 퐶 퐸2퐵2, = = 푥 = 퐵푘휌 푥 훾 d퐸 d푥 d퐸 d푥 퐸 퐸 d푃 푃 = 2 1 + 푘휌퐷 d퐸 퐸 푥 • Substituting this in to the equation on the previous slide d푈 2푈 1 퐷 1 ׯ 푃 푥 2푘 + d푠 + 0 = d퐸 퐸 푐퐸 휌 휌2 퐶 퐸4 1 퐶 퐸4 1 Using P = 훾 , therefore U = ׯ 푃d푠 = 훾 ׯ d푠 • 푒2푐2 휌2 0 푐 푒2푐3 휌2 1 1 ׯ 퐷푥 2푘+ d푠 d푈 푈0 휌 휌2 = 2 + 1 d퐸 퐸 ׯ d푠 휌2 • Energy oscillations exponentially damped 푒−휁푠푡 with the form 1 1 ׯ 퐷푥 2푘+ d푠 1 d푈 푈0 휌 휌2 휁푠 = = 2 + 풟 , 풟 = 1 2푇0 d퐸 2푇0퐸 ׯ d푠 휌2 K.
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