SLAC-PUB-16450

Synchrotron Radiation Kent Wootton SLAC National Accelerator Laboratory

US Particle Accelerator School Fundamentals of Accelerator Physics 02/02/2016 University of Texas Austin, TX

This work was supported in part by the Department of Energy contract DE-AC02-76SF00515. Key references

• D. A. Edwards and M. J. Syphers, An Introduction to the Physics of High Energy Accelerators, Wiley, Weinheim, Germany (1993). • DOI: 10.1002/9783527617272 • H. Wiedemann, Particle Accelerator Physics, 4th ed., Springer, Heidelberg, Germany (2015). • DOI: 10.1007/978-3-319-18317-6 • E. J. N. Wilson, An Introduction to Particle Accelerators, Oxford University Press, Oxford, UK (2001). • DOI: 10.1093/acprof:oso/9780198508298.001.0001

2 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). 3 Circular electron colliders – SR is a nuisance

• “… the radiative energy loss accompanying the Source: Australian Synchrotron circular motion.”

= 3 2 4휋 푒 4 • Limits훿훿 max energy훾 푅 • Damps beam size

Source: CERN D. Iwanenko and I. Pomeranchuk, Phys. Rev., 65, 343 (1944). J. Schwinger, Phys. Rev., 70 (9-10), 798 (1946). 4 ‘Discovery machines’ and ‘Flavour factories’

• Livingston plot • Proton, electron colliders • Protons, colliding partons (quarks) • Linear colliders Linear colliders proposed

Source: Australian Synchrotron 5 Future energy frontier colliders

FCC-ee/ FCC/ SPPC d=32 km

LEP/

Source: Fermilab 6 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 – Overview of Light Sources

7 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

8 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

9 Electric field of an accelerated charge ( = . )

휷 ퟎ ퟏ

http://www.shintakelab.com/en/enEducationalSoft.htm 10 Electric field of an accelerated charge ( = . )

휷 ퟎ ퟗ

Detector

http://www.shintakelab.com/en/enEducationalSoft.htm 11 Electric field, synchrotron radiation

• Acceleration of a relativistic electron

– –

• Do virtual photons become real photons? • How much more power from transverse acceleration?

12 Power – linear acceleration

= (charge rest frame) 2 1 푒 푐 2 2 2 푃푅 6휋휖0 푚푐 푎 = (lab frame) 2 1 푒 푐 d푝 2 1 d퐸 2 2 2 2 Using푃퐿 6휋 휖0 =푚푐 d휏 +− 푐 ,d d휏 d it can be shown that 2 2 2 2 2 0 = � 푚 푐 푝= 푐 푡 → 훾 휏 (lab) 2 2 1 푒 푐 d푝 2 1 푒 푐 d푝 2 2 2 2 2 Using푃퐿∥ 6휋휖0=푚푐 , 훾d휏 6휋휖0 푚푐 d푡 d푝 d퐸 d푡 d푠 = (lab) 2 1 푒 푐 d퐸 2 2 2 푃 퐿∥ 6휋휖0 푚푐 d푠

J. Schwinger, Phys. Rev., 75, 1912 (1949). 13 Power – transverse acceleration

0 = (lab) 2 1 푒 푐 d푝 2 1 d퐸 2 2 2 2 푃퐿 6휋휖0 푚푐 d휏 − 푐 d휏 = (lab) 2 1 푒 푐 d푝 2 2 2 Using푃퐿⊥ 6휋d 휖0 푚푐d , d휏

= 푡 → 훾 휏 (lab) = 2 2 1 푒 푐 2 d푝 2 2 2 퐿⊥ 퐿∥ 푃퐿⊥ 6휋휖0 푚푐 훾 d푡 푃 훾 푃 Using = , 2 2 d푝 4 퐸 2 = d푡 훽 (lab)휌 2 4 4 푒 푐 4 훾 2 훾 푃퐿⊥ 6휋휖0 훽 휌 푃 ∝ 2 J. Schwinger, Phys. Rev., 75, 1912휌 (1949). J. Schwinger, ‘On Radiation by Electrons in a Betatron’, (2000), LBNL-39088. 14 Power – summary

= 2 4 푒 푐 4 훾 • Transverse a factor푃 of퐿⊥ larger훽 than2 longitudinal 6휋휖0 휌 • 2 훾 • Factor4 of 10 for / (10 for / ) 푃 ∝ 훾 • Proton (or muon)13 circular− + colliders9 −for high− energy 푒 푝 푒 휇 • , large bending radius beneficial 1 2 푃 ∝ 휌

Source: Australian Synchrotron 15 Energy loss per turn

• Energy loss per turn = d

• Isomagnetic lattice (only푈0 one∮ 푃 퐿bending⊥ 푡 radius ), d = 2휋� = 휌 푡 훽� 2 4 푒 3 훾 0 GeV푈 =3휖0 훽 휌 , 4 퐸 GeV 푈=0 8.85 × 10퐶훾 휌mm GeV −5 −3 퐶훾

16 Spectrum – Universal function of synchrotron radiation

• Critical photon energy 3 = 2 3 ℏ푐훾 • For electrons휀푐 ≡ ℏ휔푐 휌 GeV keV = 2.218 m 3 � 휀푐 = 0. GeV [T] 휌 • Synchrotron radiation spectrum2 - half the power 665� 퐵 above, half below 푐 d 9 3 휔 = d = d ∞ 푃훾 푃훾 휔 푃훾 휔 � 퐾5 푥 푥 푆 • 휔Universal휔푐 8휋 function휔푐 푥0 3 휔푐 휔푐 H. Wiedemann, (2007) Particle Accelerator Physics, Springer-Verlag, p. 823 푆 17

Angular distribution

Spatial and spectral flux distribution of photon flux

= , 2 / / ph 2 d 푁 2 Δ휔 휔 훾 2 2 2 5 2 d휃d휓 퐶Ω� 퐼 휔 휔푐 1+훾 휃 퐾2 3 휉 퐹 휉 휃 photons = = 1.3255 × 10 4 s rad GeV A 3훼 22 퐶Ω 2 is the modified2 Bessel’s function,2 2 휋 푒 푚푒푐

퐾 / , = 1 + 1 + 1 +2 2 2 2 2 2 훾 휃 퐾1/3 휉 퐹 휉 휃 훾 휃 2 2 2 훾 휃 퐾2 3 휉

H. Wiedemann, Particle Accelerator Physics, Springer-Verlag, (2007). 18 Angular distribution – relativistic acceleration

• Narrow cone of radiation 1 • Tangential to the direction 훾of acceleration

D. H. Tomboulian and P. L. Hartman, Phys. Rev., 102, 1423 (1956). 19 Polarisation

( ) 퐵 휋

( )

휎 • Synchrotron radiation highly polarised • Linear polarisations denoted with respect to magnetic field • Rings usually have a vertical main dipole field • – ‘senkrecht’, German for perpendicular (horizontal) • – parallel (vertical) 휎 • Power휋 = , = 7 1 푃휎 8 푃 푃휋 8 푃 20 Polarisation – angular distribution

(horizontal) (vertical)

휎 휋

• Typical electron, = 3 GeV, = 1.2 T, = 7.2 keV • Horizontal polarisation in orbit plane 푐 • Vertical polarisation� above and퐵 below the휀 orbit plane

H. Wiedemann, (2007) Particle Accelerator Physics, Springer-Verlag, p. 814 21 Accelerator physics implications

Protons are like elephants…

… electrons (in rings) are like goldfish

Source: http://www.mrwallpaper.com

Source: http://www.theresponsiblepet.com

22 Liouville’s theorem

• Liouville’s theorem • Emittance conserved under acceleration • SR introduces damping

푥푥

푥

23 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). 24 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

25 SR damping – vertical

• Electron performing vertical betatron oscillations • SR photons emitted in a very narrow cone, essentially parallel to electron trajectory with momentum

• Photon with momentum carries away0 transverse momentum of electrons 푝 푝훾 • RF system replenishes only longitudinal momentum • Transverse momentum asymptotically damps

26 Damping rate – vertical

• Consider betatron oscillation = cos , = sin ( ) 퐴 ′ • Oscillation푦 amplitude퐴 휙 푦 훽 푠 휙 = + • Electron loses2 momentum2 to ′SR,2 gains only 퐴 푦 훽 푠 푦 energy at cavity = 훿� • Change of amplitude′ ′ 훿푦 −푦 퐸 = 2 푑� 2 ′ ′ 2퐴 푑=푦 훽 푠 푦 훿� • Averaging over all betatron2 ′2phases, 퐴�퐴 −훽 푠 푦 퐸 = sin d = 2 2 2 ′2 퐴 2휋 2 퐴 0 M. 훽Sands,푠 푦 ‘The Physics2휋 ∫ of Electron휙 휙 Storage2 Rings: An Introduction’, Stanford Linear Accelerator Center, Menlo Park, CA, SLAC-R-121 (1970). 27 Damping rate – vertical

• Change in action = 2 퐴 훿� • Therefore = 퐴� 퐴 − 2 퐸 훿� 훿� • Summing all퐴 the− energy2퐸 losses around the ring must equal the total energy loss , therefore for one turn

0 1 0 = 푈 = = 푇 Δ퐴 푈0 Δ퐴 푑� 푈0 • Exponential damping− → with damping− coefficient 퐴 2� 퐴푇0 퐴 푑� 2�푇0 −휁푦푡 푒 = = 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). 28 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

29 Damping rate – longitudinal

• An off-energy particle traverses a dispersive orbit

d = 1 + = 1 + d ′ Δ푥 퐷푥 Δ퐸 휌 • This particle radiates energy휌 over 휌one퐸 turn d 푠 푑� 푑� 푠 Δ푥 = d = d =푈� 1 + d 푠 � ′ 1 1 퐷푥 Δ퐸 • Differentiating푈 ∮ 푃 with푡 respect∮ 푃 푐 푠 to� energy푐 ∮ 푃 휌 퐸 푠 = + + d d푈 1 d푃 퐷푥 d푃 Δ퐸 푃 • The averaged 퐸energy푐 ∮ offsetd퐸 휌 d퐸 should퐸 퐸 be푠 zero, therefore Δ퐸 = 퐸+ d d푈 1 d푃 퐷푥 푃 K. Wille, The Physics of Particled퐸 Accelerators:푐 An dIntroduction,퐸 휌 Oxford퐸 University푠 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). 30 Damping rate – longitudinal

• Using = , = = = 2 2 d퐵 d퐵 d푥 d퐵 퐷푥 퐷푥 푃 퐶훾� 퐵 d퐸= 2 d푥 1d퐸+ d푥 퐸 퐵퐵� 퐸 d푃 푃 • Substituting this in to the equation on the previous slide d퐸 퐸 𝑘퐷푥 = + + d d푈 2푈0 1 퐷푥 1 2 P = d퐸 , 퐸 푐� ∮U푃=휌 2� d 휌= 푠 d • Using 4 therefore 4 퐶훾 퐸 1 퐶훾퐸 1 2 2 2 2 3 2 푒 푐 휌 0 푐 ∮ 푃 푠 푒 푐 ∮ 휌 푠 = 2 + 1 1 푥 d푈 푈0 ∮ 퐷 휌 2푘+휌2 d푠 1 • Energy oscillationsd퐸 exponentially퐸 damped∮휌2d푠 with the form −휁푠푡 = = 2 + , = 푒1 1 푥 1 d푈 푈0 ∮ 퐷 휌 2푘+휌2 d푠 1 푠 0 0 K. Wille휁, The Physics2푇 ofd Particle퐸 Accelerators:2푇 퐸 An Introduction,풟 풟Oxford University Press,∮휌2d Oxford,푠 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). 31

SR damping – horizontal

• It can be shown that … • Horizontal oscillations exponentially damped with the form −휁푥푡 푒

= 1 , = 1 1 푥 푈0 ∮ 퐷 휌 2푘+휌2 d푠 1 푥 0 휁 2푇 퐸 − 풟 풟 ∮휌2d푠

32 Robinson criterion and damping summary

1 = = , = 1 2 0 푥 푈 푥 푥 휁 푥 1 0 풥 풥 − 풟 =휏 =푇 � , = 1 2 0 푦 푈 푦 푦 휁 1 푦 0 풥 풥 = 휏= 푇 � , = 2 + 2 푈0 휁푠 풥푠 풥푠 풟 휏푠 푇0� = 1 1 ∮ 퐷푥휌 2푘+휌2 d푠 1 풟 + +∮휌2d푠= 4

풥푥 풥푦 풥푠

K.W. Robinson, Phys. Rev., 111, 373 (1958). 33 Consequences for electron rings

• If no vertical bending magnets, = 1 • True for most rings, bending only in one plane 풥푦 • If no quadrupole gradient in main bending magnets, 0, 1, 2 • Most FODO synchrotron (collider) lattices are like this 풟 → 풥푥 ≈ 풥푠 ≈ • Most new storage ring light sources are not

34 What happens to the emittance?

• Emittance damps to zero? • No! • Average energy loss per turn from SR results in damping • SR photons are both waves 푥푥 and quanta • Random quantised SR emission results in random excitation of electrons 푥 • Balance leads to equilibrium

35 Equilibrium emittances – vertical

• Damps to almost zero • Cone of synchrotron radiation, random vertical emission of photons • Limit from opening angle of synchrotron radiation • Typically much larger, arising from uncorrected betatron coupling with horizontal plane • Emittance ratio = • Arises from misalignment of quadrupole, sextupole 휀푦 휅휀푥 centres on the order of ±20 μm.

36 Equilibrium emittances – energy spread, horizontal

• Damps to equilibrium energy spread 1/ d = 2 2 1/ 3d 휎퐸 훾 ∮ 휌 푠 = 3.84퐶푞 × 10 2m � 풥푠 ∮ 휌 푠 • Damps to equilibrium horizontal −emittance13 퐶푞 / d = 2 1/ 3d ℋ 휌 푠 푥 푞 훾 ∮ =휀 퐶 + 2 2+ 풥푥 ∮ 휌 푠 • Curly-H function? Next ′2lecture! 2 ℋ 훽푥휂 훼푥휂휂� 훾푥휂

37 Summary

• SR the main difference between electron, proton rings • Significant only for acceleration of electrons transverse to their velocity • Determined energy loss per turn, characteristics of SR • Average energy loss per turn from SR results in damping • Random quantised SR emission results in random excitation of electrons • Balance results in an equilibrium emittance

This work was supported in part by the Department of Energy contract DE-AC02-76SF00515. 38