Principles of Synchrotron Radiaon

Boaz Nash Accelerator Source Division ESRF X-Ray and Neutron Science Summer Program August 30, 2016 Properes of radiaon

spectrum flux

(photons/second)

coherence polarization (single wavefront? “directionality” of radiation field ability to make an interference pattern) linear, circular partial/full polarization brightness Synchrotron light sources give some control over flux divided by source size all these properties, in may cases providing the only such source for particular parameters. Other lectures tell you why x-rays are useful.

Here, I will talk about where x-rays come from! Outline

(1) X-rays from electrons radiation

electricity and magnetism some relativity +

(2) Where do the electrons come from? synchrotron

particle accelerators = electron storage ring beam dynamics synchrotron radiation PART 1

radiation Radiaon from charged parcles- generalies

Maxwell’s equations: In vacuum, one derives the wave equation, where one gets plane waves representing electromagnetic radiation.

A source for such radiation requires a time dependent current.

Accelerating charged particles will thus provide a source of radiation. Dipole radiaon

Example of radiation source:

Consider a charge oscillating in sinusoidal motion. ! x θ r It produces a radiation pattern: sin2 (θ) S ∼ rˆ r2 ω The radiation will have a frequency f = 2π x(t) = Acos(ωt) c and wavelength λ = remember c = λ f f Polarization is linear in direction ˆ θ c = 2.99792e8 m/sec = speed of light Now, consider a moving, wiggling charge

x

s λu

Observed from a distance in the plane of oscillation, this looks (almost) like the oscillating dipole again!

The motion creates two important differences: 1) wavelength shifted by Doppler effect, and in case of relativistic speed, there is a time contraction effect. Net effect: λ λ = u γ 2 2) Pattern of radiation gets distorted from motion. For high energy gets bent into cone of angle 1 γ Undulator magnet causes electron wiggle

magnetic array or alternating field direction

N S N S N S N S N S Brief review of relavity

electron moving with velocity v v define β = c 1 speed of light γ = 2 1− β E[MeV ] E = γ mc2 γ = = 1.96 × E[MeV ] 1 0.5109989 β = 1− 2 γ Kinetic energy: 1 E = (γ − 1)mc2 → E = mv2 , for β << 1 k k 2 Relavity (2)

Note that for gamma>5, However, effects as gamma gets large: velocity increase becomes length contraction negligible time dilation General expression for Radiaon from a trajectory

⎛ R⎞ ∞ ! iω τ + ! ! ⎡ nˆ(τ ) ⎛ ic ⎞ ⎤ ⎝⎜ c ⎠⎟ E(R,ω) = β(τ ) − ⎜1+ ⎟ e dτ ∫−∞ ⎢ R ⎝ R⎠ ⎥ ⎣ ω ⎦ ! R 2π ω ! k = = R λ c ! nˆ = β(τ ) R Derived from Liénard-Wiechert potentials ! ! v β = c So, given the electron orbit, we can compute the radiated electric field at a given frequency Undulator/wiggler orbit

2πs eB B = B cos( ) 0λu y 0 K = 2 = 0.934B0[T ]λu [cm] λu 2πmc (simplest planar undulator) K N periods x' = max γ

λu

K ⎛ 2πs⎞ 1 K 2 K 2 4πs x' = β (s) = sin β (s) = 1− − + cos( ) x ⎜ ⎟ s 2 2 2 γ ⎝ λu ⎠ 2γ 4γ 4γ λ0 transverse velocity long. velocity modulation Single electron radiation Undulator/Wiggler spectrum

N −1 for K<1, all radiation contained in same cone inωT SN (ω) = ∑ e n=0

sin(NωT / 2) i(N −1)ωT /2 = e sin(ωT / 2)

Transition between undulator and Wiggler spectrum We also get radiaon out of a dipole magnet

By = B0

1 γ combining with time compression, we get a characteristic time ρ ρ 1 dT ' = (1+ (αγ )2 ) orbit length dL = γ 3 γ 2cγ ρ dT ' = characteristic time cγ Dipole magnet orbit

Consider electron in constant magnetic field

! β(τ )

ρ ! ⎛ βcτ ⎛ βcτ ⎞ ⎞ r(τ ) = ρ sin ,ρ 1− cos ,0 ⎜ ρ ⎝⎜ ρ ⎠⎟ ⎟ ⎝ ⎠

B=.85 T p/c=6.04 GeV Be ω = ρ = 23m 0 γ m Dipole magnet spectrum

crical frequency defined as (18.8 KeV for current ESRF ) hc (E = hf = ) λ Computing spectrum, one finds h=Planck’s constant

9 3 ∞ S(ξ) = ξ K5/3(ξ )dξ 8π ∫ξ

Note that spectrum is much broader than for the undulator. PART 2

synchrotron Parcle accelerators and storage rings

Two things to understand:

1) Single electrons: How to store an electron and what kind of orbit will it have?

2) What kind of distribution of electrons will we get in the synchrotron? How to store a high energy electron?

First accelerate: 6 GeV for ESRF

To move in a circle, we use dipole magnets

For transverse focussing/stability, use quadrupoles

To fix chromatic aberration, we need sextupoles

To give energy back lost to synchrotron radiation, and to provide longitudinal stability, use RF cavities ESRF Acceleraon Complex

1− β = 3.6 × 10−9

E 0.025eV Ek = 200MeV E = 6.03GeV k = Ek = 100keV Ek = 11MeV k γ = 1 γ = 1.2 γ = 22.5 γ = 391 γ = 11,800

electron buncher linac TL1 booster TL2 gun prebuncher storage ring Electron Gun and pre-buncher

impulse to gun determines bunch shape and length

pre-buncher does not accelerate

100 keV triode gun γ = 1.2 gun is triggered either at 10 Hz or at 1 Hz Storage ring components

dipole

quadrupole

sextupole

RF cavity Quadrupoles for strong focusing of electrons

¼ of an ESRF quadrupole

B Apply Lorentz force law 1 Opposite signs! kx = − Field in body and we get focal strengths Bρ Requires clever given by quad placement B ! k = 1 and polarity to get B B (y xˆ x yˆ) y overall focussing! = 1 + Bρ Sextupoles

Sextupoles may be used to correct energy effect from quadrupoles (chromaticity). Then causes additional stability problems which need to be corrected! ! Beam lifetime and dynamic aperture for injection B B (xy xˆ (x2 y2 )yˆ) hard problem in non-linear = 2 + − dynamics! RF cavity

Gives energy back that was lost from radiation and provides longitudinal focussing. Most of the ESRF energy use (around 1.5 MW of power) is in these cavities; What happens to stored electrons? Phase space

x

x=0 time

px,y = γ mvx,y configuration space x vs. time for electron, we normalize with

phase P0 = γ mvs space x vs. p and use p dx x' = x = P0 ds What kind of distribuon of electrons will we have in a storage ring?

several stored electrons with different amplitudes.

x’

Another effect: radiation

x

4 4π r m cC e −5 γ E Cγ = 2 3 = 8.85 *10 3 radiated power 3 (mc ) GeV Pγ = 2 in a dipole: 2π ρ radiation constant

Higher energy radiates more Lower energy radiates less: Radiation Damping! Radiaon effect on Longudinal dynamics

E = E0 + ΔE

E = E0

E = E0 − ΔE

bending RF cavity magnets energy • Higher energy radiates more, lower energy less. Causes damping towards reference energy.

position along ring Radiaon damping

All electrons damp towards the same orbit!

What sets the size of the electron beam? Where does the electron beam size come from? quantum excitation Graininess of photon emission

Two sources of randomness: emission time of photons are random: Poisson process

Energy emitted is also a random process, with the power spectrum as the probability distribution for each photon.

For ESRF, only about 800 photons per turn! Or, about 1 photon emied per meter! (approx. 12 photons per dipole)

This quantum mechanical diffusion process accounts for the size of the electron beam, which (usually) determines the size of the x-ray beam! Where does the electron beam size come from? quantum excitation Quantum fluctuaon effect on electron dynamics

without QFluct with QFluct damping

synchrotron oscillations

Electron motion and without quantum fluctuations. Result of damping/diffusion

The electron beam reaches a unique Gaussian distribution– independent of how one injects into the ring. This is a major difference between electron synchrotrons and proton synchrotrons (e.g. LHC)

By careful choice of where the dipoles and quadrupoles are, one can reduce the size of this equilibrium beam size (emittance = beam size in phase space). So called “Low emittance ring design”

In fact, due to developments in lattice design, ESRF is completely replacing the storage ring in 2018 to reduce the electron beam emittance. 4nm -> 150 pm Coherent versus incoherent SR

In storage ring, average interparticle spacing is larger than radiation wavelength, thus each electron emits independently. Power scales with N, number of electrons.

An FEL (Free Electron Laser) is a different kind of accelerator where the electron beam is designed to have a microbunching structure on the scale of the radiation. This gives coherent radiation with a power proportional to N^2. There are several XFEL projects, e.g. LCLS (US), European XFEL (Germany) , SACLA (Japan), and more Because these are single pass (vs. storage ring) the energy requirement and repetition rate is typically much lower than storage rings. It looks like both storage ring synchrotron sources and FEL sources will fulfill different requirements and can complement each other. Further References

Synchrotron Radiation

Elleaume, Onuki, “Undulators, Wigglers and their Applicaons”, Taylor and Francis (2003)

Electron Storage rings

Ma Sands, “The Physics of Electron Storage Rings” Slac report 121 (1970)

General accelerator physics references Helmut Wiedemann, ‘Particle Accelerator Physics’, 1&2, Springer-Verlag, 1993, (1999) S.Y. Lee, ‘Accelerator Physics’, World Scientific, (1999) Klaus Wille, ‘The Physics of Particle Accelerators’, Oxford University Press, (1996) Thank you for your attention!! Extra Slides Single electron radiation Time compression factor

B C

A α

Electron moves at speed β, emitting wavefront at A, B, C Δt time between (c − v) v wavefronts Δt = Δt ' = βe cosα c c t (1 cos ) t ' t ' time for Δ = − βe α Δ Δ electron to emit 1+ (αγ )2 Δt ≈ Δt ' 2γ 2 dt = (1− β cosα) continuous generalization dt '