Principles of Synchrotron Radia4on Boaz Nash ESRF - PowerPoint PPT Presentation

Principles ¡of ¡Synchrotron ¡ Radia4on ¡ Boaz ¡Nash ¡ ESRF ¡ ESRF/ILL X-Ray and Neutron Science Summer Program September 5, 2017

Proper4es ¡of ¡radia4on ¡ spectrum flux (photons/second) coherence polarization (single wavefront? ability to make an interference “directionality” of radiation field pattern) linear, circular partial/full polarization brightness Synchrotron light sources give some control over flux divided by source size all these properties, in many 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!

Radia4on ¡from ¡charged ¡par4cles-­‑ ¡ generali4es ¡ 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 ¡radia4on ¡ Example of radiation source: Consider a charge oscillating in sinusoidal motion. θ � r x It produces a radiation pattern: S ∼ sin 2 ( θ ) ˆ r r 2 The radiation will have a frequency f = ω 2 π x ( t ) = A cos( ω t ) and wavelength λ = c c = λ f remember f Polarization is linear in direction ˆ θ c = 2.99792e8 m/sec = speed of light

Now, ¡consider ¡a ¡moving, ¡wiggling ¡ charge ¡ x λ u s 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 dilation effect. λ = λ u Net effect: 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 ¡rela4vity ¡ electron moving with velocity v β = v define c speed of light 1 γ = 1 − β 2 [ ] γ = E MeV [ ] 0.5109989 = 1.96 × E MeV E = γ mc 2 1 − 1 β = γ 2 Kinetic energy: → E k = 1 2 mv 2 , for β << 1 E k = ( γ − 1) mc 2

Rela4vity ¡(2) ¡ Note that for gamma>5, However, effects as gamma gets large: velocity increase becomes length contraction negligible time dilation

General ¡expression ¡for ¡Radia4on ¡ from ¡a ¡trajectory ¡ � � � ⎛ ⎞ n ( τ ) ⎡ ⎤ i ω τ + R ⎛ ⎞ β ( τ ) − ˆ 1 + ic ∞ ⎜ ⎟ d τ ∫ ⎝ ⎠ R , ω ) = c ⎜ ⎟ E ( e ⎢ ⎥ ⎝ ⎠ ω R ⎣ ⎦ −∞ R � k = 2 π λ = ω R � c R n = � ˆ 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 ¡ K = eB 0 λ u B y = B 0 cos(2 π s 2 π mc 2 = 0.934 B 0 [ T ] λ u [ cm ] ) λ u (simplest planar undulator) x ' max = K N periods γ λ u ⎛ ⎞ γ sin 2 π s 4 γ 2 cos(4 π s x ' = β x ( s ) = K 2 γ 2 − K 2 4 γ 2 + K 2 1 β s ( s ) = 1 − ) ⎜ ⎟ ⎝ λ u ⎠ λ 0 long. velocity modulation transverse velocity

Single electron radiation Undulator/Wiggler ¡spectrum ¡ for K<1, all radiation contained in same cone N − 1 ∑ S N ( ω ) = e in ω T n = 0 ( ) = sin N ω T / 2 e i ( N − 1) ω T /2 ( ) sin ω T / 2 Transition between undulator and Wiggler spectrum

¡ ¡We ¡also ¡get ¡radia4on ¡out ¡of ¡a ¡ dipole ¡magnet ¡ B y = B 0 1 γ combining with time compression, we get a characteristic time ρ dT ' = 2 c γ 3 (1 + ( αγ ) 2 ) dL = ρ 1 orbit length γ γ dT ' = ρ characteristic time c γ

Dipole ¡magnet ¡orbit ¡ Consider electron in constant magnetic field � β ( τ ) ρ ⎛ ⎞ r ( τ ) = ρ sin β c τ ρ , ρ 1 − cos β c τ ⎛ ⎞ � ⎟ ,0 ⎜ ⎜ ⎟ ρ ⎝ ⎠ ⎝ ⎠ B=.85 T p/c=6.04 GeV ω 0 = Be ρ = 23 m γ m

Dipole ¡magnet ¡spectrum ¡ cri4cal ¡frequency ¡defined ¡as ¡ (18.8 ¡KeV ¡for ¡current ¡ESRF ¡) ¡ ( E = hf = hc λ ) Computing spectrum, one finds h=Planck’s constant S ( ξ ) = 9 3 ∞ ∫ 8 π ξ K 5/3 ( ξ ) d ξ ξ Note that spectrum is much broader than for the undulator.

Par4cle ¡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 ¡Accelera4on ¡Complex ¡ 1 − β = 3.6 × 10 − 9 E k = 200 MeV E k = 6.03 GeV E k = 0.025 eV E k = 100 keV E k = 11 MeV γ = 391 γ = 11,800 γ = 1 γ = 1.2 γ = 22.5 electron buncher linac TL1 booster TL2 gun storage ring prebuncher

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

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 x + ( x 2 − y 2 ) ˆ B = B 2 ( xy ˆ hard problem in non-linear y ) 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 p x , y = γ mv x , y configuration space x vs. time for electron, we normalize with 0 = γ mv s P phase space x vs. p and use x ' = p x = dx P ds 0

What ¡kind ¡of ¡distribu4on ¡of ¡electrons ¡ will ¡we ¡have ¡in ¡a ¡storage ¡ring? ¡ several stored electrons with different amplitudes. x’ Another effect: radiation x γ = cC γ E 4 C γ = 4 π r m radiated power ( mc 2 ) 3 = 8.85 *10 − 5 e P GeV 3 3 2 π ρ 2 in a dipole: radiation constant Higher energy radiates more Lower energy radiates less: Radiation Damping!

Radia4on ¡effect ¡on ¡Longu4dinal ¡ dynamics ¡ E = E 0 + Δ E E = E 0 E = E 0 − Δ E energy bending RF cavity magnets • Higher energy radiates more, lower energy less. Causes damping towards reference energy. position along ring

Radia4on ¡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 ¡emi^ed ¡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 ¡fluctua4on ¡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