1. Electron Rings 2
Eduard Pozdeyev

2. Outline Introduction: Electron rings and their applications
Transverse (Betatron) motion in a rings
Longitudinal motion in rings
Chromatism of betatron oscillation and its compensation
Brief overview of nonlinear effects
Synchrotron radiation
Damping
Quantum nature of synchrotron radiation 
Electron rings as Synchrotron Light Sources
E. Pozdeyev, Electron Linacs

3. Chromatism of Betatron Oscillations [1]x
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4. Chromatism of Betatron Oscillations [2]
Beneficial to install Sextupoles
At locations with a large
beta function
𝛽 𝑠
𝛽 𝑞
E. Pozdeyev, Electron Linacs

5. Non-Linear Dynamics and Its Treatment
Nonlinear elements can severely affect beam dynamics in the rings
Cause fast beam losses and beam quality degradation
Limit beam lifetime in an accelerator
Limit suitable selection of betatron tunes
Accurate treatment of nonlinear motion still is not possible. There is no mathematical apparatus that would allow to that in a general case (except some specific cases)
Iterative perturbation analysis and averaging are used and produce good results. However, this treatment is beyond the scope of the course (although it is not too complicated and relies on analysis of corresponding Hamiltonian Functions. It is just time consuming.)
We study a simple model numerically to get a qualitative picture
E. Pozdeyev, Electron Linacs

6. Numerical Model and Motion Far From Resonances
Step 1 – one tern transformation, linear optics
Step 2 – thin sextupole and octupole transformations
𝜈= far from resonances motion with nonlinearities is perturbed but not dramatically. Linear motion shows no perturbations (ellipse). 3 2 2
Linear
With Nonlinearities
S= 0.05, O= -0.01
𝜕𝜈 𝜕 𝐴 2 >0
for O < 0
Tune shift is positive for large amplitudes
E. Pozdeyev, Electron Linacs

7. 𝝂=𝒒/𝟑 Resonance (in horizontal x-x’ phase space plane)
Linear motion, sext = 0, oct = 0 – no phase space perturbation
𝜈=0.31
𝜈=0.32
𝜈=0.33
𝜈=0.34
Non-linear motion, sext = 0.05, oct = – strong perturbation of phase space.
Particles become unstable (Amplitude →∞), causing losses in a few turns
𝜈=0.30
𝜈=0.31
𝜈=0.32
𝜈=0.33
𝜈=0.34
Particles with larger amplitudes get have a higher frequency, see previous slide
E. Pozdeyev, Electron Linacs

8. 𝝂=𝒒/𝟒 Resonance (in horizontal x-x’ phase space plane)
Linear motion, sext = 0, oct = 0 – no phase space perturbation
𝜈=0.23
𝜈=0.245
𝜈=0.27
Non-linear motion, sext = 0.05, oct = – strong perturbation of phase space
𝜈=0.23
𝜈=0.24
𝜈=0.245
𝜈=0.25
𝜈=0.27
E. Pozdeyev, Electron Linacs

9. Tune Diagram with Resonances
In general, the resonances happen when tunes satisfy equation
𝑘𝜈 𝑥 + 𝑙𝜈 𝑦 =𝑚
𝑘,𝑙,𝑚 −𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠
The strength of the resonances and their destructive effects reduce with the resonance order
Resonances higher than 4th order rarely cause instantaneous beam loss but can cause emittance increase and beam quality reduction.
Resonance harmonics equal to machine periodicity can be particularly strong
Tune Diagram between 2 and 3 for 𝑘,𝑙 ≤4
E. Pozdeyev, Electron Linacs

10. Tune Diagram with Resonances
In general, the resonances happen when tunes satisfy equation
𝑘𝜈 𝑥 + 𝑙𝜈 𝑦 =𝑚
𝑘,𝑙,𝑚 −𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠
The strength of the resonances and their destructive effects reduce with the resonance order
Resonances higher than 4th order rarely cause instantaneous beam loss but can emittance increase and beam quality reduction.
Resonance harmonics equal to machine periodicity can be particularly strong
Red circles show approximate area typically used by electron ring synchrotrons for operations.
E. Pozdeyev, Electron Linacs

11. Synchrotron Radiation
Particles moving with acceleration radiate. Bending is acceleration
(change of velocity vector)
SR is quantum effect
Classical E&M theory
gives good approximation
for most estimates
Loss per turn per electron in GeV
Same loss per turn in keV
E. Pozdeyev, Electron Linacs

12. Spectrum of Synchrotron Radiation
Characteristic frequency of
SR spectrum
Spectrum of SR
from a 0.5T magnet
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13. Damping of Vertical Oscillationsdp
Friction
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14. Damping of Synchrotron Oscillations
Energy transformation after
1 turn for electron with energy
deviated from the synchronous
energy
Energy and phase of synchronous
particle
For small oscillations
E. Pozdeyev, Electron Linacs

15. Theorem of the Sum of Decrements
For most modern large scale machines
𝐷≈ 𝛼 𝑝 𝑅 𝑟 ≪1
𝑅 is the average machine radius
p is the compaction factor
r is the magnet radius
E. Pozdeyev, Electron Linacs

16. Quantum Nature of Synchrotron Radiation and Equilibrium Emittances
Number of photons emitted per turn 𝑁≈𝛼𝛾= 𝛾 137
𝛼- is the fine-structure constant
Emission of a quantum appears
as a change in an equilibrium orbit causing oscillations around that new orbit.
Multiple emissions behave like Brownian motion causing increase of emittance.
Quantum oscillations ultimately limit the equilibrium emittance.
The equilibrium emittance is defined by the dumping rate and and
by the growth rate caused by random emissions of light quanta.
E. Pozdeyev, Electron Linacs

17. Homework Problem 1 VEPP-3 (Novosibirsk) Energy, E = 2 GeV
Circumference, C = 74.4 m
Bending radius: r = 10 m
Beam current: I = 100 mA
Calculate:
Energy loss by an electron per turn
Total power of synchrotron radiation
Damping time of vertical, horizontal, and synchrotron oscillations
E. Pozdeyev, Electron Linacs

18. Electron Rings As Light Sources
Ring parameters can be adjusted to meets specific requirements.
Presented parameters are typical operational parameters.
E. Pozdeyev, Electron Linacs

19. Choice of Beam Energy Advantages of higher electron beam energy:
Easier to produce high-energy photons (hard x-rays).
Better beam lifetime.
Easier to achieve higher current without encountering beam instabilities.
Disadvantages of higher beam energy:
Higher energy beams have larger emittances (reduced brightness) for a given lattice.
Stronger (more expensive) magnets are needed to steer and focus the beam.
Larger rf system needed to replace synchrotron radiation energy losses.
Many modern machines settle around GeV
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20. Insertion Devices [1]
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21. Insertion Devices [2] LCLS PM Undulator Prototype
SRC University of Wisconsin Madison
EM Wiggler
SLAC
PM Undulator
DESY
PM Undulator
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22. Insertion Devices Wigglers and Undulators
The wavelength of the undulator radiation (observed along the axis of the undulator):
K is the undulator parameter. K/γ is the maximum angle of the particle trajectory with respect to the undulator axis
An insertion device with K ≤ 1 is called an undulator. The radiation from an undulator has bandwidth ∆ω/ω = 1/2Nu (where Nu is the number of periods in the undulator), and is emitted in a cone with opening angle 1/γ.
An insertion device with K > 1 is called a wiggler. Synchrotron radiation from wigglers is similar to synchrotron radiation from dipoles: the spectrum is broad compared to an undulator, and the radiation is emitted in a wider fan than in an undulator (with opening angle K/γ).
The choice of insertion devices (and associated synchrotron light beam lines) depends on the user community.
E. Pozdeyev, Electron Linacs

23. Typical Applications of Light Sources Protein Crystallography
SPRING 8
Japan
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24. Preparation and Execution Of Experiments
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25. Result Processing Diffraction Image
Software-reconstructed electron density
3D visualization
Of electron density
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3D Protein Structure

26. Aknowledgements
Some material (mostly pictures) were “borrowed” from the USPAS school course “Design of Electron Storage and Damping Rings” by Andy Wolski and David Newton, USPAS, Fort Collins, Colorado, 2013
SPRING 8 informational video available on YouTube
E. Pozdeyev, Electron Linacs