Electron Rings 2 Eduard Pozdeyev

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

Chromatism of Betatron Oscillations [1] x E. Pozdeyev, Electron Linacs

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

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

Numerical Model and Motion Far From Resonances Step 1 – one tern transformation, linear optics Step 2 – thin sextupole and octupole transformations 𝜈=0.171 - 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

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 = -0.01 – 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

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 = -0.01 – strong perturbation of phase space 𝜈=0.23 𝜈=0.24 𝜈=0.245 𝜈=0.25 𝜈=0.27 E. Pozdeyev, Electron Linacs

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

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

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

Spectrum of Synchrotron Radiation Characteristic frequency of SR spectrum Spectrum of SR from a 0.5T magnet E. Pozdeyev, Electron Linacs

Damping of Vertical Oscillations dp Friction E. Pozdeyev, Electron Linacs

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

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

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

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

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

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 3 - 5 GeV E. Pozdeyev, Electron Linacs

Insertion Devices [1] E. Pozdeyev, Electron Linacs

Insertion Devices [2] LCLS PM Undulator Prototype SRC University of Wisconsin Madison EM Wiggler SLAC PM Undulator DESY PM Undulator E. Pozdeyev, Electron Linacs

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

Typical Applications of Light Sources Protein Crystallography SPRING 8 Japan E. Pozdeyev, Electron Linacs

Preparation and Execution Of Experiments E. Pozdeyev, Electron Linacs

Result Processing Diffraction Image Software-reconstructed electron density 3D visualization Of electron density E. Pozdeyev, Electron Linacs 3D Protein Structure

Aknowledgements Some material (mostly pictures) were “borrowed” from the USPAS 2013 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