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Electron Rings

This article provides an introduction to electron rings and their applications, including transverse and longitudinal motion, analysis of nonlinear perturbations, synchrotron radiation, damping, excitation, and effects determining lifetime in rings.

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Electron Rings

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  1. Electron Rings Eduard Pozdeyev

  2. Outline • Introduction: Electron rings and their applications • Transverse (Betatron) motion in a rings • Longitudinal motion in rings • Analysis of nonlinear perturbations • Synchrotron radiation • Damping • Exitation • Steady emittance size and  • Effects determining lifetime in rings E. Pozdeyev, Electron Linacs

  3. Introduction: Electron Rings • A ring is a an electromagnetic system with a closed particle orbit. • The closed orbit is a natural choice of the reference orbit in rings. The motion of particles typically is described relatively to the closed orbit. • We will be interested in systems with a stable orbit. That is, particles with a small enough deviation from the closed orbit are stable in respect to the closed orbit. • Electrons in circular accelerators can make many turns and interact with accelerating RF many times, reaching high energy over an extended period of time. • In linacs, this happens only once or several times (recirculating linacs). • Also, rings can store electrons (and positrons) for significant amount of time (hours), providing unique experimental capabilities as colliders and synchrotron light sources. E. Pozdeyev, Electron Linacs

  4. Introduction:Electron Synchrotron Boosters • Electron synchrotron boosters accelerate electrons to a specific energy to inject them into other accelerators. Electron linacs are frequently used as injectors to boosters: sourcelinacbooster ringexperimental ring • Historically, boosters were used for fixed target experiments. However, those machines have been decommissioned long time ago The first synchrotron to use the "racetrack" design with straight sections, a 300 MeV electron synchrotron at University of Michigan in 1949, designed by Dick Crane. E. Pozdeyev, Electron Linacs

  5. Introduction:Electron-Electron, Electron-Positron Colliders Large Electron–Positron Collider (LEP) Operational: 1989 - 2000 Tunnel was used for LHC after LEP was decommissioned VEP-1, 1963 Russia, Novosibirsk Particles: electron – electron Collision energy: 160 MeV Luminosity: 1028 1/(cm2s) Rings size: two 1m x 1m Particles: electron – positron Collision energy: 100 GeV Luminosity: 1032 1/(cm2s) Circumference: 27 km E. Pozdeyev, Electron Linacs

  6. Introduction: Light SourcesMain Application of Modern Electron Rings Particles: electrons Energy: 3 GeV Beam current: 0.5 A Circumference: 792m Number of bunches: 1056 Beam size (v/h): 3-13 m / 30-150 m Experimental beamlines: 58 E. Pozdeyev, Electron Linacs

  7. Transverse Motion in Electron Rings E. Pozdeyev, Electron Linacs

  8. Simple Electron Ring Latticeand Typical Subcomponents • Basic subcomponents of electron rings: • Bending magnets or electrostatic bends - dipoles • Focusing magnets – quads (can be incorporated into dipoles) • Multiple magnets to achieve specific beam dynamics characteristics • RF cavities to accelerate or compensate losses due to synchrotron losses and keep beam bunched • Injection/extraction systems • Simplified lattice example • Bend • FODO doublet (Qx, Qy) • Sextupoles to compensate tune chromatism (Sx, Sy) Dipole Qy Sy Sx Qx E. Pozdeyev, Electron Linacs

  9. Equations of Motion and Hill Equation - small deviations from the reference particle s is the independent variable instead of t (s=v*t) - Dipole magnet with gradient focusing n is the field index - Quadrupole - Drift - General Hill equation with periodic focusing E. Pozdeyev, Electron Linacs

  10. Example: Weak Focusing Azimuthally Symmetric Field with Gradient Solution easily obtainable Current is phase space density times area Increase density Increase aperture Increase focusing Increasing focusing in both planes is Impossible. Need other focusing Mechanism (strong focusing) E. Pozdeyev, Electron Linacs

  11. Strong Focusing Strong focusing can be achieved by introducing variable focusing as function of s. However, stability and properties of such motion needs to be investigated. E. Pozdeyev, Electron Linacs

  12. Linear Betatron Motion Linear motion can be described by vectors and matrices E. Pozdeyev, Electron Linacs

  13. Stability of Betatron Motion [1] - Eigen vectors of M (basis) with eigen values and . - initial vector - after a turn - after N turns For the motion to be stable E. Pozdeyev, Electron Linacs

  14. Stability of Betatron Motion [2] Matrices T and M are Wronskians => det(T) – constant. det(T) = det(M) = 1 – obtain from initial conditions is the betatron phase advance per turn E. Pozdeyev, Electron Linacs

  15. Twiss Parametrization Twiss parametrization Because det(M)=1 E. Pozdeyev, Electron Linacs

  16. Evolution of Particle Coordinatesat Specific Location s is the betatron phase advance per turn E. Pozdeyev, Electron Linacs

  17. Courant-Snyder Ellipse At Specific Location s E. Pozdeyev, Electron Linacs

  18. Particle Motion Along AcceleratorEquations for and E. Pozdeyev, Electron Linacs

  19. Particle Motion Along AcceleratorPhase Advance - Betatron tune E. Pozdeyev, Electron Linacs

  20. ExampleSmall Focusing Perturbation Thin, weak lens added to a ring with the one-turn matrix M0 Find new betatron tune and at the location of the lens For small E. Pozdeyev, Electron Linacs

  21. Homework Problem 1 E. Pozdeyev, Electron Linacs

  22. Motion of Particle With Energy Deviation Search for solution in this form Equation for dispersion function E. Pozdeyev, Electron Linacs

  23. Azimuthally Symmetric Case E. Pozdeyev, Electron Linacs

  24. Energy-Phase Motion with RF [1] - sine dependence! E. Pozdeyev, Electron Linacs

  25. Energy-Phase Motion with RF [2] Assumes slow Change per turn W0 is energy loss per turn to radiation For synchronous phase, W0 is exactly compensated by energy gain Equations of energy-phase motion E. Pozdeyev, Electron Linacs

  26. Small Oscillations For small deviations in phase from the synchronous phase Equations of energy-phase motion for small amplitudes - synchrotron frequency - normalized synchrotron tune E. Pozdeyev, Electron Linacs

  27. Homework Problem 2 Synchrotron Frequency in VEPP-3, Novosibirsk Find synchrotron frequency and revolution frequency and show that the assumption of slow synchrotron oscillations is correct E. Pozdeyev, Electron Linacs

  28. Hamiltonian of Energy-Phase Motion with RF E. Pozdeyev, Electron Linacs

  29. Energy-Phase Motion with RF with Arbitrary Amplitudes E. Pozdeyev, Electron Linacs

  30. Homework Problem 3 E. Pozdeyev, Electron Linacs

  31. To be continued E. Pozdeyev, Electron Linacs

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