Electron Rings Eduard Pozdeyev This material is based upon work supported by the U.S. Department of Energy Office of Science under Cooperative Agreement DE-SC0000661, the State of Michigan and Michigan State University. Michigan State University designs and establishes FRIB as a DOE Office of Science National User Facility in support of the mission of the Office of Nuclear Physics. 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, Slide 2
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, Slide 3
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, Slide 4 Introduction:
Electron-Electron, Electron-Positron Colliders VEP-1, 1963 Russia, Novosibirsk Particles: electron electron Collision energy: 160 MeV Luminosity: 1028 1/(cm2s) Rings size: two 1m x 1m Large ElectronPositron Collider (LEP) Operational: 1989 - 2000
Tunnel was used for LHC after LEP was decommissioned Particles: electron positron Collision energy: 100 GeV Luminosity: 1032 1/(cm2s) Circumference: 27 km E. Pozdeyev, Electron Linacs, Slide 5 Introduction: Light Sources
Main 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, Slide 6 Transverse Motion in Electron Rings E. Pozdeyev, Electron Linacs, Slide 7 Simple Electron Ring Lattice and 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) Sx Qx Dipole Qy Sy
E. Pozdeyev, Electron Linacs, Slide 8 Equations of Motion and Hill Equation - small deviations from the reference particle s is the independent variable instead of t (s=v*t) 0 - Dipole magnet with gradient focusing = n is the field index
( ) - Quadrupole - Drift - General Hill equation with periodic focusing
E. Pozdeyev, Electron Linacs, Slide 9 0 Example: Weak Focusing Azimuthally Symmetric Field with Gradient =
Solution easily obtainable Current is phase space density times area 1. Increase density 2. Increase aperture 3. Increase focusing Increasing focusing in both planes is Impossible. Need other focusing
Mechanism (strong focusing) E. Pozdeyev, Electron Linacs, Slide 10 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, Slide 11 Linear Betatron Motion Linear motion can be described by vectors and matrices E. Pozdeyev, Electron Linacs, Slide 12 Stability of Betatron Motion 
- 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, Slide 13 Stability of Betatron Motion 
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, Slide 14 Twiss Parametrization Twiss parametrization Because det(M)=1
2+ =1 E. Pozdeyev, Electron Linacs, Slide 15 Evolution of Particle Coordinates at Specific Location s is the betatron phase advance per turn E. Pozdeyev, Electron Linacs, Slide 16
Courant-Snyder Ellipse At Specific Location s E. Pozdeyev, Electron Linacs, Slide 17 Particle Motion Along Accelerator Equations for and E. Pozdeyev, Electron Linacs, Slide 18
Particle Motion Along Accelerator Phase Advance - Betatron tune E. Pozdeyev, Electron Linacs, Slide 19 Example Small 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, Slide 20 Homework Problem 1 E. Pozdeyev, Electron Linacs, Slide 21
Motion of Particle With Energy Deviation Search for solution in this form Equation for dispersion function E. Pozdeyev, Electron Linacs, Slide 22 Azimuthally Symmetric Case
E. Pozdeyev, Electron Linacs, Slide 23 Energy-Phase Motion with RF  - sine dependence! E. Pozdeyev, Electron Linacs, Slide 24 Energy-Phase Motion with RF  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, Slide 25
Small Oscillations For small deviations in phase from the synchronous phase 1 Equations of energy-phase motion for small amplitudes - synchrotron frequency
- normalized synchrotron tune E. Pozdeyev, Electron Linacs, Slide 26 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, Slide 27
Hamiltonian of Energy-Phase Motion with RF E. Pozdeyev, Electron Linacs, Slide 28 Energy-Phase Motion with RF with Arbitrary Amplitudes E. Pozdeyev, Electron Linacs, Slide 29
Homework Problem 3 E. Pozdeyev, Electron Linacs, Slide 30 To be continued E. Pozdeyev, Electron Linacs, Slide 31
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