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Focusing and Bending Or How to Build Your Own Storage Ring

Focusing and Bending Or How to Build Your Own Storage Ring

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Focusing and Bending Or How to Build Your Own Storage Ring

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  1. Mike Forster 11 December 2006 Focusing and Bending Or How to Build Your Own Storage Ring

  2. Lorentz Force Equation

  3. Make a Ring Electrostatic plates or Dipoles? B E

  4. Magnet Choices • Permanent Magnets • Fixed field • Ferrites typically up to .05-.10 T • Rare earth alloys typically up to 1.2 T or more • Iron-Copper Electromagnets • Copper coils around iron laminations • Superconducting Magnets • Cryogenically cooled • Conventional or Superferric

  5. Beam Coordinates y Actual Particle Trajectory r Reference Particle Trajectory x s

  6. Describing the Motion

  7. Multipole Expansion The field around the beam can be seen as a sum of multipoles. Dipole Sextupole Octupole Quadrupole

  8. Focusing Methods • Dipole Beam Loss • Horizontally stable • Vertical unstable • Weak Focusing • n = - (dB/B)/(dr/r) • Stability in both planes requires 0<n<1 • Gains vertical focusing at the expense of horizontal • Combined Function Magnets • More than one multipole term • Focusing to Bending Ratio is fixed B

  9. Focusing Methods • Strong or Alternate Gradient Focusing (Christofilos 1950, Courant et al. 1952) • First used operationally at Cornell in 1954 • Alternating combined function magnets "If you can't do two things together, you just do one after the other - that's all there is to it!" Ernest Courant, Brookhaven

  10. Quadrupoles • Hyperbolic shaped poles • Focusing in one transverse plane, defocusing in the other From J. Crittenden

  11. From Good field region

  12. Trajectory Equations First assume on energy particles: Dp/p = 0. Look at horizontal motion in a horizontally defocusing quad (k > 0). Solutions are: Use x0 and x’0 as initial conditions for the constants of integration.

  13. Particle Trajectories and Transfer Matrices Which can be put into matrix notation as: M M is the transfer matrix for a defocusing quadrupole. Build up a toolbox of transfer matrices for various elements!

  14. Trajectory Tracking • Through a section of elements: • X1 = Mdrift·MQF·Mdrift·MQD·X0 • Or build up to a complete revolution of the ring • Combine transverse planes

  15. Dispersion • Realistically p/p  0. • Only significant when 1/R  0, i.e. in a bend. • Solve for a special trajectory, (s) and ’(s) , when p/p = 1. • A particle with momentum offset p/p will have a horizontal position of x(s) + (s) (p/p) • Likewise, the angle will be: x’(s) + ’(s) (p/p)

  16. Beta Functions and Betatron Oscillations Need to describe a beam of particles. Defines motion of transverse oscillation about the orbit called the betatron oscillation. Trial solution: x(s) = A u(s) cos( (s) +  ) After constants of integration are worked out:

  17. Beta Functions and Betatron Oscillations (s) is the beta function or amplitude function.  is the emittance. (s) is the phase. The square root of (s) defines transverse size of beam at any point s.

  18. Courant-Snyder Parameters or Twiss Functions These functions combine to describe an ellipse in x-x’ phase space:

  19. x’ x Beam Phase Space • Particles can be characterized by their position, x, and angle, x’ about the reference orbit. • The area of the ellipse remains constant and equals the emittance. (Satisfies Liouville’s Theorem.)

  20. From D. Robin USPAS lecture

  21. Beam cross section • Electrons and positrons are well approximated by a Gaussian charge distribution: • The standard is to use the phase ellipse of the particle at 1 to define the beam emittance. • Generally look for 10 clearance for good beam lifetime.

  22. Phase Space Propagation • Twiss parameters can also be propagated through optics elements with matrices. • Magnet focusing strengths are set to achieve the desired twiss properties at arbitrary points in the ring. • Typically done in “cells” like FODO cells: • Focusing Quad, drift or bend, Defocusing Quad, drift or bend • Designed so Twiss functions at the end of the cell match the beginning From J.Rossbach CERN Accelerator School lecture

  23. Periodicity • FODO Cells • Many large accelerators combine magnet power supplies • Independent magnet control at CESR • Flexibility has contributed to long term viability

  24. Tunes • The amount of phase advance through one complete revolution is call the Tune. • Integer tunes are trouble. • In fact, a lot of other tunes are trouble: m Qx + n Qy = p Where m, n, p are integers. • These are optical resonances • |m| + |n| is the order of the resonance

  25. Resonances • Linear Resonances (Integer and half integer) • Non-linear resonances • Coupling Resonance • Synchro-Betatron Resonance • Tune Plane From A. Temnykh talk at Frascati 2005

  26. Chromatic Errors • Focusing errors due to energy differences From D. Robin USPAS lecture

  27. Chromatic Errors • Correct with sextupoles From D. Robin USPAS lecture

  28. Insertion Regions Match twiss conditions at ends and insert: • RF Cavities • Wigglers • Separators • Transfer Lines • Detector

  29. Interaction Point Focusing • SCIR quads • Permanent magnet quads

  30. One beam will not make luminosity • Horizontal Separators • Horizontal tune of 10 • Pretzel Orbit • Lab ingenuity J. Crittenden 2004

  31. Detector Solenoid and Compensation • Solenoid coupling • Compensation with: tilted SCIR quads • Fixed 4.5 degree rotation • SCIR and normal conducting skew quads • Superconducting Anti-solenoids • Work to reduce any coupling between horizontal and vertical motion

  32. Cesr Optics

  33. Measurement and Correction

  34. See the real stuff!