Mike Forster 11 December 2006 Focusing and Bending Or How to Build Your Own Storage Ring
Lorentz Force Equation Wikipedia.org
Make a Ring Electrostatic plates or Dipoles? B E
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
Beam Coordinates y Actual Particle Trajectory r Reference Particle Trajectory x s
Multipole Expansion The field around the beam can be seen as a sum of multipoles. Dipole Sextupole Octupole Quadrupole
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
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
Quadrupoles • Hyperbolic shaped poles • Focusing in one transverse plane, defocusing in the other From J. Crittenden
From circe.lnl.gov Good field region
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.
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!
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
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)
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:
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.
Courant-Snyder Parameters or Twiss Functions These functions combine to describe an ellipse in x-x’ phase space:
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.)
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.
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
Periodicity • FODO Cells • Many large accelerators combine magnet power supplies • Independent magnet control at CESR • Flexibility has contributed to long term viability
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
Resonances • Linear Resonances (Integer and half integer) • Non-linear resonances • Coupling Resonance • Synchro-Betatron Resonance • Tune Plane From A. Temnykh talk at Frascati 2005
Chromatic Errors • Focusing errors due to energy differences From D. Robin USPAS lecture
Chromatic Errors • Correct with sextupoles From D. Robin USPAS lecture
Insertion Regions Match twiss conditions at ends and insert: • RF Cavities • Wigglers • Separators • Transfer Lines • Detector
Interaction Point Focusing • SCIR quads • Permanent magnet quads
One beam will not make luminosity • Horizontal Separators • Horizontal tune of 10 • Pretzel Orbit • Lab ingenuity J. Crittenden 2004
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
See the real stuff! http://www.lns.cornell.edu/~hoff/LECTURES/07S_488/index.html