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## Focusing and Bending Or How to Build Your Own Storage Ring

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**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