Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 1 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Mike Forster 11 December 2006 Focusing and Bending Or How to Build Your Own Storage Ring

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 2 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Lorentz Force Equation Wikipedia.org

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 3 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Make a Ring Electrostatic plates or Dipoles? B E

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 4 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Magnet Choices Permanent Magnets –Fixed field –Ferrites typically up to 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

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 5 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Beam Coordinates x y s  Reference Particle Trajectory Actual Particle Trajectory

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 6 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Describing the Motion

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 7 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Multipole Expansion Dipole Quadrupole Sextupole Octupole The field around the beam can be seen as a sum of multipoles.

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 8 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster 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

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 9 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster 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

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 10 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Quadrupoles Hyperbolic shaped poles Focusing in one transverse plane, defocusing in the other From J. Crittenden

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 11 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster From circe.lnl.gov Good field region

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 12 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Trajectory Equations First assume on energy particles:  p/p = 0. Look at horizontal motion in a horizontally defocusing quad (k > 0). Solutions are: Use x 0 and x’ 0 as initial conditions for the constants of integration.

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 13 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster 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!

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 14 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Trajectory Tracking Through a section of elements: –X 1 = M drift ·M QF ·M drift ·M QD ·X 0 Or build up to a complete revolution of the ring Combine transverse planes

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 15 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster 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)

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 16 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster 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:

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 17 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster 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.

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 18 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Courant-Snyder Parameters or Twiss Functions These functions combine to describe an ellipse in x-x’ phase space:

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 19 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster 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.) x’ x

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 20 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster From D. Robin USPAS lecture

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 21 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster 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.

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 22 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster 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

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 23 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Periodicity FODO Cells Many large accelerators combine magnet power supplies Independent magnet control at CESR Flexibility has contributed to long term viability

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 24 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster 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

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 25 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Resonances Linear Resonances (Integer and half integer) Non-linear resonances –Coupling Resonance –Synchro-Betatron Resonance Tune Plane From A. Temnykh talk at Frascati 2005

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 26 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Chromatic Errors Focusing errors due to energy differences From D. Robin USPAS lecture

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 27 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Chromatic Errors Correct with sextupoles From D. Robin USPAS lecture

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 28 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 29 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Insertion Regions Match twiss conditions at ends and insert: RF Cavities Wigglers Separators Transfer Lines Detector

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 30 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Interaction Point Focusing SCIR quads Permanent magnet quads

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 31 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster One beam will not make luminosity Horizontal Separators Horizontal tune of 10 Pretzel Orbit Lab ingenuity J. Crittenden 2004

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 32 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster 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

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 33 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Cesr Optics

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 34 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster Measurement and Correction

Wilson Lab Tour Guide Orientation 11 December 2006 CLASSE 35 Focusing and Bending Wilson Lab Tour Guide Orientation M. Forster See the real stuff!