1. Spectrometer Optics John J. LeRose

2. The Basics Charged particles moving through static magnetic fields.  Magnetic Rigidity Local radius of curvature Bend Angle

3. References: K.L. Brown, D.C. Carey, C. Iselin and F. Rothacker, Designing Charged Particle Beam Transport Systems, CERN 80-04 (1980) K.L. Brown, SLAC Report-75 (http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-r-075.pdf) …... TRANSPORT formalism x y  Reference Trajectory Arbitrary Trajectory Magnetic Midplane

4. All trajectories are characterized by their difference from a reference trajectory* x z y z *”The Central Trajectory” z x x x l = length difference between trajectory and the reference trajectory

5. General Solution of the equation of motion: Each component can be expressed as a Taylor series around the Central Ray:

6. The first order transfer matrix: For static magnetic systems with midplane symmetry:

7. Generalized Transfer Tensor and its inverse:

8. Calibrations for normal running In general one wants to determine the tensor elements, D ijkl, T ijkl, P ijkl, Y ijkl Start from the last best known values –Previous run –Calculated from SNAKE output (new tunes) Use your favorite polynomial fitting routine Collect calibration data: –The extent of the calibration data taken depends on how well you need to measure things –Elastic scattering with and without sieve Delta scans “Optics” target (segmented in y 0 ) Optimize the tensors Pointing survey is also a good idea

9. Sample Sieve Slit Spectrum

10. 10 12 C(e,e’) @ 6° and 2 GeV 10 -4 (FWHM) all peaks

12. Hall A workshop December 10,2002 Some examples:

13. Hall A workshop December 10,2002 Matrix for a string of elements is the product of the individual matrices: Drift L 1 Quad L Drift L 2

14. Hall A workshop December 10,2002 1st order resolving power: Parallel to point focus: Point to point focus:

15. Hall A workshop December 10,2002 Focal Plane: 

16. Hall A workshop December 10,2002 Multipoles: Multipole strength per unit length: Multipole strength of entire magnet: Dipoles: Sextupole: Quadrupole:     =1/h R1R1 R2R2 x x

17. Hall A workshop December 10,2002 The equation of motion:

18. Hall A workshop December 10,2002 The solution: Coupling to magnetic elements (k n ) can be determined by differentiating: 1st order terms drive the higher order terms. Higher order field components don’t contribute to lower order matrix elements.

19. Hall A workshop December 10,2002 Going back to focal plane rotation:

20. Hall A workshop December 10,2002 Tools: TRANSPORT: Goes up to 3rd order. Multiplies successive matrices (tensors). Allows the user to select magnetic parameters to be varied and properties to be achieved. Good place to start. COSY: Goes up to arbitrary order. (not sure what this means) Evaluates the transfer tensor for each element. Multiplies successive matrices (tensors). (I think) Very powerful, but difficult. The novice user should be very careful. (My opinion!) Raytracing Codes (SNAKE, RAYTRACE, and GEANT): Integrates the equation of motion particle by particle through the magnetic system Should be as good as the magnetic field description. Slow (but with today’s computing power not really a limitation anymore). Matrices/Tensors are inferred from the results of the tracings. (e.g. fit with MUDIFI)