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Parameter Derivation for Zgoubi Simulation in PAMELA Lattice and Magnets

Deriving parameters for Zgoubi in PAMELA using Matlab script based on Zgoubi manual equations for closed orbit, Twiss parameters, lattice, and magnet parameters. Calculations include cell packing factor, triplet length, cell count, and magnet field calculations. Detailed steps for obtaining Zgoubi parameters are provided, along with suggested references and magnet design considerations.

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Parameter Derivation for Zgoubi Simulation in PAMELA Lattice and Magnets

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  1. PAMELA lattice and magnets - Deriving the parameters needed for Zgoubi Kai Hock 28/4/13

  2. PAMELA Closed orbit and Twissparameters for a 100 MeVproton, computed using a Matlab script based on equations from the Zgoubimanual. These must be obtained using Zgoubi first, before moving on to dynamic aperture calculation.

  3. Lattice parameters 1 cell Packing factor, a = 0.48 No. of cells, Ncell = 12 r0 = 6.251 m Obtained from PAMELA papers. Lcell L L L L = 0.314 m L L Triplet length = 5L = aLcell F D F r0 q q = 2p/Ncell

  4. Suggested reference path for Zgoubi Likely to be closer to closed orbit because of symmetry. Start closed orbit search here. New cell F D F New reference r0 q q = 2p/Ncell

  5. Lattice parameters needed for Zgoubi d1 = 5L/2 q1 = tan-1(d1/r0) r1 = r0/cosq1 q2 = p/Ncell – q1 d2 = r1cos q2 d1 q1+q2 q1+q2 F D F d2 d2 r1 r0 q2 q1

  6. Magnet parameters k = 38 B0 = 1.67 T for F magnet B0 = -2.44 T for D magnet r = r0+x Along radial direction in each magnet, By = B0(r/r0)k.

  7. Field created by multipole expansion Taylor expand about r=r0: To obtain Bx, replace each term by multipole: Check that real part agrees with previous equation for By. This works because each multipole term satisfies Maxwell’s equations. Since x, y << r0, it may be possible to truncate the series. This graph compares N = 3 with the actual field.

  8. Magnet parameters for Zgoubi Zgoubi requires the magnetic field (magnitude) at pole tip. To find this, we first write down an expression for a multipole term. Comparing with a sum of multipole fields Bn: the nth order multipole field is given by: Consider a pole tip on the x axis at distance R0 from reference path. This pole tip is at x=R0, y=0. So at the pole tip, the field is:

  9. For future comparison: These results are obtained using a multipole expansion up to N=3, and for a 100 MeV proton.

  10. References PAMELA Design Report http://www.hep.manchester.ac.uk/u/hywel/papers/proton/PamelaPDR.pdf S. Sheehy, et al, “PAMELA: LATTICE DESIGN AND PERFORMANCE”, Proceedings of PAC09, Vancouver, BC, Canada http://accelconf.web.cern.ch/AccelConf/PAC2009/papers/fr5pfp001.pdf H. Witte, et al, “PAMELA MAGNETS - DESIGN AND PERFORMANCE”, Proceedings of PAC09, Vancouver, BC, Canada http://accelconf.web.cern.ch/AccelConf/PAC2009/papers/mo6pfp073.pdf S. Sheehy, Design of a Non-Scaling Fixed Field Alternating Gradient Accelerator for Charged Particle Therapy, PhD Thesis, Oxford, 2010. http://ora.ox.ac.uk/objects/uuid:d9cd977c-35db-45cc-ad33-67710fc3e82f

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