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

PAMELA Closed orbit and Twiss parameters for a 100 MeV proton, computed using a Matlab script based on equations from the Zgoubi manual. These must be obtained using Zgoubi first, before moving on to dynamic aperture calculation.

L = m L cell Packing factor,  = 0.48 No. of cells, N cell = 12 r 0 = m   N cell FFD L L L L L Lattice parameters Triplet length = 5L =  L cell 1 cell r0r0 Obtained from PAMELA papers.

  N cell FFD Suggested reference path for Zgoubi r0r0 New cell New reference Likely to be closer to closed orbit because of symmetry. Start closed orbit search here.

 FFD Lattice parameters needed for Zgoubi r0r0 d1d1 d2d2  r1r1     d2d2 d 1 = 5L/2  1 = tan -1 (d 1 /r 0 ) r 1 = r 0 /cos  1  2 =  /N cell –  1 d 2 = r 1 cos  2

Magnet parameters Along radial direction in each magnet, B y = B 0 (r/r 0 ) k. k = 38 B 0 = 1.67 T for F magnet B 0 = T for D magnet r = r 0 +x

Field created by multipole expansion Taylor expand about r=r 0 : To obtain B x, replace each term by multipole: Check that real part agrees with previous equation for B y. This works because each multipole term satisfies Maxwell’s equations. Since x, y << r 0, it may be possible to truncate the series. This graph compares N = 3 with the actual field.

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 B n : the n th order multipole field is given by: Consider a pole tip on the x axis at distance R 0 from reference path. This pole tip is at x=R 0, y=0. So at the pole tip, the field is:

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

References PAMELA Design Report S. Sheehy, et al, “PAMELA: LATTICE DESIGN AND PERFORMANCE”, Proceedings of PAC09, Vancouver, BC, Canada H. Witte, et al, “PAMELA MAGNETS - DESIGN AND PERFORMANCE”, Proceedings of PAC09, Vancouver, BC, Canada S. Sheehy, Design of a Non-Scaling Fixed Field Alternating Gradient Accelerator for Charged Particle Therapy, PhD Thesis, Oxford,