1. UKNF WP1 meeting, 12/11/08 Zgoubi development David Kelliher ASTeC/STFC/RAL

2. Contents Zgoubi introduction (2) Python interface (4) Fringe field studies in MULTIPOL (6)

3. UKNF WP1 meeting, 12/11/08 Zgoubi Zgoubi is a ray-tracing code that computes particle trajectories in arbitrary magnetic and/or electric field maps or analytical models. The code is a compendium of numerical recipes for simulation of mode optical elements encountered in beam optics Uses an integration method founded on stepwise resolution of the Lorentz equation using a technique based on Taylor series C.R. Prior, “Start to end simulation of a Neutrino Factory”, IDS-NF-003

4. UKNF WP1 meeting, 12/11/08 Integration method Use truncated Taylor series for position and normalised velocity Calculating R to the sixth order requires 4 th derivative of B(s) Obtain velocity terms via Lorentz equation

5. Python interface

6. UKNF WP1 meeting, 12/11/08 Running Zgoubi zgoubi.datRun zgoubizgoubi.res At present running Zgoubi can be tedious as much work needs to be done “by hand”. Two interfaces being developed  C++/Root (S. Sheehy/R. Fenning)  Python (S. Tygier)

7. UKNF WP1 meeting, 12/11/08 Input file emma 'OBJET' -3.501946e+01 2 1 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 1.000000e+00 ' ' 1 'PARTICUL' 5.109989e-01 -1.602177e-19 1.159652e-03 0.000000e+00 0 'DRIFT' ld 1.050000e+01 'CHANGREF' 0.000000e+00 0.000000e+00 8.571429e+00 'CHANGREF' 0.000000e+00 -3.404800e+00 0.000000e+00 'QUADRUPO' defoc 0 7.569900e+00 5.300000e+00 2.493120e+00 0.000000e+00 0 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 #20|20|20 0 0.000000e+00 0.000000e+00 0.000000e+00 'CHANGREF' 0.000000e+00 3.404800e+00 0.000000e+00 'DRIFT' sd 5.000000e+00 'CHANGREF' 0.000000e+00 -7.514000e-01 0.000000e+00 'QUADRUPO' foc 0 5.878200e+00 3.700000e+00 -2.477150e+00 0.000000e+00 0 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 #20|20|20 0 0.000000e+00 0.000000e+00 0.000000e+00 'CHANGREF' 0.000000e+00 7.514000e-01 0.000000e+00 'END'

8. UKNF WP1 meeting, 12/11/08 Python interface Zgoubi input file created and may be easily modified. Elements such as QUADRUPO and MULTIPOL included. Still working on DIPOLES and FFAG. A utility to find the closed orbit, the tune (from the MATRIX result) and the dynamic aperture have been written. Utilities currently being developed - calculate dynamic aperture over tune space - add magnet misalignments and check effect on dynamic aperture

9. UKNF WP1 meeting, 12/11/08 EMMA example ob = OBJET2()‏ emma.add(ob)‏ emma.add(ELECTRON())‏ emma.add(DRIFT('ld', XL=ld*cm_/2))‏ emma.add(CHANGREF(ALE=angle))‏ emma.add(CHANGREF(YCE=d_offset*cm_))‏ emma.add(QUADRUPO('defoc', XL=dq*cm_, R_0=dr*cm_, B_0=db*kgauss_, XPAS=xpas))‏ emma.add(CHANGREF(YCE=-d_offset*cm_))‏ emma.add(DRIFT('sd', XL=sd*cm_))‏ emma.add(CHANGREF(YCE=f_offset*cm_))‏ emma.add(QUADRUPO('foc', XL=fq*cm_, R_0=fr*cm_, B_0=fb*kgauss_, XPAS=xpas))‏ emma.add(CHANGREF(YCE=-f_offset*cm_))‏ emma.add(DRIFT('ld', XL=ld*cm_/2))‏ emma.add(FAISCNL(FNAME='zgoubi.fai'))‏ emma.add(REBELOTE(K=99, NPASS=10))‏ emma.add(END())‏

10. Fringe field studies

11. Zgoubi fringe fields - Multipol MULTIPOL Uses the multipole fringe field with Enge fall off as described earlier. A rectangular type magnet is assumed. Four terms in multipole expansion are included. e.g. Vertical dipole field expansion

12. Zgoubi fringe fields – DIPOLES DIPOLE/DIPOLE-M/DIPOLES Provides a model of a dipole field in a radial magnet geometry. Terms up to octupole can be included. Multipole expansion is ignored – instead a smooth Enge fall off is assumed at every position. In polar coordinates

13. Dipole fringe field in MULTIPOL Assume 1m MULTIPOL magnet with 1T dipole field only. The fringe field fall off extent λ is set to 20cm. Calculate vertical fringe field in Zgoubi by tracking a particle through the magnet at various horizontal locations (5cm, 10cm, 20cm) from that magnetic axis. (Switch off bending)

14. Mathematica study of MULTIPOL fringe field Replicate result in Mathematica. First four terms in dipole fringe field included. Plot B z longitudinally at 5cm, 10cm and 20cm. Fringe field extent is 20cm

15. MULTIPOL fringe field details 5 cm10 cm20 cm

16. UKNF WP1 meeting, 12/11/08 MULTIPOL summary There is some radius inside which the multipole equations converge. Beyond that strong oscillations in the fringe field appear. Create a new “MULTIPOL” object – a rectangular magnet with arbitrary number of multipoles, each of which fall off smoothly at all radii, just as in the case of radial magnets Rather than using analytic equations directly, instead create “ideal” field map in which to track (JAI).

17. UKNF WP1 meeting, 12/11/08 Conclusions Python interface to Zgoubi being developed. The interface will facilitate the use of Zgoubi in ongoing studies of muon acceleration. The problem of fringe field oscillations in the Zgoubi description of rectangular magnets is being tackled.