Tracking with tiny arcs

1. Tracking with tiny arcs Kai Hock, Chris Edmonds University of Liverpool and Cockcroft Institute FFAG11 Workshop, 11 – 16 September 2011

2. Overview • Description of the arc method • Quick comparison with Zgoubi • Trying out on a spiral FFAG • Conclusions

3. Description of the arc method • When the velocity of a charged particle is perpendicular a uniform magnetic field, the path is a circular arc. • By joining many tiny arcs together, the path through a nonuniform field can be obtained. • This method has been used in the 1980s in PSI to track protons in cyclotrons. The code developed was called TRACK.

4. Exact solution in a uniform field • In a uniform field, the result is exact. • So the computation time needed is negligible. • In a nonuniform field, the arcs must be short enough.

5. Quick comparison with Zgoubi • In a nonuniform field, a tiny arc is roughly the same as a second order term in a Taylor expansion to calculate the path. • Since Zgoubi uses higher order terms, we would expect that the required steps in the arc method are much shorter. • So it is of practical interest to compare the computation time for the two methods. • We made a quick comparision using a 10 MeV proton through a FODO lattice.

6. A FODO lattice with 100 cells • The thickened parts of the trajectories show where the quadrupoles are. • When the arc length is less than 1/10 the step size in Zgoubi, the trajectories look the same. • For comparison, we define the error as the deviation from the correct answer of the y value for the endpoint of the calculated trajectory.

7. Comparing computation times • Since we do not know the correct answer, we take this as the average of the endpoints of the two methods for the smallest step sizes used. • If we need to track for many turns, much shorter arcs would be needed, and the arc method would be slower than Zgoubi.

8. Trying out on a spiral FFAG S. Antoine et al, NIM A (2009) • To try out the arc method, we used the spiral FFAG lattice of the RACCAM project. • As we do not have the detailed field maps, the objective is to develop the codes and to get approximate agreements with published results.

9. Field in the spiral magnet Field measured in the RACCAM prototype magnet Simple model that we used S. Antoine et al, NIM A (2009) • The vertical magnetic field with varies with radial distance r from the ring centre as r5. • For the azimuthal variation in each magnet, we used the simple model shown above on the right.

10. Closed orbits in the spiral FFAG Orbits from the arc method Published orbits, from Zgoubi S. Antoine et al, NIM A (2009)

11. Closed orbits in the spiral FFAG Apertures from the arc method Published apertures, from Zgoubi S. Antoine et al, NIM A (2009) • The shapes and positions of the phase trajectories are sensitive to the location in the lattice. • The size and positions of the apertures are not far off, but it would be nice to get them to agree exactly using the actual field map.

12. Conclusion • The arc method is simple to implement. We have written the codes in C. • It would be slower than Zgoubi if we need to track for many turns. • The phase space trajectory in a small lattice like the spiral FFAG takes a few minutes to compute. • If the full power of Zgoubi is not needed, the arc method could be a useful alternative.