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The Arc Method

The arc method is a simple way to calculate the path of a charged particle in an accelerator, that is to track the particle.

At present, we have only implemented this method for tracking through a magnetic field. The idea is that a charged particle moving perpendicular to a uniform magnetic field travels in a perfectly circular arc. In a nonuniform field, the magnetic field would still be approximately constant over a short distance. So by joining many tiny arcs together, we can plot the path of the particle.

In general, the particle can move at any angle to the magnetic field. The full description of the method is given in the following paper:

K. M. Hock and C. S. Edmonds, "Tracking protons in an FFAG and a synchrotron with tiny arcs," Nuclear Instruments and Methods A, vol. 664, (2012), pp. 140-147.

We have implemented this using Octave. The functions needed to computed the trajectory through a quadrupole field can be downloaded here: arc_function.zip. The main function is described here: arc_function_guide.pdf.

To use these codes, we suggest the steps below. These steps are based on the lattice for the EMMA accelerator, in which the only magnets are quadrupoles. You can find a desciption of EMMA here:

R. Edgecock, et al, "The EMMA Non-scaling FFAG," Proceedings of IPAC’10, Kyoto, Japan (2010).

and details of the magnets here:

The EMMA Lattice or here.

EMMA Baseline Parameters or here.

The steps are as follows:

1. Visualise the lattice

• Specify the position of a quadrupole.
• Make up a rectangular outline to represent a view of the quadrupole from above. Compute the coordinates of this outline. The size can be approximate, but the entrance and exit faces must be accurate.
• Plot the outline coordinates to visualise the quadrupole.
• Write a code to put quadrupole outline at any position.
• Write a code to generate the positions of all the quadrupoles in EMMA.
• Plot the outline of all the quadrupoles to visualise EMMA.

2. Track an electron

• Read the arc method paper.
• Use the arc function to track an electron through a single quadrupole.
• Repeat with different initial conditions to see the effect of the quadrupole.
• Do this for the focussing and defocusing quadrupoles separately.
• Do it for with two quadrupoles together.
• Do it for the EMMA lattice.