An application: The rf Paul Ion Trap

The rf Paul ion trap, a configuration of hyperbolic electrodes, and similar applications, have been extensively studied (see the report on the rf Paul ion trap for more details). For a much more professional relaxation calculation for such traps made using hyperbolic electrodes, the reader is encouraged to study Gaerald Gabrielse's paper [5].

We set our algorithm to create a pseudo cross section of an rf-Paul trap. The upper electrodes are held at -1 Volt, and the middle ring is held at 1 Volt. We proceed with an SOR iteration with a tolerance of 0.0001 (defining the maximal difference between iterations of any point on the grid) of a 1000 by 1000 grid. Our results are presented below. As expected, an unstable equilibrium is found at the center. It took 1318 iterations, and though we lacked the patience to give the Jacobi method a chance to run this scenario, we suspect it would have taken much much longer.

Figure 5.1: The pseudo rf-Paul trap, as modeled using SOR