Laboratory experiments with ion traps require expensive equipment and highly sophisticated detection techniques . An accurate, fast numerical simulation would provide a useful supplement to laboratory observation of ion trapping. Prior research on ion trapping simulations includes a comparison of order/chaos transitions in computer simulation of two trapped ions with experimental results , a simulation of 100-1000 ions in a Penning trap , and an exploration of parameters for which an ion's motion can be modeled by the first approximation of the Mathieu differential equation . The general form of the Mathieu differential equation is
The Mathieu equation has both stable and unstable solutions as a function of and , corresponding to trapping and escaping ions . When the parameters and are sufficiently small, the stable periodic motion can be represented by the first approximation of the solution to the Mathieu differential equation . However, in the stable domain for which and are not sufficiently small, perturbative solutions are no longer valid .
The proposed work will explore the computational power of High Performance Fortran (HPF) on a parallel machine to model behavior of a large number of ions and to explore parameter realms for which perturbative mathematical solutions are not valid.