Laboratory experiments with ion traps require expensive equipment and highly sophisticated detection techniques [12]. 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 [8], a simulation of 100-1000 ions in a Penning trap [5], and an exploration of parameters for which an ion's motion can be modeled by the first approximation of the Mathieu differential equation [3]. 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 [3]. 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 [11]. However, in the stable domain for
which
and
are not
sufficiently small, perturbative solutions are no longer valid [3].
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.