In the Mathieu Regime, the interparticle spacing is large and
the Coulomb interactions are negligible. Large distances between particles
imply large displacements from the center of the trap; large displacements from the center of the trap translate into a large amount of potential energy. Consequently, the Mathieu
regime is a high-energy regime where damping--- which would decrease the particles' distances from the trap center and from each other, rendering
nonlinear Coulomb terms significant--- is minimal. Thus, the equations of motion for **N** ions in the Mathieu regime are well-approximated by **N**
uncoupled, undamped equations of the form Eq. , which, as
discussed earlier, can be understood in terms of the Mathieu equation.

The hallmarks of the Mathieu regime are demonstrated by the configuration of file ` MReg5A.trp`, which depicts **5** undamped Mg ions Paul-trapped in
stability region A. The temperature is derived from the average kinetic energy:

where **k** is Boltzmann's constant and is the time-averaged kinetic energy. For this simulation, the temperature is roughly **23866** K. With an average interparticle separation of m, the Coulomb interaction between particles is negligible. The simulation of ` MReg5A.trp` demonstrates that a
collection of repulsive ions can be confined in a Paul-trap * without* damping, since there is no gain of kinetic energy or increase in root-mean-square distance from the trap center (either mechanism would eventually ``boost'' ions out of the potential well).

Fri May 12 10:36:01 EDT 1995