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Mathieu Regime

  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. gif, 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.gif 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).

Wolfgang Christian
Fri May 12 10:36:01 EDT 1995