Micromotion Scaling

When the micromotion scaling option is selected, TrapApp sets the time step to a fraction of the period of the ``fastest'' motion associated with the configuration:

where c is a proportionality constant, is the period of the micromotion, and is the (angular) frequency of the micromotion. In Paul trap simulations, the frequency of the micromotion is taken as the frequency of the oscillating electric potential. In Penning trap simulations, the cyclotron motion, whose unperturbed frequency is given by Eq. , is interpreted as the micromotion.

Although one needs only a few time steps per oscillation for stable finite-difference propagation,[20] TrapApp chooses a conservatively small time step because:

1. The fastest motion is not always obvious or even well-defined, as in the case of region B Paul-trapping simulations for which higher-order harmonics of the oscillating electric potential's frequency are excited.
2. Coulomb interaction and other interparticle interactions may produce accelerations greater than those associated with the ``fast'' motion induced by the trap.

Recall that simulations of the Lennard-Jones interaction use a different system of reduced units than do non-Lennard-Jones simulations. The length unit, , is set to the equilibrium distance, i.e., the particle spacing for which the potential energy is minimized. The energy unit, , is set to the depth of the potential well, where the zero of potential energy is taken to be at infinite separation.

This system of units simplifies the task of finding an appropriate time step, since the reduced unit conversion factor for time is derivable in terms of system-dependent quantities:

Given this system of reduced units, a generally acceptable value for dt, according to Haile,[19] is . This time step corresponds to s for an argon simulation ( meV, Angströms).