Device-dependent Equations of Motion of Charged Fine Particles in an AC Trap

Device-dependent equations of motion for charged fine particles in an AC trap, constructed as a simplified implementation of the Paul trap, were derived within the framework of classical dynamics. Two characteristic length scales are relevant to the system: the device-dependent scale set by the radius of the charged ring and the characteristic collective length associated with Coulomb interactions. When the collective Coulomb length scale is sufficiently small compared with the ring radius, the Mathieu–Coulomb equation, obtained from the quadratic approximation to the trapping potential near the trap center, provides an accurate description of the particle dynamics. Numerical solutions of the device-dependent equations were compared with those of the Mathieu–Coulomb equation. For the five- and fifty-particle cases examined here, the two descriptions show good agreement and reproduce regular or chaotic motion and lattice configurations. For the ten-thousand-particle cases examined here, substantial differences appear when the particle distribution extends beyond the near-center region, and particle loss is observed in the device-dependent dynamics. To isolate the dependence on the Coulomb-interaction parameter β, we additionally investigated a two-particle system while keeping the other parameters fixed. The numerical results show the scaling R 0 ∝ β for the mean interparticle distance R 0 in the Mathieu–Coulomb equation, whereas the full device-dependent dynamics deviate from this simple scaling as β increases. A first-order harmonic-balance analysis reproduces the R 0 ∝ β scaling of the Mathieu–Coulomb equation and shows how the leading quartic correction breaks this scaling.