We examine the motions of particles in quadrupole ion traps as a function of
damping and trapping forces, including cases where nonlinear damping or
nonlinearities in the electric field geometry play significant roles. In the
absence of nonlinearities, particles are either damped to the trap center or
ejected, while their addition brings about a rich spectrum of stable closed
particle trajectories. In three-dimensional (3D) quadrupole traps, the extended
orbits are typically confined to the trap axis, and for this case we present a
1D analysis of the relevant equation of motion. We follow this with an analysis
of 2D quadrupole traps that frequently show diamond-shaped closed orbits. For
both the 1D and 2D cases we present experimental observations of the calculated
trajectories in microparticle ion traps. We also report the discovery of a new
collective behavior in damped 2D microparticle ion traps, where particles
spontaneously assemble into a remarkable knot of overlapping, corotating
diamond orbits, self-stabilized by air currents arising from the particle
motion
