August 2025 - Electrodynamic Ion Traps

I found a teaching guide on Ion Traps by Newtonian Labs, which I might as well read in its entirety, even though my primary interest lies in the Quadrupole traps. According to Earnshaw, construction of stable ion traps through electrostatic fields alone is impossible. This is a problem to be circumvented in the construction of Ion Traps, though doing so need not be complicated. An example using a uniform field uses an oscillating potential between two electrode plates. The equation of motion follows the classical construction m∂²_tz = qE₀cos(ωt). The field is uniform between the electrodes at all times, and to keep the particle stable, the oscillation behavior has to be tuned to the charge and mass of the particle. mω²A = -qE₀. Introducing a field gradient will essentially begin from the same place, though rewriting the energy as E = (E₀ + E'z)cos(ωt).

A simple quadrupole trap is built by two spherical electrodes in a grounded box, so that the field lines of both electrodes repel one another and instead diverge toward the surrounding box. Similarly, the second electrode could be replaced by a ring electrode for a similar result. The electric fields are given by E_z = -∂_zV = -4A₂z cos ωt, E_r = -2E_z. Adding more dimensions will give very similar fields for the other coordinates, depending on orientation.

Introducing a damping constant γ changes the equation of motion

In practice, this might occur through Stokes Damping, derived from the Reynolds number.

A particle in a 3D quadrupole trap, where the oscillation amplitude is derived from the applied voltage and geometry, and z_eff is the spacing between trap electrodes, and applying an external force to balance the trapping force,