An rf multipole trap uses a combination of static direct current (dc) and radio frequency (rf) oscillating electric fields to trap ions.
The electrical potential of such a trap is determined by the geometry, dimension of it's rf electrodes and the applied voltages. To model
the electrical and mechanical potential of a trap we make use of the boundary element method (BEM).
To understand the effects of systematic changes of trap parameters (dimensions and shape of the electrodes) we express the solution by an appropriate
multipole expansions. As a result, the quality of a trap can be characterized by a small set of parameters.
Through this approach one can find the ''optimum'' geometry and alignment tolerances to reduce undesired effects which e.g. could heat up the stored ion cloud.
One advantage of BEM is the possibility to set up realistic 3D trap-models with finite electrodes. Therefore the effect of misalignments or undesired manufacturing
tolerances of electrodes to the potential can be investigated. Also geometries with a broken rotational symmetry can be simulated reliably.

Boundary Element Method (BEM)
To find the electric potential of an rf multipole trap we have to find a solution for the ''Laplace'' equation:
ΔΦ=0
The movement of a particle in a rapidly oscillating field can described by an effective trapping potential. It is calculated by
taking the time-average over one period of the fast oscillation rf field .
This effective potential can be expressed as:
U_{effective} = qΦ_{dc}+ q²⁄(4mΩ²)·(∇Φ_{rf})²
with Φ_{dc} the static part of the potential, Φ_{rf} the rf part, Ω the rf frequenz and m the mass of the particle.
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Recent Results

As an effect of the applied dc voltage at the entrance and exit lenses of an 22 pole ion trap, a minima in the simulated effective potential at r≈3mm appears around the middle of the trap.
Simulations of a misalignment of only 0.15mm of 11 rods of a 22 pole trap shows that 10 minimas appear around a the center of the trap.