  Radio-frequency linear multipole traps have been shown to be very sensitive
to mis-positioning of their electrodes, which results in a symmetry breaking
and leads to extra local minima in the trapping potential \cite{pedregosa17}
disturbing the operation of the trap. In this work, we analytically describe
the RF-potential of a realistic octupole trap by including lower order terms to
the well-established equation for a perfectly symmetric octupole trap. We
describe the geometry by a combination of identified defects, characterised by
simple analytical expressions. A complete equation is proposed for a trap with
any electrode deviation relying on a combination of the simple cases where the
defects are taken individually. Our approach is validated by comparison between
analytical and numerical results for defect sizes up to 4\% of the trap radius.
As described in \cite{pedregosa18}, an independent fine-tuning of the amplitude
of the RF voltage applied on each electrode can be used to mitigate the
geometrical defects of a realistic trap. In a different way than in
\cite{pedregosa18}, the knowledge of an analytical equation for the potential
allows to design the set of RF-voltages required for this compensation, based
on the experimental measurement of the ion position in the trap, without
information concerning the exact position of each electrode, and with a small
number of iterations. The requirements, performances and limitations of this
protocol are discussed via comparison of numerical simulations and analytical
results.

Quantum Physics