Approximate multipole coefficients of RF ion traps as functions of aperture size

Abstract

In this study we present approximate analytical expressions for estimating the variation in multipole expansion coefficients as a function of the size of the apertures in the electrodes in axially symmetric (3D) and two-dimensional (2D) ion trap ion traps. Following the approach adopted in our earlier studies which focused on the role of apertures to fields within the traps, here too, the analytical expression we develop is a sum of two terms, A n,noAperture , the multipole expansion coefficient for a trap with no apertures and A n,dueToAperture , the multipole expansion coefficient contributed by the aperture. A n,noAperture has been obtained numerically and A n,dueToAperture is obtained from the n th derivative of the potential within the trap. The expressions derived have been tested on two 3D geometries and two 2D geometries. These include the quadrupole ion trap (QIT) and the cylindrical ion trap (CIT) for 3D geometries and the linear ion trap (LIT) and the rectilinear ion trap (RIT) for the 2D geometries. Multipole expansion coefficients A 2 to A 12, estimated by our analytical expressions, were compared with the values obtained numerically (using the boundary element method) for aperture sizes varying up to 50% of the trap dimension. In all the plots presented, it is observed that our analytical expression for the variation of multipole expansion coefficients versus aperture size closely follows the trend of the numerical evaluations for the range of aperture sizes considered. The maximum relative percentage errors, which provide an estimate of the deviation of our values from those obtained numerically for each multipole expansion coefficient, are seen to be largely in the range of 10–15%. The leading multipole expansion coefficient, A 2, however, is seen to be estimated very well by our expressions, with most values being within 1% of the numerically determined values, with larger deviations seen for the QIT and the LIT for large aperture sizes.