An analytical approximation for the field in toroidal ion trap mass analysers having circular slits on endcap electrodes

In this study, an analytical approximation that can be used to estimate the change in the electric field caused by the introduction of circular slits on endcap electrodes of toroidal ion trap mass analysers will be developed. A theory has been developed to estimate the contribution of circular slits on the electrodes of a toroidal ion trap. In this theory, the first potential due to an arbitrary aperture on an infinite ground plane will be derived. This will be specialised to obtain potential due to a circular slit on an infinite ground plane. The potential due to a circular slit on an infinite ground plane will be used to evaluate the field within the toroidal traps having circular slits. The approximation developed in this paper has been verified on two arbitrarily selected toroidal traps. One of these arbitrary traps has circular-shaped electrodes, and the other trap has asymmetric hyperbolic-shaped electrodes. Fields estimated by this approximation agree well with those obtained using the boundary element method. Additionally, the study reveals an interesting correlation between the secular frequency of ion motion and the width of the circular slit. The secular frequency reduces as the slit width is increased. The reduction in secular frequency is proportional to the square of the slit width.