Third order transfer matrices for real Wien filters with homogeneous main fields

The determination of the ion optical aberrations of the Wien filters operated independently or in more complex mass analyzing or charged particle transport systems needs the calculation of the trajectories in a third order approximation. The trajectory calculations for crossed electric and magnetic fields generate unmanageably long formulas for the matrix elements. The reduction of the extent of the matrix elements to usable size can be reached by restricting calculations to less complex configurations as the homogeneous main fields Wien filters, which are probably the most often used crossed field velocity filters.Analytic expressions for transfer matrix elements for homogeneous main fields Wien filters were determined up to third order accuracy level. The elements of three pairs of matrices, those of entry fringing fields, those of the main fields and those of the exit fringing fields were derived for the plane of mass dispersion (radial) and for a direction normal to it (axial). The third order accuracy was kept over all the calculated contributions to the ion trajectory. The first order elements, to be multiplied by first order small quantities were calculated in a second order approximation, those of second order with a first order accuracy, while those of third order were not allowed to contain small quantities in their expressions. The matrix elements describing the ion position, trajectory slope, ion energy and mass were accounted for. In the fringing field matrix elements the effect of the electric and magnetic field boundary curvature radii were included.In a brief application, on one spectrometer geometry, the calculated matrices were used to determine the effect of third order angular aberration generated by the Wien filter alone on the resolution when the second order aberration has been cancelled.