An investigation is made of the focusing properties of axially symmetric inhomogeneous magnetic sector fields with arbitrary circular boundaries. In first-order approximation the field is assumed to vary as r~" (0^n< 1). The equations of motion for the ion trajectories are developed from a least-action principle and solved through second-order approximation. Suitable expressions are derived for the horizontal and vertical focus positions, which are located outside the field boundaries. The mass dispersion and resolving power are found to vary as (1 — n)~ and so are considerably increased over the corresponding values for the homogeneous field (n = 0). Expressions for the second-order radial and vertical aberrations are derived. It is shown that the second-order radial aberration may be eliminated by proper shaping of the magnetic field and/or field boundaries. An equation is developed for the profile of the pole faces required to produce the desired field. The results are finally modified to account for the defocusing action of the magnetic fringing field. A numerical example is presented for a symmetrical spectrometer having a 90° sector field with n = 1/2. For this case simultaneous double directional and second-order radial focusing are possible.
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