Abstract
General expressions for the aberration coefficients are derived in a form in which derivatives of the magnetic gradient do not appear. Specializations to a segment of the optic axis on which the gradient is uniform and to the rectangular model are given. A segment on which the gradient varies is treated by expansion in a power series; the expansion coefficients depend on form-factors similar to those occurring in the first-order theory. Momentum-dependent aberrations and the transit-time are handled in the same way. In all cases laws are given for combining the effects of two adjacent segments. The calculations do not support the idea that hyperboloidal pole-ends result in reduced aberrations.
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