ABSTRACT
Fringe effects, with the notable exception of vertical focusing, are usually neglected in rings. We will see that they are connected to the small angle approximation—where this approximation is very good, we often find these fringe effects to be negligible. The crucial question is this: under the Talman's philosophy of tracking we are replacing a single ideal quadrupole by a collection of tiny quadrupoles closely bunched together. The expression derived in this section is to leading order in the field and all orders in the other variables. The formula applies exactly to a bend with an infinitely long face in the transverse x-direction. The final expression depends on powers of the momentum in all planes and is small if the small angle approximation is valid. This explains why the fringe field of sector bends as well as the fringe field of straight elements has been neglected in simulations where large machine (small angle) codes were used.
