Healy's modular approach to the computation of the general bending magnet map applied to the quadratic part of the Hamiltonian which is exact in Delta p/p(sub 0).

摘要

In the context of large machines the approximate Hamiltonian K, which is obtained by expanding to second order in the transverse variables while keeping the (delta)-dependence ((delta) = p (minus) p(sub 0)/p(sub 0)) exact, is often a legitimate representation of circular machines. One nebulous element has always been the general bending magnet. With the exception of the sector bend, many knowledgeable accelerator physicists have accepted the postulate that the parallel face bends can be obtained as P = QSQ, where Q are quadrupoles, and S is the sector bend matrix. This result is correct for (delta)independent linear maps. However, in this paper we will show how one proceeds to a correct solution using the Hamiltonian K and in particular, we will show that the trick of quadrupole edge breaks down in a proper (delta)-independent treatment. We will derive a slightly extended Maxwellian fringe field effect from what Dragt used in his paper on the chromaticity of small rings. Also, we will provide a (delta)-dependent matrix for the body of a parallel face bend. The approach we will follow relies heavily on the work of Liam Healy, who solved the general bend problem in the context of the code MARYLIE.