Symmetry corrected second order achromat

It is noted that, in some systems, depending upon beam emittance and energy spread, it is important to control centroid shifting aberrations, as well as second-order effects. It is shown that a second-order achromat provides a good starting point for the control of geometric and chromatic aberrations. However, when used in a ring with sections outside the achromat, these elements add chromatic aberrations not fully controllable with sextupoles inside the second-order achromat. It is suggested that, by taking advantage of the appropriate first-order orbit symmetry, it is possible to eliminate the centroid shifting chromatic aberrations, while still having full control over the second-order machine functions such as chromaticity. Furthermore, it is possible to minimize the number of sextupoles required for chromatic correction with this scheme. Application to the Bates South Hall Ring is considered