Higher Order Chromatic Aberration of Betatron Function of Circular Accelerators

The chromaticity control is very important ingredient in the operation of a high energy ring accelerator. The strong nonlinear magnetic field is then necessary to correct the large chromaticity, which inevitably enhances the nonlinearity of the beam dynamics of the circular accelerator. Hence, for the purpose of precisely controlling the chromaticity, one should understand the higher order behavior with respect to the momentum deviation. In the preceding paper [1] we derived the higher order formula of the nonlinear dispersion function of a circular accelerator by means of the perturbative method. Based on the formulation the perturbative formula of the nonlinear chromaticity is derived in [2]. In these proceedings we develop the formulation of the chromaticity to the local one, i.e. the chromatic aberration of the betatron function. Using the local information of the chromaticity, one can control the global one in detail over the wide momentum range.