Application of the Green's Function Method to Some Nonlinear Problems of an Electron Storage Ring. Part II. Checking the Method by a Quadrupole Pertubation

摘要

In the first part of this work I described a general approach to some storage ring problems. The basic concept and the main equations were developed there. The next natural step in this work should be an application of the developed technique to some particular nonlinear problem. Instead I found it useful to apply the method to the case of a field gradient perturbation first. This happened to be plausible not only as a check of the method, but, more importantly, as a way to resolve several troublesome difficulties encountered in subsequent calculations. I consider here the effect of a global gradient error in a weak-focusing machine and the effect of a local gradient error in a strong focusing machine. The distribution function of a particle bunch for a perturbed lattice in these cases can be written explicitly. The expansion of this distribution function as a series of perturbation terms produces the first and the second order corrections to the unperturbed function. From them we calculate then corresponding second moments. A similar expansion of the distribution function is found by the Green's function method. The second moments are found independently with the help of this distribution function. This comparison is done for the first and second order perturbations, i.e., up to the order which is sufficient for subsequent applications. (ERA citation 08:044504)