The Z4 orbit code and the focusing bar fields used calculations for superconducting cyclotrons

A discussion is presented of the Z4 orbit code that was developed to investigate nonlinear effects in the extraction systems of superconducting cyclotrons where such effects are especially strong. These systems make extensive use of sets of focusing bars which, because of their small aperture, are responsible for a large part of the nonlinearity. The Z4 orbit code uses exact equations of motion together with a special process for calculating all of the components of the main magnetic field correctly to fourth order in z. In addition, the fields of all of the focusing bars are calculated “exactly” using simple analytical formulas, and these fields are combined with the main field to obtain the total field acting on the particle at each point along its computed trajectory. As an aid to understanding the results of the orbit computations, a multipole analysis of the focusing bar fields was carried out, and the results are presented together with plots of the multipole coefficients as a function of various parameters.