Mathematical modelling of non-linear magnetic quadrupole systems for shaping ion beams of minimum spot size with a given beam current on target

Different magnetic quadrupole focusing systems for an ion microprobe consisting of doublets, triplets and Russian quadruplets are numerically investigated and compared. All geometrical aberrations up to third order are being taken into account. The systems with a crossover inside the system and with no crossover are studied. The matrizant of the third order (or transfer matrix) is found, which is used for the optimization of the system. Several components of particular interest (lens strengths, spherical aberrations and demagnification) are plotted. The minimum spot size for different emittances, working distances and total length of the system is found.