New classes of uniform distributions for charged particles in longitudinal magnetic field

The problem of stationary self-consistent distributions for charged particles beam in a longitudinal magnetic field was considered in various works. The simplest known distribution is Brillouin flow. Another simple case is the Kapchinskij-Vladimirskij distribution. The supports of these distributions in the phase space of transverse configurations and velocities have zero volume. The distributions with non-zero phase volume for beams with constant cross-section radius were also obtained previously. In the present report a more general case is investigated when the radius of beam cross-section, longitudinal velocity, and magnetic field change along the longitudinal axis. Wide classes of new stationary axially symmetric self-consistent distributions are found. New distributions have uniform charge density in the beam cross-section and, generally speaking, nonzero phase volume in the phase space of transverse motion. In a particular case of longitudinal uniformity they coincide with known ones. Such distributions can be applied for modelling of the beam which is nonuniform along its axis magnetic field with particles moving with different velocities in various cross-sections