The problem of constructing self-consistent stationary particle distributions in four-dimensional phase space is considered for an azimuthally symmetric charged particle beam in a longitudinal magnetic field. In the general case of a longitudinally nonuniform beam, it is assumed that the magnetic field and the radius of the beam cross-section can slowly vary in the axial direction. The simplest case of a longitudinally uniform beam is studied in more detail. The approach applied here is to analyze the particle density in the space of integrals of motion. The relations between this density, the phase density, and the density in the configuration space are obtained. The set of admissible values of integrals of motion for a radially confined beam is examined. The construction of new self-consistent distributions consists in the specifying of some function defined on this set. Wide classes of new distributions are found. In particular cases, some of these distributions are identical to those known before, for example, the Kapchinsky-Vladimirsky distribution.
Self-consistent description of charged particle beam propagation in terrestrial magnetic field
