The formation of axisymmetric electron beams by a quasi-periodic magnetic field

Solutions are constructed in the smooth approximation for the internal and external fields of a dense electron beam formed by a quasi-periodic sequence of short magnetic lenses. Relations are obtained for the synthesis of the smooth transitions of the beam from the gun to an equilibrium state in the magnetic focus channel, as well as relations for the smooth transitions between the equilibrium states in the focus channels with various parameters in the absence of beam pulsation.