A. Dymnikov et R. Hellborg, THE INFLUENCE OF SPACE-CHARGE ON THE MOTION OF A BUNCH OF PARTICLES IN AN ION-OPTICAL SYSTEM, Nuclear instruments & methods in physics research. Section B, Beam interactions with materials and atoms, 83(3), 1993, pp. 407-412
A general formalism in four-dimensional space-time is developed to obt
ain the electromagnetic field of a three-dimensional, elliptical bunch
of particles in the first approximation. The paraxial or Gaussian equ
ations of an impulse space-charge beam in the beam control system are
presented and a method of solving these equations is proposed.